9156
J. Phys. Chem. B 1999, 103, 9156-9160
Theory of Diffusion-Assisted Reactions on Micelle Surfaces: Photoinduced Electron Transfer Followed by Back Transfer A. V. Barzykin, K. Seki, and M. Tachiya National Institute of Materials and Chemical Research, Tsukuba, Ibaraki 305-8565, Japan ReceiVed: February 22, 1999; In Final Form: May 17, 1999
A general theory of diffusion-assisted irreversible reactions on micelle surfaces is presented. An exact solution for the experimental observables is given in a matrix form in terms of the eigenmodes of the diffusion operator in the same geometry but without interaction potential and reaction. Photoinduced electron transfer followed by back transfer is considered in detail as a practically important example of reaction. Excellent agreement of the matrix solution with the numerical solution of the corresponding partial differential equations via a standard discretization procedure is observed for typical sets of parameter values.
1. Introduction Reactions in restricted geometries such as micelles, microemulsions, zeolites, and thin films have attracted growing research interest for several decades.1-9 It has been realized that proper organization of reactants by their solubilization in suitable microheterogeneous environments enables one to catalyze and control a wide variety of practically important reactions. On the other hand, simple indicator reactions such as luminescence quenching have proven to provide a useful tool to investigate the embedding structures themselves. Theoretical formulation of reaction kinetics in micelles and related structures is well established.4-9 Micelles form finitevolume domains where reactants are confined. The number of reactants in each domain is small. Therefore, one has to deal with a discrete statistical distribution of reactants among the micelles instead of conventional concentrations. The overall kinetics in the ensemble of micelles is obtained by averaging the microscopic intramicellar kinetics with a given number of reactants over the occupancy distribution. The number of reactants in a given micelle fluctuates with time as a result of various intermicellar migration processes. These fluctuations are slow, so that reaction inside the micelle can normally be treated as kinetically independent of the migration.10,11 In analyzing intramicellar reaction kinetics, the binary approximation is used, that is, all pairs of reactants in a micelle are assumed to react independently and contribute multiplicatively to the kinetics. Monte Carlo simulations have proven that indeed, as far as reactions in micelles with not so high average occupancies are concerned, the binary approximation works quite well and thus the intramicellar kinetics can be described in terms of the pair survival probability.12-15 The bimolecular reaction rate depends on various factors, such as local properties of the surrounding solvent, but most importantly, it depends on the distance between reactants. Therefore, translational diffusion can play a significant role in influencing the reaction kinetics. Two limiting cases are considered in the literature. Short-range reactions, such as electron transfer and energy transfer by exchange mechanism, are usually assumed to be diffusion-controlled, occurring at the encounter of reactants.16-20 To a good approximation, the pair survival probability decays exponentially in this case. On the other hand, long-range incoherent electronic energy transfer in
microheterogeneous systems is usually assumed to be static.21-25 The resulting decay is strongly nonexponential. Early experimental studies on luminescence quenching in micelles found reasonable agreement with either of the two approximations. However, recent numerical analyses and accurate experiments have revealed their breakdown under certain conditions.13-15 Theoretically, bridging of the above two limiting cases is achieved by defining the pair survival probability as a solution to the diffusion equation in a given geometry with a distancedependent reaction sink. Because of the lack of an analytical solution for the arbitrary functional form of the rate constant, direct numerical integration of the corresponding partialdifferential equation was originally performed.13-15,26,17 Recently, we have shown that the solution of the diffusionreaction equation in confined systems with an arbitrary sink can be obtained in a closed matrix form in terms of the eigenmodes of the diffusion operator in the absence of reaction.28 This method has been applied to irreversible reactions of neutral particles on micelle surfaces, providing a simple way to calculate the pair survival probability.29 In this paper, we will include the interaction potential between reactants and prove that the pair survival probability in this case can also be presented in a simple matrix form. We will show that our approach can be used even for multistage reactions and consider photoinduced electron transfer followed by back transfer as an experimentally relevant example.
