4064
J. Phys. Chem. 1996, 100, 4064-4070
Transient Effect in Fluorescence Quenching by Electron Transfer. 3. Distribution of Electron Transfer Distance in Liquid and Solid Solutions Shigeo Murata* and M. Tachiya* Department of Physical Chemistry, National Institute of Materials and Chemical Research, Tsukuba, Ibaraki 305, Japan ReceiVed: September 18, 1995; In Final Form: NoVember 30, 1995X
We have calculated the distribution Y(r) of electron transfer distance which is defined in such a way that Y(r) dr represents the probability of electron transfer over a distance between r and r + dr. The effect of ∆G on the Y(r) distribution was evaluated using the following parameters: the transfer integral J0 at 6 Å, its attenuation coefficient β with distance, and the diffusion coefficient D. These parameters have already been determined experimentally (J0 and β) or empirically (D) in a previous paper (J. Phys. Chem. 1995, 99, 5354). The parameter values were then changed to include the more commonly used solvent, acetonitrile, and a wide range of electron transfer rates. These results indicate that the distribution is not only a function of ∆G but also a function of the quencher concentration. These results cannot be obtained simply by referring to the distance dependence of the rate constant of the reaction. The distribution considered here has experimental significance, since it is directly related to the yield of free ions.
1. Introduction Photoinduced electron transfer is one of the simplest chemical reactions and plays important roles in many chemical and biological processes. Since the excited state of either electron donor (D) or acceptor (A) is involved in it, it can be studied by measuring the quenching of fluorescence. In previous papers,1,2 the authors measured the transient effect in fluorescence quenching at quencher concentrations of 0.11 and 0.2-0.4 M2 in solution. Analysis of the fluorescence decay curves was made by the method based on the Collins-Kimball model1 and by the method which uses the Marcus equation of electron transfer and the solution to the diffusion equation.2 It has been demonstrated that the latter method gives the decay curves which are in better agreement with experiment compared with the former. By comparing the calculated and the experimental fluorescence decay curves, the parameters involved in the Marcus equation were determined for several D-A pairs in ethylene glycol (EG) solvent. Similar studies using an exponentially decaying function of r instead of the Marcus equation have been reported by Eads et al.3 and by Song et al.4 Using the parameters thus determined, we can calculate the magnitude and the distance dependence of the electron transfer rate for each D-A pair in the solvent employed. Although the electron transfer rate is well characterized by its magnitude and distance dependence, other quantities are also sometimes useful in studying electron transfer in more detail. Analysis based on the Collins-Kimball model gives the reaction radius which makes the physical images of the reaction clearer. However, as described above, this method is less accurate than that based on the Marcus equation. In the Collins-Kimball model it is assumed that electron transfer occurs only at the reaction radius. However, in real systems it occurs over a range of donoracceptor distances. In this paper we evaluate the distribution of electron transfer distance in liquid and solid solutions. In comparison with the reaction radius in the Collins-Kimball model, this quantity can characterize the reaction more accurately, since it is based on more appropriate expressions of the distance dependence of the reaction rate. In addition, this X
Abstract published in AdVance ACS Abstracts, February 1, 1996.
0022-3654/96/20100-4064$12.00/0
distribution has experimental significance, since it is directly related to the yield of free ions. Similar quantities have been discussed by Burshtein et al.5,6 In the calculations of the distribution of electron transfer distance in EG solution, we used the parameter values that we had determined experimentally in a previous paper. In addition, calculations with different parameter values were also carried out in order to investigate the dependence of the distribution on these parameters. 2. Method of Calculation First, the method of calculating the fluorescence decay curve in the presence of a quencher is briefly summarized.2 Suppose electron donor (D) and acceptor (A) molecules are dispersed in a medium (liquid or solid) and fluorescence is quenched by electron transfer. Here, for convenience, we assume that A is excited and its fluorescence quenched by D. The fluorescence decay P(t) is theoretically expressed by eq 1.7
P(t) ) exp(-t/τ0 - 4πc0∫d [1 - U(r,t)]r2 dr) ∞
(1)
Here, τ0 is the fluorescence lifetime in the absence of the quencher, c0 the quencher concentration, and d the sum of the molecular radii of D and A. U(r,t) stands for the survival probability of a D-A* pair at time t which was initially (at time 0) separated by distance r. In solid solution where diffusion of D and A can be neglected, U(r,t) is simply given by eq 2.8
U(r,t) ) exp[-k(r)t]
(2)
where k(r) is the first-order quenching rate constant. In liquid solution D and A undergo mutual diffusion, and in that case it has been shown that U(r,t) satisfies the diffusion equation of the following form:9
(
)
∂U(r,t) ∂2 2 ∂ U(r,t) - k(r) U(r,t) )D 2+ ∂t r ∂r ∂r
(3)
where D is the sum of the diffusion coefficients of D and A. Equation 3 must be solved under the following initial and boundary conditions: © 1996 American Chemical Society
Fluorescence Quenching by Electron Transfer
[
J. Phys. Chem., Vol. 100, No. 10, 1996 4065
U(r,0) ) 1
(4)
U(∞,t) ) 1
(5)
]
(6)
∂U(r,t) ∂r
r)d
2π 2 J exp{-β(r - r0)} p 0
1
x4πλkBT
×
[
(∆G + λ) 4λkBT
]
)(
1 1 1 2 e2 1 + 2 op s a b r
)
(7)
(8)
where op and s are the optical and static dielectric constants and a and b are the molecular radii of D and A. We see from eqs 1-8 that the parameters determining P(t) are J0, β, and D in the case of liquid solution. In the previous work, eq 3 coupled with eqs 4-8 was solved by numerical calculation, and the solution was used to calculate P(t) using eq 1. P(t) thus calculated was compared with experiment to determine J0 and β values corresponding to the best fit. The comparison was made by nonlinear least-squares method. Using these values, we can calculate from eq 7 the rate constant of electron transfer as a function of D-A separation in EG solution (see Figure 1). In their experimental studies of electron transfer in solid solutions, Miller et al.11,12 have shown that the results can be very well analyzed by assuming the following form for k(r).
