Exciplex Formation Accompanied with Excitation Quenching - The

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J. Phys. Chem. A 2010, 114, 4558–4569

Exciplex Formation Accompanied with Excitation Quenching Stanislav G. Fedorenko Institute of Chemical Kinetics and Combustion, Russian Academy of Sciences, NoVosibirsk, Russia

Anatoly I. Burshtein* Weizmann Institute of Science, RehoVot 76100, Israel ReceiVed: January 4, 2010; ReVised Manuscript ReceiVed: February 17, 2010

The competence of the reversible exciplex formation and parallel quenching of excitation (by electron or energy transfer) was considered using a non-Markovian π-forms approach, identical to integral encounter theory (IET). General equations accounting for the reversible quenching and exciplex formation are derived in the contact approximation. Their general solution was obtained and adopted to the most common case when the ground state particles are in great excess. Particular cases of only photoionization or just exciplex formation separately studied earlier by means of IET are reproduced. In the case of the irreversible excitation quenching, the theory allows specifying the yields of the fluorescence and exciplex luminescence, as well as the long time kinetics of excitation and exciplex decays, in the absence of quenching. The theory distinguishes between the alternative regimes of (a) fast equilibration between excitations and exciplexes followed by their decay with a common average rate and (b) the fastest and deep excitation decay followed by the weaker and slower delayed fluorescence, backed by exciplex dissociation. I. Introduction The irreversible quenching of the excited donor D* by electron acceptor A is studied in the wide class of transfer reactions, D* + A f D+ + A-, either exergonic or endergonic. Moreover, the famous Rehm-Weller dependence1 of the stationary quenching rate ki on the free energy of electron transfer, ∆GI greater than or less than 0, was finally plausibly explained2 within the differential encounter theory (DET).3 However, this theory does not work in a narrow strip near the resonance where ∆GI(σ) ) 0, provided the electron tunneling V(σ) ) V0 at contact distance σ is rather strong. The exciplex E ) [Aδ-Dδ+] (with partial charge separation δ) is formed there from the contact pair of neutral reactants [AD*], while the full electron transfer occurs between the space separated reactants. It proceeds with the distancedependent rate W(r) which is usually specified by the secondorder perturbation theory with respect to V(r). The quasiresonance case is an exceptional one due to the reversibility of the electron transfer that cannot be accounted for with DET but only with IET (integral encounter theory) or its modified version (MET).

The quasi-resonant electron transfer results in either the exciplex or the free ion (A- · · · D+) formation (Figure 1) according to Scheme 1. The rates of the forward and backward electron transfer obey the detailed balance principle

WF(r) ) WB(r) exp(-∆GI /T)

(1.2)

as well as the rates of association and dissociation of exciplex compounds

ka /kd ) V exp(-∆GE /T)

(1.3)

Here the Boltzmann constant kB ) 1, V is the reaction volume, and the free energy of the resonant exciplex formation is known to be4-6 ∆GE ) -Vc, where Vc is the electron coupling between the resonant states D*A and D+A- at contact. The exciplex formation and dissociation is contact by definition, but the electron transfer (especially proceeding in the normal Marcus region) is often fashioned as an exponential one3,7,8

SCHEME 1

10.1021/jp100056a  2010 American Chemical Society Published on Web 03/15/2010

Exciplex Formation with Excitation Quenching

J. Phys. Chem. A, Vol. 114, No. 13, 2010 4559 tion quenching. Their relative yields and kinetics are subjected to analytical investigation in section IV. Until now the energy (fluorescence) quenching by either electron transfer or exciplex formation was considered only separately according to the reduced reaction schemes

A- + D+

WF

{\} WB

VWR

Figure 1. The space dependence of ionization free energy ∆GI ) rc/σ - rc/r at resonant exciplex formation. The spheres positioned on the abscises present the reactants separated by distance r which become the counterions (shown above) as a result of the electron transfer at this distance. The excited donor and the exciplex formed at contact decay with the times τD and τE, respectively. The rates of reversible free ions formation are WI and WB, while WR is the rate of ion recombination to the reactants ground state. The rate constants of exciplex association and dissociation are ka and kd, respectively.

(

W(r) ) W0 exp -

2(r - σ) l

)

(1.4)

All the parameters could be different for WF, WB, and WR except the common contact radius σ. Moreover, for the sake of simplicity the electron transfer is sometimes considered as a contact one.3,7 In this approximation it is accounted for (via boundary conditions) by the corresponding kinetic rate constants

kf )

∫ WF(r) d3r

kb )

∫ WB(r)er /r d3r

kr )

∫ WR(r)er /r d3r

(1.5)

c

c

where rc ) e2/εT is the Onsager radius of the Coulomb attraction. According to Figure 1 the ionization preceding exciplex formation is an endergonic one and therefore essentially reversible (kb > kf). However, the quenching of the excitation by dipole-dipole (Foerster) or exchange (Dexter) mechanisms is usually irreversible.9 The exchange rate has an exponential shape like (1.4) and therefore is often considered as a contact one. The doublechannel reaction including the irreversible energy quenching in line with reversible exciplex formation proceeds according to the scheme:

