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Reversible energy quenching and conservation
Chemical Physics 370 (2010) 208–214
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Reversible energy quenching and conservation S.G. Fedorenko a, A.I. Burshtein b,* a b
Institute of Chemical Kinetics and Combustion, Novosibirsk, Russia Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 28 August 2009 In final form 1 February 2010 Available online 6 February 2010
The kinetics of reversible energy transfer from photo-excited donors to energy acceptors is studied at arbitrary concentrations of both and any relationship between the decay-times of the reactants. The backward reaction of transfer products in a bulk is included in the consideration. Its contribution to delayed fluorescence, resulting from the energy conservation on the long-lived acceptors, is specified. Ó 2010 Elsevier B.V. All rights reserved.
Keywords: Reversible energy transfer Integral encounter theory Dipole–dipole interaction Molecular wires
1. Introduction
For the short range interaction it is obtained from the last expression by the limiting transition
The reversible energy transfer from the excited donor D to the energy acceptors is given by the simplest reaction scheme WF
D þ A D þ A ;
ð1:1Þ
WB
# sD
# sA
where sA and sB are the life-times of excited reactants. The transfer reactions in diluted solutions is the usual subject of the Encounter Theory that enables specifying the reaction kinetics and the quantum yields of the luminescence and transfer products. The input data are the forward and backward transfer rates, W F and W B , as well as encounter diffusion coefficient D. For a forward dipole–dipole energy transfer the position dependent rate is given by the Förster formula [1]:
W F ðrÞ ¼
a r6
ð1:2Þ
:
It decreases as a power of distance r between the donor and acceptor of energy. If the transfer is performed by the exchange interaction of Dexter type then [2]
r r W F ðrÞ ¼ W 0 exp L
ð1:3Þ
where r is a contact distance and L is the characteristic transfer length. In line with the true space dependent rates, there is also the popular contact approximation for them:
W F ðrÞ ¼ kf dðr rÞ where kf ¼
Z
1
W F ðrÞ4pr 2 dr
0
* Corresponding author. Tel.: +972 8934 3708; fax: +972 8934 4123. E-mail address: [email protected] (A.I. Burshtein). 0301-0104/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2010.02.002
ð1:4Þ
L ! 0;
W 0 ! 1;
while kf ¼ const:
This approximation is not applicable to static quenching, but is very often used (from Smoluchowski times) for a description of contact diffusional quenching. The kinetic scheme for reversible inter-molecular transfer (1.1), includes the backward rate W B that relates to a forward one, via equilibrium constant c:
W F kf 1 ¼ ¼ ¼ expðDG=TÞ; W B kb c
ð1:5Þ
where DG is the difference between the free energy of the products and initial reactant states (the Boltzmann constant kB is set to 1). The irreversible analog of this process, performing the quenching of initial excitation at W B ¼ 0 is perfectly described by means of differential encounter theory (DET) [3]. In solid solutions (in absence of encounter diffusion) the ‘‘static quenching” kinetics obtained with DET reproduces the exact multi-particle solution of the problem which is valid at any acceptor concentration c ¼ ½A. However, DET can do nothing with a reversible transfer when W B – 0 and the backward process plays an important role in energy quenching or conservation. On the contrary, the relatively new integral encounter theory (IET) describes the reversible transfer reactions as well as irreversible ones but only in the lowest order concentration approximation (with respect to c) [4,5]. As a result, IET is inapplicable to the long time quenching which is a multi-particle one, although it allows to describe reasonably (in the lowest concentration approximation) the preceding diffusion–accelerated reaction. More accurate
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description of the latter is given by the modified encounter theory (MET) [6,7]. Taking more consistently all the binary terms, MET corrects the kernel of IET, improving the kinetics of diffusion accelerated process and avoiding the false long time behavior of IET. Using both theories which are identical in static stage, we will indicate the diffusion–accelerated one, where they become different, giving preference to MET. In the next section, we will present the formalism of both theories, MET and IET, using the position dependent rates of dipole– dipole and ‘‘exchange” energy transfer, which are the alternatives to the contact one studied earlier [8–10]. They will be used to study the decay of donor excited state population, N ðtÞ, as well as the quantum yield of fluorescence quenching:
g¼
Z
1
e ð0Þ=sD : N ðtÞdt=sD ¼ N
Here we imply for simplicity that the diffusion coefficients for the forward and backward reactions are the same and there is no inter–particle interaction. The equations for modified encounter theory (MET) are almost the same, except the kernel:
N_ ðtÞ ¼ c
Z
t
ds Rm ðsÞ N ðt sÞ Z t N ðtÞ þ cq ds Rm ðsÞ N A ðt sÞ sD 0 Z t ds Rm ðsÞ N ðt sÞ N_ A ðtÞ ¼ c 0 Z t N ðtÞ ds Rm ðsÞ N A ðt sÞ A ; cq 0
0
In Section 3 we will consider the static quenching and then turn to the diffusion–accelerated one (Section 4). The kinetics of the reversible energy transfer, immediately following the d-pulse excitation, will be analyzed at different ratio of forward and backward rates and the arbitrary relationship of excitation life-times. Then the same will be done in a long run taking into account the subsequent backward reaction in the bulk, proportional to the donor concentration q ¼ ½D. The latter results in equilibration of the excited states whose final decay is the same in both theories. Their analysis and comparison is performed with a numerical program developed in Ref. [11] and the results are summarized in Section 5.
ð2:7Þ
sA
0
ð1:6Þ
ð2:6Þ
The modified kernel of the integral MET equations has the following definition
Z e m ðsÞ ¼ s þ 1 þ c Rð0Þ e ~m ðr; sÞ W B ðrÞl ~ m ðr; sÞd3 r; ½W F ðrÞm R
sD
ð2:8Þ via the position dependent rates and the pair correlation functions. The latter obey the set of auxiliary equations:
h
i
e m_ m ðr; tÞ ¼ W F ðrÞmm þ W B ðrÞlm þ DDmm s1 D þ c Rð0Þ mm ; ð2:9Þ h
i
e l_m ðr; tÞ ¼ W F ðrÞmm W B ðrÞlm þ DDlm s1 A þ cq Rð0Þ lm ; ð2:10Þ
2. Integral encounter theories of energy transfer Owing to a request for a detailed balance (1.5), the kernels of IET equations for inter-molecular energy transfer (see Eq. (3.14) in review [4]) relate to each other as SðtÞ ¼ cRðtÞ [12]. This relationship allows representing these equations in the following form:
Z
N_ ðtÞ ¼ c
t
ds RðsÞ N ðt sÞ þ cq
0
N_ A ðtÞ ¼ c
Z
Z
t
ds RðsÞ NA ðt sÞ
0
N ðtÞ
ð2:1Þ
sD
t
e ðsÞ ¼ N
ds RðsÞ N ðt sÞ cq
0
Z
t
ds RðsÞ NA ðt sÞ
0
NA ðtÞ
sA
;
Z 1 ~ðr; sÞ W B ðrÞl ~ ðr; sÞd3 r; ½W F ðrÞm sþ
sD
ð2:3Þ
via position dependent rates and the pair correlation functions. The latter obey the set of auxiliary equations:
m_ ðr; tÞ ¼ W F ðrÞm þ W B ðrÞl þ DDm mðr; 0Þ ¼ 1;
@ m ¼0 @r r¼r
l_ ðr; tÞ ¼ W F ðrÞm W B ðrÞl þ DDl lðr; 0Þ ¼ 0;
@ l ¼0 @r r¼r
1 sþs
1 D
e ðsÞ þR
;
e ðsÞ ¼ N A
m ; sD
1
s þ sA
e ðsÞ R
; e s þ s1 D þ R ðsÞ ð2:11Þ
ð2:2Þ
where N ðtÞ ¼ ½D t =½D 0 and NA ðtÞ ¼ ½A t =½D 0 – the survival probabilities for the time t of excited donors and acceptors in the bulk. Such equations have been already used, employing the contact approximation: Eq. (3.127) in the same review [4]. Extending them to a distant transfer, we preserve the original IET definition of the kernel
e RðsÞ ¼
where the initial and boundary conditions remain the same as in IET (see Eqs. (2.4) and (2.5)). These results follow from the general matrix formulation of MET [13,14]. They make the applicability of the so modified integral theory to a higher acceptor concentration than IET. The difference between them indicates the scale of corrections at any time, giving preference to MET where they are large. Making the Laplace transformation of Eqs. (2.1) and (2.2) or e ðsÞ, e ðsÞ and N (2.6) and (2.7), one can resolve them relative to N A getting the following results:
e where the ‘‘mass operator” can be defined through the kernels RðsÞ e (or R m ðsÞ) of the corresponding integral equations in the following way:
e ðsÞ ¼ R
e c RðsÞ
e 1 þ cq RðsÞ= s þ s1 A
ð2:12Þ
Note that the numerator of this expression represents the initial stage of the reaction (geminate stage) following the light excitation. It corresponds to the first terms in the rhs of Eqs. (2.1) and (2.2) and (2.6) and (2.7) while the term in the denominator takes into account the subsequent reaction in the bulk represented by the second terms of the same equations. 3. Static quenching
ð2:4Þ
l ; sA ð2:5Þ
Let us turn to the solid or highly viscous solutions, where diffusion is frozen ðD ¼ 0Þ. Generally speaking, IET is not applicable to such systems because the transfer there is essentially multi-particle (to all partners simultaneously), while IET is only the lowest order (binary) approximation with respect to the partner concentration c, reproducing only the short time kinetics of irreversible
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quenching. The reversible transfer kinetics in IET is different but reduces to the irreversible one when sA ! 0. These conclusions are reached by analyzing the kernel of static quenching:
10
-5
ð3:1Þ
10
-6
It was obtained by the Laplace transformation of Eq. (2.3) and the set (2.4), (2.5) at D ¼ 0. We will consider first the irreversible energy quenching ðW B ¼ 0Þ and then its reversible analog, responsible for the delayed fluorescence.