2. Irreversible Reaction Consider a bimolecular irreversible reaction of a pair of particles diffusing on a spherical surface of a micelle. It is convenient to choose z ) -cos θ as a coordinate, where θ is the difference between the polar angles corresponding to the centers of the reactants. The pair survival probability, Φ(z, t), that is, the probability for a pair with initial relative separation z to stay unreacted by time t, satisfies the following differential equation,30
∂ Φ(z, t) ) L †z Φ(z, t) - k(z)Φ(z, t) ∂t
10.1021/jp9906278 CCC: $18.00 © 1999 American Chemical Society Published on Web 07/02/1999
(1)
Diffusion-Assisted Reactions on Micelle Surfaces
J. Phys. Chem. B, Vol. 103, No. 43, 1999 9157
where
L
† z
)
∂ ∂ ∂ (1 - z2) - (1 - z2)V′(z) ∂z ∂z ∂z
Therefore, we can use the eigenfunctions φn(z) of L basis to express the solution of eq 1
(2)
is the adjoint of the Smoluchowski diffusion operator, time is in units of τd ) R2/D, R is the distance between the centers of the micelle and either reactant, D is the mutual diffusion coefficient, V(z) is the interaction potential in units of kBT, k(z) is the dimensionless distance-dependent reaction rate constant, and the prime sign denotes the derivative. We assume that reactants do not penetrate each other and reflect at encounter so that z ∈[z0, 1] and
|
∂ )0 Φ(z, t) ∂z z)z0
(11)
b(t) ) exp(-tQ)‚b(0)
(12)
where
and the matrixes are defined by
Q)Λ+W+V
(13)
Λn,m ) λnδn,m
(14)
∫z1 dz φn(z) k(z) φm(z)
(15)
∫z1 dz φn(z) (1 - z2) V′(z) φ′m(z)
(16)
Wn,m ) c Vn,m ) c
0
0
(4)
∫z1 dz u(z)Φ(z, t)
δn,m denoting the Kronecker symbol. The initial condition, eq 4, results in
bn(0) ) δn,0
(17)
(5)
0
In calculating the elements of the matrix V it is convenient to use the following relationship:
If the reaction starts from thermal equilibrium,
ueq(z) )
∑ bn(t) φn(z)
n)0
(3)
Our goal is the pair survival probability averaged over the distribution of initial pair separation, u(z), i.e.,
Φ(t) )
as a
∞
Φ(z, t) )
where z0 ) - cos θ0 ) d2/2R2 - 1, d being the contact distance. The second boundary condition for eq 1 is that Φ(1, t) is finite. The initial condition is given by
Φ(z, 0) ) 1
(0) z
exp[-V(z)]
∫z
1
d (1 - z2) Pν(z) ) ν [Pν-1(z) - zPν(z)] dz
(6)
dz exp[-V(z)]
(18)
0
In our original formulation, we have started with the Smoluchowski equation for the pair probability density instead of the differential equation for the pair survival probability.28,29 Both approaches are equivalent, of course. The matrix solution relies on knowing the eigenmodes of the diffusion operator without reaction, Lz, and those are not available in the presence of an arbitrary interaction potential. But we know the eigenmodes of L (0) ) (∂/∂z)(1 - z2)(∂/∂z) in the absence of z interaction potential. The eigenfunctions are defined by
φn(z) ) anPνn(z)
(7)
where
a-2 n )c
∫z1 dz P2ν (z) 0
n
(8)
|
(9)
numbered in increasing order, n ) 0, 1, ..., ∞, Pν(z) is the Legendre function. The lowest root is zero, i.e., ν0 ) 0, corresponding to the uniform distribution in the absence of interaction potential. The associated eigenvalues are given by
λn ) νn (νn + 1)
Φ(t) ) UT‚(e-tQ)‚U0
(19)
∫z1 dz φn(z) u(z)
(20)
where
Un )
0
and [U0]n ) δn,0, corresponding to the equilibrium distribution without interaction potential, i.e., u(z) ) u(0) eq (z) ) c in eq 20. In the case of no interaction potential and an initial equilibrium distribution of the pair separation, Q ) Q0 ) Λ + W, U ) U0 and eq 19 is simplified into
Φ(t) ) [exp(-tQ0)]0,0
c ) (1 - z0)-1 and νn are the roots of
∂ )0 P (z) ∂z ν z)z0
Finally, we obtain for the space-averaged pair survival probability,
(10)
What is important is that the interaction potential does not enter the boundary condition, eq 3, for the survival probability.
