J. Phys. Chem. 1994,98, 7319-7324
7319
Binary Photoionization Followed by Charge Recombination A. I. Burshtein' Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel
E. Krissinel Institute for Water and Environmental Problems, Barnaul656099, Russia
M. S. Mikbelashvili Department of Chemical Physics, The Hebrew University, Jerusalem, 91904 Israel Received: February 3, 1994'
The initial distribution of charges arising in the course of bimolecular ionization as well as subsequent ion recombination and separation is calculated. The ionization is shown to be contact only in the kinetic control (fast diffusion) limit while in the course of diffusion-controlled and static reactions the majority of ions are born far from each other. At moderate and slow diffusion, spatial dispersion of the electron transfer rates and self-generated initial conditions for recombination must be taken into account to find the real quantum yield as well as the kinetics of charge separation.
Introduction Remote electron transfer is widely studied as an efficient mechanism of energy quenching in solution. Following light excitation of donor (D), the ionization proceeds according to the kinetic scheme' D*
+A
-
[D+-.A-]
(1)
where A is the acceptor of electrons. At sufficiently low concentration of acceptors, c, they may be considered as point particles. In this case thedifferential encountertheory, developed in refs 2-5, best describes the quenching kinetics and quantum yield of luminescence.6 When ionization is diffusion controlled or static, the best fit to experimentaldata shows that the effective ionization radius significantly exceeds the closest approach distance &?-lo Contrary to this conclusion, in any experimental study of back electron transfer the recombination rate is calculated within a primitive "exponential model" which implies the counterionsare created in contact and then react accordingto the kinetic scheme11 [D-*A] + [D+-*A-]
-+
D+
+ A-
(2)
Much more appropriate is the so called "contact approximation", based on the Smoluchowski or the Collins and Kimball approach, which also assumes that recombination occurs at contact but leaves open the question of where ions 0riginate.12-~5 It has been shown recently that the contact approximation is restricted to relatively fast diffusion. In a quasistatic limit, the positiondependent recombination rate must be used with reflecting boundary conditions.16 Another disadvantage of contact approximation is the uncertainty of initial conditions. The arbitrary choice of initial charge separation is only eliminated when ion pairs are produced by light excitation of donor-acceptor complexes. If they actually exist, the ions are created just at contact, and their subsequent recombination and separation may be approximately described even by an exponential model." However, the same model is often applied toquitedifferent systems or solutionswherecomplex formation is completely excluded.10 The free reactants perform a random walk in solution, and ions arise during their encounters
at such distance where the forward electron transfer happens. Since the rate of ionization is distant dependent and the distance is modulated by diffusion, the initial distribution of ions and its subsequent transformationmust be established within generalized encounter theory. A number of attempts have been made to build a theory for the joint analysis of back and forward electron transfer in liquid solutions.18-20 We consider thelast oneas a final version although the authors of ref 21 tried to do more. They took into account the finite size of acceptors whose positions around the donor become correlated due to the excluded volume of the particles. This is a multiparticlecorrection that contributesonly to nonlinear terms in concentration expansion of the quenching rate. The generalizationof quenching kinetics used in ref 21 to account for this effect is well known to be exact for solid solutions where there is no diffusion.22 It remains exact for liquids only for the case when donors are immobile.23 Since in reality both reactants are participating in encounter diffusion, the accuracy of all nonlinear terms in concentration expansion of the ionizationrate is questionable." To avoid any complications of this sort, we restrict ourselves to the conventional encounter theory,m which reduces to the binary approximation when applied to liquid solutions. Our main goal is to put in order different situations that arise when numerous parameters of the system are changed. We intend to pickout their universal combination that establishes the regimes of ionization varying from the static to the diffusion and the kinetic control. In each of them wedetermine initial distributions of ions that differ qualitatively and are noncontact at slow diffusion. Then the quantum yield and the kinetics of charge separation at arbitrary diffusion rate will be calculated to show a significant difference in results between the present approach and the contact approximation used before.
Initial Distribution of Charged Products After pulse excitation of the sample, the total number of excited donors Nobeys the conventional kineticequationof thedifferential non-Markovian encounter theory:4.5
*Abstract published in Advance ACS Absrrucrs. July 1, 1994.