10
-7
e ¼ RðsÞ
Z
3
d r
1
s þ sD
s þ s1 W F ðrÞ s þ s1 D A
1 1 s s þ s1 þ s þ W W F ðrÞ B ðrÞ þ s þ sA D A
3.1. Irreversible quenching ðW B ¼ 0Þ If the state accepting the energy decays immediately no backward transfer is possible. This makes the forward transfer irreversible as if W B ¼ 0. As a result the general reaction scheme (1.1) reduces to a simplified one WF
D þ A ! D þ A;
ð3:2Þ
# sD The solution of this problem following from Eq. (2.1) at c ¼ 0 is universal:
N ðtÞ ¼ expðt=sD Þ N0 ðtÞ;
ð3:3Þ
where N0 ðtÞ known as ‘‘static quenching kinetics” is given by its Laplace transformation:
e ðsÞ ¼ N 0
1
; e ir s s1 s þ cR D
ð3:4Þ
where
e ir ðsÞ ¼ s þ s1 R D
Z
3
d r
W F ðrÞ s þ W F ðrÞ þ s1 D
ð3:5Þ
3.1.1. Dipole–dipole mechanism Using here the dipole–dipole rate (1.2), we get for the strong quenching (by extending the integral in (3.5) from 0 to infinite r):
e ðsÞ ¼ N 0
1 pffiffiffiffiffiffi ; s þ D ps
ð3:6Þ
pffiffiffi
where D ¼ p c a. Taking the inverse Laplace transformation of this result, carried out in our earlier work [15], we obtain: 2 3
3=2
pffiffiffiffiffiffiffiffiffiffiffi N0 ðtÞ ¼ expðpD2 tÞerfc p D2 t
ð3:7Þ
Using the cumulant expansion at short times we obtain from here:
N0 ðtÞ ¼ e2D
pffi t þðp2ÞD2 t23ð4pÞðD2 tÞ3=2
at D2 t 1;
ð3:8Þ
while the exact Förster solution valid at any times and concentrations is
N0 ðtÞ ¼ e2D
pffi t
1 pffiffiffiffiffiffiffiffiffi 1 þ D psD
ð3:9Þ
0.01
0.02
0.03
Fig. 1. The kinetics of irreversible static energy quenching by dipole–dipole (blue) and exchange (magenta) mechanisms. The IET results (solid lines) are compared with their exact analogies (dashed lines) obtained with the Förster and Dexter rates. The rest of the parameters are c = 0.3 M, a = 5 108 Å6/ns, W0 = 9 104 ns1, r = 5 Å, L = 0.9 Å. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
As well as quenching kinetics, this result obtained with IET coincides with the exact one (see (Eq. (5.18)) in review [18]) only in the lowest order concentration approximation when g 1 pffiffiffiffiffiffiffiffiffi D psD . 3.1.2. Exchange mechanism The exchange interaction (1.3) can be subjected to a similar analysis. Using it in Eq. (3.5) we obtain for strong transfer after space integration:
! 2 2
e ir s s1 ¼ s 4p R3 þ p L R0 ; R D 0 3 4
at
L R0 1 2
ð3:10Þ
where the radius of strong quenching
R0 ¼
L Wc ln ; 2 s
ð3:11Þ
e ir ðsÞ from (3.10) in the general formula where W c ¼ W 0 er=L . Using R (3.4) we get:
" !!#1 2 2 e ðsÞ ¼ s 1 þ 4p c R3 þ p L R0 N : 0 0 3 4
ð3:12Þ
Making the inverse Laplace transformation, we obtain in the lowest order (linear) approximation with respect to c:
N0 ðtÞ ¼ 1 cVðtÞ;
ð3:13Þ
where the quenching volume
VðtÞ ¼
As can be seen the IET result for dipole–dipole quenching coincides with an exact one only initially: at times smaller than sF ¼ 1=4D2 , while the longer time tail is the ‘‘false asymptote” of the binary approximation (Fig. 1). If the quantum yield of fluorescence is measured stationary the whole quenching kinetics contributes to the value defined in Eq. (1.6). From this definition and Eq. (3.6) the yield for the dipole–dipole static quenching is
e s1 =sD ¼ g¼N 0 D
0.00
pL3 6
3
2
½ln W c t þ A1 ln W c t þ A2 ln W c t þ A3
extends with time. Here A1 ¼ 3C ¼ 1:7316; A2 ¼ 3C 2 þ p2 =2 ¼ 5:9343; A3 ¼ C 3 þ C p2 =2 w00 ð1Þ ¼ 5:4449; C – Euler constant, w00 ðxÞ tetra-gamma function [16]. The result (3.13) coincides with the exact one obtained in Ref. [17]
N0 ðtÞ ¼ ecVðtÞ in the lowest order approximation with respect to c. The coincidence occurs even in more limited region of the shortest times ðcVðtÞ 1Þ but the relative reduction is evidently smaller than in dipole–dipole case (Fig. 1).
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3.2. Energy conservation ðsA ¼ 1Þ
stationary solutions of Eqs. (2.1), (2.2), (2.6) and (2.7), provided
sA ¼ sD ¼ 1: Let us consider the opposite case, when the energy is temporarily conserved on a stable partner but finally returns back to the donor state. This phenomenon manifests itself via delayed fluorescence of the donor but the energy quenching, measured stationary or in indefinite time after the d-pulse, does not occur at all. That is the quantum yield (1.6), calculated with Laplace transformation of the set (2.1), gives
g ¼ 1 at sA ¼ 1
ð3:14Þ
Although there is no quenching in this case the kinetics of donor decay says a lot about the conservation and dissipation of energy defined in Eq. (2.11). The corresponding mass operator (at sA ¼ 1) appears to be:
e ðsÞ ¼ R
e c RðsÞ ¼ e 1 þ cq RðsÞ=s
(
e c RðsÞ
at short times ðsmall qÞ sc=cq at long times ðlarge qÞ ð3:15Þ
The short time decay is actually a donor quenching by energy acceptors which are initially almost empty. Due to further reversible transfer the equilibration of energy distribution is finally reached and the quasi-stationary partner populations decay simultaneously further on, via donor channel. Such an asymptotic decay is a universal one (independent of the particular quenching mechanism) but never considered earlier, since the bulk reactions were usually ignored setting q ! 0 [8–10]. 3.2.1. Dipole–dipole energy transfer Taking the integral in Eq. (3.1), we obtain for the dipole–dipole transfer at sA ¼ 1:
e ¼D c RðsÞ
pffiffiffiffi
p
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s s þ s1 D 1 : s þ c s þ sD
ð3:16Þ
e For short times c RðsÞ is identical to the mass operator (3.15), that being used in (2.11) gives the Laplace transformation of the initial transfer kinetics:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "
#1 pffiffiffiffi s s þ s1 1 e D 1
N ðsÞ ¼ s þ sD þ D p s þ c s þ sD
ð3:17Þ
As a matter of fact, the inverse transformation of this result describes only the geminate reaction immediately following the pulse excitation but completely ignores the bulk reactions as well as states equilibration. It is easy to show that this response to d-excitation has also two stages:
8 qffiffiffiffiffiffi < exp 2D t t=sD ; D2 t 1; 1þ c N ðtÞ ¼ : 2 pffiffiffi 3=2 DsD =2 c t ; ct D2 s2D
Neq A ¼
c ; c þ cq
N eq ¼
cq c þ cq
ð3:19Þ
:
The excitation decay after equilibration of population distribution proceeds as follows from the conventional chemical kinetics:
t N ðtÞ N eq exp Neq :
ð3:20Þ
sD
The initial static quenching gives way to this universal (mechanism independent) decay of the equilibrated system, which starts earlier the larger is the donor concentration, q. The latter is shown by dashed lines in Fig. 2. As soon as the kinetic (solid) curves 1, 2 and 3 approach the dashed ones the results become exact. They are also exact at the very beginning, where the same curves approach that for the irreversible static quenching of Förster’s type (blue). The latter gives place to the Dexter’s type kinetics if the transfer is an exchange one. It can be easily specified by integrating exchange rate in Eq. (3.1) at sA ¼ 1 that provides the mass operator for large s (small times) which specify the initial kinetics of reversible excitation decay i. e. exchange analog of Eq. (3.18):
N ðtÞ exp
c Vð½1 þ ctÞ t=sD ; 1þc
cVð½1 þ ctÞ 1; ð3:21Þ
where VðtÞ is a quenching volume defined in Eq. (3.13). The interpolation between specific static quenching and final (mechanism independent) equilibrated decay is questionable (due to the power time asymptote of the post-static initial quenching). However, this intermediate region reduces with donor concentration and at the highest q is actually eliminated (see curve 3 in Fig. 2). The static quenching almost immediately turns to be a universal one. Thus the application of the integral theory to solid solutions is reasonable for relatively high q. The situation is much better in liquid solutions due to the acceleration of the energy transfer by encounter diffusion. It makes the IET work well and MET even better in just the intermediate time region (Section 4).