k(r) ) ν exp(-βr)
(9)
Equations 7 and 9 are formally similar: eq 9 has a large number of parameters combined into the constants ν and β. We hereafter assume eq 9 for the rate constant in solid solution. Equations 1, 2, and 9 show that the parameters determining P(t) in solid solution are ν and β. Let us consider the A molecule excited at t ) 0. Electron transfer occurs over a range of D-A* distance, and its firstorder rate constant is given by eq 7 or 9. Let Y(r) dr denote the probability that electron transfer occurs over a distance between r and r + dr in the time interval from zero to infinity. Here we present a method to calculate Y(r). The total probability T of electron transfer from D to A* is given by eq 10.
T ) ∫d Y(r) dr ∞
(10)
On the other hand, T can be derived in the following way. The second-order rate constant k(t) is given using k(r).13,14
k(t) ) ∫d 4πr2k(r) q(r,t) dr ∞
(12)
where [A*]0 is the concentration of A* at t ) 0. P(t) is the fluorescence decay function given by eq 1. Integration of s(t) over t gives the concentration of A* which decayed by electron transfer. Consequently, the total probability (or the quantum yield) T of electron transfer is given by ∞
where J0 is the transfer integral at r ) r0 ()6 Å in this paper), β is its attenuation coefficient, and ∆G is the free energy change of the reaction. The solvent reorganization energy λ is also a function of r:
(
) (∫d 4πr2k(r) q(r,t) dr)c0P(t)[A*]0
T ) (∫0 s(t) dt)/[A*]0
2
exp -
λ)
s(t) ) k(t)[D][A*] ∞
)0
It is now necessary to assume a functional form of k(r). For electron transfer in liquid solution, we assume the Marcus equation10 for k(r):
k(r) )
case of solid solution and eq 3 in the case of liquid. The electron transfer rate s(t) is given using eq 11.
(11)
where q(r,t) ) c(r,t)/c0; c(r,t) is the concentration of D (quencher) at distance r from A* at time t, and c0 is its bulk concentration. It can be shown7 that q(r,t) satisfies eq 2 in the
) ∫d (4πr2c0k(r)∫0 q(r,t) P(t) dt) dr ∞
∞
(13)
By comparing eqs 10 and 13, we get
Y(r) ) 4πr2c0k(r)∫0 q(r,t) P(t) dt ∞
(14)
Y(r) is hereafter referred to as the distribution of electron transfer distance. We see from eq 14 that the distribution of electron transfer distance is a function of the first-order rate constant k(r), the distribution q(r,t) of quencher molecules, and the fluorescence decay function P(t). This indicates that the distribution depends on the parameters J0, β, and D in the case of liquid solution and on ν and β in the case of solid solution. If the upper limit of integration is replaced by t in eq 14, it gives the distribution accumulated up to time t.
