Dissipation - The Journal of

Oct 5, 2010 - The natural extension and reformulation of the unified theory (UT) proposed here makes it integro-differential and capable of describing...
0 downloads 0 Views 591KB Size
Download PDF
11506

J. Phys. Chem. A 2010, 114, 11506–11512

Unified Theory of the Exciplex Formation/Dissipation Svetlana S. Khokhlova Volgograd State UniVersity, UniVersity AVenue 100, Volgograd 400062, Russia Weizmann Institute of Science, RehoVot 76100, Israel

Anatoly I. Burshtein* Weizmann Institute of Science, RehoVot 76100, Israel ReceiVed: May 23, 2010; ReVised Manuscript ReceiVed: September 15, 2010

The natural extension and reformulation of the unified theory (UT) proposed here makes it integro-differential and capable of describing the distant quenching of excitation by electron transfer, accompanied with contact but reversible exciplex formation. The numerical solution of the new UT equations allows specifying the kinetics of the fluorescence quenching and exciplex association/dissociation as well as those reactions’ quantum yields. It was demonstrated that the distant electron transfer in either the normal or inverted Marcus regions screens the contact reaction of exciplex formation, especially at slow diffusion. I. Introduction The unified theory1 is a natural extension of the differential encounter theory (DET) of elementary transfer reactions in liquids, which began in the works of Smoluchowski and Collins and Kimball.2,3 It allows tracing the further evolution of the reaction products from where they were created by the precursor reaction. On the other hand, DET together with UT can consider only the irreversible sequential reactions. In this way important success was achieved by fitting the theory to the reactions of any complexity and taking into account the spin states of the reactants, their charges, and vibrations.1,4-6 However, the treatment of the reversible reactions is the privilege of integral theories (IET, MET, SCRTA, MPKs, and their analogs).7 IET is known to be the lowest concentration approximation in encounter theory.6,7 There were a few unsuccessful attempts to extend DET for the reversible reaction A + B h C. They are known as the “modified” or “phenomenological” theory of Lee and Karplus8 and the modified rate equation (MRE) approximation of Szabo.9 Making another attempt to extend UT for the present problem, we combine the DET description of the contact energy quenching by reversible exciplex formation with not a phenomenological but IET description of the reverse reaction of exciplex dissociation. The total reaction scheme of this reaction looks as follows:

where ka is the rate constant of the reactants association into the exciplex and kd is the exciplex dissociation rate, while τD and τE are the excited donor and exciplex lifetimes, respectively. As far as we know, the kinetics of reversible complex dissociation, [DA] h D + A, was first considered by Berg.10 It was proved to obey the integral kinetic equation that was later on shown to be identical to that of IET11 and works equally well for exciplex dissociation for either charged or neutral components.6,11 The corresponding IET equations for reaction 1.1 accounting not only for dissociation but also for the association, were proposed much later:7,12,13

dN* ) -cka dt

dNE ) cka dt

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

(1.2)

∫0t dτ Σ(t - τ)N*(τ) kd

∫0t dτ Σ(t - τ)NE(τ) -

NE τE

(1.3)

Here N* is the excited state population of electron donor, NE is that of the exciplex and c ) [A] is the concentration of the electron acceptors. The kernel of these equations in highly polar solvents (assuming an Onsager radius rc ) 0) has the following Laplace transformation:

ka 1 )1+ ˜Σ(s) kD(1 + √τd(s + τD-1))

(1.4)

where kD ) 4πσD is the diffusional rate constant, τd ) σ2/D is the encounter time (σ is the contact radius; D is the encounter diffusion coefficient). The fluorescence quantum yield (in the absence of internal conversion) is

η)

˜

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

(1.5) where ηE is the quantum yield of exciplexes. The fluorescence quantum yield obeys the Stern-Volmer law, which can be easily specified by the Laplace transformation of the set (1.2) and (1.3). Its Stern-Volmer constant κ appears to be6,7

10.1021/jp1047216  2010 American Chemical Society Published on Web 10/05/2010

Exciplex Formation/Dissipation

J. Phys. Chem. A, Vol. 114, No. 43, 2010 11507

ka

κ) 1+

ka kD(1 + √τd /τD) )

+ kdτE

{

ka /(1 + kdτE) at D f ∞ at D f 0 4πσ2√D/τD

(1.6)