10
-5
10
-6
10
-7
ð3:18Þ
The first coincides with the exact reversible static quenching at the very beginning and p differs ffiffiffiffiffiffiffiffiffiffiffiffi from the Förster kinetics only by the scale coefficient 1= 1 þ c , while the last one is recognized as a false power time asymptote of IET which is well-known in diffusion–accelerated quenching [4]. The rate of the reversible static transfer decreases with time and finally supports delayed fluorescence by backward transfer from remote acceptors. However, it can happen out of the validity limit of the static approximation (at t sF ) where the false (power time) IET asymptotic behavior take place. Fortunately, it follows from Eq. (3.15) that at any finite q the longest time behavior is controlled by a common exponential decay of the excited state populations, after their shares reach the final equilibrated values. These values can be found from the
0
10
20
30
40
Fig. 2. The kinetics of reversible static dipole–dipole fluorescence quenching followed by delayed fluorescence (solid lines) at different donor concentrations q: (1) 103 M, (2) 102 M, (3) 101 M. Their long time asymptotes is given by dashed lines, in comparison to a natural decay (green straight line) and the irreversible static quenching (the lowest blue line). Acceptor concentration c ¼ 0:1 M; sD ¼ 6 5 ns; sA ¼ 1; c ¼ 1; a ¼ 5 108 Å =ns. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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3.3. Reversible–irreversible transition with shortening
sA
Using Eqs. (2.11) and (2.12) in the general definition (1.6), we obtain that the yield of static quenching obeys the Stern–Volmer law,
1 1 g¼ ¼ ; e 1 þ cjsD 1 þ R ð0ÞsD (
e Rð0Þ e 1 þ cqsA Rð0Þ
-5
10
-6
10
-7
ð3:22Þ
where the Stern–Volmer constant
e ð0Þ=c ¼ j¼R
10
e Rð0Þ;
e 1 cqsA Rð0Þ e 1=ðcqsA Þ; cqsA Rð0Þ 1 ð3:23Þ
There are only two opposite limits. Either the quenching is cone or, alternatively, the nartrolled by energy transfer having j ¼ Rð0Þ row throat is the weighted natural decay of equilibrated excited states, when the Stern–Volmer constant, j ¼ ½cqsA 1 does not depend of the transfer mechanism at all. If this mechanism is a dipole–dipole one the former limit is presented by
cj ¼ D
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 1 p ; sD þ csA
ðsmall qÞ
ð3:24Þ
calculated from the general formula (3.1). In the opposite limit
j¼
1
cqsA
ðlarge qÞ
ð3:25Þ
at any transfer mechanism. For the exchange or contact mechanisms only the result (3.24) is different although follows from the same Eq. (3.1). At sA ¼ 0 from Eqs. (3.22)–(3.24) follows the earlier obtained result for the irreversible dipole–dipole transfer, Eq. (3.9). In the opposite case of reversible transfer to a stable partner ðsA ¼ 1Þ, it follows from Eq. (3.25) that there is no quenching at all ðj ¼ 0Þ, i.e. g ¼ 1. Eq. (3.23) allows tracing the gðsA Þ dependence from its minimal to maximum value. Turning now to the dipole–dipole quenching kinetics, we obtain from Eqs. (2.11) and (2.12)
N ðtÞ ¼ expðt=sA ÞN ðtÞ;
e ðsÞ ¼ N e ðs þ 1=sA Þ N
ð3:26Þ
The Laplace transformation of N ðtÞ is given by
e ðsÞ ¼ N
1 e e s þ s1 þ c R ðsÞ=½1 þ cq R ðsÞ=s
;
ð3:27Þ
while that of the kernel is
e ðsÞ ¼ D cR
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi sðs þ s1 Þ ; p s þ c s þ s1
ð3:28Þ
where 1 1 s1 ¼ sD sA :
Note that the structure of Eqs. (3.27) and (3.28) is identical to the Eqs. (3.16) and (3.17), so for the positive value of s ðsA > sD Þ both limits considered in (3.18) are also peculiar to N ðtÞ with the replacement sD ! s . Therefore it can be shown that the geminate reaction kinetics at sA > sD is also a two stage process. The initial exponential stage remains the same as in Eq. (3.18) but the following power time dependence is multiplied by an exponent:
Ds 2 N ðtÞ pffiffiffiffi 3=2 expðt=sA Þ; 2 pt
ct 1 D2 s2
0
5
10
15
20
25
30
Fig. 3. The transition from reversible (1) to irreversible (5) static quenching by shortening the life-times of the acceptor: sA ¼ 1ð1Þ; sA ¼ 100ð2Þ; sA ¼ 10ð3Þ; 6 sA ¼ 5ð4Þ; sA ¼ 0ð5Þ ns (at c ¼ 0:1 M; q ¼ 103 M; c ¼ 1; a ¼ 5 108 Å =ns and sD ¼ 5 ns).
in (3.3) (Fig. 3). The border between them is indicated by curve 4 ðsD ¼ sA Þ that asymptotically approaches the natural decay. However, the geminate stage is not the final stage of the reversible reaction: eventually the equilibration of excitations occurs and the earlier the larger is q. Hence, the geminate stage gives way to the bulk one:
eq 1 t : N ðtÞ N eq exp Neq s1 D þ N A sA
ð3:30Þ
This is the universal asymptotic decay of excitations that proceeds through both available channels, D and A. Note that in a borderline case ðsA ¼ sD ¼ sÞ this asymptotic degenerates into the kinetics of natural decay: N ðtÞ N eq expðt=sÞ. 4. Diffusion accelerated energy transfer 4.1. Geminate stage Let us first consider the geminate stage of the reaction that takes place first (after d-excitation) and remains the single one in very dilute solutions. Setting q ! 0 in Eqs. (2.1) and (2.2) we lose the inter-molecular backward transfer but the reduced equations
Z t N ðtÞ N_ ðtÞ ¼ c ds RðsÞ N ðt sÞ sD 0 Z t N ðtÞ ds RðsÞ N ðt sÞ A ; N_ A ðtÞ ¼ c
ð4:1Þ ð4:2Þ
sA
0
still account for the ‘‘micro-reversibility” in the kernel, RðsÞ, via W B in Eqs. (2.4) and (2.5). They should be solved now taking into account the encounter diffusion ðD – 0Þ. For contact reactivity this was done a few times analytically but using here the dipole–dipole transfer rates in both directions, Eqs. (1.2) and (1.5), we have to solve them once again. e ðsÞ ¼ c RðsÞ e As q ¼ 0 the IET ‘‘mass operator” is R and the kinetics of geminate energy dissipation and accumulation are given by the simple Laplace transformations:
e ðsÞ ¼ N
1 e s þ s1 D þ c RðsÞ
;
e ðsÞ ¼ N A
sþs
1 A
e c RðsÞ ;
e s þ s1 D þ c RðsÞ
ð3:29Þ
ð4:3Þ
Changing sA we can trace the transition from the absolutely reversible limit ðsA ¼ 1Þ to the irreversible one ðsA ¼ 0Þ described
following from Eq. (2.11). The quantum yield calculated from here obeys the Stern–Volmer law, g ¼ 1=ð1 þ cjg sD Þ, but the geminate Stern–Volmer constant,
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e jg ¼ Rð0Þ:
ð4:4Þ
In the very popular contact approximation of transfer rates (1.4) it is known to be [4,8]
1
jg
¼
" # 1 1 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ c pffiffiffiffiffiffiffiffiffiffiffiffi ; þ kf kD 1 þ sd =sD 1 þ sd = sA
ð4:5Þ
1
where sd ¼ r2 =D is the encounter time and kD ¼ 4prD is the diffusion rate constant in the contact model. The analogous results for the exponential transfer rate (1.3) were obtained a number of times when the reversible electron transfer, executed by such a rate, was studied (see Chapter VIII in Ref. [4] or pp. 3.3 and 3.4 in the recent review [5]). Therefore, we concentrate later on only the dipole–dipole mechanism of reversible transfer. It is well-known from DET that the diffusion accelerates the irreversible quenching of donor by the remote energy transfer. After short static quenching and subsequent transient kinetics the donor decays exponentially with a ‘‘quasi-stationary” rate constant,
kD ¼ 4pRQ D; where RQ is the quenching radius to be calculated [3–5]. In IET the short static quenching also precedes the fast (diffusion accelerated) exponential quenching. When the backward energy transfer is accounted for, the kinetics of the geminate stage of the reaction remains qualitatively the same, but proceeds slower. The same has been done with MET by substituting Rm ðtÞ for RðtÞ in Eqs. (4.1) and (4.2). Fortunately, the very effective program for solving either IET or MET equations at any c and arbitrary relationship between the life-times was developed in Ref. [11]. Using it to specify the role of encounter diffusion in geminate quenching we obtain a noticeable difference between the IET and MET results for the irreversible and less for the reversible reactions even at low donor concentrations (Fig. 4). Only the initial static stage of geminate quenching is the same in both theories while the diffusion–accelerated quenching is better described by MET that eliminates the false IET tails. The kinetics of either irreversible (blue) or reversible (magenta) quenching develops faster than the static one but the difference between IET and MET (considered as more accurate) is more pronounced when the quenching is irreversible.