0022-3654/94/2098-73 19304.50/0 0 1994 American Chemical Society
(3)
7320 The Journal of Physical Chemistry, Vol. 98, No. 30, 1994
Here
TD
If there is no diffusion, i.e. the ionization is static (D = 0), then it follows from eq 5 that
is the lifetime of the excitation and
k I ( t )=
J WI(r) n(r,t) d3r
(4)
is the nonstationary rate constant of binary ionization which proceeds with a position-dependent rate WI. To find kl,one has to solve the auxiliary kinetic equation for the pair distribution function of reactants
h = DAn - W,(r)n
(5)
with the initial and the reflecting boundary conditions
where D is the encounter diffusion coefficient of the neutral reactants. Actually n(r,t) is the distribution of excitationsaround a single acceptor. The electron transfer burns the hole in its center that becomes deeper and wider with time. The spread of the hole ceases when its radius reaches a maximum value equal to theeffective radius of quasistationary ionizationRQ(see Figure 2 in ref 20). Binary approximation demands RQto be much less than the average distance between acceptors. If back electron transfer is forbidden then the number of ions P(t) produced in reaction 1 increases with time according to an equation supplementary to eq 3:
P = kI(t)cN
(7)
Taking N(0) = 1, we may consider P ( t ) as the total probability of forward electron transfer at time t and p(r,t) as the density of products (counterions) around a single donor. In the binary approximation P(t) must be linear in acceptor concentration c, and therefore
P ( t ) = Jp(r,t) d3r = cJm(r,t) d3r
(8)
p(r,O) = m(r,O) = 0
(9)
where
If ions are immobile, one obtains from eqs 7, 8, and 4
m = WI(r) n(r,t) N ( t )
Burshtein et al.
t 10)
This is the equation for the charge accumulation that comes to an end at time t = (ck ~ / T D ) - ' where k = ~TRQD is the final (stationary) reaction constant. Soon after this time, the total number of charges reaches its maximal value, which is the quantum yield of ionization +:
= e-W(r)r and mo(r) = WIKe-WIrN(t) dt = v( W,(r))
(14)
is thelocal quantum yield of the products ~ ( W Ijust ) , in thevicinity of point r. In the opposite limit (D =), the ionization is kinetic controlled. This means that
-
n(r,t) = 1 at any r
(15)
mo(r) = W , ( r ) K N ( t )dt = WI(r);
(16)
and
where is the luminescence lifetime. Between these extremes there is the intermediate region of moderate diffusion. When the ionization is diffusion-controlled, the effective radius of reaction sphere RQ exceeds the closest approach distance &, and the quenching kinetics N ( t ) is a multistage process. During the initial static and the subsequent nonstationary quenching, the kinetics of the process is nonexponential. However, the majority of donors are ionized during the least quasistationary stage when
N=
" 30
k = 41rRqD at t >> T , = -
(17)
During two initial stages all the acceptors inside the reaction sphere of radius RQ have been charged already, and the rest are charged at the moment they reach its border in the course of encounter d i f f ~ s i o n . ~Therefore ~ the majority of them are accumulated near the reaction surface, and the distribution of ions has a maximum at r = RQ. To proceed further, we have to choose now the particular shape of the ionization rate Wl(r) that determines &(D) dependence. There are a few analytically solvable models,4.* but we prefer to start with thesimplest one, peculiar to the Marcus "normal region" of ionizationz6and well known as the "exchange transfer":
+
P(-) = cJim,(r) d3r = # where m,(r) = m(r,m) = WI(r)gn(r,t) N ( t ) dt
(12)
This charge distribution is kept frozen if ions do not move. Alternatively,it can serveas an initial condition for the subsequent transformation to m(r,t) if there is encounter diffusion of ions and/or backelectron transfer. The normalized initial distribution of charges is given by the following formula:
The shape of the distribution depends crucially on how fast the encounter diffusion of reactants is.