(21)
This is the case we have analyzed previously.29 Important conclusions are: (i) Although matrices of infinite dimensions are involved, they can be truncated at a certain finite dimension N which should be chosen depending on the sink strength; rapidly increasing eigenvalues guarantee convergence of the truncated matrix solution. (ii) After the matrices are truncated, the kinetics can be represented by a superposition of N exponentials with the decay constants determined by the roots ˜ 0(s)| ) 0, where Q ˜ 0(s) ) sI + Q0 sj of the secular equation, |Q and I denotes the identity matrix. All sj < 0. We have found that often only a few modes are sufficient to attain excellent accuracy.
9158 J. Phys. Chem. B, Vol. 103, No. 43, 1999
Barzykin et al.
It can be readily shown that the above conclusions also hold in a general case with interaction potential. In this case, the solution is written in the following form:
∑ j,n
Φ(t) )
µn,0(sj)Un ∆j(sj)
exp(sjt)
(22)
where sj are now the roots of
|Q ˜ (s)| ) 0
(23)
with Q ˜ (s) ) sI + Q, µn,0(s) denotes the minor of [Q ˜ (s)]n,0, and
|Q ˜ (s)| s - sj
∆j(s) )
(24)
An alternative way of calculation is just by numerically evaluating the matrix exponential in eq 19. In what follows we will be mainly interested in photoinduced electron transfer. We will assume the widely accepted Marcus form for the distance dependence of the rate constant31
k(r) ) ×
2π 2 J exp[-β(r - d)] p 0
1
x4πλrkBT
[
exp -
]
(∆G + λr)2 4λrkBT
(25)
where
λr )
(
)(
e2 1 1 2 op s
)
1 1 2 + rd ra r
(26)
is the solvent reorganization energy, ∆G is the free energy change due to the transfer, op and s are the optical and static dielectric constants, respectively, J0 and β are parameters that characterize the magnitude and attenuation of the transfer integral, rd and ra are the radii of the reactants (donor and acceptor, respectively), r is the distance between their centers, and e, p, kB, and T have their usual meaning. Dimensionless k(z) in eq 1 is obtained by substituting r ) R (2(1 + z))1/2 and multiplying k(r) by τd. We will also assume the Coulombic potential, if any, between donor and acceptor ions
V(r) ) r0/r
(27)
where r0 stands for the Onsager length and is negative for attractive interaction. The above expression for the electron transfer rate constant is derived under the assumption of a homogeneous dielectric continuum model for the solvent. It has to be modified to account for the microheterogeneous nature of the micelle.27 However, these effects are not important for the present study and we will use the classical Marcus expression. This is what Weidemaier and Fayer used in their numerical analysis of photoinduced electron transfer on micelle surfaces, and we are going to compare our results with theirs.14 Before we proceed further with our analysis, let us show how the experimental observables are determined in terms of the pair survival probability Φ(t). In the case of photoinduced electron transfer, one measures the excited state decay. Experimentally, the excitation efficiency is normally sufficiently low so that no more than one excited-state donor molecule is contained in a micelle. The number of acceptors can be arbitrary to a certain extent. As mentioned in Introduction, to a good
Figure 1. Excited-state survival probability, Φex(t), for a repulsive potential (Rc ) 60 Å), no potential, and an attractive potential (Rc ) -60 Å) between the donor-acceptor pair, top to bottom. The initial distribution of the pair separation is assumed to be equilibrium in all cases. The electron transfer parameters for all curves are J0 ) 100 cm-1, β ) 1.0 Å-1, ∆G ) -1.5 eV, op ) 2.0, and s ) 10.0; the micelle radius is R ) 20 Å, the donor-acceptor contact distance is d ) 8 Å, and the mutual diffusion coefficient is D ) 10 Å2/ns. Donor self decay lifetime is not included to emphasize the electron transfer event. The solid lines represent the matrix solution and the circles are the results of numerical solution.