10
-5
10
-6
10
-7
0
1
2
3
e The Stern–Volmer constant in MET, jm g ðc; qÞ ¼ R m ð0Þ, accounts for the concentration dependence of this quantity which is missed e m ðs; c; qÞ is originated from in IET. The concentration dependence R the shortening of the life-times of both partners quenched by the energy transfer to ‘‘outsiders” presented around in concentration c and q. For instance, in the contact approximation
4
5
Fig. 4. Irreversible (blue; sA ¼ 0) and reversible (magenta; sA ¼ 1) dipole–dipole geminate quenching in MET (solid lines) and IET (dashed lines) at 6 c ¼ 0:1 M; q ¼ 103 M; c ¼ 1; D ¼ 2:6 105 cm2 =s; a ¼ 5 108 Å =ns and sD ¼ 5 ns. The straight green line represents the natural (exponential) decay of the donor. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
jmg
" # 1 1 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ kf kD 1 þ sd ð1=sD þ cjg Þ 1 þ sd ð1=sA þ cqjg Þ ð4:6Þ
As is seen, MET accounts for the shortening of the donor (acceptor) life times due to their acceptor (donor) surroundings. The effect which is larger in more concentrated systems accelerates the diffusional transfer extending the transient contribution. As a result the geminate Stern–Volmer constant in MET slightly increases with both concentrations (Fig. 5), while that of IET is a true constant. 4.2. Including the bulk stage If the pulse experiment is carried on with a highly concentrated solution of donors and the fluorescence is registered long enough 2.2x10
-11
2.0x10
-11
1.8x10
-11
1.6x10
-11
1.4x10
-11
1.2x10
-11
1.0x10
-11
8.0x10
-12
6.0x10
-12
4.0x10
-12
0.0
0.1
0.2
0.3
0.4
0.5
Fig. 5. The Stern–Volmer constant dependence on acceptor concentration c at different concentrations of donors, q, obtained with MET in the contact approximation at (1) q ¼ 103 M, (2) q ¼ 102 M, (3) q ¼ 101 M. The horizontal dashed line is that of IET. The rest of the parameters are kF ¼ 1013 ðM sÞ1 ; kD ¼ 5 1010 ðM sÞ1 ; sd ¼ sD ¼ 5 ns; sA ¼ 10 ns; c ¼ 1.
10
-5
10
-6
10
-7
10
-8
0
10
20
30
Fig. 6. The long time kinetics of the dipole–dipole quenching (including the bulk stage) at the same parameter and color definitions as in Fig. 4. The thin solid magenta line represents the asymptotic exponential decay of the equilibrated system at c ¼ 0:1 M; q ¼ 103 M; c ¼ 1; sA ¼ 1. The dotted line represents the static quenching. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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S.G. Fedorenko, A.I. Burshtein / Chemical Physics 370 (2010) 208–214
1.2x10
-10
1.0x10
-10
8.0x10
-11
6.0x10
-11
4.0x10
-11
2.0x10
-11
To calculate the quantum yield from Eq. (1.6) one should know N ðtÞ for infinitely long times. In the contact approximation this can be done analytically and the corresponding Stern–Volmer constant appears to be
1
j
0.000
0.002
0.004
0.006
0.008
0.010
Fig. 7. The stationary (red) and geminate (blue) Stern–Volmer constants, as functions of donor concentration q at c ¼ 0:01 M. Solid lines – MET, dashed lines – IET. Other parameters are the same as in Fig. 5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
compared to the donor life-time, then the bulk stage of the reaction cannot be ignored and the theoretical description of the system response should be based on the general equations of IET or MET given in Section 2. The results of their numerical solution for the dipole–dipole rates (1.2)–(1.4) and (1.5) are presented in Fig. 6. There is a new feature compared to the geminate stage shown in Fig. 4. The results of IET coincide with those of MET not only at the shortest time where quenching is static but also at the longest ones where the excitation distribution is equilibrated and decays with the universal rate (3.30). In between where the completely reversible reaction ðsA ¼ 1Þ is diffusion accelerated, the IET and MET curves (solid and dashed magenta lines) differ just a bit. This difference is more pronounced at a finite acceptor life-time and never disappears at sA ¼ 0 (blue lines) when the transfer (quenching) is irreversible and never becomes the equilibrated one. These conclusions are in excellent agreement with those obtained for inter-molecular transfer between two excited states catalyzed by encounters with other solutes. They follow from an analysis of eclectic collection of Brownian dynamics simulations presented in Ref. [19], that was reproduced in color (with critical comments) in a recent review (Fig. 7 in Ref. [5]). In original article authors compared IET, MET and other multi-particle theories with BD simulations at different excitation times relationship providing either quenching or conservation of energy. In accordance with what was established above (in Figs. 4 and 6) the difference between IET and MET decay kinetics is pronounced when the transfer is irreversible and almost invisible in case of reversible transfer (conservation). In the latter case the discrepancy between them is covered by the delayed fluorescence from equilibrate state. The deviations from BD results are not seen at all at reasonable concentration presented in the left panels of the figure under discussion 3 ðcB ¼ cD ¼ 104 Å ¼ 0:16 MÞ. And only in the limit of the very 3 high concentration ðcB ¼ cD ¼ 103 Å ¼ 1:6 MÞ presented in the right panels of the same figure it is seen where it should be expected: only IET being zero order approximation deviates from BD calculated irreversible transfer kinetics. No such a demerit is inherent to MET collecting all binary terms.
( ¼
e IET 1= Rð0Þ þ cqsA ¼ 1=jg þ cqsA ; m e 1= R m ð0Þ þ cqsA ¼ 1=jg þ cqsA ; MET
ð4:7Þ
Absolutely the same can be obtained from the stationary solution of the integral equations if the permanent light pumping is accounted for. Adding its rate I0 q to the rhs of Eq. (2.6) one can obtain the stationary population, N s , and using it in an alternative definition of quantum yield, g ¼ N s =ðI0 qsA Þ, confirm the result (4.7). This is the stationary fluorescence quantum yield which is easier to measure than to calculate it from the time resolved data. Unlike the geminate one the stationary j decreases with donor concentration, due to the backward energy transfer in the bulk, Fig. 7. 5. Results Using the simplest and modified integral theories, IET and MET, we considered the reversible inter-molecular energy transfer conducted by remote (dipole–dipole) and short range interactions. The initial static stage of quenching kinetics has an exact description common for both theories, as well as the final stage following complete equilibration of state populations. In between, in a geminate stage of the diffusion accelerated reaction, the MET provides a better description then IET but the difference increases with donor and acceptor concentrations. We calculated the kinetics of donor decay, after d-pulse excitation and the fluorescence quantum yield in diluted and concentrated solutions. The geminate Stern–Volmer constant obtained from the short time system response and that found from the stationary detected fluorescence were shown to be different. They depend differently on both donor and acceptor concentrations and these dependencies were specified with both theories using the popular contact model of transfer rates. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
Th. Förster, Ann. Phys. 2 (1948) 55. D.I. Dexter, J. Chem. Phys. 21 (1953) 36. A.I. Burshtein, Adv. Chem. Phys. 114 (2000) 19. A.I. Burshtein, Adv. Chem. Phys. 129 (2004) 105. A.I. Burshtein, Adv. Chem. Phys. Hindawi Publishing Corporation, 34 p., doi:10.1155/2009/214219. I.V. Gopich, A.A. Kipriyanov, A.B. Doktorov, J. Chem. Phys. 110 (1999) 10888. K.L. Ivanov, N.N. Lukzen, A.B. Doktorov, A.I. Burshtein, J. Chem. Phys. 114 (2001) 1763. N.N. Lukzen, A.B. Doktorov, A.I. Burshtein, Chem. Phys. 102 (1986) 289. A.I. Burshtein, N.N. Lukzen, J. Chem. Phys. 103 (1995) 9631. A.I. Burshtein, N.N. Lukzen, J. Chem. Phys. 105 (1996) 588. P.A. Frantsuzov, O.A. Igoshin, E.B. Krissinel, Chem. Phys. Lett. 317 (2000) 481. K.L. Ivanov, N.N. Lukzen, A.B. Doktorov, A.I. Burshtein, Phys. Chem. Chem. Phys. 114 (2001) 1764. K.L. Ivanov, N.N. Lukzen, A.A. Kipriyanov, A.B. Doktorov, J. Chem. Phys. 6 (2004) 1706. K.L. Ivanov, N.N. Lukzen, A.A. Kipriyanov, A.B. Doktorov, J. Chem. Phys. 6 (2004) 695. S.G. Fedorenko, A.I. Burshtein, J. Chem. Phys. 97 (1992) 8223. Handbook of Mathematical Functions, in: M. Abramovitz, I.I. Stegun (Eds.), Dover, New York, 1965. M. Inokuti, F. Hirayama, J. Chem. Phys. 43 (1965) 1978. A.I. Burshtein, Avtometriya (Russian Review) 65 (1978) 76. S. Park, K.J. Shin, A.V. Popov, N. Agmon, J. Chem. Phys. 123 (2005) 034507.