Y(r,t) ) 4πr2c0k(r)∫0 q(r,t) P(t) dt t
(15)
The calculation of eq 14 for solid solution (in which eqs 1, 2, and 9 are used) can be made by numerical integration using the double-exponential formula.15 To calculate Y(r) for liquid solution, eq 3 coupled with eqs 4-8 must be solved numerically. It was solved using the program prepared previously.2 P(t) and q(r,t) thus obtained were used to calculate Y(r) by the trapezoidal rule. 3. Results and Discussion The reactant radii were obtained following the method described in literature.16 They are (in angstroms) as follows: CA, 3.6; DCA, 3.7; ANS, 3.1; DMA, 3.2; ANL, 2.9. (See Table 1 for abbreviations.) The values of D were calculated using the Stokes-Einstein equation.2 A. Liquid Solution. A. 1. Ethylene Glycol (EG). In the case of liquid solutions, the parameters needed in the calculation of Y(r) are J0 and β. These values have already been determined experimentally in EG for some D-A pairs in a previous paper,2 and those used in the present paper are summarized in Table 1. Since we could not determine the parameter values corresponding to ∆G ) -2 eV, the same values as those for ∆G ) -1.26 eV were assumed. Figure 1 shows the distance dependence of the reaction rate calculated from eq 7. It is seen that the r dependence of k(r) is nearly exponential, except for ∆G ) -2.0 eV. In Figure 2 is shown Y(r) for four different ∆G values. The encounter distance d is different for different pairs, since the molecules have different sizes. We can see from this figure that electron transfer occurs not at a fixed distance but over a range of distance. This is quite natural, but the conventional method based on the Collins-Kimball model does not account for this effect. Figure 2 also shows that the distribution of
4066 J. Phys. Chem., Vol. 100, No. 10, 1996
Murata and Tachiya
TABLE 1: Parameter Values Obtained Experimentally2 in EG Solution and Used in This Papera ∆G (eV) -0.73 -1.02 -1.26 -2.0
A
D
CA DCA DCA
DMA ANL ANS
J0 (cm-1)
β (Å-1)
τ0 (ns)
D (10-5 cm2 s-1)
25 30 40 40
0.9 0.9 1.1 1.1
14.8 13.1 13.1 13.1
0.094 0.098 0.094 0.094
a The values corresponding to ∆G ) -2 eV were assumed to be the same as those corresponding to ∆G ) -1.26 eV. J0 is the transfer integral at 6 Å. CA ) 9-cyanoanthracene, DCA ) 9,10-dicyanoanthracene, DMA ) N,N-dimethylaniline, ANL ) aniline, and ANS ) p-anisidine. τ0 is the fluorescence lifetime of the A molecule, and the diffusion coefficient D was calculated using the Stokes-Einstein equation.
Figure 1. First-order electron transfer rate in liquid EG (curves with ∆G values) and in solid solution calculated from eqs 7 and 9. The ∆G values correspond to the D-A pairs in Table 1.
Figure 3. Time variation of q(r,t) in EG solution. c0 ) 0.3 M. The three curves in the same set correspond, from left to right, to t ) 0.05, 0.5, and 5 ns, respectively. The ∆G values correspond to the D-A pairs in Table 1.
already been qualitatively predicted from the different r dependence of k(r) for different ∆G values1,17,18 but is shown quantitatively for the first time in Figure 2. It is interesting to note that the curves corresponding to ∆G ) -1.02 and -1.26 eV have a maximum at r ) 8.6 and 9.1 Å, respectively, and the curve corresponding to ∆G ) -0.73 eV has a shoulder at r ) 7.3 Å. The appearance of these maxima and shoulder cannot be explained simply by referring to the distance dependence of the rate constant; in these three cases, the rate constant monotonously decreases with increasing r as shown in Figure 1. The appearance of these maxima and shoulder can be explained by considering the fate of D and A* molecules which come close to each other by diffusion from r ) ∞. Since the rate constant rapidly decreases with r, the rate of electron transfer is very low at larger r. As the D-A* pair comes closer, the rate of electron transfer increases. Consider D-A* pairs which have arrived at a region r ∼ r + ∆r without suffering electron transfer. Some fraction of the pairs will suffer electron transfer there and will be lost, while the other fraction will survive against electron transfer and continue to approach each other. Since the average time for which the pairs stay in the region r ∼ r + ∆r is roughly given by (∆r)2/D where D is the diffusion coefficient, the probability that the pairs will suffer electron transfer in the region r ∼ r + ∆r is given by k(r)(∆r)2/D. This probability is very low at large r and increases with decreasing r. Once this probability becomes significant at certain position r, namely,
k(r)(∆r)2/D ≈ 1
Figure 2. Distribution of electron transfer distance for different ∆G values in EG solution. c0 ) 0.3 M.
electron transfer distance is different for reactions with different ∆G values. In particular, the distribution shifts toward larger r values on going to larger -∆G values. The median s of the transfer distance which is defined by eq 16
∫dsY(r) dr ) ∫s∞Y(r) dr
(16)
is 8.1, 9.1, 9.6, and 10.9 Å for -∆G ) 0.73, 1.02, 1.26, and 2.0 eV, respectively. This shift is consistent with the previous observation1 that the reaction radius in the Collins-Kimball model increases as the value of -∆G increases. This shift has
(17)
a large fraction of pairs will be lost there by electron transfer and only a small fraction is left for further approach. The maximum and shoulder in the above three cases correspond to this situation. The choice of the value of ∆r is somewhat arbitrary. If one takes ∆r as 1 Å by referring to the widths of the peaks in the curves corresponding to ∆G ) -1.02 and -1.26 eV, the above equation predicts that the maxima and shoulder occur at the position where k(r) becomes ≈1010 s-1. This prediction is more or less consistent with results shown in Figures 1 and 2. In the case of ∆G ) -2.0 eV the maximum in Figure 2 roughly corresponds to the maximum of the rate constant shown in Figure 1. We have described above that the maxima in Figure 2 for reactions with ∆G ) -1.02 and -1.26 eV appear because a large fraction of D-A* pairs reacts around the maxima, and only a small fraction is left for further approach. Figure 3 shows this more quantitatively: for the reaction with ∆G ) -1.26 eV, q(r,t) values at t g 0.5 ns are small at r ≈ 9 Å and approach 0 at smaller r (
