J. Phys. Chem. 1993,97, 196-202
196
Generalizations of the Stern-Volmer Relation Nicholas J. B. Green Department of Chemistry, King’s College, Strand, London WC2R 2LS, U.K.
Simon M. Pimblott‘ Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556
M. Tachiya Division of Basic Research, National Chemical Laboratory for Industry, Tsukuba, Ibaraka 305, Japan Received: September 15, 1992
The familiar Stern-Volmer relationship of photochemistry is generalized to describe systems in which the decay under investigation is nonexponential and systems in which the quenching or scavenging is inherently transient and must be described with a time-dependent rate constant.
1. Introduction The Stern-Volmer analysis is an elementary method of photochemistry and is discussed in many standard physical chemistry and kinetics texts.I.2 The technique is used toinvestigate the kinetics in competition experiments where a pseudo-firstorder quenching or scavenging reaction competes with a firstorder decay process such as fluorescence. Generally, the ratio of the radiative time constant and the quenching rate constant is obtained from the quencher concentration dependence of the fluorescence. If one of these parameters, usually the radiative time constant, is already known, then the other can be inferred. Alternatively, when the influence of quencher concentration on the radiative lifetime is measured, then the Stern-Volmer analysis provides both the radiative time constant and the quenching rate constant. In many systems of experimental interest the decay process under investigation is not a simple exponential, or the kinetics of the quenching (or scavenging) process have to be described by a time-dependent rate coefficient; consequently, the basic SternVolmer relation is not appropriate. This paper describes the generalization of the Stern-Volmer relationship to encompass these more complex systems. The methods presented can be used to investigate either the kinetics of the underlying decay or the kinetics of the quenching or scavenging process. All the generalizations lead to Laplace transform type relationships between the natural kinetics and the quenching or scavenging yield. Geminate recombination and other nonhomogeneous kinetic processes in photochemistry3~~ and radiation chemistrysq6 are inherently nonexponential and transient. It is not always possible (or convenient) to investigate such processes in real time, and so alternative indirect methods are sometimes necessary. One suitable method is through the concentration dependence of the quenching or scavenging In section 2 this type of study is discussed in detail and is demonstrated to be a generalization of the Stern-Volmer method. Nonexponential decay kinetics can also arise from a formation process which is extended over time; this might be the result of employing a time-dependent light pulse for excitation or the result of the formation of an excited state by ion recombination. The experimental decay kinetics are then a convolution of the underlying exponential decay over the formation time profile, and so deconvolution is necessary to extract the underlying decay constant. In section 2 it is shown that the yield of quenching or scavenging in such a system is independent of the rate of the formation process, so 0022-3654/ 58/2097-0196$04.00/0
that the Stern-Volmer method gives a window on the underlying rate, and deconvolution is not necessary. Quenching processes in many commonly occurring photochemical systems are transient, and their kinetics are conventionally described using a time-dependent rate coefficient.1° The nature of the time dependence of the rate coefficient depends on its origins. Several quenching mechanisms have been proposed in which the quenching rate coefficient depends upon the distance between theexcited state and thequencher.I1J2 Each mechanism gives rise to a characteristic time-dependent rate coefficient.13 When the quenching is diffusion-limited, another type of time dependence is found.I4J5 The transient nature of the quenching rate coefficient leads to a modification of the relationship between the quenching yield and the concentration of quencher which should be taken into account in any analysis of experimental data or in any attempt toreconstitute theunderlying quenching kinetics from such data. The necessary modifications, which are derived and discussed in section 3, constitute a second generalization of the Stern-Volmer relation. In some systems both the underlying decay and the