In section II the IET will be employed to specify the relaxation kinetics that will be used later for comparison with the analogous results of the unified theory (UT) of the phenomenon. The latter is suggested in section III, where the DET description of exciplex formation is naturally combined with IET description of its dissociation. In section IV the remote excitation quenching preceding the contact one (by exciplex formation) will be taken into consideration. It will be demonstrated that the latter screens the former when the solvent viscosity increases, retarding the encounter diffusion. II. Relaxation Kinetics The Laplace transformation of eqs 1.2 and 1.3 turns them into algebraic equations for the Laplace transformations of reactant populations, N˜*(s) and N˜E(s) at initial condition N*(0) ) 1 and NE(0) ) 0. Solving them, we obtain

s + kdΣ˜ + 1/τE N˜*(s) ) [s + ckaΣ˜ + 1/τD][s + kdΣ˜ + 1/τE] - ckakd (2.1a)

Figure 1. Kinetics of excitation decay and exciplex accumulation (solid lines) at (a) an exothermic reaction, γ ) 0.1, and (b) quasi-resonance reaction, γ ) 1.0. The analytical approximations shown by dashed lines are obtained in ref 14 at τD ) 100 ns. The other parameters are c ) 0.1 M, τd ) 6 ns, and σ ) 6 Å.

N˜C(s) )

cka s{s[1 + ka /kD(1 + √sτd)] + kd + cka} (2.2b)

As time increases, the populations approach their equilibrium values:14

˜ E(s) ) N

ckaΣ˜ [s + ckaΣ˜ + 1/τD][s + kdΣ˜ + 1/τE] - ckakd (2.1b)

One can make numerically the inverse Laplace transformation of these quantities, getting N*(t) and NE(t). The kinetics of the excitation quenching and exciplex accumulation/dissipation obtained by the inverse Laplace transformation of eqs 2.1 at τE ) ∞ is compared with their asymptotic expressions found analytically in ref 14 at the same conditions. The coincidence is quite satisfactory at long times but not at the very beginning, since the transient effects that dominate there are fully ignored in the asymptotic solutions (Figure 1). A. Equilibration of Stable Reactants. If there are not excited but only stable reactants (τD ) τE ) ∞), the problem is reduced to the complex formation D + A h C, resulting in a gradual approach to the equilibrium. The time dependent densities of the reactants, N(t) and NC(t), approach with time their permanent populations N ) N(t ) ∞) and NC ) NC(t ) ∞). Provided initially there were only D molecules (N(0) ) 1 while NC(0) ) 0), the relaxation kinetics of both are given by the following Laplace transformations:

scka 1 )s+ ˜N(s) s[1 + ka /kD(1 + √sτd)] + kd

(2.2a)

N ) lim sN˜(s) ) sf0

1 1+γ

NC ) lim sN˜C(s) ) sf0

γ 1+γ

(2.3) where the equilibrium constant γ ) kd/cka. There is also an alternative way of solving this problem. One can get Σ(t) by the inverse Laplace transformation of Σ˜ (s) specified in eq 1.4, employing the result for the numerical solution of IET eqs 1.2 and 1.3 at τD ) τE ) ∞. The kinetics of the excitation dissipation and exciplex accumulation obtained in this way is shown in Figure 2. It can be equally well specified by the inverse Laplace transformation of the final IET results given in eqs 2.2. The exact coincidence of both the results approves the further implication of the developed numerical algorithm for similar problems considered later. B. Exciplex Decay. If initially there are no excited reactants (N*(0) ) 0) but only a single exciplex that experiences the dissociation accompanied by numerous recombination, the ˜ E(s) kinetics of its decay is given by the Laplace transformation N obtained as a solution of eq 1.3 at NE(0) ) 1:

N˜E(s) )

1 ˜ s + kdΣ(s) + 1/τE

(2.4)

where Σ˜ (s) was specified in eq 1.4. C. Berg Modification of IET. The original Berg equation extended to exciplex dissociation,

11508

J. Phys. Chem. A, Vol. 114, No. 43, 2010

Khokhlova and Burshtein

Figure 2. Kinetics of excitation dissipation and exciplex accumulation obtained by a numerical solution of eqs 1.2 and 1.3 (solid lines), which are compared with those calculated by the inverse Laplace transformation of eq 2.2a and 2.2b (points). The results are the same within the accuracy of numerical computations. The rest of the parameters are σ ) 6 Å, D ) 6 Å2/ns, c ) 0.1 M, ka ) 1500 Å3/ns, and kd ) 0.15 ns-1.

dNE ) -kdNE + kd dt

∫0t dτ F (t-τ) NE(τ) -

NE τE

Figure 3. Kinetics of exciplex decay obtained by the inverse Laplace transformation of eq 2.4 (points) and the straightforward numerical solution of eq 2.5 (solid line). The results are the same within the accuracy of numerical computations. The rest of the parameters are σ ) 6 Å, D ) 6 Å2/ns, c ) 0.1 M, ka ) 817 Å3/ns, kd ) 1.8 ns-1, τE ) 15 ns, and τD ) 20 ns.