AD

kf

79

A + D*

ka

{\}

Aδ-Dδ+

kd

VτD

(1.6)

VτE

This is the simplest particular case of two competing contact reactions, reversible exciplex formation and irreversible excita-

A + D*

A + D*

ka

{\}

Aδ-Dδ+

kd

VτD

(1.7)

VτD

(1.8)

VτE

The former was considered first by means of conventional IET accounting for the space dependence of all the transfer rates but only for the geminate reaction (neglecting charge recombination in the bulk).10 In the next article it was taken into consideration11 but for only the irreversible ionization (WB ) 0). The general IET equations obtained and studied in ref.12 account for both: the reversible geminate photoionization and the bulk reaction of charged products. The latter is responsible for the delayed fluorescence and spin conversion effects that were studied in ref 13 using the contact approximation in the basic equations. All these results are reviewed in ref 7. The reversible dissociation of the stable complex (τE ) ∞) was first studied by Berg.14 The extension of this theory to the reversible exciplex dissociation made in ref 15 showed its identity with the IET analogue of the monomolecular exciplex dissociation: from right to left of reaction scheme 1.8. The IET equations for reverse reaction of bimolecular exciplex formationsfrom left to right of reaction scheme 1.8swere obtained in ref 16 and reproduced numerously. Then the theory was extended to account for both: the reversible exciplex dissociation and the backward bimolecular exciplex formation.18 The same results reviewed in refs 7 and 8 were obtained independently by other methods in refs 20-24. The original method proposed in the last work allows the easiest extension of the theory to the general reaction (Scheme 1) that appears in the next section. There the non-Markovian kinetic equations for all the particle concentrations and for the π-forms of reacting pairs will be obtained in the contact approximation. Their general solution in the case of equal particle mobilities and zero Coulomb fields will be presented in section III. It was shown to obey the corresponding IET equations whose kernels are specified via reaction constants and the diffusion coefficient. In the particular cases (1.7) and (1.8) these equations reduce to already known ones. In the last section we concentrate on the reaction 1.6 looking for the fluorescence yields and decay kinetics of the excited donor and exciplex.

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J. Phys. Chem. A, Vol. 114, No. 13, 2010

Fedorenko and Burshtein

II. Kinetic Equations A. Reaction under Kinetic Control. In the kinetic limit (at the highest diffusion) one can use the conventional set of chemical kinetic equations for particle concentrations

d[D*]t 1 ) -(kf + ka)[D*]t[A]t + kb[D+]t[A-]t + kd[E]t - [D*]t dt τD

(

)

d[A]t 1 [E]t ) -(kf + ka)[D*]t[A]t + (kb + kr)[D+]t[A-]t + kd + dt τE d[D+]t d[A-]t ) ) kf[D*]t[A]t - (kb + kr)[D+]t[A-]t dt dt d[D]t 1 1 ) kr[D+]t[A-]t + [D*]t + [E]t dt τD τE

(

(2.1) (2.2) (2.3) (2.4)

)

d[E]t 1 [E]t ) ka[D*]t[A]t - kd + dt τE

(2.5)

The conservation laws for the particles participating in these reactions demands that

[D*]t + [D+]t + [D]t + [E]t ) constant,

[A-]t + [A]t + [E]t ) constant

(2.6)

Therefore we have to keep only three independent equations, (2.1), (2.3), and (2.5), omitting all the rest. B. Non-Markovian Equations Based on π-Forms. In the case of moderate mobilities the multiparticle correlations have to be taken into consideration. Formally exact kinetic equations for the concentrations [Xi]t should account for the π-forms,22 Pij(r,t) ) Fij(r,t) - [Xi]t[Xj]t, i.e., for the deviations of the pair distribution functions Fij(r,t) from the products of the corresponding concentrations24

d[Xi]t Φimn([Xm]t[Xn]t + Pmn(R, t)) )dt mgn

∑

∑ (Ωin + Qin)[Xn]t

(2.7)

n

Here the first sum collects all the quadratic in concentration terms while the second contains the linear terms: dissociation {Ωin} and relaxation {Qin} (N × N) matrixes, provided N is a number of reactants. In the simplest approximation corresponding to the integral encounter theory π-forms satisfy the following matrix reactiondiffusion equations

∂P ) L(DP + PD) - QP - PQT ∂t

(2.8)

here D is the diagonal matrix of the diffusion coefficients and Lij are the stochastic motion operators, while

[ ] [

0 0 0 Ω) 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 -kd 0 -kd 0 0 0 0 0 0 0 kd

τD-1

0 0 0 0

0 0 Q) 0 -τD-1

0 0 0 0

0

0 0 0 0

0 0 0 0

0

0 -τE-1 0 0 0 0 0 -τE-1

0 0 0 0

τE-1

]

(2.9)

