8114
J. Phys. Chem. 1994, 98, 81 14-81 17
Charge Transfer in Restricted Geometries: Binary Approximation M. S. Mikhelashvili' and A. M. Michaeli Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem 91 904, Israel Received: February 23, 1994; In Final Form: May 13, 1994"
W e study the time dependence of donor-acceptor forward and back electron transfer in solids with restricted geometries in binary approximation. W e focus on the time dependence of the donor cation state probability, P ( t ) within spherical and cylindrical volumes. Deviations from the known Forster type behavior are observed due to the geometrical restrictions. The effects of the finite volumes are pronounced when thevolume dimensions are comparable to the critical radius for charge transfer. The space restriction slows down the charge separation. The results are good in the limit of low concentrations of acceptors.
1. Introduction Electron transfer (ET)-one of the most important molecular elementary reactions-has been the subject of lively investigation for the past years.1-4 Recent theoretical and experimental advances have provided an understanding of kinetics of the forward and back charge transfer in solids and liquids in binary approximation.- Separated ions are extremely reactive; however, back ET prevents their effective separation. In refs 5-7 the ensemble average for a donorxation probability P ( t ) has been derived for solid and liquid solutions in the infinite volume. These studies as well as lots of studies of electron transfer in solutions8 have focused on the infinite volume systems with an infinite number of molecules. There is a variety of experimental situations in which the electron energy or the charge transport occurs where the volume or the number of molecules involved may be too small to allow the application of the infinite volume solution theories. Examples of it, such as micellar systems, polymers in solution, porous silica gels, zeolites, and complex assemblies of biological interest such as membranes and cells, are both of technological and of scientific interest.gJ0 Photochemical reactions and electron excitation or chargetransfer processes in finite volumes display characteristics which distinguish them from comparable reactions in homogeneous solvents. These processes in finite volumes can give information on dimensions of volumes of micelles, information on their structures and dynamics, and the methods for obtaining this information.9-i7 In ref 14 the authors concentrate on the relaxation of an excited donor due to the direct energy transfer to randomly distributed acceptors in model systems for which the spatial geometry is restricted. In this study we discuss the forward and back chargetransfer kinetics in a restricted space in binary approximation6V8 (thelow concentration of acceptors) on the basis of theory? which is sufficiently simple for analysis and for computations. It is possible to solve the problem of the restricted geometries considered here also on the basis theorysbq6with a novel averaging procedure. The results depend on the location of the donor and on underlying geometry.gJ4 We focus on two examples, involving spherical and cylindrical volumes, which are the conventional models for simple pores, and may also describe the micellar and zeolite casesg We show that there are significant deviations in the behavior of finite volume systems compared to those of infinite volume systems. This occurs as the dimensions of the finite volume become comparable to the distances associated with charge transfer;'$ the reaction space restriction essentially influences the charge separation. The results are good in the limit of low concentrations of acceptors. Experimental studies of photoinduced charge-transfer phenomena in some confined systems (micelles, vesicle, high polymers @
Abstract published in Advance ACS Abstracts. July 1 , 1994.