scavenging are nonexponential, and a further generalization of the SternVolmer relation is needed. This final generalization of the SternVolmer relation, presented in section 4, is a fusion of those presented in sections 2 and 3. It may be particularly important for geminate recombination, which is inherently nonexponential, and where the transient part of any competing scavenging reaction should be taken into account in analyzing the concentration dependence of the scavenging yield. In a recent paper a method for reconstructing the kinetics from the scavenging yield in such a case was suggested.16 It must be emphasized at this stage that, for the analysis presented in this paper to hold, both the kinetics of the recombination or decay process and the kinetics of the quenching or scavenging process must be described by linear rate equations. Formally, this excludes any second-order process except under pseudo-first-order conditions. Additional problems are encountered in trying to apply these methods to second-order reactions or to more complex stochastic systems such as spur recombination in radiation tracks.’6*l7
2. Nonexponentinl Decay Kinetics The first generalization of the Stern-Volmer relation identified in the Introduction comes about when the decay of interest is nonexponential yet the rate equation governing thedecay is linear. This situation can arise for several possible reasons, for example, 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 1 , 1993 197
Generalizations of the Stern-Volmer Relation a time-dependent source term or a time-dependent decay rate coefficient. In photochemistry a time-dependent source term can result either from the time dependence of the light source or as a result of alternative routes leading to the excited state (e.g., delayed fluorescence or recombination fluorescence). The second situation, where the decay rate can be described by a timedependent first-order rate constant, is observed in the case of geminate recombination. As the quenching or scavenging yield is unaffected by the time dependenceof the source term, but does depend on the natural time dependence of the transient, the two situations will be dealt with separately. 2.1. Time-Dependent Source Term. Suppose that a species with concentration c ( t ) is formed with a given rate per unit volume g(t) and decays with a first-order rate constant k, in which the contribution from fluorescence is kf. If a quencher is also present and quenching is pseudo-first-order with rate constant k,c,, then c ( t ) obeys the rate equation dc/dr = g(t) - (k
+ k,c,)c(t)
(1)
which has the elementary solution
c(t) = exp[-(k
+ k,c,)f](c(O) + Jofg(f?exp[(k + kqCq)'l dt') (2)
The overall fluorescence yield is the integral over all time of the instantaneous fluorescence rate, i.e.
(3) and is easily shown to be (4)
where G = j'i g(t) dt. This is the Stern-Volmer relation with the simple modification that c(0) has been replaced by c(0) G. There is clearly no information on the nonexponential time dependence of c ( t ) from the quenching yields. The reason for this is essentially trivial: the exponential decay and the pseudofirst-order quenching start for each excited molecule from the moment of its formation. Each system is independent and has the same ultimate fluorescence probability kf/(k + k,c,), and the total number of independent systems produced per unit volume is simply c(0) G. In other words, the fluorescence quantum yield does not depend on the shape of the light pulse. This apparent inability of the fluorescence yield to probe the nonexponential source term is, of course, a great advantage of the method, as it probes the underlying kinetics (extraction of the ratio k,/k is easy), and it is independent of experimental artifacts such as pulse shape. The fluorescence yield automatically performs the deconvolution that would otherwise have been necessary. Although this result appears to be trivial and is wellknown in photochemistry, it is equally true for the more complex situations considered in the remainder of this paper, a fact which does not seem to have been widely appreciated. 2.2. Transient Decay Kinetics. The Stern-Volmer relation is truly generalized when the natural decay rate is not exponential. This scenario was first analyzed by Hummel and by T a ~ h i y a , ~ who considered the recombination and scavenging kinetics of radiation induced geminate ion pairs in low-permittivity solvents. Geminate recombination is usually described using a diffusion equation,"J and the most convenient formalism for this purpose is the Kolmogorov backward equation, which describes the dependence of the survival probability of the pair, Q(r,t),on the initial separation distance r.18 If the potential energy of interaction of the pair in units of keT is U then the appropriate backward