In the extremely fast diffusion limit, the ionization is kinetic controlled and results in the initial distribution (1 6) that exactly reproduces the exponential function (18). The latter is rather narrow and looks like the straight line in the coordinates of Figure 1. On the contrary, the static initial distribution (14) is the broadest though aIso monotonous. Only those created by diffusion-controlled ionization have a maximum at r = &.The maximum moves away with a decrease in diffusion. The normalized charge distributions for these three regimes, shown in Figure 2, were used to calculate the root mean square (rms) distances between them, R, by the conventional formula R2 = Ji?flr)4r?
--
dr
-
+
As has been expected, RQ changes between R& & L in the kinetic control limit ( D =) and R , that is attained in the opposite static limit ( D 0). Between these limits shown by horizontal lines in Figure 3 the ionization is controlled by the diffusion, and the rms distances R, shown by stars, practically coincide with positions of maxima that are given by reaction
The Journal of Physical Chemistry, Vol. 98, No. 30, 1994 7321
Binary Photoionization and Charge Recombination
limit where RQ > R because in this region the ionization is nonbinary in principle. At the same time, the encounter theory of remote transfer is much better than that of contact ionization. The latter leads to the well-known Collins and Kimball result shown by the dashed line in Figure 3: RQ
10-5
t
20
30
-
7,
1
Figure 1. Initial distributions of charged acceptors around donor at TD = and different D = m, lP3,l e , ..., lv,10-1*, 10-13, 10-'6, Ocm2/s. K, kinetic, D, diffusional, and S, staticreactions (W, = IO3ns-I,
L
0.75 A, Ro
5
A, c
0.1 M).
lh 1-4 0.005
t'
"" " "
' I
"
0'0041h
' I
k0
Here kD = 4.lrRaD is diffusional rate constant while the kinetic one is
\\\Y 10
= 4-
" " "" " "
'
" I '
'
'
"
'
IK
"
'
1
0.003
i
J
The contact estimate of RQ from eq 21 coincides with the right one, resulting from eq 20, just in the case of the kinetic control regime (ko y&Z,(2&)
In Figure 3 this estimate is shown by a solid line connecting triangles. It is valid everywhere except the slow diffusion (static)
n(ro,t) = JiG(r,t;ro,0 ) 4 d d r
(29)
This function describes the recombination kinetics as well as the quantum yieldof chargeseparationIP(r0)= n(ro,=). In the contact
Burshtein et ai.
1322 The Journal of Physical Chemistry, Vol. 98, No. 30, 1994
charge separation in the static limit is evidentlyzero at any remote electron transfer.16 The appropriate description of remote recombination is given by the equation
approximation one has12J4
where
m = -WR(r)m
+ -?a?-a3e-v$vm ar
(38)
with the reflecting boundary condition and (39)
- -
is the rate constant of diffusion-controlledreaction accompanied by Cculomb interaction between reactants (at r, 0 ED 4?rR&). If the ions were born at contact (ro = Ro), the quantum yield would coincide with that given by the primitive 'exponential model":
Though this is never a case, the model is widely used in experimental works" to determine its single parameter k, which is actually related to a contact kinetic constant (27):
The survivingprobability of ions initially separated by distance r may be obtained from the equation conjugate to eq 38,13J6s20
with the corresponding boundary condition
and initial condition
n(r,o) = 1
The difference in absolute values of k, and W,is not of great importance because the major goal of the experimental study was a check-up of the "free energy gap law". This is the W,(AG) dependence that is the same for k,. Unfortunately, even a semiquantitative examination of normal-inverted turnover executed this way is questionable. As was shown, the contact creation of ions assumed in theexponential model is not confirmed by a consistent theory of bimolecular ionization. The relation between contact and remote recombination is seen from the form of the numerator in eq 3 1, which is actually the kinetic rate constant