0022-365419412098-8 114$04.50/0
or monolayers) have been important subjects of recent investigations (see, for example, ref 9a, p 23 (Noburo Mataga and references therein)). 2. Forward and Back Charge Transfer
In this section we get a general expression of the donor-cation state probability for the donor molecule due to the direct charge transfer to acceptors which occupy some of the sites of a given structure in finite space. We take into account the consequence of back charge transfer to the same donor. In order to show the logical extension of infinite volume systems, we repeat the initial steps of Klafter and Blumen in ref 14. We follow the methods of Antonov-Romanovskii and Galanin18J9to analyze the ensemble averaged donor-cation state probability in finite space; we use the two-particle approximation. We consider a solid solution containing a certain concentration of excited donor molecules of type S*. The deactivation of the S* excited molecule occurs due to charge transfer to another acceptor molecule A and due to its spontaneous transition to the ground state. The acceptors are the initially unexcited and randomly distributed in finite space, but otherwise identical, molecules. 18920 Fluorescence quenching of molecular excitations due to the charge transfer among randomly distributed donors and acceptors is similar to the linear energy transfer, which is well-known and is being widely studied nowadays.8J8-22 A common difficulty of all the quantitative analyses mentioned is that one has to take into account the multiparticle correlations. The Galanin equationsl8 describing the resonance energy-transfer cases take into account only two-particle correlations, Le. interaction of the closest partners in pairs. So the excitation decay of the donor molecule due to the charge transfer to an acceptor to some site is assumed to be unaffected by the presence of an acceptor at some other sites. This allows one to reduce the n particle problem of the donor decay to a superposition of two-particle problems. The number of acceptors in any local region is not strictly fixed; the density of acceptors can vary from one region to another. This is a necessary requirement while dealing with restricted geometries.Il At low acceptor concentrations, the lattice site occupancy is small. In this case the summation over lattice points can be replaced by an integration over space, and we can use the method of AntonovRomanovskii and Galaninl8-25 which was first suggested for electronic energy transfer in solid solution. We use the auxiliary function-the donor-noncation state ensemble average probability:7 P J t ) = 1 - P(t)
whereobviously P ( t ) is thedonor-cation state average probability, 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 33, 1994 8115
Charge Transfer in Restricted Geometries From this follows the equation for the derivatives:
P ( t ) = -Pnc(r) This expression simplifies the solution of our problem, so it reduces the problem to determination of Pnc(t). For this function, in the absence of the back charge transfer, Galanin’s’* theory and its equations describing the donor excited-state relaxation in solutions~*J9are valid. However, obviously, this approximation is valid and also correct for the account of the back charge transfer in our case of only two-particle correlations. The back charge transfer probably occurs to the primary or initial partner S. The authors of refs 5 and 6 use another method for deriving the donor-cation state probability for solutions. The results of refs 5-7 coincide for low acceptor concentrations.” More detailed analysis and comparison of these theories of the reversible charge transfer are presented in ref 7d. Here our discussion is completed on the basis of t h e ~ r y which ,~ is simple for analysis and for computations. According to refs 7, 18, and 19, the donor-noncation state probability is determined by dNn,(t)/dt = 47rNJt) JdC(r,t)/dt
3dr
(2)
For the donor-cation state probability we have7a
P ( t ) = 1 - exp[-JR12 C(r,t) d T]
(3)
Here 47rrZC(r,t) dr represents the average number of acceptor anions A-, which are located a t a distance between r and r + dr from the donor in the cation state S+,calculated for one of the initially excited donors. In accordance with the reaction schemez1-z2
-+ + - + -+
In eq 7 P(r,ro,t) = C(r,ro,t)/C. For eq 7 one now arrives at the Forster type decay by taking go(r) = g = constant and extending the integration over the whole space. Vis the reaction volume (of restricted space). More complex situations can be studied by introducing different forms of go(r). In the next section we study thedecay for models, when thedonor islocated on the pore surface. 3. Examples of Time Dependence of f i t ) in Regular Restricted Geometries We now consider two examples which show the role played by the geometry in modifying the regular relaxation laws for the donor-cation statedue to the reversiblecharge transfer. We study models of spherically and cylindrically shaped pores, which can be treated by choosing the appropriate forms of go(r). For each geometry one has to evaluate the following integral, which appears in the exponent of eq 3:
P ( t ) = 1 - exp[-pl(t,ro)] I(t,ro) = Jgo(r) P(r,ro,t) d r
(8)
We consider the acceptors to be distributed in a spherical pore of radius R , and we calculate the P ( t ) for a donor placed on the surface of the sphere, say on the Z-axis, at Ro = (O,O,R). In order to calculate go(r) we use spherical coordinates centered a t Ro. Since kf(r) and kb(r) depend only on r, we need the r-dependence of go(r). For this weevaluate thevolume comprised between two spherical shells centered a t Ro: go(r) d r = g 3 d r s,’”dy
g m s i n t9 dt9 = 2 r 3 ( 1 - r/2R)g d r
(9)
S* + A
S+ A-
with the rate constant kf(r)
with Omax = arccos(r/2R). Inserting eq 9 into eq 8, we obtain
S+ A-
S A
with the rate constant kb(r)
Z(t,Ro)= 27rg r 3 P ( r , r o , t ) d r - ( 7 r g / R ) r r3P(r,r0,f) d r
S*
S
with the rate constant 1 / ~ (4) hv the concentration C(r,t) is determined by the expression’
If R
-
m,
(10) one can neglect the second term and then we get Z(t,R,) = 2 7 r g r 3P(r,ro,t) d r
exP[-(kf(r) + 1 / d t I l ( 5 ) In eq 3 Rlz is the sum of the radii of molecules S and A. We see that at the time t = 0 [C(r,O) = 01 the probability P(0) = 0. The rate constants for the forward and back electron transfer depend on the electronic coupling between the donor and acceptor (or their ions) which in turn exponentially depend on the intermolecular distance r (see ref 2, chapter 3, and ref 22): kf(r) = (1/7) exp[-(r - Rf)/af]
forward transfer
back transfer (6) kb(r) = ( 1 / ~ )exp[-(r - R,)/a,] Rf and Rb are characteristic parameters of the theory;z1.22af and ab characterize the exponential decrease with distance of the electronic wave function of the neutral and ionic states of the donor and acceptor. The space restriction, obviously, must influence only the integral in eq 3. By introducing a site-density function go(r),14we obtain in place of (3) the expression
where the index 0 in go(r) acts as a reminder that ro is a given donor location. The multiplierp before the integral in eq 7 is the probability of a substitutional occupancy of sites by acceptors.