approximation all acceptors act independently and contribute multiplicatively to the excited-state deactivation. Thus, the survival probability of an excited donor with n cohabitant electron acceptors is Φn(t). Hereafter we omit the self-decay, focusing purely on the electron transfer dynamics. The selfdecay would appear as a multiplicative factor exp(-t/τ), where τ is the excited-state lifetime. If the intermicellar migration does not occur on the time scale of the transfer, each micelle acts as a cage and we can obtain the macroscopically observable excited-state decay function by averaging Φn(t) over the equilibrium occupancy distribution of acceptors. In the simplest case of noninteracting particles, the distribution is Poissonian and one obtains
〈Φ(t)〉 ) exp{-nj[1 - Φ(t)]}
(28)
where nj is the average number of acceptors per micelle. The equilibrium distribution is different for charged species, but there is a well-defined recipe for calculating this distribution using the standard formalism of statistical mechanics.32,33 In other words, there is a well-defined recipe for calculating 〈Φ(t)〉 in terms of Φ(t).4,7-9 Figure 1 illustrates the space-averaged excited-state survival probability Φ(t) for a donor-acceptor pair undergoing diffusionassisted electron transfer on the surface of a spherical micelle. The parameters are given in the figure caption and were chosen in accordance with ref 14. Excellent agreement of the matrix solution with the results of numerical solution of eq 1 obtained via a standard discretization procedure is observed. Clearly, electron transfer occurs much faster in the case of attractive interaction between donor and acceptor (negative r0). 3. Photoinduced Electron Transfer Followed by Back Transfer Now that we have understood how to treat simple irreversible reactions, let us consider a specific example of a multistage reaction, namely, photoinduced electron transfer followed by
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J. Phys. Chem. B, Vol. 103, No. 43, 1999 9159
back transfer to the ground state. We denote the rate constant for the forward electron transfer from the excited donor-acceptor state by kf(r) and the rate constant for the back transfer to the ground donor-acceptor state by kb(r) and supply all the corresponding parameters with appropriate indices. The self-decay is omitted, as in the preceding section. Following Weidemaier and Fayer, we also assume for simplicity that there is no Coulombic interaction between the donor and acceptor prior to the forward transfer event.14 Experimentally observable are the excited-state survival probability, 〈Φex(t)〉, and the ion pair state survival probability, 〈Φip(t)〉, both averaged over spatial configurations and the occupancy distribution. Since back electron transfer to the excited donor state is neglected in this scheme (on quite reasonable grounds), 〈Φex(t)〉 can be calculated using the results of section 2, i.e., eqs 21 and 28. The ion-pair-state survival probability, Φip(z, t), satisfies eq 1 with the boundary and initial conditions given by eqs 3 and 4. The rate of creation of geminate ion pairs is determined by the pair probability density Pex(z, t) satisfying the forward diffusion equation with the equilibrium initial condition and reflecting boundary condition at encounter. Since no interaction potential is assumed in the initial state, Lz ) L †z ) L (0) z in this case, and thus Pex(z, t) ) Φex(z, t) and Pex(t) ≡ ∫z10 Pex(z, t) ) Φex(t). If an excited donor shares a micelle with n acceptors, the ion pair state survival probability averaged over spatial configurations, Φip(n, t), is given by14
Φip(n, t) ) n
Figure 2. Ion pair state survival probability, 〈Φip(t)〉, for a repulsive potential (Rc ) 60 Å), no potential, and an attractive potential (Rc ) -60 Å), top to bottom. The electron transfer parameters for all curves are J0f ) 100 cm-1, βf ) 1.0 Å-1, ∆Gf ) -1.0 eV, J0b ) 100 cm-1, βb ) 1.0 Å-1, ∆Gb ) -1.5 eV, op) 2.0, and s) 10.0; the micelle radius is R ) 20 Å, the donor-acceptor contact distance is d ) 8 Å, and the mutual diffusion coefficient is D ) 10 Å2/ns. Poisson occupancy statistics are assumed with the average number of acceptors per micelle nj ) 9. Donor self decay lifetime is not included to emphasize the electron transfer event. The solid lines represent the matrix solution and the circles are the results of numerical solution. The insert shows a wider time window in the semilog scale.
