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Theoretical Study of Catalytic Effects in Micellar Solutions J. Resˇcˇicˇ* and V. Vlachy Faculty of Chemistry and Chemical Technology, University of Ljubljana, 1000 Ljubljana, Slovenia
L. B. Bhuiyan Laboratory of Theoretical Physics, Department of Physics, University of Puerto Rico, San Juan, Puerto Rico 00931-3343
C. W. Outhwaite School of Mathematics, University of Sheffield, Sheffield S3 7RH, U.K. Received March 18, 2004. In Final Form: September 1, 2004 The catalytic effect of charged micelles as manifested through the increased collision frequency between the counterions of an electrolyte in the presence of such micelles is explored by the Monte Carlo simulation technique and various theoretical approaches. The micelles and ions are pictured as charged hard spheres embedded in a dielectric continuum with the properties of water at 298 K with the charge on micelles varying from zero to zm ) 50 negative elementary charges. Analytical theories such as (i) the symmetric Poisson-Boltzmann theory, (ii) the modified Poisson-Boltzmann theory, and (iii) the hypernetted-chain integral equation are applied and tested against the Monte Carlo data for micellar ions (m) with up to 50 negative charges in aqueous solution with monovalent counterions (c; zc ) +1) and co-ions (co; zco ) -1). The results for the counterion-counterion pair correlation function at contact, gcc(σcc), are calculated in a micellar concentration range from cm ) 5 × 10-6 to 0.1 mol/dm3 with an added +1:-1 electrolyte concentration of 0.005 mol/dm3 (for most cases), and for various model parameters. Our computations indicate that even a small concentration of a highly charged polyelectrolyte added to a +1:-1 electrolyte solution strongly increases the probability of finding two counterions in contact. This result is in agreement with experimental data. For low charge on the micelles (zm below -8), all the theories are in qualitative agreement with the new computer simulations. For highly charged micelles, the theories either fail to converge (the hypernetted-chain theory) or, alternatively, yield poor agreement with computer data (the symmetric Poisson-Boltzmann and modified Poisson-Boltzmann theories). The nonlinear PoissonBoltzmann cell model results yield reasonably good agreement with computer simulations for this system.
1. Introduction Polyelectrolyte solutions made up of synthetic or naturally occurring polyelectrolytes, globular proteins, and ionic surfactants above the critical micellar concentration are all examples of highly asymmetric electrolytes. Their properties are strongly influenced by the high asymmetry in both charge and size between the macroions (micelles) and the microions of opposite charge, called counterions, which are also present in the solution. Due to the consequent strong electrostatic attraction between the macroions and counterions, the mobility and activity values of the latter species are reduced significantly below their corresponding bulk values. This results in the thermodynamic and transport properties of polyelectrolyte solutions being appreciably different in many aspects from those of the well-known and more common low-molecular electrolytes.1,2 In particular, even for very dilute solutions with respect to macroions, the polyelectrolyte solutions exhibit large deviations from ideality.3 * To whom correspondence should be addressed. (1) Schmitz, K. S. Macroions in Solution and Colloidal Dispersion; VCH Publishers Inc.: New York, 1993. (2) Dautzenberg, H.; Jaeger, W.; Ko¨tz, J.; Philipp, B.; Seidel, C.; Stcherbina, D. Polyelectrolytes: Formation, Characterization and Application; Hanser Publ.: Munich, Germany, 1994. (3) Dolar, D. In Polyelectrolytes; Selegnyi, E., Mandel, M., Strauss, U. P., Eds.; D. Reidel: Dordrecht, The Netherlands, 1974; Vol. 1, p 97.