+
+
equation is
where D' is the coefficient of relative diffusion of the pair. Chemical reaction can be incorporated either via a reactive boundary condition at the encounter distance R or via a distancedependent reaction rate k(r)Q,which is subtracted from the righthand side of eq Sa4 The outer boundary condition is usually taken to be Q 1 as r m, which is a natural boundary in three dimensions (attracting but ~nattainable)'~ for all potential energy functions of physical significance. The introduction of a scavenger leads to the depletion of pairs independently of interparticle distance with a pseudo-first-order rate constant k,c,. Thus, an extra reaction term k,c,Q must be subtracted from eq 5 . (Here it is assumed that only one particle in the pair is scavengable. The case where both particles can be scavenged is similar and has been reported e l s e ~ h e r e . ~ ~ It . l ~is) straightforward toshow that theresultingequation with thesame reactive boundary condition and the outer boundary condition Q(r,t,c,) exp(-k,c,t) as r m has the solution
- -
-
-
Q(r,t,c,)= a(r,t,O)exp(-k,c,t) (6) Since the instantaneous scavenging rate is k,c,Q, the ultimate scavenging yield is y,(c,) = Jmkscsn(r,t,cs) dt = Jomk,c,O(r,t,O) exp(-k,c,t) dt
(7) which is kscstimes the Laplace transform of Q(r,t,O),the survival probability in the absence of scavenger, where k,c, is the Laplace variable conjugate to t. This analysis can be generalized very simply to the case where the initial pair separation is given by a probability density p ( r ) rather than being fixed. The resulting survival probability in the absence of scavenger is given by
and the scavenging yield is given by eq 7 in which Q(r,r,O) is replaced by II(t). Thus, in principle, the geminate recombination kinetics can be reconstitured from the scavenging yield by inverting the Laplace transform. Of course, this cannot be done exactly unless an analytical solution for the scavenging yields is known or else unless the scavenging yield is known over an infinite range of k,c,. Operationally, analytical functions are usually fitted to experimental data and then i n ~ e r t e d . ~ This - ~ procedure has been discussed elsewhere,I6whereit was shown to be accurate to within experimental error if the initial yield of pairs and the low scavenger concentration limit of the scavenging probability are both known. Once again a time-dependent source term can be added: if the initial concentration of the pairs is c(0) and the probabilitydensity of the interparticle distance is p(r) and if pairs are formed with a time-dependent rate g(r) with the same relative distribution ~ ( 4then , c ( t ) = c(0)
n(t)exp(-k,c,r) + Jofg(t3 n(t-r')exp(-k,c,(t-t'))
dt' (9)
where II is given in eq 8. The ultimate scavenging yield is (c(0) + G)k,c,II(k,c,),where II(k,c,) is the Laplace transform of l l ( t ) , giving exactly the same dependence on c, as before. Again the results do not depend on the shape of the source term, only on its integral G = J,"At)dt. The interpretation of this observation is exactly the same as that for the first-order decay with a source which was discussed in section 2.1.
Green et al.
198 The Journal of Physical Chemistry, Vol. 97,No. I, 1993
3. Transient Quenching A second generalization of the Stern-Volmer relation is necessary in the case where quenching occurs with a transient rate and competes with a truly first-order decay, such as fluorescence.20,21In this case the survival probability obeys a phenomenological rate equation of the
dQ/dt = -kQ - kq(t)cqQ
(10)
to a good approximation when there are weak long-range forces between the particles, for example, for reactions between ions in high-permittivity solvents or where the Coulomb potential is screened.23~2~For systems in which quenching is diffusion-limited, the time scale function 0 is given by kactkdiff
=
kact
+ kdiff
[t + -
2 f i -CY&
A(
whose solution is
Q(r)
‘act(
kdiff
1 - exp(a2t) erfc ( a f i ) ) ) ] (16)
= exp[-(kt
a
+ Jotkq(t?c, dr?]