For the reaction under study there are only 3 independent equations from the set 2.7:

d[D*]t 1 ) -(kf + ka)([D*]t[A]t + PD*A(σ, t)) + kb([D+]t[A-]t + PD+A-(σ, t)) + kd[E]t - [D*]t dt τD d[D+]t d[A-]t ) ) kf([D*]t[A]t + PD*A-(σ, t)) - (kb + kr)([D+]t[A-]t + PD+A-(σ, t)) dt dt d[E]t 1 [E]t ) ka([D*]t[A]t + PD*A(σ, t)) - kd + dt τE

(

)

(2.10) (2.11) (2.12)

Since initially the particles are assumed to be randomly distributed, the deviations Pij(r,t ) 0) ) 0. Accordingly, the correlations become negligible for the large distances: Pij(rf∞,t) ) 0. Due to the symmetry relationship, Pij(r,t) ) Pji(r,t), the matrix eq 2.8 consist actually of 21 equations for independent π-forms but only three of them are coupled and necessary for solving the present problem


J. Phys. Chem. A, Vol. 114, No. 13, 2010 4561

∂PD*A 1 1 ) DD*ALD*APD*A - PD*A + PED* ∂t τD τE

(2.13)

∂PD+A) DD+A-LD+A-PD+A∂t

(2.14)

∂PED* 1 1 ) DED*LED*PED* - PED* - PED* ∂t τD τE

(2.15)

It is assumed here that all the reactions occur only at contact radius σ and the corresponding reaction fluxes are incorporated into the boundary conditions for the reacting pairs

DD*A4πσ2

∂PD*A ∂r

DD+A-4πσ2e-βU(σ)

|

r)σ

) (kf + ka)([D*]t[A]t + PD*A(σ, t)) - kb([D+]t[A-]t + PD+A-(σ, t)) - kd[E]t

|

∂ βU(r) PD+A) (kb + kR)([D+]t[A-]t + PD+A-(σ, t)) - kf([D*]t[A]t + PD*A(σ, t)) e ∂r r)σ

(2.16) (2.17)

where U(r) is the Coulomb potential acting between the counterions. For the unreactive ED* pair there is the reflecting boundary condition

DED*4πσ2

∂PED* ∂r

|

r)σ

)0

(2.18)

It is easy to see that eq 2.15 with the boundary condition (2.18) has zero solution. So we can omit the last term of the right-hand side of eq 2.13 and retain only two coupled equations (2.13) and (2.14). C. Reactions in Highly Polar Solvents. In solvents of high polarity one can set U(r) ) 0 and, assuming for simplicity equal mobilities in any reactive pairs, introduce the following simplification

DD*A ) DD+A- ) D,

LD*A ) LD+A- ) ∇2

(2.19)

∂ 1 P (r, t) ) D∇2PD*A(r, t) - PD*A(r, t) ∂t D*A τD

(2.20)

∂ (r, t) ) D∇2PD+A-(r, t) P ∂t D+A-

(2.21)

1 sP˜D*A(r, s) ) D∇2P˜D*A(r, s) - P˜D*A(r, s) τD

(2.22)

sP˜D+A-(r, s) ) D∇2P˜D+A-(r, s)

(2.23)

After that we have the following equations

From their Laplace transformation we get

while the boundary conditions are

D4πσ2

∂P˜D*A(r, s) ∂r

D4πσ2

|

r)σ

) (kf + ka)(F˜D*A + P˜D*A(σ, s)) - kb(F˜D+A- + P˜D+A-(σ, s)) - kdE˜(s)

∂P˜D+A-(r, s) |r)σ ) (kb + kr)(F˜D+A- + P˜D+A-(σ, s)) - kf(F˜D*A + P˜D*A(σ, s)) ∂r

(2.24)

(2.25)

where F˜D*A, F˜D+ A-, and E˜(s) are the Laplace images of [D*]t[A]t, [D+ ]t[A- ]t, and [E]t, correspondingly. So the formulation of the problem for a double channel contact reaction includes eqs 2.10-2.12 for the concentrations and (corresponding to the accuracy of IET) eqs 2.20-2.25 for the π-forms of the reactive pairs. In the next section we will solve the problem in general and illustrate it by reactions of separate formation of ions and exciplex (1.7) and (1.8) that have been well studied previously.3,7,8

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III. Competition of Photoionization and Exciplex Formation A. General Solution. Now we start to investigate the general reaction (Scheme 1) solving eqs 2.22 and 2.23 with the boundary conditions (2.24) and (2.25). Thus we obtain π-forms at the contact distance

Here

A)-

P˜D*A(R, s) ) AF˜D*A + BE˜(s) + CF˜D+A-

(3.1)

P˜D+A-(R, s) ) A1F˜D*A + B1E˜(s) + C1F˜D+A-

(3.2)

[

(3.3)

kd(sk˜D(s) + kb + kr) 1 , Z sk˜D(s)s1k˜D(s1)

B) C)

]

kf + ka kfkr + ka(kb + kr) 1 + , ˜ s1kD(s1) sk˜D(s)s1k˜D(s1) Z

kb

1 , s1k˜D(s1) Z

C1 ) -

[

B1 )

kf 1 ˜ skD(s) Z

A1 )

kdkf 1 Z ˜ ˜ skD(s)s1kD(s1)