It has been known for some time4-11 that the addition of a polyelectrolyte to an electrolyte solution substantially influences the reaction rate between the other charged particles. As an example, we can cite the work by Morawetz and Vogel5 on some reactions of hydrolysis where a small addition of a polyelectrolyte was seen to cause a dramatic acceleration of the chemical reaction between equally divalent counterions in solution. A similar behavior was observed in the case of some electron-transfer reactions (see, for example, ref 7 and the review8). The experimental results were rather successfully analyzed in the framework of the cylindrical or spherical cell model and the related nonlinear Poisson-Boltzmann (PB) equation.10,11 In principle, the effect of the polyelectrolyte on ion-ion collision frequencies can also be used to probe the distribution of ions around the macroion. In this regard, mention can be made of the works by Meares and co-workers,12,13 who (4) Morawetz, H.; Shaffer, J. A. J. Phys. Chem. 1963, 67, 1293. (5) Morawetz, H.; Vogel, B. J. Am. Chem. Soc. 1969, 91, 563. (6) Morawetz, H. Acc. Chem. Res. 1970, 3, 354. (7) Morawetz, H. J. Polym. Sci., Part B 2002, 40, 1080. (8) Baumgartner, E.; Fernandez-Prini, R. In Polyelectrolytes; Frisch, K. C., Klempner, D., Patsis, A. V., Eds.; Technomic: Westport, CT, 1976; p 1. (9) Mita, K.; Kunugi, S.; Okubo, T.; Ise, N. J. Chem. Soc., Faraday Trans. 1 1975, 69, 936. (10) Ishikawa, M. J. Phys. Chem. 1979, 83, 1576. (11) Rodenas, E.; Dolcet, C.; Valiente, M. J. Phys. Chem. 1990, 94, 1472. (12) Wensel, T. G.; Meares, C. F. Biochemistry 1983, 22, 6247.
10.1021/la049285+ CCC: $30.25 © 2005 American Chemical Society Published on Web 12/04/2004
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probed the electrostatic properties of myoglobin12 and DNA13 using diffusion-enhanced energy transfer. Recent studies in this direction are due to Tapia and coworkers.14,15 As discussed by Keizer,16,17 the rapid bimolecular chemical reaction is influenced by chemical and physical factors. The chemical part is determined by a molecular mechanism and involves knowledge of electron redistribution during the reaction and therefore requires a quantum-chemical approach, while the physical factor, which we aim to discuss in this study, is related to the rate at which potentially reactive encounters occur. It is conceivable, of course, that the measured rate of rapid reaction may depend on both factors in a rather complicated way. In the approximate statistical-mechanical approach used here, the physical factor is closely related to the two-particle correlation function of the reactive particles. For the reaction to occur, the two reactant molecules have to to be at a given separation and orientation. This is exactly the information provided by the two-particle correlation function. For spherically symmetric interactions and with the further assumption that the interacting particles are in close contact, the physical factor may be related to the value of the pair correlation function of the reacting particles s and t, at contact, namely, gst(σst). When a polyelectrolyte is added to a low-molecular electrolyte, the probability of finding two counterions near each other grossly increases. The counterions (here, those ions with the sign of charge opposite to that of the macroion) are strongly attracted to macroions: their concentration near the macroion surface is several orders of magnitude larger than their average concentration in the pure solution. In addition to the counterions and macroions, co-ions are also present in the solution. The latter ions being of the same sign as the macroion are repelled from the vicinity of the macroions. The effect of the addition of polyelectrolyte to electrolyte solution can be measured by the following ratio:4,18
possible configuration set of particles localized at r1, r2, ... , rN and ZN the standard configurational integral.19,20 As usual, β ) 1/(kBT), where kB is the Boltzmann constant and T the absolute temperature. For two interacting counterions near a macroion surface, a large enhancement in k (k . k0) is found,5 whereas, for the potentially reacting co-ions, only a weak increase in the reaction rate may be anticipated. The latter effect is closely related to the fact that the co-ions are likely to find themselves much further away from macroions where the influence of macroions is weaker. Finally, for a reacting ion pair having opposite charges, a decrease in k is obtained. This inhibition arises because upon addition of a polyelectrolyte the counterions and co-ions of an electrolyte get spatially separated.4 In this study, we will use various theoretical approaches to predict the catalytic effect of the addition of a spherical polyelectrolyte to an electrolyte solution. To this end, we have employed the following methods: (a) the potential theories such as the symmetric Poisson-Boltzmann (SPB) and modified Poisson-Boltzmann (MPB) theories and an integral equation approach based on the hypernettedchain (HNC) approximation and (b) Monte Carlo (MC) computer simulations. For highly charged macroions with |zm| > 13, the HNC equation does not converge, while the potential theories are inaccurate. At higher magnitudes of the macroion charge, isotropic and cell model results are compared and they are also compared with the nonlinear PB cell model predictions.