Unfortunately, analytic inversion of 6 for t is not straightforward. However, the most important part of the transient is the long time asymptotic behavior in which the final term in eq 16, 1 - exp(cr2t) erfc ( a d t ) ] is , negligible. The function 0 then has the same structure as the totally diffusion controlled limit and can be inverted analytically, as is discussed below. When quenching is fully diffusion controlled (kat, >> kdiff), the time-dependent rate coefficient reduces to the familiar Smoluchowski form
The overall quenching yield is therefore
This integral may be simplified by a change of time scale
T = e(?) = Jotks(f’)dt’ which gives
kdc(t)
in which the inversion of the function, t = & I ( T),is unambiguous because 6 is a strictly increasing function (k, > 0). The time scale function 6(r) has units of (concentration)-’ while & I ( T ) has units of time. It is immediately obvious from eq 14 that the quenching yield is cq times the Laplace transform of exp(-k&I( T)), where the Laplace variable is c,. This observation can be used in two ways. If the function 6 ( t ) is known, then the lifetime of the underlying exponential decay can be obtained; furthermore, if the functional form of 6 is known but the values of the various parameters are not, then certain combinations of these parameters can also be obtained from the observed quenching yield. However, perhaps the most interesting and potentially useful application is that if the concentration dependence of the quenching yield is known over a sufficiently wide range, then a numerical or approximate inverse Laplace transform can be performed to give the function exp(-k&’( T)),from which the time dependence of the quenching rate constant can be reconstituted if the original data are good enough. We do not recommend the use of this procedure without extensive numerical checks on the sensitivity of the results to errors in all the time ranges considered; numerical inversion of Laplace transforms is a notoriously ill-conditioned numerical problem.22 It is frequently better to choose an interpolation function with sensible asymptotics to fit the data and then to invert that function. 3.1. Diffusion-Limited Quenching. A variety of different mechanisms that can give rise to a transient quenching rate have been postulated. The most familiar of these is diffusion-limited quenching. The most general form of the rate constant for such a process isl4-15 kactkdiff
+
kdl(t) = kact kdif[
+
k,,, exp(a2t) erfc ( a f i ) ] kdi,
( 1 5)
= kdiff(l +
(17)
where @ = R / ( ~ T D ~and ~ / the ~ , time scale function simplifies to edc(t)
= kdiff(1 -k 2@.\/i)
In this case the inverse function
&I(
T ) is given by
An alternative definition of the time scale functions, 8d{ and &, is to factor out the steady-state rate coefficient (that is kactkdiff/ (kaci + kdiff) or kdiff) and to subsume it into the Laplace variable. The advantage of this alternative definition is that the resulting time scale function is a true time; however, if the steady-state rate constant kq(-) is not known in advance (which is usual in photochemistry), it is much more convenient to regard the quenching yield as a function of the quencher concentration than of the composite variable kq(=)cq. Furthermore, the other mechanisms leading to transient quenching do not in general have nonzero steady-state rate coefficients. 3.2. Quenching in Frozen Systems. Diffusion is not the only mechanism which can lead to time-dependent rate constant^.^^^^^^^^ Energy-transfer or electron-transfer reactions frequently take place in systems where the particles are static or diffuse very slowly, and the rate for such a reaction typically depends on the separation between the reactive particles. The time-dependent rate coefficient for a distance-dependent reaction when the reactants are uniformly distributed in three dimensions is given by the integral
k ( t ) = 4nJamr24r) exp(-r(r)f) d r
a=
a ( k a c + ~ kdiff) Rkdiff
Here kac, and kdiff (=477D‘R) are the rate coefficients for the activated quenching process and for the diffusive encounter process a t a separation R,respectively, and D’is the coefficient of relative diffusion of the reacting pair. It is also known that time-dependent rate constants of the same functional form as eq 15 can be used
(20)
where a is the minimum separation possible for the particles and K(r) is the distance-dependent rate constant. It is obvious from eq 20 that the resulting time scale function is O ( r ) = 4sJOmr2( 1 - exp(-K(r)t)) d r
with
(18)
(21)
3.2.1. Multipolar Mechanism. The two mechanisms most commonly invoked for these distance-dependent processes are the multipolar mechanism” and the exchange mechanism.IZ In the multipolar mechanism the rate constant takes the form