k, = J;W,(r)e-'/'
d3r = J;WR(r)erc/r4~? d r (34)
This can be proved by using eq 26 with sufficiently small 1 to calculate
i;, = k o e r c l ~at I 1 by ion diffusion to the reaction sphere. At x the recombination becomes static and cannot be described by contact approximation.16 The latter leads to the physically wrong result QJ
+om= limcp(ro) = 1 x-c'
1 - exp(-r,/ro) 1 - exp(-r,/Ro)
-1--
similar to that in eq 6. The solution to this problem may be used to find the total fraction of ions surviving at time t, provided they were initially distributed with a normalized space density f l r ) : (43) Whenflr) is identical to m(r,O) given in eq 28, the result depends on the initial separation ro as before. However, one can also use in eq 43 the initial distribution of ions created by preliminarily processing the sample if it is accomplished long before the recombination begins. For example, ions may be generated by radiation of solid solution, but their motion may be unfrozen much later by a sudden temperature jump. As seen from eqs 42 and43, R(r) monotonouslydecreasesfrom1 at t = 0 toa minimum value (44)
(35)
The single parameter
-
(42)
RO
that is the quantum yield of charge separation at any initial conditions. To find recombination kinetics in the fast ionization limit (when recombination lasts much longer,*O the initial distribution Ar) from eq 13 may be used in eq 43. It is clear that
where k = kI(-). This is also a monotonouslydecreasingfunction that tends to a finite limit P ( m ) = 4, which is the quantum yield of photoionization
atrc+O
'0
(37)
At any ro # Ro, cp- # 0 while actually the quantum yield of
Unfortunately, the time separation of back and forward electron transfer is the exclusion rather than the rule. Generally speaking,
The Journal of Physical Chemistry, Vol. 98, No. 30, I994 7323
Binary Photoionization and Charge Recombination the binary ionization and recombination must be considered together, as they often w u r on the same time scale.
1.00
' ... '.'
----.'--e:.-'
.
0.98
Photoionization Accompanied by Recombinrtion To take simultaneously into account the back and forward electron transfers, one has to sum on the right-hand side of the following kinetic equation the arrival term from eq 10 and departure terms from eq 38:
'
'
,
'
'
'
'
'
'
'
'
\.
\
0.96
'\
'\,
1
The solution must be obtained with the initial condition of eq 9 and the boundary condition of eq 39. Being used in eq 8, it leads to the following:zO
P ( t ) = cJ;d3r
Wl(r)JomR(r,t- t? n(r,t? N(t? dt'
(48)
In the fast ionization limit P(t) initially increases due to ionization but begins to decrease as soon as recombination starts to prevail. After it passes the maximum (P- S (t), the surviving probability goes down to a constant value which is the separation quantum yield P(m) = 4. One may approximately describe this kinetics setting Q(t - t') = Q(t), that leads to the expression
P ( t ) = eJ;O(r,r)
d3r WI(r)Jo'n(r,t? N(t? dt'
IO-^
+ CkTD)
I 1.0 0.9
(50)
-
where iit(r,t) is the solution of eq 10: (52)
The quantum yield of photoionization is evidently
where (t is defined in eq 11,f(r) in eq 13, and example,
TD
>>
0.0
'
-I----:-
IO.?, ,
10"
,
,
'
'
'
'
'
'
'
,
,
,
,
,
,
,
loo
lo1
,
lo2
,, lo3
LIkO
In the slow ionization limit the kinetics of photoionization is qualitatively different. There is no descending branch. P(t) is a monotonously increasing function of time from 0 to 4, obtained from eq 48 with 52 ~ ( r ) : ~ ~
fi(r,t) = m&) - WI(r)Jmn(r,r) N ( t ) dt
104
t, N
O 2I 0.
P(r) rS: CJin(r,t) mo(r) d3r at t >> TD/(1
lo2 103
loo lo1
Figwed. Kineticsof photoionization (solid lines) at R = 7.3 Aand k,/kD = 0 . 2 ( W r = 3 . 9 6 n s - 1 ) , b = D = 10-scm2/s,c=0.1 M , L = I = 0 . 7 5 , & = 5 A, r, = 0. The dashed linea are kinetics of recombination from initial distance ro = 6 , 7.3, 10 A (from bottom to top).