〈gst(σst)〉 k ) k0 〈g°st(σst)〉
LB r g σst ust(r) ) r r < σst ∞
2.1. Primitive Model. The polyelectrolyte-electrolyte solution is pictured in accordance with the primitive model: the three ionic speciessmacroions or micelles (m), counterions (c), and co-ions (co)sare represented by spherical rigid ions of arbitrary diameters, σs (s ) m, c, or co), embedded in a continuum dielectric with relative permittivity, r. The interaction pair potential for two species of valences zs and zt, separated by a distance, r, is assumed to be
{
(1)
where k and gst(σst) are the reaction rate and the contact value of the pair correlation function for the two electrolyte species s and t in contact for the polyelectrolyte-electrolyte mixture, respectively, while k0 and g°st(σst) denote the corresponding values for the pure electrolyte solution. The angular brackets denote canonical average in the form19,20
〈f〉 )
2. Models and Methods
∫ ‚‚‚ ∫f(r1, r2, ... , rN) ×
1 ZN
exp[(-βU(r1, r2, ... , rN))] dr1 dr2 ... drN (2) where U(r1, r2, ... , rN) is the configurational energy of one (13) Wensel, T. G.; Meares, C. F.; Vlachy, V.; Matthew, J. B. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 3267. (14) Tapia, M. J.; Burrows, H. D.; Azenha, M. E. D. G.; Miguel, M. G.; Pais, A. A. C. C.; Sarraguca, J. M. G. J. Phys. Chem. B 2002, 106, 6966. (15) Tapia, M. J.; Burrows, H. D. Langmuir 2002, 18, 1872. (16) Keizer, J. J. Phys. Chem. 1982, 86, 5052. (17) Keizer, J. Acc. Chem. Res. 1985, 18, 235. (18) Resˇcˇicˇ, J.; Vlachy, V. In Macro-ion Characterization; Schmitz, K. S., Ed.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994; Vol. 548, p 24. (19) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 1986. (20) Allen, M. P.; Tildesley, D. J. Computer simulations of liquids; Clarendon Press: Oxford, U.K., 1989.
zszt
LB )
(3)
βe2 4π0r
(4)
where σst ) (σs + σt)/2. As usual, LB is the Bjerrum length, 0 is the vacuum permittivity, zs is the valency of ion species s, and e is the proton charge. The counterions and co-ions are assumed to be singly charged in the present study. 2.2. Isotropic Model Theories. In the SPB theory, the pair correlation function gst(r) appearing in the classical, standard PB theory is symmetrized with respect to the interchange of the indices s and t for asymmetric systems21-23 so that the Onsager condition gst(r) ) gts(r) is satisfied. In the SPB theory, the (symmetric) gst(r) function is given by
1 gst(r) ) g0st exp - β(es(ψt(r) + ψ0t (r)) + 2
[
]
et(ψs(r) + ψ0s (r))) (5) where es ) zse, ψs(r) is the mean electrostatic potential at (21) Outhwaite, C. W. Chem. Phys. Lett. 1978, 53, 599. (22) Outhwaite, C. W. J. Chem. Soc., Faraday Trans. 2 1987, 83, 949.