Contents lists available at ScienceDirect
Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
Reversible energy quenching and conservation S.G. Fedorenko a, A.I. Burshtein b,* a b
Institute of Chemical Kinetics and Combustion, Novosibirsk, Russia Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 28 August 2009 In final form 1 February 2010 Available online 6 February 2010
The kinetics of reversible energy transfer from photo-excited donors to energy acceptors is studied at arbitrary concentrations of both and any relationship between the decay-times of the reactants. The backward reaction of transfer products in a bulk is included in the consideration. Its contribution to delayed fluorescence, resulting from the energy conservation on the long-lived acceptors, is specified. Ó 2010 Elsevier B.V. All rights reserved.
Keywords: Reversible energy transfer Integral encounter theory Dipole–dipole interaction Molecular wires
1. Introduction
For the short range interaction it is obtained from the last expression by the limiting transition
The reversible energy transfer from the excited donor D to the energy acceptors is given by the simplest reaction scheme WF
D þ A D þ A ;
ð1:1Þ
WB
# sD
# sA
where sA and sB are the life-times of excited reactants. The transfer reactions in diluted solutions is the usual subject of the Encounter Theory that enables specifying the reaction kinetics and the quantum yields of the luminescence and transfer products. The input data are the forward and backward transfer rates, W F and W B , as well as encounter diffusion coefficient D. For a forward dipole–dipole energy transfer the position dependent rate is given by the Förster formula [1]:
W F ðrÞ ¼
a r6
ð1:2Þ
:
It decreases as a power of distance r between the donor and acceptor of energy. If the transfer is performed by the exchange interaction of Dexter type then [2]
r r W F ðrÞ ¼ W 0 exp L
ð1:3Þ
where r is a contact distance and L is the characteristic transfer length. In line with the true space dependent rates, there is also the popular contact approximation for them:
W F ðrÞ ¼ kf dðr rÞ where kf ¼
Z
1
W F ðrÞ4pr 2 dr
0
* Corresponding author. Tel.: +972 8934 3708; fax: +972 8934 4123. E-mail address: [email protected] (A.I. Burshtein). 0301-0104/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2010.02.002
ð1:4Þ
L ! 0;
W 0 ! 1;
while kf ¼ const:
This approximation is not applicable to static quenching, but is very often used (from Smoluchowski times) for a description of contact diffusional quenching. The kinetic scheme for reversible inter-molecular transfer (1.1), includes the backward rate W B that relates to a forward one, via equilibrium constant c:
W F kf 1 ¼ ¼ ¼ expðDG=TÞ; W B kb c
ð1:5Þ
where DG is the difference between the free energy of the products and initial reactant states (the Boltzmann constant kB is set to 1). The irreversible analog of this process, performing the quenching of initial excitation at W B ¼ 0 is perfectly described by means of differential encounter theory (DET) [3]. In solid solutions (in absence of encounter diffusion) the ‘‘static quenching” kinetics obtained with DET reproduces the exact multi-particle solution of the problem which is valid at any acceptor concentration c ¼ ½A. However, DET can do nothing with a reversible transfer when W B – 0 and the backward process plays an important role in energy quenching or conservation. On the contrary, the relatively new integral encounter theory (IET) describes the reversible transfer reactions as well as irreversible ones but only in the lowest order concentration approximation (with respect to c) [4,5]. As a result, IET is inapplicable to the long time quenching which is a multi-particle one, although it allows to describe reasonably (in the lowest concentration approximation) the preceding diffusion–accelerated reaction. More accurate
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S.G. Fedorenko, A.I. Burshtein / Chemical Physics 370 (2010) 208–214
description of the latter is given by the modified encounter theory (MET) [6,7]. Taking more consistently all the binary terms, MET corrects the kernel of IET, improving the kinetics of diffusion accelerated process and avoiding the false long time behavior of IET. Using both theories which are identical in static stage, we will indicate the diffusion–accelerated one, where they become different, giving preference to MET. In the next section, we will present the formalism of both theories, MET and IET, using the position dependent rates of dipole– dipole and ‘‘exchange” energy transfer, which are the alternatives to the contact one studied earlier [8–10]. They will be used to study the decay of donor excited state population, N ðtÞ, as well as the quantum yield of fluorescence quenching:
g¼
Z
1
e ð0Þ=sD : N ðtÞdt=sD ¼ N
Here we imply for simplicity that the diffusion coefficients for the forward and backward reactions are the same and there is no inter–particle interaction. The equations for modified encounter theory (MET) are almost the same, except the kernel:
N_ ðtÞ ¼ c
Z
t
ds Rm ðsÞ N ðt sÞ Z t N ðtÞ þ cq ds Rm ðsÞ N A ðt sÞ sD 0 Z t ds Rm ðsÞ N ðt sÞ N_ A ðtÞ ¼ c 0 Z t N ðtÞ ds Rm ðsÞ N A ðt sÞ A ; cq 0
0
In Section 3 we will consider the static quenching and then turn to the diffusion–accelerated one (Section 4). The kinetics of the reversible energy transfer, immediately following the d-pulse excitation, will be analyzed at different ratio of forward and backward rates and the arbitrary relationship of excitation life-times. Then the same will be done in a long run taking into account the subsequent backward reaction in the bulk, proportional to the donor concentration q ¼ ½D. The latter results in equilibration of the excited states whose final decay is the same in both theories. Their analysis and comparison is performed with a numerical program developed in Ref. [11] and the results are summarized in Section 5.
ð2:7Þ
sA
0
ð1:6Þ
ð2:6Þ
The modified kernel of the integral MET equations has the following definition
Z e m ðsÞ ¼ s þ 1 þ c Rð0Þ e ~m ðr; sÞ W B ðrÞl ~ m ðr; sÞd3 r; ½W F ðrÞm R
sD
ð2:8Þ via the position dependent rates and the pair correlation functions. The latter obey the set of auxiliary equations:
h
i
e m_ m ðr; tÞ ¼ W F ðrÞmm þ W B ðrÞlm þ DDmm s1 D þ c Rð0Þ mm ; ð2:9Þ h
i
e l_m ðr; tÞ ¼ W F ðrÞmm W B ðrÞlm þ DDlm s1 A þ cq Rð0Þ lm ; ð2:10Þ
2. Integral encounter theories of energy transfer Owing to a request for a detailed balance (1.5), the kernels of IET equations for inter-molecular energy transfer (see Eq. (3.14) in review [4]) relate to each other as SðtÞ ¼ cRðtÞ [12]. This relationship allows representing these equations in the following form:
Z
N_ ðtÞ ¼ c
t
ds RðsÞ N ðt sÞ þ cq
0
N_ A ðtÞ ¼ c
Z
Z
t
ds RðsÞ NA ðt sÞ
0
N ðtÞ
ð2:1Þ
sD
t
e ðsÞ ¼ N
ds RðsÞ N ðt sÞ cq
0
Z
t
ds RðsÞ NA ðt sÞ
0
NA ðtÞ
sA
;
Z 1 ~ðr; sÞ W B ðrÞl ~ ðr; sÞd3 r; ½W F ðrÞm sþ
sD
ð2:3Þ
via position dependent rates and the pair correlation functions. The latter obey the set of auxiliary equations:
m_ ðr; tÞ ¼ W F ðrÞm þ W B ðrÞl þ DDm mðr; 0Þ ¼ 1;
@ m ¼0 @r r¼r
l_ ðr; tÞ ¼ W F ðrÞm W B ðrÞl þ DDl lðr; 0Þ ¼ 0;
@ l ¼0 @r r¼r
1 sþs
1 D
e ðsÞ þR
;
e ðsÞ ¼ N A
m ; sD
1
s þ sA
e ðsÞ R
; e s þ s1 D þ R ðsÞ ð2:11Þ
ð2:2Þ
where N ðtÞ ¼ ½D t =½D 0 and NA ðtÞ ¼ ½A t =½D 0 – the survival probabilities for the time t of excited donors and acceptors in the bulk. Such equations have been already used, employing the contact approximation: Eq. (3.127) in the same review [4]. Extending them to a distant transfer, we preserve the original IET definition of the kernel
e RðsÞ ¼
where the initial and boundary conditions remain the same as in IET (see Eqs. (2.4) and (2.5)). These results follow from the general matrix formulation of MET [13,14]. They make the applicability of the so modified integral theory to a higher acceptor concentration than IET. The difference between them indicates the scale of corrections at any time, giving preference to MET where they are large. Making the Laplace transformation of Eqs. (2.1) and (2.2) or e ðsÞ, e ðsÞ and N (2.6) and (2.7), one can resolve them relative to N A getting the following results:
e where the ‘‘mass operator” can be defined through the kernels RðsÞ e (or R m ðsÞ) of the corresponding integral equations in the following way:
e ðsÞ ¼ R
e c RðsÞ
e 1 þ cq RðsÞ= s þ s1 A
ð2:12Þ
Note that the numerator of this expression represents the initial stage of the reaction (geminate stage) following the light excitation. It corresponds to the first terms in the rhs of Eqs. (2.1) and (2.2) and (2.6) and (2.7) while the term in the denominator takes into account the subsequent reaction in the bulk represented by the second terms of the same equations. 3. Static quenching
ð2:4Þ
l ; sA ð2:5Þ
Let us turn to the solid or highly viscous solutions, where diffusion is frozen ðD ¼ 0Þ. Generally speaking, IET is not applicable to such systems because the transfer there is essentially multi-particle (to all partners simultaneously), while IET is only the lowest order (binary) approximation with respect to the partner concentration c, reproducing only the short time kinetics of irreversible
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quenching. The reversible transfer kinetics in IET is different but reduces to the irreversible one when sA ! 0. These conclusions are reached by analyzing the kernel of static quenching:
10
-5
ð3:1Þ
10
-6
It was obtained by the Laplace transformation of Eq. (2.3) and the set (2.4), (2.5) at D ¼ 0. We will consider first the irreversible energy quenching ðW B ¼ 0Þ and then its reversible analog, responsible for the delayed fluorescence.