(3.4)

]

kb + kr kfkr + ka(kb + kr) 1 + sk˜D(s) sk˜D(s)s1k˜D(s1) Z

(3.5)

where

k˜D(s) ) and

Z)

kb sk˜D(s)

( )

kD 1+σ s

(

+ 1+

s , D

kf s1k˜D(s1)

kD ) 4πσD,

)(

1+

kr sk˜D(s)

)

+

s1 ) s + 1/τD

ka s1k˜D(s1)

(

1+

kb + kr sk˜D(s)

(3.6)

)

(3.7)

Making the inverse Laplace transformation over (3.1) and (3.2)

PD*A(R, t) ) PD+A-(R, t) )

∫0t dτA(t - τ)[D*]τ[A]τ + ∫0t dτB(t - τ)[E]τ + ∫0t dτC(t - τ)[D+]τ[A-]τ ∫0t dτA1(t - τ)[D*]τ[A]τ + ∫0t dτB1(t - τ)[E]τ + ∫0t dτC1(t - τ)[D+]τ[A-]τ

and substituting the result into (2.10)-(2.12) we obtain the integro-differential equations for the problem (Scheme 1)

d[D*]t )dt

∫0t dτR1(t - τ)[D*]τ[A]τ + ∫0t dτR2(t - τ)[E]τ + ∫0t dτR3(t - τ)[D+]τ[A-]τ -

[D*]t τD

d[D+]t t t t ) 0 dτR4(t - τ)[D*]τ[A]τ + 0 dτR5(t - τ)[E]τ - 0 dτR6(t - τ)[D+]τ[A-]τ dt d[E]t [E]t t t t ) 0 dτR7(t - τ)[D*]τ[A]τ - 0 dτR8(t - τ)[E]τ + 0 dτR9(t - τ)[D+]τ[A-]τ dt τE

∫

∫

∫

∫

∫

∫

(3.8) (3.9) (3.10)

where the kernels are defined in the Laplace representation

R˜1(s) ) (kf + ka)(1 + A) - kbA1, R˜3(s) ) -(kf + ka)C + kb(1 + C1), R˜5(s) ) kfB - (kb + kr)B1, R˜7(s) ) ka(1 + A),

R˜2(s) ) -(kf + ka)B + kbB1 + kd

(3.11)

R˜4(s) ) kf(1 + A) - (kb + kr)A1

(3.12)

R˜6(s) ) -kfC + (kb + kr)(1 + C1)

R˜8(s) ) -kaB + kd,

R˜9(s) ) kaC

(3.13) (3.14)

This is the complete collection of the contact problem solution. B. The Ground State Particles in Great Excess. For further consideration, we introduce the following dimensionless values

N*(t) ) [D*]t /[D*]0,

N((t) ) [D+]t /[D*]0 ) [D-]t /[D*]0,

NE(t) ) [E]t /[D*]0

(3.15)

and take into account that concentrations of acceptor and donors are time independent: [A]t ≈ [A]0 ) c, [D]t ≈ [D]0 ) F, provided acceptors are in great excess and the excitation is relatively low, ε ) [D*]0/[D]0 , 1. In such a case eqs 3.8-3.10 take the form

˙ *(t) ) -c ∫t dτR1(t - τ)N*(τ) + N 0 N˙((t) ) c

∫0t dτR2(t - τ)NE(τ) + εF ∫0t dτR3(t - τ)[N((τ)]2 - N*(t) τD

∫0t dτR4(t - τ)N*(τ) + ∫0t dτR5(t - τ)NE(τ) - εF ∫0t dτR6(t - τ)[N((τ)]2

(3.16) (3.17)


N˙E(t) ) c


∫0t dτR7(t - τ)N*(τ) - ∫0t dτR8(t - τ)NE(τ) + εF ∫0t dτR9(t - τ)[N((τ)]2 -

NE(t) τE

(3.18)

where the kernels in the Laplace representation after some algebra are the following

[

R˜1(s) ) (kf + ka) 1 -

]

kbkf (kf + ka)[sk˜D(s) + kr] + kakb Zsk˜D(s)s1k˜D(s1) Zsk˜D(s)

[ [ (

R˜2(s) ) kd 1 -

(kf + ka)[sk˜D(s) + kr] + kakb Zsk˜D(s)s1k˜D(s1)

]

(3.19)

(3.20)

)] )]

(kb + kr)[s1k˜D(s1) + ka] + kfka 1 kf + ka R˜3(s) ) kb 1 + Z s1k˜D(s1) sk˜D(s)s1k˜D(s1)

(3.21)

kb + kr ka(kb + kr) + kfkr 1 kf + ka R˜4(s) ) kf 1 + + Z s1k˜D(s1) sk˜D(s) sk˜D(s)s1k˜D(s1)

(3.22)

[ (

1 kfkd R˜5(s) ) Z s1k˜D(s1)