K(r) = a/s (22) where n usually has the value 6 (FBrster dipole4ipole mechanism).” The multipolar mechanism leads to a time-dependent
Generalizations of the Stern-Volmer Relation
The Journal of Physical Chemistry, Vol. 97, No. 1, 1993 199
rate coefficient
coefficient for the quenching reaction3*j2
and the time scale function
The corresponding time scale function is
emult(t)= ~[(ar)3'"y(l-3/n,ar/an) - a3(1- exp(-at/a"))l (24) where y(u,x) represents the incomplete gamma function.27 For the dipole4ipole mechanism ( n = 6 ) the time-dependent rate coefficient be~omes~*-~O
and the resulting time scale function can also be expressed in terms of the error function. Once again the time scale function f3 must be inverted numerically in general. However, all these formulas can be simplified considerably if the short time behavior of the rate is unimportant. If the dimensionless group atla" >> 1 , the timedependent rate coefficient can be expressed accurately as
which in the important dipole4ipole case ( n = 6) becomes
(33) with (34) There is no steady-state rate coefficient for kexch,which decays away to zero at long times, and is frequently approximated by
which is simply the first term in a longer series, but in many experimental situations it is sufficient. Further terms of the series are given in several references, e.g., refs 13 and 34. This is an asymptotic result, valid only for long times, and it cannot be integrated from 0 as is required in the evaluation of the time scale function; however, 0 can be extended to longer times by
e(t) = e(t,) = e(?,)
The error made by truncating the higher order terms is of order exp(-at/a"), which is generally very small since the finite encounter volume V = 4ira3/3 can be neglected under realistic experimental condition^.^^ In this approximation the time scale function is
+ St4"[ln w3t
+ %([In 3w
(tko)12d t (tk0)l3- [In (toko)13)
As before, the function 0 is not readily invertible except by numerical methods, or when kot is very small, in which case
h ( z ) = 62 - 3z2/8 When kot is large32 h ( z ) = [In ( z ) ] j
+ 0(z3)
+ 3y[ln ( z ) ] *+ 3( y2 +);
(37)
In (z).+ y3 +
57r2+ Once again, the error implicit in this approximation is of exponential order. The time scale function for the multipolar mechanism can now be inverted easily
and for the dipole-dipole mechanism t=
9 ( T + fi2 16 a 3 a
Similar results are found in systems with reduced dimensionality or where another mechanism leads to a different power law for K(r)and in turn todifferent time dependences of the rate constant; these results are summarized in the review by G13sele.I~ 3.2.2. Exchange Mechanism. The case of the exchange mechanism has also been studied inten~ively.~O-3~ This mechanism has been applied both to energy transfer and to electron transfer and is usually described by a distance-dependent reactivity K(r) which falls off exponentially with increasing distance:12
~ ( r=) k, exp(-wr) (31) In the limit of slow diffusion this leads to a time-dependent rate
(36)
X ( 3 ) (38)
where y is the Euler constant and t i s theRiemann zeta function.27 The remainder is of order exp(-kat). The function 0 can then be inverted by solving the cubic equation for In (kat), which is elementary.27 In either the multipolar or the exchange mechanism the inclusion of a nonzero diffusion rate leads to a longtime asymptotic time dependence for the rate constant of the form of eq 17 with an effective interaction distance.34 Numerical calculations show that the short-time behavior of k(r) follows the slow diffusion limit and that there is a transition to the long-time behavior when t = ( r 0 ~ / D ' ) ~where ~ 3 ~ ~ro3is~the ~ range of the interaction. 3.3. Thestern-Volmer Plot. Sections 3.1 and 3.2concentrated on the quenching yield, which has the Laplace interpretation identified in eq 14. However, it is more normal to apply the Stern-Volmer type of analysis to the fluorescence efficiency yf =
kf
-yq(cq)1 = kfJOmew[-(kt + ~ , w ) d tI (39)
which is formally kf times the Laplace transform of exp(-cq8(t)), although this is not a useful identification since the Laplace variable is the first-order decay constant k, which cannot be varied. This integral can be found analytically for some of the cases discussed in section 3.1 but is generally rather inconvenient for fitting parameters. In general, the simple expedient of plotting l / y fagainst c, is expected to give a curve. The functional form of this curve is discussed by Rice (ref 10, p 35) and by Nemzek
200
Green et al.