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a distance, r, about an ion, s, with ψ0s (r) being the corresponding discharged potential ψ0s (r) ) ψs(r; es ) 0) and g0st(r) ) gst(r; es ) et ) 0) being the exclusion volume term. The discharged potentials vanish if all the constituent components have the same size. The SPB theory is formed by utilizing the above gst(r) function in Poisson’s equation
∇2ψs(r) ) -
1
∑t etFtgst(r)
(6)
0r
where Ft is the mean number density of component t. In the Schmidt and Ruckenstein version24 of the SPB theory, g0st(r) ) 1, r > σst, and ψ0s (r) ) 0. The exclusion volume term was approximated by the Percus-Yevick (PY) hard sphere pair correlations complemented by the Verlet-Weis (r) (see, for example, ref corrections, viz., g0st(r) ) gPY+VW st 19). The interionic correlation effects, which are neglected in the SPB approximation, are incorporated into the MPB theory through the fluctuation potential terms. In the latter theory, the gst(r) function reads (see, for example, refs 23, 25, and 26)
{
1 gst(r) ) g0st(r) exp - β(es(Ls(ut) + Ls(u0t )) + et(Lt(us) + 2
}
Lt(u0s ))) (7) Lt(us) )
1 {u (r + σit) + us(r - σit) + 2r(1 + κσit) s κ
r+σ ∫r-σ
it
it
us(r) dR} (8)
u0s ) us(es)0) κ2 )
1 0r
∑s Fse2s
β
(9) (10)
where us(r) ) rψs(r), κ is the Debye-Hu¨ckel parameter, and the subscript i in σit is reserved for the smallest ion in the system. The exclusion volume term was again taken (r). The SPB and MPB theories have been to be gPY+VW st successfully applied to symmetric and asymmetric electrolytes23,25,26 as well as to electrolyte mixtures containing neutral particles.27 The basic relation in the integral equation approach is the Ornstein-Zernike equation19
hst(rst) ) cst(rst) +
∑k Fk∫csk(rsk) hkt(rkt) drk
(11)
where the summation is performed over all the components. This equation is merely a definition of the direct correlation function cst, and another relation between the (23) Outhwaite, C. W.; Molero, M.; Bhuiyan, L. B. J. Chem. Soc., Faraday Trans. 1991, 87, 3227. (24) Schmidt, A. B.; Ruckenstein, E. J. Colloid Interface Sci. 1992, 150, 169. (25) Outhwaite, C. W.; Molero, M.; Bhuiyan, L. B. J. Chem. Soc., Faraday Trans. 1993, 89, 1315. (26) Outhwaite, C. W.; Molero, M.; Bhuiyan, L. B. J. Chem. Soc., Faraday Trans. 1994, 90, 2002. (27) Bhuiyan, L. B.; Vlachy, V.; Outhwaite, C. W. Int. Rev. Phys. Chem. 2002, 21, 1.