[ ( [ (

(3.23)

ka(kb + kr) + kfkr 1 kb + kr R˜6(s) ) (kb + kr) 1 + Z sk˜D(s) sk˜D(s)s1k˜D(s1)

)]

-

ka(kb + kr) + kfkr 1 kf + ka R˜7(s) ) ka 1 + Z s1k˜D(s1) sk˜D(s)s1k˜D(s1)

[

R˜8(s) ) kd 1 -

]

kb + kr + sk˜D(s) , Zsk˜D(s)s1k˜D(s1)

R˜9(s) )

)]

kfkb Zs1k˜D(s1)

(3.24) (3.25)

kakb Zs1k˜D(s1)

(3.26)

C. Only Exciplex Formation. Since in the case of the particular reaction 1.8 kf ) kb ) kr ) 0, only four kernels remain nonzero and all of them could be expressed via the universal one, Y˜(s)

˜ (s) ) Υ

R˜1(s) R˜7(s) R˜2(s) R˜8(s) 1 ) ) ) ) ka ka kd kd 1 + ka /[s1k˜D(s1)]

(3.27)

Thus we obtain the integro-differential equations

dN* ) -cka dt dNE ) cka dt

∫0t dτΥ(t - τ)N*(τ) + kd ∫0t dτΥ(t - τ)NE(τ) - N*(t) τD

∫0t dτΥ(t - τ)N*(τ) - kd ∫0t dτΥ(t - τ)NE(τ) -

NE(t) τE

(3.28) (3.29)

The set of these equations with their kernel (3.27) are identical to IET equations derived in ref 17 (eqs 4.1 with kernels 3.15) and their analogues postulated in ref 18. They are also in agreement with some particular results obtained earlier.19-23 D. Only Photoionization. Turning to reaction 1.7 we have to set ka ) kd ) 0 and keep only the nonzero kernels defined in the Laplace representation

(

R˜*(s) ) R˜1(s) ) kf 1 + R˜†(s) ) R˜4(s) )

kf , Z

where

Z)

kb sk˜D(s)

kr

)

1 , sk˜D(s) Z

R˜#(s) ) R˜3(s) )

(

R˜‡(s) ) R˜6(s) ) kb + kr +

(

+ 1+

kf s1k˜D(s1)

)(

1+

kr sk˜D(s)

kb Z

(3.30)

)

kfkr 1 ˜ s1kD(s1) Z

(3.31)

)

(3.32)

With these kernels the IET integro-differential equations take the form common for all previous studies

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dN* ) -c dt


∫0t dτR*(t - τ)N*(τ) + εF ∫0t dτR#(t - τ)[N((τ)]2 - N*(t) τD

dN( )c dt

(3.33)

∫0t dτR†(t - τ)N*(τ) - εF ∫0t dτR‡(t - τ)[N((τ)]2

(3.34)

In particular eqs 3.33 and 3.34 identically coincide with eqs 3.358 and 3.369 of the review7 where a number of their applications are also considered. IV. Irreversible Energy Quenching and Exciplex Formation Let us consider, as the simplest application of the theory, the irreversible energy quenching competing with reversible exciplex formation, according to (1.6). In this particular situation kb ) kr ) 0 and the kernels R˜3(s), R˜9(s) turn to zero while the necessary ones are

R˜1(s) ) R˜7(s) )

ka , Y

kf + ka , Y

kd Y kd kf 1+ R˜8(s) ) Y ˜ s1kD(s1) R˜2(s) )

(

(4.1)

)

(4.2)

where

Y)1+

kf + ka s1k˜D(s1)

(4.3)

Ignoring the ground state product of energy quenching, we have to write only two kinetic equations for N*(t) and NE(t) which follow from eqs 3.16 and 3.18, after neglecting there the recombination terms

˙ *(t) ) -c ∫t dτR1(t - τ)N*(τ) + N 0 N˙E(t) ) c

∫0t dτR2(t - τ)NE(τ) - N*(t) τD

∫0t dτR7(t - τ)N*(τ) - ∫0t dτR8(t - τ)NE(τ) -

(4.4)

NE(t) τE

(4.5)

˜ *(s) N

(4.6)

Making Laplace transformations of these equations we obtain

˜ *(s) ) N

1 s + τD-1 + cΣ˜ (s)

,

˜ E(s) ) N

where the “mass operator” equals

Σ˜ (s) ) R˜1(s) -

R˜2(s)R˜7(s) kf + ka ) Y s + 1/τE + R˜8(s)

[

cR˜7(s) s + 1/τE + R˜8(s)

s + τE-1 + kdkf /(kf + ka) kdkf 1 1 s + τE-1 + + Y s1k˜D(s1) kf

A. The Luminescence Quantum Yields. The excitation fluorescence quantum yield

η)

(

)

]

˜

1 ) ∫0∞ N*(t) dt/τD ) N*(0) τD 1 + cκτD

(4.7)

(4.8)

obeys the Stern-Folmer law, whose constant κ can be easily found using eq 4.7

κ ) Σ˜ (0) ) R˜1(0) -

R˜2(0)R˜7(0) 1/τE + R˜8(0)