10
-7
e ¼ RðsÞ
Z
3
d r
1
s þ sD
s þ s1 W F ðrÞ s þ s1 D A
1 1 s s þ s1 þ s þ W W F ðrÞ B ðrÞ þ s þ sA D A
3.1. Irreversible quenching ðW B ¼ 0Þ If the state accepting the energy decays immediately no backward transfer is possible. This makes the forward transfer irreversible as if W B ¼ 0. As a result the general reaction scheme (1.1) reduces to a simplified one WF
D þ A ! D þ A;
ð3:2Þ
# sD The solution of this problem following from Eq. (2.1) at c ¼ 0 is universal:
N ðtÞ ¼ expðt=sD Þ N0 ðtÞ;
ð3:3Þ
where N0 ðtÞ known as ‘‘static quenching kinetics” is given by its Laplace transformation:
e ðsÞ ¼ N 0
1
; e ir s s1 s þ cR D
ð3:4Þ
where
e ir ðsÞ ¼ s þ s1 R D
Z
3
d r
W F ðrÞ s þ W F ðrÞ þ s1 D
ð3:5Þ
3.1.1. Dipole–dipole mechanism Using here the dipole–dipole rate (1.2), we get for the strong quenching (by extending the integral in (3.5) from 0 to infinite r):
e ðsÞ ¼ N 0
1 pffiffiffiffiffiffi ; s þ D ps
ð3:6Þ
pffiffiffi
where D ¼ p c a. Taking the inverse Laplace transformation of this result, carried out in our earlier work [15], we obtain: 2 3
3=2
pffiffiffiffiffiffiffiffiffiffiffi N0 ðtÞ ¼ expðpD2 tÞerfc p D2 t
ð3:7Þ
Using the cumulant expansion at short times we obtain from here:
N0 ðtÞ ¼ e2D
pffi t þðp2ÞD2 t23ð4pÞðD2 tÞ3=2
at D2 t 1;
ð3:8Þ
while the exact Förster solution valid at any times and concentrations is
N0 ðtÞ ¼ e2D
pffi t
1 pffiffiffiffiffiffiffiffiffi 1 þ D psD
ð3:9Þ
0.01
0.02
0.03
Fig. 1. The kinetics of irreversible static energy quenching by dipole–dipole (blue) and exchange (magenta) mechanisms. The IET results (solid lines) are compared with their exact analogies (dashed lines) obtained with the Förster and Dexter rates. The rest of the parameters are c = 0.3 M, a = 5 108 Å6/ns, W0 = 9 104 ns1, r = 5 Å, L = 0.9 Å. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
As well as quenching kinetics, this result obtained with IET coincides with the exact one (see (Eq. (5.18)) in review [18]) only in the lowest order concentration approximation when g 1 pffiffiffiffiffiffiffiffiffi D psD . 3.1.2. Exchange mechanism The exchange interaction (1.3) can be subjected to a similar analysis. Using it in Eq. (3.5) we obtain for strong transfer after space integration:
! 2 2
e ir s s1 ¼ s 4p R3 þ p L R0 ; R D 0 3 4
at
L R0 1 2
ð3:10Þ
where the radius of strong quenching
R0 ¼
L Wc ln ; 2 s
ð3:11Þ
e ir ðsÞ from (3.10) in the general formula where W c ¼ W 0 er=L . Using R (3.4) we get:
" !!#1 2 2 e ðsÞ ¼ s 1 þ 4p c R3 þ p L R0 N : 0 0 3 4
ð3:12Þ
Making the inverse Laplace transformation, we obtain in the lowest order (linear) approximation with respect to c:
N0 ðtÞ ¼ 1 cVðtÞ;
ð3:13Þ
where the quenching volume
VðtÞ ¼
As can be seen the IET result for dipole–dipole quenching coincides with an exact one only initially: at times smaller than sF ¼ 1=4D2 , while the longer time tail is the ‘‘false asymptote” of the binary approximation (Fig. 1). If the quantum yield of fluorescence is measured stationary the whole quenching kinetics contributes to the value defined in Eq. (1.6). From this definition and Eq. (3.6) the yield for the dipole–dipole static quenching is
e s1 =sD ¼ g¼N 0 D
0.00
pL3 6
3
2
½ln W c t þ A1 ln W c t þ A2 ln W c t þ A3
extends with time. Here A1 ¼ 3C ¼ 1:7316; A2 ¼ 3C 2 þ p2 =2 ¼ 5:9343; A3 ¼ C 3 þ C p2 =2 w00 ð1Þ ¼ 5:4449; C – Euler constant, w00 ðxÞ tetra-gamma function [16]. The result (3.13) coincides with the exact one obtained in Ref. [17]
N0 ðtÞ ¼ ecVðtÞ in the lowest order approximation with respect to c. The coincidence occurs even in more limited region of the shortest times ðcVðtÞ 1Þ but the relative reduction is evidently smaller than in dipole–dipole case (Fig. 1).
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3.2. Energy conservation ðsA ¼ 1Þ
stationary solutions of Eqs. (2.1), (2.2), (2.6) and (2.7), provided
sA ¼ sD ¼ 1: Let us consider the opposite case, when the energy is temporarily conserved on a stable partner but finally returns back to the donor state. This phenomenon manifests itself via delayed fluorescence of the donor but the energy quenching, measured stationary or in indefinite time after the d-pulse, does not occur at all. That is the quantum yield (1.6), calculated with Laplace transformation of the set (2.1), gives
g ¼ 1 at sA ¼ 1
ð3:14Þ
Although there is no quenching in this case the kinetics of donor decay says a lot about the conservation and dissipation of energy defined in Eq. (2.11). The corresponding mass operator (at sA ¼ 1) appears to be:
e ðsÞ ¼ R
e c RðsÞ ¼ e 1 þ cq RðsÞ=s
(
e c RðsÞ
at short times ðsmall qÞ sc=cq at long times ðlarge qÞ ð3:15Þ
The short time decay is actually a donor quenching by energy acceptors which are initially almost empty. Due to further reversible transfer the equilibration of energy distribution is finally reached and the quasi-stationary partner populations decay simultaneously further on, via donor channel. Such an asymptotic decay is a universal one (independent of the particular quenching mechanism) but never considered earlier, since the bulk reactions were usually ignored setting q ! 0 [8–10]. 3.2.1. Dipole–dipole energy transfer Taking the integral in Eq. (3.1), we obtain for the dipole–dipole transfer at sA ¼ 1:
e ¼D c RðsÞ
pffiffiffiffi
p
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s s þ s1 D 1 : s þ c s þ sD
ð3:16Þ
e For short times c RðsÞ is identical to the mass operator (3.15), that being used in (2.11) gives the Laplace transformation of the initial transfer kinetics:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "
#1 pffiffiffiffi s s þ s1 1 e D 1
N ðsÞ ¼ s þ sD þ D p s þ c s þ sD
ð3:17Þ
As a matter of fact, the inverse transformation of this result describes only the geminate reaction immediately following the pulse excitation but completely ignores the bulk reactions as well as states equilibration. It is easy to show that this response to d-excitation has also two stages:
8 qffiffiffiffiffiffi < exp 2D t t=sD ; D2 t 1; 1þ c N ðtÞ ¼ : 2 pffiffiffi 3=2 DsD =2 c t ; ct D2 s2D
Neq A ¼
c ; c þ cq
N eq ¼
cq c þ cq
ð3:19Þ
:
The excitation decay after equilibration of population distribution proceeds as follows from the conventional chemical kinetics:
t N ðtÞ N eq exp Neq :
ð3:20Þ
sD
The initial static quenching gives way to this universal (mechanism independent) decay of the equilibrated system, which starts earlier the larger is the donor concentration, q. The latter is shown by dashed lines in Fig. 2. As soon as the kinetic (solid) curves 1, 2 and 3 approach the dashed ones the results become exact. They are also exact at the very beginning, where the same curves approach that for the irreversible static quenching of Förster’s type (blue). The latter gives place to the Dexter’s type kinetics if the transfer is an exchange one. It can be easily specified by integrating exchange rate in Eq. (3.1) at sA ¼ 1 that provides the mass operator for large s (small times) which specify the initial kinetics of reversible excitation decay i. e. exchange analog of Eq. (3.18):
N ðtÞ exp
c Vð½1 þ ctÞ t=sD ; 1þc
cVð½1 þ ctÞ 1; ð3:21Þ
where VðtÞ is a quenching volume defined in Eq. (3.13). The interpolation between specific static quenching and final (mechanism independent) equilibrated decay is questionable (due to the power time asymptote of the post-static initial quenching). However, this intermediate region reduces with donor concentration and at the highest q is actually eliminated (see curve 3 in Fig. 2). The static quenching almost immediately turns to be a universal one. Thus the application of the integral theory to solid solutions is reasonable for relatively high q. The situation is much better in liquid solutions due to the acceleration of the energy transfer by encounter diffusion. It makes the IET work well and MET even better in just the intermediate time region (Section 4).