This result is similar to the result in an infinite three-dimensional space.18-20 The difference by a factor of 2 in the exponent of eqs 7 and 11 is due to the fact that the donor on the surface sees, as R 43, only half of the space occupied by the acceptors. We now consider the corrections due to the finite volume. At long times t , k(r) >> 1 for all r inside the volume V; one can neglect the exponential terms in eq 5, and thus we obtain Z(t,Ro) = 0 and the donor cation state probability P ( t ) = 0. This is the consequence of the volume restriction if the back charge-transfer rate constant kb(r) > 0. If kb(r) = 0, then in this long-time case I(t,Ro) = constant and so P ( t ) = constant > 0, as we see from eq 10 (time-independent value). In this case
-
It is seen that if Rf > R [see eqs 8-10], then the 1 / term ~ in eq 12 may be neglected and therefore Z(t,Ro) = 27rgJOzR(1 - r/2R)? d r = (4/3)7rR3g (13) is the function of the radius R, and P ( t ) is determined by the finite volume (4/3)irR3:
-
-
P ( t ) = 1 - exp[-(4/3)7rR3pgl
(14) If R m, then P ( t ) 1 in the case of k~,(r)= 0. If R > Rf, then the numerical integration of expression 8 [or P ( t ) ] shows that the probability P ( t ) is a nondecreasing function of time t
Mikhelashvili and Michaeli
8116 The Journal of Physical Chemistry, Vol. 98, No. 33, 1994 0.8
,
(7) is a linear function of time in the restricted volume and is essentially determined by molecule size. From (10) and ( 5 ) we get P(f/T
{/,...... ,
... ...................
......
.
.
~~
05
c
15 t -e
2c
25
1
3c
(t/-)
Figure 1. Time dependence of P(t) for the radius of the sphere: R = 7, 12, and 25 A; Y = t / r = 0. Solid lines: Rf = 13 A; Rb = 0; C = 1 M; Of = ab = 1 A. Dotted lines: Rf = 13 A; kb(r) = 0. I
'0 I
(15)
Here also the back charge transfer prevents the charge separation in the finite volume. The direction of the corrections, as well as in the energy-transfer case,14 is evident, since not all of the half-space seen by the donor is accessible to acceptors. The restricted geometry slows down the charge separation. The next example presents the effects of a cylindrical volume on the charge-transfer process. We assume an infinite cylinder of radius R. The acceptors are distributed within thevolume; the donor is located on the surface of the cylinder at the origin 0 = (O,O,O). The axis of the cylinder lies at distance R from the origin along the 2 direction. The site density function for this case is14
3C 0 0
0. If kb(r) = 0, then we get
3C
time (t/-)
Figure 2. Time dependence of P(t) for the radius of the cylinder: R = 10 and 15 A; Y = t / r = 0. Solid lines: Rr = 13 A; Rb = 10 A; C = 1 M; af ab = 1 A. Dotted lines: Rr = 13 A; kb(r) = 0.