(4.9)

and the quantum yield of the exciplex fluorescence can be specified using eq 4.6

˜ E(0) N φE ˜ *(0) ) N τE τE

(4.10)

N˜E(0) cR˜7(0) ) ˜ *(0) N 1/τE + R˜8(0)

(4.11)

ηE ) It is seen that the expression

φE )

is just the fraction of the exciplexes transformed from excitations. For a better understanding of the quenching phenomenon, it is useful to present eqs 4.9 in the following form



cκ ) cR˜1(0) - R˜2(0)φE

(4.12)

The first term in the right-hand side of this expression is the sum of the rates of energy transfer and irreversible exciplex formation

[

cR˜1(0) ) c

]

ka kf + , Y0 Y0

Y0 ) 1 +

kf + ka kD(1 + √τc /τD)

(4.13)

where τc ) σ2/D is the encounter time. Just this term, cR˜1(0), turns to be linear in kD when the quenching becomes diffusional

cR˜1(0) ≈ kD(1 + √τc /τD)

at

kf + ka . kD

The second term in eq 4.12 is the rate of the reverse electron transfer (from the exciplex to the excited donor) which initiates the delayed fluorescence. It is equal to the share of exciplexes φE multiplied by the rate of their dissociation into the fluorescence pair, R˜2(0). In diffusion limit the second term is quadratic in kD, i.e., rather small, but restores partially the density of the excitations reducing the Stern-Volmer constant. In the opposite kinetic limit (D f ∞, Y0 f 1) one can obtain

φE ) ckEτE

and

κ ) kf + kE

(4.14)

where the effective rate constant of the exciplex formation is

kE )

(

) {

kd 1 + τE ka ka

-1

ka

)

at

kdτE , 1

-1

KeqτE

kdτE . 1

at

(4.15)

The latter integrates the processes occurring at different times: the upper estimate relates to the initial stage of exciplex formation proceeding irreversibly while the lower one relates to the final stage of the exciplex decay, after equilibration between them and their precursors determined by the equilibrium constant Keq ) ka/kd. As can be seen from Figure 2, between the diffusional and kinetic limits of quenching, the Stern-Volmer constant is the monotonously growing function of diffusion

kD(1 + √τc /τD) e κ(D) e kf + kE

(4.16)

As well as κ(D), the exciplex yield monotonously grows with diffusion between the diffusional and kinetic limits (Figure 3)

kE ck (1 + √τc /τD)ητD , ηE , ckEητD kf + kE D

(4.17)

In the kinetic limit the exciplexes are produced with the effective rate constant kE, while in the diffusional limit kE determines just the exciplex share (kE)/(kf + kE) from the products of diffusional quenching (with the rate ckD). This share is not affected by diffusion, but this independence is a price for the contact approximation. If the quenching is distant, being diffusional, it proceeds far from the contact. In particular the remote diffusional ionization screens the contact from the excitations approaching it, thus preventing any contact reactions.26 B. Kinetics. As a rule the exciplex lives longer than the primary excitation (τE > τD), pumping the latter by dissociation and thus prolonging its delayed fluorescence. The latter follows closely the initial stage of the excitation quenching, due to exciplex association/ dissociation. To study such a process of energy dissipation and conservation, we can restrict ourself to rather long times (small s), where one can expand the kernels defined in eqs 4.1 and 4.2, leaving only the smallest nonstationary corrections. In diffusion limit they are

[ (

√τcτD s R˜j(s) ≈ R˜j(0) 1 + 2 1 + τ /τ √c D

)]

∼ R˜j(0)[1 + o(stb)]

(4.18)

where τc ) σ2/D. The correction, which is linear in s, does not influence the long time kinetics. Being small at stb , 1 it defines the boundary time tb that restricts the validity limit of such an approximation

√τcτD 1 + √τc /τD

t . tb )

(4.19)

Just in this time interval the integral eqs 4.4 and 4.5 are simplified to the differential ones with the time-independent coefficients R˜j(0). The “mass operator” (4.7) also becomes much simpler in this approximation

(

Σ˜ (s) ) R˜1(0) where

s + τE-1 + ϑ s + τE-1 + θ

)

(4.20)

4566



Figure 2. The Stern-Volmer constant diffusional dependence in the case of irreversible (upper curve, kdτE ) 0) and reversible exciplex formation: kdτE ) 1 (intermediate), kdτE ) 100 (ground one). Other parameters: τD ) 1 ns, kf ) 1660 Å3 /ns, ka ) 5000 Å3/ns.

Figure 3. The ratio of the luminescence yields changing with diffusion from the diffusional to kinetic control, at different reversibility of exciplex formation: kdτE ) 0.3 (upper curve); kdτE ) 3 (intermediate); kdτE ) 30 (ground one). The rest of parameters are: τD ) 1 ns, τE ) 10 ns, kf ) 1660 Å3/ns, ka ) 5000 Å3/ns.