10
-5
10
-6
10
-7
ð3:18Þ
The first coincides with the exact reversible static quenching at the very beginning and p differs ffiffiffiffiffiffiffiffiffiffiffiffi from the Förster kinetics only by the scale coefficient 1= 1 þ c , while the last one is recognized as a false power time asymptote of IET which is well-known in diffusion–accelerated quenching [4]. The rate of the reversible static transfer decreases with time and finally supports delayed fluorescence by backward transfer from remote acceptors. However, it can happen out of the validity limit of the static approximation (at t sF ) where the false (power time) IET asymptotic behavior take place. Fortunately, it follows from Eq. (3.15) that at any finite q the longest time behavior is controlled by a common exponential decay of the excited state populations, after their shares reach the final equilibrated values. These values can be found from the
0
10
20
30
40
Fig. 2. The kinetics of reversible static dipole–dipole fluorescence quenching followed by delayed fluorescence (solid lines) at different donor concentrations q: (1) 103 M, (2) 102 M, (3) 101 M. Their long time asymptotes is given by dashed lines, in comparison to a natural decay (green straight line) and the irreversible static quenching (the lowest blue line). Acceptor concentration c ¼ 0:1 M; sD ¼ 6 5 ns; sA ¼ 1; c ¼ 1; a ¼ 5 108 Å =ns. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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3.3. Reversible–irreversible transition with shortening
sA
Using Eqs. (2.11) and (2.12) in the general definition (1.6), we obtain that the yield of static quenching obeys the Stern–Volmer law,
1 1 g¼ ¼ ; e 1 þ cjsD 1 þ R ð0ÞsD (
e Rð0Þ e 1 þ cqsA Rð0Þ
-5
10
-6
10
-7
ð3:22Þ
where the Stern–Volmer constant
e ð0Þ=c ¼ j¼R
10
e Rð0Þ;
e 1 cqsA Rð0Þ e 1=ðcqsA Þ; cqsA Rð0Þ 1 ð3:23Þ
There are only two opposite limits. Either the quenching is cone or, alternatively, the nartrolled by energy transfer having j ¼ Rð0Þ row throat is the weighted natural decay of equilibrated excited states, when the Stern–Volmer constant, j ¼ ½cqsA 1 does not depend of the transfer mechanism at all. If this mechanism is a dipole–dipole one the former limit is presented by
cj ¼ D
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 1 p ; sD þ csA
ðsmall qÞ
ð3:24Þ
calculated from the general formula (3.1). In the opposite limit
j¼
1
cqsA
ðlarge qÞ
ð3:25Þ
at any transfer mechanism. For the exchange or contact mechanisms only the result (3.24) is different although follows from the same Eq. (3.1). At sA ¼ 0 from Eqs. (3.22)–(3.24) follows the earlier obtained result for the irreversible dipole–dipole transfer, Eq. (3.9). In the opposite case of reversible transfer to a stable partner ðsA ¼ 1Þ, it follows from Eq. (3.25) that there is no quenching at all ðj ¼ 0Þ, i.e. g ¼ 1. Eq. (3.23) allows tracing the gðsA Þ dependence from its minimal to maximum value. Turning now to the dipole–dipole quenching kinetics, we obtain from Eqs. (2.11) and (2.12)
N ðtÞ ¼ expðt=sA ÞN ðtÞ;
e ðsÞ ¼ N e ðs þ 1=sA Þ N
ð3:26Þ
The Laplace transformation of N ðtÞ is given by
e ðsÞ ¼ N
1 e e s þ s1 þ c R ðsÞ=½1 þ cq R ðsÞ=s
;
ð3:27Þ
while that of the kernel is
e ðsÞ ¼ D cR
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi sðs þ s1 Þ ; p s þ c s þ s1
ð3:28Þ
where 1 1 s1 ¼ sD sA :
Note that the structure of Eqs. (3.27) and (3.28) is identical to the Eqs. (3.16) and (3.17), so for the positive value of s ðsA > sD Þ both limits considered in (3.18) are also peculiar to N ðtÞ with the replacement sD ! s . Therefore it can be shown that the geminate reaction kinetics at sA > sD is also a two stage process. The initial exponential stage remains the same as in Eq. (3.18) but the following power time dependence is multiplied by an exponent:
Ds 2 N ðtÞ pffiffiffiffi 3=2 expðt=sA Þ; 2 pt
ct 1 D2 s2
0
5
10
15
20
25
30
Fig. 3. The transition from reversible (1) to irreversible (5) static quenching by shortening the life-times of the acceptor: sA ¼ 1ð1Þ; sA ¼ 100ð2Þ; sA ¼ 10ð3Þ; 6 sA ¼ 5ð4Þ; sA ¼ 0ð5Þ ns (at c ¼ 0:1 M; q ¼ 103 M; c ¼ 1; a ¼ 5 108 Å =ns and sD ¼ 5 ns).
in (3.3) (Fig. 3). The border between them is indicated by curve 4 ðsD ¼ sA Þ that asymptotically approaches the natural decay. However, the geminate stage is not the final stage of the reversible reaction: eventually the equilibration of excitations occurs and the earlier the larger is q. Hence, the geminate stage gives way to the bulk one:
eq 1 t : N ðtÞ N eq exp Neq s1 D þ N A sA
ð3:30Þ
This is the universal asymptotic decay of excitations that proceeds through both available channels, D and A. Note that in a borderline case ðsA ¼ sD ¼ sÞ this asymptotic degenerates into the kinetics of natural decay: N ðtÞ N eq expðt=sÞ. 4. Diffusion accelerated energy transfer 4.1. Geminate stage Let us first consider the geminate stage of the reaction that takes place first (after d-excitation) and remains the single one in very dilute solutions. Setting q ! 0 in Eqs. (2.1) and (2.2) we lose the inter-molecular backward transfer but the reduced equations
Z t N ðtÞ N_ ðtÞ ¼ c ds RðsÞ N ðt sÞ sD 0 Z t N ðtÞ ds RðsÞ N ðt sÞ A ; N_ A ðtÞ ¼ c
ð4:1Þ ð4:2Þ
sA
0
still account for the ‘‘micro-reversibility” in the kernel, RðsÞ, via W B in Eqs. (2.4) and (2.5). They should be solved now taking into account the encounter diffusion ðD – 0Þ. For contact reactivity this was done a few times analytically but using here the dipole–dipole transfer rates in both directions, Eqs. (1.2) and (1.5), we have to solve them once again. e ðsÞ ¼ c RðsÞ e As q ¼ 0 the IET ‘‘mass operator” is R and the kinetics of geminate energy dissipation and accumulation are given by the simple Laplace transformations:
e ðsÞ ¼ N
1 e s þ s1 D þ c RðsÞ
;
e ðsÞ ¼ N A
sþs
1 A
e c RðsÞ ;
e s þ s1 D þ c RðsÞ
ð3:29Þ
ð4:3Þ
Changing sA we can trace the transition from the absolutely reversible limit ðsA ¼ 1Þ to the irreversible one ðsA ¼ 0Þ described
following from Eq. (2.11). The quantum yield calculated from here obeys the Stern–Volmer law, g ¼ 1=ð1 þ cjg sD Þ, but the geminate Stern–Volmer constant,
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e jg ¼ Rð0Þ:
ð4:4Þ
In the very popular contact approximation of transfer rates (1.4) it is known to be [4,8]
1
jg
¼
" # 1 1 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ c pffiffiffiffiffiffiffiffiffiffiffiffi ; þ kf kD 1 þ sd =sD 1 þ sd = sA
ð4:5Þ
1
where sd ¼ r2 =D is the encounter time and kD ¼ 4prD is the diffusion rate constant in the contact model. The analogous results for the exponential transfer rate (1.3) were obtained a number of times when the reversible electron transfer, executed by such a rate, was studied (see Chapter VIII in Ref. [4] or pp. 3.3 and 3.4 in the recent review [5]). Therefore, we concentrate later on only the dipole–dipole mechanism of reversible transfer. It is well-known from DET that the diffusion accelerates the irreversible quenching of donor by the remote energy transfer. After short static quenching and subsequent transient kinetics the donor decays exponentially with a ‘‘quasi-stationary” rate constant,
kD ¼ 4pRQ D; where RQ is the quenching radius to be calculated [3–5]. In IET the short static quenching also precedes the fast (diffusion accelerated) exponential quenching. When the backward energy transfer is accounted for, the kinetics of the geminate stage of the reaction remains qualitatively the same, but proceeds slower. The same has been done with MET by substituting Rm ðtÞ for RðtÞ in Eqs. (4.1) and (4.2). Fortunately, the very effective program for solving either IET or MET equations at any c and arbitrary relationship between the life-times was developed in Ref. [11]. Using it to specify the role of encounter diffusion in geminate quenching we obtain a noticeable difference between the IET and MET results for the irreversible and less for the reversible reactions even at low donor concentrations (Fig. 4). Only the initial static stage of geminate quenching is the same in both theories while the diffusion–accelerated quenching is better described by MET that eliminates the false IET tails. The kinetics of either irreversible (blue) or reversible (magenta) quenching develops faster than the static one but the difference between IET and MET (considered as more accurate) is more pronounced when the quenching is irreversible.