The Journal of Physical Chemistry, Vol. 97, No. 1, 1993
and Ware.j7 If data are available over a sufficiently wide concentration range, then it is desirable to fit them using this full formula. However, in many cases the whole curve is not available over a wide concentration range; rather, the quantities of most interest are the intercept and limiting slope a t low concentrations. The behavior a t low concentrations can be found by expanding the second exponential in the integrand of eq 39:
yf(Cq)= k f K e x p ( - k t ) [ l - c,$
+ 0 ( c q 2 ) ]dt
(40)
Reciprocating and expanding in powers of c,, we find
+
-=
( k / k f ) [ l c,kJomexp(-kt)8(t) d t
+ O(c;)]
Yf(Cq)
(41) Thus, the Stern-Volmer plot has an intercept of k/kf and a slopeto-intercept ratio of k8(k), where 8 is the Laplace transform of 8. It is generally simpler to express the slope-to-intercept ratio in terms of the time-dependent rate coefficient k,(t), which is related to 8 by eq 13. Integrating by parts
radiative lifetime of 38.5 ns they found an encounter distance of 9 A with a relative diffusion coefficient of 6.3 X lo-" m2 s-l. These parameters imply that the [ s / i ]ratio of the Stern-Volmer plot is about 57% higher than the corresponding ratio using the steady-state quenching rate coefficient. Using 1,2-benzanthracene quenched by carbon tetrabromide in mineral oil a t 298 K, the corresponding parameters are lifetime 42.8 ns, encounter radius 10.5 A, and relative diffusion coefficient 2.2 X 10-l' m2 s-1, which result in a [s/i] ratio 109% higher than would be expected. These results are rather extreme, since the systems were chosen to test for the presence of transient effects; however, they serve to illustrate the magnitude of the errors that can be incurred by failing to recognize that the slope-to-intercept ratio contains a contribution from the transient part of the quenching rate coefficient. 3.3.2. Multipolar Mechanism. The general form of the timedependent rate coefficient is given in eq 23, whose Laplace transform is found to be b / ~ l m u l t=
[ s / i ] = k&k) = Jomkq(t)exp(-kt) dt = k,(k)
4ra3
n-3
a
(
*F1 1,1,2-3/n; -
(42)
The slope-to-intercept ratio in the classical Stern-Volmer plot is therefore the Laplace transform of the time-dependent quenching coefficient. In the following subsection each of the models discussed in section 3.1 is considered, and it is shown how this limiting slope-to-intercept ratio can be calculated and what information it contains. 3.3.1. Diffusion-Limited Quenching. Here the time-dependent rate coefficient is given in general by eq 15, and the slope-tointercept ratio is
a
+ ka"
=-E n(m + 4ra
OD
(-a/ka")"
(45) kak3 m=O 1) -3 where 2 F l ( u , b , ~represents ;~) the hypergeometric function.27The summation form for eq 45 converges for all positive values of k. For the important case of dipoledipole energy transfer, the hypergeometric function simplifies considerably:
In the simpler case where the quenching is fully diffusion controlled, the time-dependent rate coefficient takes the familiar Smoluchowski form given in eq 17, and the ratio is
If a can be estimated, the ratio a / k can be found from the slopeto-intercept ratio. As might be anticipated, if the asymptotic forms of the timedependent rate coefficient can be used (eqs 26 and 27), Le., if atla" is large for the entirety of the fluorescence, the equation for the limiting slope-to-intercept ratio becomes much simpler
(44)
(47)
It is particularly interesting to note that the limiting Stern-Volmer slope-to-intercept ratio is not simply the ratio of the steady-state quenching rate constant to the radiative time constant k , ( - ) / k . The transient part of the rate coefficient contributes not only to the curvature of the plot but also to the limiting slope. Mathematically, this finding is not surprising, since the slopeto-intercept ratio is the Laplace transform of the time-dependent rate coefficient, and the Laplace transform reflects the whole of the time dependence. This means that diffusion-limited rate constants extracted from Stern-Volmer plots should be corrected for the effect of the transient on the limiting slope. The importance of this correction depends not only on the properties of the quenching reaction but also on the radiative lifetime of the excited molecule, governed by k,and increases in significance with k. However, it seems that such corrections are not made routinely in photochemistry-Ricelo comments on the correction and states that it is generally less than 5%. However, it is clear that it could be important if the radiative lifetime is short or if the diffusion coefficient is small. As an example it is instructive to consider the experimental data of Nemzek and Ware, who analyzed their results using the full expression rather than simply the slopeto-intercept ratio.37 These authors used systems with rather long fluorescence lifetimes because of the light pulse availabile; in order to find transient effects, they also used viscous solutions so that the diffusion was slowed down. For naphthalene quenched by carbon tetrabromide in propane-1,Zdiol a t 298 K, with a
which becomes
In either case it is simple to extract the ratio a / k and hence to estimate a using the known radiative lifetime. 3.3.3. Exchange Mechanism. The final case considered is the most problematical. In the Dexter exchange mechanism the timedependent rate coefficient for quenching is given by eq 32, and evaluating the integral for the slope-to-intercept ratio gives the series solution
Unfortunately, this series only converges if ko/k 5 1. However, the function defined within this circle of convergence is analytic in the whole complex plane cut along the negative real axis from -1 to --. It is discussed briefly in Erdelyi,3* who shows that it can be analytically continued into the remainder of the complex plane by
\d"I
where the summation evidently converges in the complementary
The Journal of Physical Chemistry, Vol. 97, No. 1, 1993 201
Generalizations of the Stern-Volmer Relation circle k/ko I 1. Notice that the parameters in the distancedependent rate constant K(r) separate neatly since the form of the slope-t+intercept ratio is u-3times a function of ko/k, Le., F(ko/ k)/u3. If one of the parameters is known or can be estimated, then the other parameter can be determined as long as a suitable algorithm for evaluating F is available. It is also possible to use the asymptotic long-time form of the time scale 6 (eq 38). However, the resulting formula for the slope-to-intercept ratio
4r [~/ii,,,~(k)= ;;;j[r2 in ( k o / k )+ (1n ( ~ , / w ) ~ I(51) is only accurate if k ko,the formula is negative because it is dominated by the times where the asymptotic form of the time-dependent rate coefficient is negative. Since the underlying distance-dependent rate constant K(r) depends on two parameters ko and w (see eq 31), it is hardly surprising that they cannot be separated by simply measuring the slope-to-intercept ratio. However, given a suitable algorithm for evaluating the function F,it is clear that if one of these parameters is known, the other may be estimated.