ϑ)

kdkf , kf + ka

θ ) R˜8(0) )

(

kd kf 1+ Y0 kD(1 + √τc /τD)

)

(4.21)

So in a time interval t . τb, the general results obtained in eqs 4.6 can be represented as the sum of the common fractions

˜ *(s) ≈ N

A1 A2 + , s - s1 s - s2

N˜E(s) ≈

cR˜7(0) s + τE-1 + θ

˜ *(s) N

(4.22)

This can be easily inverted analytically

N*(t) ≈ A1 exp(s1t) + A2 exp(s2t)

NE(t) ≈

cR˜7(0) [exp(s1t) - exp(s2t)] (s1 - s2)

(4.23)

(4.24)

Hence, the time dependence of the survival probabilities N*(t) and NE(t) has the biexponential form specified by the following coefficients



τE-1 + θ + s1 A1 ) , s1 - s2

τE-1 + θ + s2 A2 ) s2 - s1

(4.25)

and

1 s1,2 ) (-b ( √b2 - 4d) 2

(4.26)

are the roots of the corresponding quadratic equation which has the following coefficients

b)

1 1 + + θ + cR˜1(0), τD τE

d)

(

)

(

)

1 1 1 + θ + cR˜1(0) +ϑ τD τE τE

1. Only Exciplex Formation. Let us illustrate these results by the simplest particular case of reversible exciplex formation without any additional quenching (kf ) 0). Then

R˜1(0) )

ka ≡ R, Y0

ϑ ) 0,

θ)

kd Y0

It is also convenient to introduce the new notations for this particular problem

δ)

1 1 - , τD τE

γ)

kd , cka

λ ) c(1 + γ)R

(4.27)

1 δ - λ ( √(δ + λ)2 - 4γcRδ + τD 2

(4.28)

In these notations the roots (4.26) take the following form

s1,2 ) -

From now on one has to distinguish between two alternative limits: 1. Fast exciplex/excitation equilibration compared to natural decay (λ . δ ≈ 1/τD). 2. Fast natural decay compared to association and dissociation (λ , δ ≈ 1/τD) In the former case one can expand expressions 4.28 into the small parameter δ/λ , 1

s1 ≈ -

(

1 1 γ 1 γ δ2 + 1 + γ τE 1 + γ τD 1+γ λ

s2 ≈ -λ -

)

(

)

(

)

(4.29) 2

( 1 +1 γ ) τ1 - ( 1 +γ γ ) τ1 - ( 1 +γ γ ) δλ D

(4.30)

E

Correspondingly, the weight coefficients Ai are found to be

A1 ≈

( 1 +γ γ )[1 + 2( 1 +1 γ ) δλ ],

A2 ≈

( 1 +1 γ )[1 - 2( 1 +1 γ ) δλ ]

(4.31)

Note, that in the case of the stable reaction partners, τE ) τD ) ∞, their populations after equilibration gain the equilibrium values, N*(∞) ) N and NE(∞) ) NE, which are

N)

γ , 1+γ

NE )

1 1+γ

(4.32)

Making use of these definitions and omitting the small corrections in δ/λ in the eqs 4.29-4.31, we obtain the results for the expressions 4.23 and 4.24:

N*(t) ≈ Ne-((N/τD)+(NE/τE))t + NEe-(λ+((NE/τD)+(N /τE))t

(4.33)

NE(t) ≈ NE[e-((N/τD)+(NE/τE))t - e-(λ+((NE/τD)+(N/τE))t]

(4.34)

The last terms in these expressions account for the fast equilibration during encounters. When they vanish the remaining terms determine the final decay of the excited state and exciplex with one and the same average rate

w0 )

NE N + τD τE

though with different amplitudes, N and NE. As these populations are equilibrated

(4.35)

4568



Figure 4. Slow donor decay τD ) 1 ns and fast equilibration cR ) 10 ns-1 (black curves) compared to exciplex accumulation (red curves) whose lifetime is τE ) 10 ns. (a)The irreversible reaction (solid curves), proceeding at γ ) 0, and weakly reversible (dashed curves), corresponding to γ ) 0.1. (b) The same for the larger reversibility, γ ) 1 (solid lines) and for γ ) 5 resulting in pronounced delayed fluorescence (dashed lines).

N ) γNE,

γ)

exp(∆GE /T) cυ

where γ is a measure of the reaction reversibility. Figure 4 demonstrates the typical kinetics of fast equilibration followed by delayed fluorescence with an average rate (4.35). In the opposite limiting case of fast excitation decay, λ , δ, it follows from eq 4.28

s1 ≈ -cR -

1 λ + γcR , τD δ

A1 ≈ 1 - γcR

s2 ≈ -γcR λ , δ2

A2 ≈ γcR

1 λ - γcR τE δ

λ δ2

(4.36) (4.37)

Thus the kinetics of the donor and exciplex dissipation takes the following form

N*(t) ≈ A1e-((1/τD)+cR)t + A2e-((1/τE)+γcR)t

(4.38)

cR -((1/τE)+γcR)t - e-((1/τD)+cR)t] [e δ

(4.39)

NE(t) ≈

The fast and deep initial decay of the donor occurs with the rate

wD )

1 + cR τD

(4.40)

which is slightly accelerated by irreversible exciplex formation. It then gives way to a weak delayed fluorescence whose decay rate

wE )

1 + γcR τE

is also slightly accelerated by the irreversible exciplex dissociation (Figure 5).