∫z1 dz ∫0t dt′ Φip(z, t - t′) 0
× kf(z) Pex(z, t′) Pn-1 ex (t′)
(29)
We have to average Φip(n, t) over the equilibrium occupancy distribution in order to get 〈Φip(t)〉. In the case where acceptors are initially neutral, this distribution is to a good approximation Poissonian and one obtains
〈Φip(t)〉 ) nj
∫z1 dz ∫0t dt′ Φip(z, t - t′) 0
× kf(z) Pex(z, t′) exp{-nj[1 - Pex(t′)]}
(30)
Other statistical distributions are possible, particularly when acceptors are initially charged, but these cases will not be considered here for simplicity. Using the eigenmode expansion, we can rewrite eq 30 in the matrix form,
〈Φip(t)〉 ) nj
∫0t dt′ Y(t, t′) exp{-nj[1 - Pex(t′)]}
(31)
where
Y(t, t′) ) [e-t′Qf‚Wf‚e-(t-t′)Qb]0,0
(32)
Now we can either evaluate the evolution matrices numerically or represent them as a superposition of exponentials following the procedure outlined in section 2. Both algorithms are very fast, particularly in comparison with numerical solution of the corresponding partial differential equations. Figures 2 and 3 show excellent agreement of the matrix solution with numerical results for a typical set of parameter values given in the figure caption and chosen from ref 14. For
Figure 3. Ion pair state survival probability, 〈Φip(t)〉, in the absence of interaction for fast (1) and slow (2) ionization. The set of parameters for the case of fast ionization is given in the caption to Figure 2. For the case of slow ionization, the electron transfer parameters are J0f ) 20 cm-1, βf ) 0.7 Å-1, ∆Gf ) -0.5 eV, J0b ) 600 cm-1, βb ) 1.1 Å-1, and ∆Gb ) -2.0 eV. Other parameters are the same. Donor self decay lifetime is not included. The solid lines represent the matrix solution and the circles are the results of numerical solution.
all of the parameter sets tested, no more than 10 eigenmodes were required to reach convergence. 4. Concluding Remarks In our previous work, we have shown that in the absence of interaction potential between reactants undergoing an irreversible reaction on a micelle surface, an exact solution for the pair survival probability with an arbitrary distance-dependent reaction term can be written in a simple matrix form using the eigenmodes of the diffusion operator without reaction. In this
9160 J. Phys. Chem. B, Vol. 103, No. 43, 1999 paper, we have considered the most general case of an irreversible reaction with interaction potential. In principle, exactly the same procedure could be used if we only knew the eigenmode expansion of the diffusion operator involving an arbitrary interaction potential with reflecting boundary condition, which is not trivial. We have found a way of bypassing this problem and expressed the pair survival probability in terms of the eigenmodes of the diffusion operator without interaction potential. The resulting matrix expression is simple and shows rapid convergence upon truncating the matrices, offering a great computational advantage over numerical integration of the corresponding partial differential equation. We have also considered a practically important example of a multistage reaction, photoinduced electron transfer followed by back transfer, and presented matrix expressions both for the excitedstate and the ion pair state survival probabilities. It should be emphasized that virtually any reaction in any geometry, even if it involves multiple stages and reverse channels, can be formulated in terms of matrices, provided the system is confined. We only need to somehow calculate the corresponding eigenmodes. This offers a possibility to analyze systems close to real, taking into account diffusion, arbitrary distance dependence of the reaction rate, interaction potential, and all that by rather simple numerical means. References and Notes (1) Fo¨rster, T.; Selinger, B. K. Z. Naturforsch. Teil A 1964, 19, 38. (2) Infelta, P. P.; Gra¨tzel, M.; Thomas, J. K. J. Phys. Chem. 1974, 78, 190. (3) Tachiya, M. Chem. Phys. Lett. 1975, 33, 289. (4) Tachiya, M. In Kinetics of Nonhomogeneous Processes; Freeman, G. R., Ed.; Wiley: New York, 1987; p 575. (5) Kalyanasundaram, K. Photochemistry in Microheterogeneous Systems; Academic Press: Orlando, 1987.