total correlation function hst ) gst - 1 and cst is needed. This second relation reads
loge[hst(r) + 1] ) -βust(r) + hst(r) + Bst(r) - cst(r) (12) The term Bst(r), known in the literature as the bridge graph term, cannot be written as a closed form function of the distribution functions hst(r) and cst(r). Bst(r) is often set to zero in what is called the hypernetted-chain (HNC) approximation which has been quite successful in describing the properties of charged fluids (see, for example, refs 28 and 29). The details of the numerical procedure used in the present work in solving the SPB, MPB, and HNC equations have been explained elsewhere (see, for example, refs 22, 30, and 31). 2.3. The Cell Model Approach. Another model that has been utilized in the present work is the spherical cell model of colloids and polyelectrolyte solutions. The model takes advantage of the high asymmetry in charge and size between the micelles and the small, simple ions, where for low micellar concentrations each micelle is essentially surrounded by its own charge cloud of simple ions. The cell model in spherical symmetry has been used by numerous authors (see, for example, refs 32-35). The macroion is located at the center of the spherical cell, and the cell volume is V ) (4π/3)R3cell; consequently, the cell radius, Rcell, is determined by the macroion concentration, cm ) (NAV)-1, where NA is Avogadro’s number. The counterions and co-ions from an added salt are distributed within the cell, and as before, the solvent is treated as a dielectric continuum. We have treated this spherical geometry by the MC simulation technique34,35 and the PB theory. The Poisson-Boltzmann equation for the spherical symmetry reads
Fe 1 d 2 dψ r )2 dr dr 0r r
(
)
(13)
In this equation, ψ(r) is the mean electrostatic potential around the macroion of charge zme, uniformly distributed over the surface, and Fe is the charge density defined by
Fe(r) )
∑s zseFs(Rcell)e-z eβψ(r) s
(14)
where Fs(Rcell) is the number concentration of ionic species s at the cell radius, Rcell. The PB equation needs to be solved numerically subject to the boundary conditions given by the Gauss law.32,34 As a result, we obtain the mean electrostatic potential, ψ(r), around the macroion, relative to the potential at the cell radius, Rcell, where the zero of the potential is chosen. The PB theory does not provide any information about the pair correlation between the two small ions (and consequently about gst(r)) in the electrical double-layer around a macroion. Within the PB approach, it is therefore (28) Rasaiah, J. C. In The Liquid State and its Electrical Properties; Kunhardt, E. E., Christophorou, L. G., Luessen, L. H., Eds.; NATO ASI Series B; Plenum Press: New York, 1988; Vol. 193. (29) Vlachy, V. Annu. Rev. Phys. Chem. 1999, 502, 145. (30) Ichiye, T.; Haymet, A. D. J. J. Chem. Phys. 1990, 93, 8954. (31) Vlachy, V.; Ichiye, T.; Haymet, A. D. J. J. Am. Chem. Soc. 1991, 113, 1077. (32) Gunnarsson, G.; Jo¨nsson, B.; Wennerstro¨m, H. J. Phys. Chem. 1980, 84, 3114. (33) Linse, P.; Jo¨nsson, B. J. Chem. Phys. 1983, 78, 3167. (34) Bratko, D.; Vlachy, V. Colloid Polym. Sci. 1985, 263, 417. (35) Rebolj, N.; Kristl, J.; Kalyuzhnyi, Yu. V.; Vlachy, V. Langmuir 1997, 13, 3646.
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consistent to use the equation derived by Morawetz6 to calculate the catalytic effect given by k/k0:
∫
V-1 vFt(r) Fs(r) dV k ) k0 V-1 F (r) dV V-1 F (r) dV v t v s
∫
∫
(15)
In this equation,
Ft(r) ) Ft(Rcell)e-βψ(r)zte
(16)
where V is the volume of the spherical cell. The electrostatic potential, ψ(r), has been obtained from the numerical solution of the nonlinear PB equation, eq 13 (for more details, see refs 32-34). 2.4. Monte Carlo Simulations. The MC simulations were performed for both the isotropic and the cell model in the canonical ensemble using the Metropolis sampling20 using the MOLSIM integrated MC/MD/BD simulation package.36 The same code has been used before in several studies of asymmetric electrolytes (see, for example, refs 35 and 37). Averages were taken over 3 × 105 attempted moves per particle (passes). At least 6 × 104 passes were performed for the equilibration run. The number of macroions included in the simulation box was at least 20. In the isotropic model calculations, the Ewald summation method20 was used to account for the effects of a finite number of particles. The MC values of the pair correlation function in contact, gst(σst), were evaluated by extrapolation over the last 30 points in the ion-ion pair correlation function collected in the form of a histogram. Histograms were collected on a 0.05 nm grid. For each bin, a separate statistic was calculated. For example, the relative standard deviation of the first bin for zm ) -50 was 1) and 0.34 and 1 (