(4.41)


J. Phys. Chem. A, Vol. 114, No. 13, 2010 4569 To account for this reversibility, DET should be replaced by the IET or any approach which is equivalent to it. This is even more so when the distant quenching or ionization is reversible as well as shown in Scheme 1. This is what we plan to do next. Acknowledgment. The authors gratefully acknowledge the chairman of the Physical Chemistry Department of WIS, Professor Vega for support and hospitality during Dr. Fedorenko’s research stay in Rehovot, where this work was performed. One of the authors (S.G.F.) is also grateful to the Russian Foundation of Basic Research for financial support (Project 0903-00456). References and Notes

Figure 5. Fast donor decay τD ) 0.1 ns, cR ) 0.5 ns-1 (black curves) compared to slow mixing, resulting in weakly accumulated exciplexes (red curves) whose lifetime is τE ) 2 ns (a) at γ ) 1 (solid lines) and (b) at γ ) 5 (dashed lines).

V. Concluding Remarks The pioneering investigation of the distant (exponential) quenching parallel to the contact one25 was made with the DET valid for only irreversible transfer WF

ka

rA + D*f VτD

(5.41)

Moreover, all the results obtained there were derived within Markovian approximation ignoring the transient effects. Just recently the non-Markovian extension of this very theory was proposed26 which stresses the determinative role of the distant and especially nonstationary (non-Markovian) quenching in screening the contact reaction. The same should take place when the contact reaction is actually the reversible exciplex formation WF

ka

Aδ-Dδ+ r A + D* a kd VτD VτE

(5.2)

(1) Rehm, D.; Weller, A. Isr. J. Chem. 1970, 8, 259. (2) Burshtein, A. I.; Ivanov, A. I. Phys. Chem. Chem. Phys. 2007, 9, 396. (3) Burshtein, A. I. AdV. Chem. Phys. 2000, 114, 419. (4) Foll, R. E.; A.Kramer, H. E.; Steiner, U. E. J. Phys. Chem. 1990, 94, 2476. (5) Gould, I. R.; Young, R. H.; Mueller, L. J.; Albrecht, A. C.; Farid, S. J. Am. Chem. Soc. 1994, 116, 8188. (6) Dossot, M.; Allonas, H.; Jacques, P. Chem.sEur. J. 2005, 11, 1763. (7) Burshtein, A. I. AdV. Chem. Phys. 2004, 129, 105. (8) Burshtein, A. I. AdV. Phys. Chem. 2009; 214219. DOI 10.1155/ 2009/214219. (9) Burshtein, A. I. SoV. Phys. Usp. 1984, 27, 579. (10) Burshtein, A. I.; Frantsuzov, P. A. J. Chem. Phys. 1997, 106, 3948. (11) Burshtein, A. I.; Frantsuzov, P. A. J. Chem. Phys. 1997, 107, 2872. (12) Burshtein, A. I.; Ivanov, K. L. J. Phys. Chem. A 2001, 105, 3158. (13) Burshtein, A. I.; Neufeld, A. A.; Ivanov, K. L. J. Chem. Phys. 2001, 115, 2652. (14) Berg, O. G. Chem. Phys. 1978, 31, 47. (15) Burshtein, A. I. J. Chem. Phys. 2002, 117, 7640. (16) Naumann, W. J. Chem. Phys. 1999, 111, 2414. (17) Ivanov, A. I.; Burshtein, A. I. J. Phys. Chem. A 2008, 112, 11547. (18) Popov, A. V.; Burshtein, A. I. J. Phys. Chem. A 2003, 107, 9688. (19) Naumann, W. J. Chem. Phys. 2004, 120, 9618. (20) Yang, M.; Lee, S.; Shin, K. J. J. Chem. Phys. 1998, 108, 117. (21) Sung, J.; Chi, J.; Lee, S. J. Chem. Phys. 1999, 111, 804. (22) Kipriyanov, A. A.; Igoshin, O. V.; Doktorov, A. B. Physica A 1999, 268, 567. (23) Ivanov, K. I.; Lukzen, N. N.; Doktorov, A. B. Phys. Chem. Chem. Phys. 2004, 6, 1719. (24) Gopich, I. V.; Szabo, A. J. Chem. Phys. 2002, 117, 507. (25) Philling, M. J.; Rice, S. A. J. Chem. Soc., Faraday Trans. 2 1975, 71, 1563. (26) Khokhlova, S.; Burshtein, A. I. Phys. Chem. Chem. Phys., in press (27) Doktorov, A. B.; Burshtein, A. I. SoV. Phys. JETP 1975, 41, 671. (28) Gladkikh, V. S.; Angulo, G.; Burshtein, A. I. J. Phys. Chem. A 2007, 111, 3458.

JP100056A

Exciplex Formation Accompanied with Excitation Quenching - The

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