10
-5
10
-6
10
-7
0
1
2
3
e The Stern–Volmer constant in MET, jm g ðc; qÞ ¼ R m ð0Þ, accounts for the concentration dependence of this quantity which is missed e m ðs; c; qÞ is originated from in IET. The concentration dependence R the shortening of the life-times of both partners quenched by the energy transfer to ‘‘outsiders” presented around in concentration c and q. For instance, in the contact approximation
4
5
Fig. 4. Irreversible (blue; sA ¼ 0) and reversible (magenta; sA ¼ 1) dipole–dipole geminate quenching in MET (solid lines) and IET (dashed lines) at 6 c ¼ 0:1 M; q ¼ 103 M; c ¼ 1; D ¼ 2:6 105 cm2 =s; a ¼ 5 108 Å =ns and sD ¼ 5 ns. The straight green line represents the natural (exponential) decay of the donor. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
jmg
" # 1 1 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ kf kD 1 þ sd ð1=sD þ cjg Þ 1 þ sd ð1=sA þ cqjg Þ ð4:6Þ
As is seen, MET accounts for the shortening of the donor (acceptor) life times due to their acceptor (donor) surroundings. The effect which is larger in more concentrated systems accelerates the diffusional transfer extending the transient contribution. As a result the geminate Stern–Volmer constant in MET slightly increases with both concentrations (Fig. 5), while that of IET is a true constant. 4.2. Including the bulk stage If the pulse experiment is carried on with a highly concentrated solution of donors and the fluorescence is registered long enough 2.2x10
-11
2.0x10
-11
1.8x10
-11
1.6x10
-11
1.4x10
-11
1.2x10
-11
1.0x10
-11
8.0x10
-12
6.0x10
-12
4.0x10
-12
0.0
0.1
0.2
0.3
0.4
0.5
Fig. 5. The Stern–Volmer constant dependence on acceptor concentration c at different concentrations of donors, q, obtained with MET in the contact approximation at (1) q ¼ 103 M, (2) q ¼ 102 M, (3) q ¼ 101 M. The horizontal dashed line is that of IET. The rest of the parameters are kF ¼ 1013 ðM sÞ1 ; kD ¼ 5 1010 ðM sÞ1 ; sd ¼ sD ¼ 5 ns; sA ¼ 10 ns; c ¼ 1.
10
-5
10
-6
10
-7
10
-8
0
10
20
30
Fig. 6. The long time kinetics of the dipole–dipole quenching (including the bulk stage) at the same parameter and color definitions as in Fig. 4. The thin solid magenta line represents the asymptotic exponential decay of the equilibrated system at c ¼ 0:1 M; q ¼ 103 M; c ¼ 1; sA ¼ 1. The dotted line represents the static quenching. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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S.G. Fedorenko, A.I. Burshtein / Chemical Physics 370 (2010) 208–214
1.2x10
-10
1.0x10
-10
8.0x10
-11
6.0x10
-11
4.0x10
-11
2.0x10
-11
To calculate the quantum yield from Eq. (1.6) one should know N ðtÞ for infinitely long times. In the contact approximation this can be done analytically and the corresponding Stern–Volmer constant appears to be
1
j
0.000
0.002
0.004
0.006
0.008
0.010
Fig. 7. The stationary (red) and geminate (blue) Stern–Volmer constants, as functions of donor concentration q at c ¼ 0:01 M. Solid lines – MET, dashed lines – IET. Other parameters are the same as in Fig. 5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
compared to the donor life-time, then the bulk stage of the reaction cannot be ignored and the theoretical description of the system response should be based on the general equations of IET or MET given in Section 2. The results of their numerical solution for the dipole–dipole rates (1.2)–(1.4) and (1.5) are presented in Fig. 6. There is a new feature compared to the geminate stage shown in Fig. 4. The results of IET coincide with those of MET not only at the shortest time where quenching is static but also at the longest ones where the excitation distribution is equilibrated and decays with the universal rate (3.30). In between where the completely reversible reaction ðsA ¼ 1Þ is diffusion accelerated, the IET and MET curves (solid and dashed magenta lines) differ just a bit. This difference is more pronounced at a finite acceptor life-time and never disappears at sA ¼ 0 (blue lines) when the transfer (quenching) is irreversible and never becomes the equilibrated one. These conclusions are in excellent agreement with those obtained for inter-molecular transfer between two excited states catalyzed by encounters with other solutes. They follow from an analysis of eclectic collection of Brownian dynamics simulations presented in Ref. [19], that was reproduced in color (with critical comments) in a recent review (Fig. 7 in Ref. [5]). In original article authors compared IET, MET and other multi-particle theories with BD simulations at different excitation times relationship providing either quenching or conservation of energy. In accordance with what was established above (in Figs. 4 and 6) the difference between IET and MET decay kinetics is pronounced when the transfer is irreversible and almost invisible in case of reversible transfer (conservation). In the latter case the discrepancy between them is covered by the delayed fluorescence from equilibrate state. The deviations from BD results are not seen at all at reasonable concentration presented in the left panels of the figure under discussion 3 ðcB ¼ cD ¼ 104 Å ¼ 0:16 MÞ. And only in the limit of the very 3 high concentration ðcB ¼ cD ¼ 103 Å ¼ 1:6 MÞ presented in the right panels of the same figure it is seen where it should be expected: only IET being zero order approximation deviates from BD calculated irreversible transfer kinetics. No such a demerit is inherent to MET collecting all binary terms.
( ¼
e IET 1= Rð0Þ þ cqsA ¼ 1=jg þ cqsA ; m e 1= R m ð0Þ þ cqsA ¼ 1=jg þ cqsA ; MET
ð4:7Þ
Absolutely the same can be obtained from the stationary solution of the integral equations if the permanent light pumping is accounted for. Adding its rate I0 q to the rhs of Eq. (2.6) one can obtain the stationary population, N s , and using it in an alternative definition of quantum yield, g ¼ N s =ðI0 qsA Þ, confirm the result (4.7). This is the stationary fluorescence quantum yield which is easier to measure than to calculate it from the time resolved data. Unlike the geminate one the stationary j decreases with donor concentration, due to the backward energy transfer in the bulk, Fig. 7. 5. Results Using the simplest and modified integral theories, IET and MET, we considered the reversible inter-molecular energy transfer conducted by remote (dipole–dipole) and short range interactions. The initial static stage of quenching kinetics has an exact description common for both theories, as well as the final stage following complete equilibration of state populations. In between, in a geminate stage of the diffusion accelerated reaction, the MET provides a better description then IET but the difference increases with donor and acceptor concentrations. We calculated the kinetics of donor decay, after d-pulse excitation and the fluorescence quantum yield in diluted and concentrated solutions. The geminate Stern–Volmer constant obtained from the short time system response and that found from the stationary detected fluorescence were shown to be different. They depend differently on both donor and acceptor concentrations and these dependencies were specified with both theories using the popular contact model of transfer rates. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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