when Y = 1 / ~ - 0 , for any kb(r) > 0 (Figures 1 and 2). Therefore thedecrease of P ( t ) is determined not by the back charge-transfer [kb(r) = 01 but by the spontaneous decay of the excited donor up to the time t / T = 3. On the other hand, if kb(r) = 0, then P ( t ) is an increasing function of the time as per formulas 7-1 1, so the results of works 5 and 6 (and ref 7b) and their meanings are very close. Moreover it is interesting to note, that the law of decay of the donor excited state in solids for the long time t / T >> 1 is determined by the number of acceptors in the volume VO= (4/ ~ ) T R The ~ ~ cation . ~ ~ state probability for the f / r >> 1 is determined by the acceptor numbers in thevolume V, (restricted space). This vague circumstance requires further elucidation. So, result 10 is a function of the acceptor's concentration and molecular constants Rfand af.It is a linear time-dependent value, as well as the respective result for the direct excitation energy transfer from donors to acceptors, for which the result is Z(t,Ro) = gVand the decay F - exp(-gv), and therefore gets a nonzero value.14 Here, we also receive thedecay for the half-space (derived for R a),corrected so as to obtain a slower decay at moderate times tk(r) < 1. There exists a crossover in the time-dependent-type behavior of the donor-cation state probability [for tk(r) > 11 due to the space restriction. If kb(r) = 0, then the probability P(r) is a nondecreasing function of the time for any volume. But, if k&) > 0, then in this case P ( t ) 0 at the time t / T >> 1, and this reduction is more sharp in the finite space. For the small times f / T 0, and the back charge transfer is sensitive to the volume alteration. For R > Rf this influence of the volume on the kinetics of P ( t ) is not essential; the value of R determines in this case only the maximum of P ( t ) and does not influence the curves slope. The same character of the curves descent for P ( t ) takes place also for the cylinder, although for this case the influence of R (at the condition R < Rf) is visibly less than that for the sphere. It is, obviously, due to the increase of total volume of the reaction space in the case of the cylinder. One interesting effect of the volume restriction is the independence of the probability P ( t ) on the rate constant k f ( r ) ,if Rf > Rb > R. In this case in eqs 9 and 10 one can neglect the terms kb(r) and t / r and also the secondexponential term in the brackets under the integral; thus the probability P ( t ) is independent of Rf. A real interest is present also in the influence of the overlap of wave functions and the solution properties on the above
Charge Transfer in Restricted Geometries
The Journal of Physical Chemistry, Vol. 98, No. 33, 1994 8117
0.28 I
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Acknowledgment. Our thanks are due to Professor J. Klafter and Professor D. Avnir for helpful discussions. We thank M. Dodu for computer simulations. This work was supported by Ministry of Science and Technology of Israel (Grant No. = 032.7290.132).
0.8
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location and that of the acceptors. They show richer patterns than that obtained for regular infinite structures. For all the pore models studied we find slower charge separation than that for the infinite structures. Moreover, in the finite volume the maximum value of P ( t ) is limited by the back charge-transfer rate constant and does not increase with unlimited increasing of the characteristic distance Rfof the forward charge-transfer rate constant (Figure 3). It is possible also to solve the problem for the restricted geometries considered here on the basis the theory of refs 5b and 6. The theoretical analysis and computer simulations show the qualitative analogy of our results and the results of works 5 and 6 for solid solutions in finite and infinite spaces; quantitatively these results are very close for low concentrations of acceptor^.'^ The obtained results are good in the limit of low concentrations of acceptors. So, the space restriction essentially influences the maximum of the donor-cation state probability, the charge separation fate, and its time dependence.
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References and Notes
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Figure 3. Time dependence of P ( t ) for the characteristic distance: Rf = 7,10,and>20A. (a,top) RadiusofthesphereR= IOA. (b,bottom) Radius of the cylinder R = 10 A; C = 1 M; af = 1 A; kb = 0; Rf = 20, 40, and >60 A.
mentioned volume effects. We hope to discuss these aspects in our further work.
4. Conclusion The efficiency of the back charge transfer increases with a decrease in the reaction volumes (Figures 1-3). The radius of the sphere influences the long-time behavior of the donor-cation state probability P ( t ) more strongly than one of the cylinder's radii. The volume's restriction prevents charge separation. For all the pore models studied (sphere and cylinder), we find crossoven in the behavior of between the P ( t ) forms for different time domains. The short-time ( t / T