4. Transient Scavenging The final generalization of the Stern-Volmer relation considered is where transient scavenging competes with another transient process of the type considered in section 2.2, i.e., a process such as geminate recombination, which also has to be described with a time-dependent rate coefficient. It should be recognized that the purpose of such a study is rather different from the systems discussed in section 3. In section 3 the underlying radiative lifetime is usually known, and the aim is to extract information about the transient quenching process. In this section the steady-state scavenging rate constant can usually be measured from the long-time homogeneous phase of the photochemical or radiation chemical reaction, and the aim is to extract information about the unknown natural decay process, which is nonexponential. The generalization needed to allow for a transient scavenging rate coefficient has been suggested previously in a numerical study of the Laplace transform relationship.16 Using the same notation as in sections 2.2 and 3, the final scavenged yield is
3 Once again, if the substitution T = e(?) is made, the equation becomes
(53) which is a Laplace transform of the recombination kinetics with a transformed time scale. Like eq 14, this equation can be used in two ways. If the recombination kinetics n(t)are known in the absence of scavenger, then the function e(?) can be isolated from the inverse Laplace transform of the experimental scavenging yield, and hence the time-dependent scavenging rate coefficient can be found. Secondly, if the time-dependent scavenging rate coefficient is known in advance, for example from the Smoluchowski theory, and is integrated to give the function e(?), then the geminate survival probability can be reconstituted from the inverse Laplace transform of the experimental scavenging yield. The second method is the usual application of the technique. However, both methods rely on having a reliable numerical or approximate method of finding the inverse Laplace transform. The usual approach is to fit a suitable invertible function to the observed Numerical studies9J6J9 have shown that for the radiation chemical and photochemical systems where this method is usually applied, it is generally accurate to within
experimental error if both short-time and long-time limits of the decay kinetics are known in advance. This is a generalization of the Laplace transform method usually employed in radiation chemistry to obtain the time-dependent kinetics of the radiation-induced transients from observable product yields. In such problems it is customary to describe scavenging with a time-independent rate constant so that the recombination kinetics can be obtained directly from the inverse Laplace transform of the scavenging yield. However, it is clear from eq 52 that the function actually obtained by this method is in fact the function n(F(t)); in other words, thecorrect kinetics are obtained from this procedure, but on a transformed time scale. In order to correct the procedure for the time dependence of the rate constant, it is simply necessary to modify the time scale of the inferred kinetics. Thus, if the inversion of the Laplace transformgivesadecayfunction Q(t) = I I ( P ( t ) ) then , thecorrect recombination kinetics are given by II(t) = Q(O(t)). The two time dependences for 6 which are most appropriate for the type of geminate problem considered here are the diffusionlimited case and the diffusion-controlled case, where the 6 functions are given in eqs 16 and 18. The correction is therefore straightforward. Stochastic IRT simulations have shown that in the majority of experiments performed on aqueous solutions the effect of assuming a time-independent rate coefficient is not significant; however, this will not necessarily be true for lowpermittivity solvents.39
5. Conclusions In this article three generalizations of the Stern-Volmer relation have been presented, which can be applied when the natural decay of the system is nonexponential and/or the scavenging/quenching which competes with this decay is transient. Section 2.1 considered briefly the simple case where the decay kinetics are naturally exponential but where a time-dependent source term renders the observed decay nonexponential. Here the original Stern-Volmer relationship is unaffected by the source term, and in consequence it is not necessary to deconvolute the results. This observation is simply a reflection of the independence of quantum yields of the time of formation. All the more complicated