(49)
The descending branch is given by an even simpler expression identical to eq 45:
IO-[
in eq 44. For
= R i / D (54)
where ns(r) = n(r,=) is a distribution of reactants in a quasistationary diffusion-controlled reaction whose kinetics is given by eq 17. For this particular case we have from eqs 51-53
If ionization is neither fast nor slow, one has to solve numerically eq 47 together with the supplementary eqs 3-6. We used the numerical procedure based on an expanded DCR programz7that was successfully employed earlier.14 From a simultaneously calculated initial distribution, we found R as a rough estimate
Figure 5. Separation quantum yield within remote transfer theory (solid line) and within thecontact approximation for ro = 5,6,7.3, lOA (dashed lines from bottom to top) as functions of x a 1/b. The reat of the parameters are the same as in Figure 4.
of initial distance between ions that may be compared with ro when contact approximation is used. This comparison is shown in Figure 4 for the case when (t = 1 since 71) = =. Under this condition the difference between P, and 1 testifies how far we are from the fast diffusion limit when P1 . To find ourself just in the intermediate region, we put D = D and L = 1, keeping the rest parameters close to a previous choice. As seen from Figure 4, the maximum is rather high and just slightly exceeds the separation quantum yield 4. This is alwa s the case when recombination is kinetic-controlled, i.e. x = I,D/ 1 this approximationis unable to predict the separation quantum yield at any reasonable choice of ro, not to mention the kinetics of the process. To make it clear, we compare in Figure 5 the photoionization yield 4 (solid line) with the recombination yield cp obtained in contact approximation for some initial charge separations ro (dashed lines). As seen from the inset at small x, both 4 and q decrease similarly, provided ro is properly chosen. However, at high x (slower diffusion) the significant discrepancy appears, that becomes qualitative at x m. This is the static limit where contact approximation leads to the unphysical results (37) seen as plateau &x) = qpo.Contrary to these expectations, the real photoionization results in limp- 4 = 0. Hence, for slow
E
-
1324 The Journal of Physical Chemistry, Vol. 98, No. 30, 1994 diffusion (viscous solutions) there is no alternative to the remote transfer theory presented above. Acknowledgment. The authors gratefully acknowledge the Ministry of Science and Technology of Israel for support of this research including the invitation from Dr. E. Krissinel for collaboration. Dr. Krissinel would also like to thank the Fundamental Research Foundation of Russia for Grant 93-035038 that partially supported this research. References and Notes (1) Rehm. D.; Weller, A. Isr. J. Chem. 1970,8, 259. (2) (a) Tunitskii, N. N.; Bagdasar’yan, Kh. S.Opt. Spectrosc. (USSR) 1963, 15, 303. (b) Kilin, S.F.; Mikhelashvilli, M. S.;Rozman, I. M. Opt. Specrrosc.(USSR)1964,16,576. (c)Vasil’ev,I.I.;Kiftanov,B.P.;Kronpw, V. A. Kinet. Kurd 1964,5,792. (c) Steinberg, I. Z.; Katchalsky, E.J. Chem. Phys. 1968,48,2404. (d) Wilemski, G.; Fixman, M. J . Chem. Phys. 1973, 58, 4009. (3) Pilling, M. J.; Rice, S.A. J . Chem. SOC. Furuduy Truns. 2 1975,71, 1563. (4) Doktorov, A. B.; Burshtein, A. I. Sou. Phys. JETP 1975, 41, 671. (5) Kipriyanov, A. A.; Doktorov, A. B.; Burshtein, A. I. Chem. Phys. 1983, 76, 149. (6) (a) Markus, R. A,; Siders, P. J. Phys. Chem. 1982, 86, 622. (b) Szabo, A. Ibid. 1989,93,6929. (c) Eads, D. D.; Dismer, B. G.; Fleming, G. R. J. Chem. Phys. 1990,93, 1136. (d) Song, L.; Dorfman, R. C.; Swallen, S. F.; Fayer, M. D. J. Phys. Chem. 1991, 95, 3454. (7) (a) Burshtein, A. I.; Kapinus, E. I.; Kucherova, I. Yu.;Morozov, V. A. J . Lumin. 1989.43.291. Ib) . , Burshtein..A. I.:.Morozov. V. A. Chem.Phvs. Lett. 1990, 165, 432.’ (8) Burshtein, A. I.; Frantsuzov, P. A. J . Lumin. 1992, 51, 215. (9) Tachiya, M.; Murata, S.J. Phys. Chem. 1992, 96, 8442. (10) Kikuchi, K.; Niwa, T.; Takahashi, Y.;Ikeda, H.; Miyashi, T. J. Phys. Chem. 1993, 97, 8070.
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