Barzykin et al. (6) Gra¨tzel, M. Heterogeneous Photochemical Electron Transfer; CRC Press: Boca Raton, 1989. (7) Almgren, M. In Kinetics and Catalysis in Microheterogeneous Systems; Gra¨tzel, M., Kalyanasundaram, K., Eds.; Marcel Dekker: New York, 1991; p 63. (8) Gehlen, M. H.; De Schryver, F. C. Chem. ReV. 1993, 93, 199. (9) Barzykin, A. V.; Tachiya, M. Heterog. Chem. ReV. 1996, 3, 105. (10) Almgren, M.; Grieser, F.; Thomas, J. K. J. Am. Chem. Soc. 1979, 101, 279. (11) Barzykin, A. V.; Tachiya, M. J. Phys. Chem. B 1998, 102, 1296. (12) Rothenberger, G.; Gra¨tzel, M. Chem. Phys. Lett. 1989, 154, 165. (13) Weidemaier, K.; Fayer, M. D. J. Chem. Phys. 1995, 102, 3820. (14) Weidemaier, K.; Fayer, M. D. J. Phys. Chem. 1996, 100, 3767. (15) Matzinger, S.; Weidemaier, K.; Fayer, M. D. Chem. Phys. Lett. 1997, 276, 274. (16) Tachiya, M. Chem. Phys. Lett. 1980, 69, 605. (17) Sano, H.; Tachiya, M. J. Chem. Phys. 1981, 75, 2870. (18) Hatlee, M. D.; Kozak, J. J.; Rothenberger, G.; Infelta, P. P.; Gra¨tzel, M. J. Phys. Chem. 1980, 84, 1508. (19) Van der Auweraer, M.; Dederen, J. C.; Gelade´, E.; De Schryver, F. C. J. Chem. Phys. 1981, 74, 1140. (20) Barzykin, A. V.; Tachiya, M. J. Chem. Phys. 1993, 99, 7762. (21) Ediger, M. D.; Fayer, M. D. J. Chem. Phys. 1983, 78, 2518. (22) Baumann, J.; Fayer, M. D. J. Chem. Phys. 1986, 85, 4087. (23) Blumen, A.; Klafter, J.; Zumofen, G. J. Chem. Phys. 1986, 84, 1397. (24) Berberan-Santos, M. N.; Prieto, M. J. E. J. Chem. Soc., Faraday Trans. 2 1987, 83, 1391. (25) Barzykin, A. V. Chem. Phys. 1991, 155, 221. (26) Weidemaier, K.; Tavernier, H. L.; Fayer, M. D. J. Phys. Chem. B 1997, 101, 9352. (27) Tavernier, H. L.; Barzykin, A. V.; Tachiya, M.; Fayer, M. D. J. Phys. Chem. B 1998, 102, 6078. (28) Seki, K.; Barzykin, A. V.; Tachiya, M. J. Chem. Phys. 1999, 110, 7639. (29) Barzykin, A. V.; Seki, K.; Tachiya, M. J. Phys. Chem., in press. (30) Sano, H.; Tachiya, M. J. Chem. Phys. 1979, 71, 1276. (31) Marcus, R. A. J. Chem. Phys. 1956, 24, 966. (32) Barzykin, A. V. Chem. Phys. 1992, 161, 63. (33) Bales, B. L.; Stenland, C. J. Phys. Chem. 1993, 97, 3418.