relationships considered in the remainder of the article have the same property, i.e., are independent of the source term. In section 2.2 the case of geminate recombination, where the natural decay kinetics are nonexponential but the scavenging reaction is pseudo-first-order, was discussed. Here the basic Stern-Volmer relation no longer holds, but a modification derived by Hummel and by Tachiya’ can be used. In this generalization of the Stern-Volmer relationship, eq 7, the yield of scavenged product is proportional to the Laplace transform of the natural decay kinetics in the absence of scavenger. This method is aimed at using the scavenging yield to reconstitute the natural underlying decay kinetics and is typically applied by fitting the scavenging yield to an analytic function which is a Laplace transform; the inversion can then be performed analyti~ally.~-~ It is generally found that this procedure is accurate to within experimental errors.16 A second generalization of the Stern-Volmer relation is identified in section 3, which deals with systems in which the natural decay is exponential but the quenching reaction is transient, and has to be treated using a time-dependent rate coefficient kq(t), This type of system is typical in photochemistry. By a simple transformation of time scale 6(t) = Jkkq(t’) dt’, it is shown in eq 14 that the quenching yield is proportional to the Laplace transform of exp(-kF(T)). The function 6(t) is evaluated explicitly for several commonly used models: partially and totally diffusion-controlled quenching (eqs 16 and 18) and quenching in frozen systems by the multipolar mechanism (eq 24) and by the exchange mechanism (eq 33). The functional dependence of 6 on time is too complicated for convenient use in
202 The Journal of Physical Chemistry, Vol. 97, No. I , 1993 several of these systems, and useful long time asymptotic approximations are identified for the diffusion-limited case (eq 18), the multipolar mechanism (eq 28), and the exchange mechanism (eqs 36 and 38). Inversion of the function 0 is performed analytically for these asymptotic forms (eqs 19, 29, and 30) except for theexchange mechanism where inversion simply requires solution of a cubic equation. It is recognized that the yield of quenching is not directly measurable. The quantity most usually determined in such photochemical experiments is the fluorescence efficiency, whose reciprocal is graphed against the quencher concentration in a Stern-Volmer plot. In general, all these transient quenching models predict curvature in the Stern-Volmer plot, but it is possible to obtain exact formulas for the intercept and for the limiting slopeofthelineat low concentrations, whichare the twoquantities most readily accessible from experiment. The slope-to-intercept ratio of the Stern-Volmer plot is shown to be the Laplace transform of the time-dependent quenching coefficient (eqs 4042) and is found explicitly for each of the mechanisms considered (eqs 43-46,49, and 50). The most important qualitative finding is that the slope-to-intercept ratio contains a contribution from the transient part of the quenching rate coefficient which can be a significant correction to the rate constant deduced from the graph. For the multipolar and exchange mechanisms the solutions can be simplified by using the asymptotic forms of 0 (eqs 47,48, and 51)), but these forms can only be used when the natural decay time scale is sufficiently long to sample only the long-time behavior of the quenching rate coefficient. In any specific case this should be tested. This proviso is particularly important for the exchange mechanism where using the asymptotic form outside its range ofvaliditycan lead to unphysical negative rateconstants. Finally, in section 4, the case of transient scavenging in competition with transient recombination is considered. Here the same change of time scale that was used in section 3 leads to the conclusion that the scavenging yield, which is directly measurable in these systems, is proportional to the Laplace transform of the underlying recombination kinetics on a transformed time scale, l l ( & l ( t ) ) , eq 53. Thus, the decay kinetics inferred from an inverse Laplace transform of the scavenging yield, Q(r), can be corrected by a simple change of time scale
Q(Nt>>. Acknowledgment. The research described herein was supported by the Office of Basic Energy Sciences of the US.Department of Energy. This is contribution NDRL-353 1 of the Notre Dame Radiation Laboratory.
Green et al.
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