dNE ) ckaN*-cka dt

∫0t dτ F (t-τ) N*(τ) - kdNE +

(2.5)

where NE(0) ) 1. This differs from the IET equation (1.3) (at N* ) 0) by only the integral kernel. The Laplace transformations of these kernels are actually related to each other as follows:6

Σ(s) ) 1 - F (s)

∫0t dτ F (t-τ) NE(τ) -

F (s)

kD(1 + √(s + 1/τD)τd) )1+ ka

1

(2.6)

NE τE

(2.8)

where the contact kernel F is given by its Laplace transformation presented in eq 2.6. For the stable reactants (τD ) τE ) ∞) the Berg kernel (2.6) is greatly simplified:

i.e.

1 ~

kd

~

)1+

F (s)

kD(1 + √sτd) ka

(2.9)

and even an analytic expression for F (t) was obtained:12 Equation 2.5 with kernel F(t) is more convenient for a numerical solution than its IET analog (1.3) with kernel Σ(t) due to an absence of δ(t) included in Σ(t) (see next subsection). The obtained result is actually identical to that obtained by the inverse Laplace transformation of eq 2.4 (see Figure 3). This is one more justification for the validity and efficiency of the algorithm of numerical investigation used here and everywhere later on. D. Exciplex Formation/Dissociation. To subject the set of eqs 1.2 and 1.3 to the Berg modification, one has just to use there the substitution

Σ(t) ) δ(t) - F(t)

F (t) ) -k˙c(t)/ka

The DET rate constant of the contact association is known to be1

(

kc(t) ) k0 1 +

∫0t dτ F (t-τ) N*(τ) + kdNE t N* kd ∫0 dτ F (t-τ) NE(τ) (2.7) τD

dN* ) -ckaN* + cka dt

)

ka x e erfc √x kD

(2.11)

where x ) (1 + ka/kD)2Dt/σ2 and the permanent asymptotic value,

k0 ) lim kc(t) ) tf∞

identical to its Laplace presentation in eq 2.6. By this substitution we obtain the Berg modification of the conventional IET equations, which account for the exciplex creation in line with its dissipation:

(2.10)

kakD ka + kD

(2.12)

is actually the stationary rate constant. ~

The alternative way to get the kernel F (t) is to make the ~

inverse Laplace transformation of F (s). Using this approach and employing the result in the algorithm of the numerical solution of the set eqs 2.7 and 2.8, we obtained IET. III. Unified Theory In original UT the irreversible forward reaction is described by DET but the further evolution of its products is also taken

Exciplex Formation/Dissipation

J. Phys. Chem. A, Vol. 114, No. 43, 2010 11509

into consideration, assuming that it is irreversible as well.1 Here we avoid the last assumption using the Berg description of reversible exciplex formation. If the latter is accompanied by distant energy quenching, the UT equations are

dN* ) -ck(t)N*+kdNE - kd dt

∫0t dτ F (t-τ) NE(τ) N*/τD (3.1)

dNE ) ckan(σ,t)N*-kdNE + kd dt

∫0t dτ F (t-τ) NE(τ) NE /τE (3.2)

where the lifetimes of the excitation and exciplex are also taken into account, as in IET eqs 2.7 and 2.8. There were the precedents of making such a compilation of DET and IET15,16 so the kernel was there given a different definition. It should be stressed that, by making such a compilation, one neglects the possible interference of competing reactions that may affect their rate constants as well as the integral kernels. This is just the simplest way to obtain the screening effect, which is of main interest here. The general definition of the bimolecular reaction constant in present UT remains the same as in DET:1,6

k(t) ) kan(σ,t) +

∫ W(r) n(r,t) d3r

(3.3)

Here the first term accounts for the contact exciplex creation while the other one takes into account the distant reaction of energy or electron transfer with the rate W(r), responsible for the remote quenching of excitation. The pair correlation function of neutral reactants obeys the kinetic equation

n˙ ) -W(r) n(r,t) +

D ∂ 2 ∂n r r2 ∂r ∂r

(3.4)

If there is an exciplex formation or other contact association, this equation should be solved with the following boundary condition:

4πDr2

∂n ∂r

|

r)σ

) kan(σ,t)

(3.5)

The initial condition n(r,0) ) 1 implies the uniform distribution of reactants at the beginning. Provided there is only the contact reaction (W(r) ) 0) the corresponding reaction constant has been known for a long time: k(t) ≡ kc(t) is given by expression (2.11). In this section we will restrict ourselves to this particular case. A. Irreversible Contact Quenching. Assuming association (exciplex formation) to be irreversible (kd ) 0) let us compare the UT and IET solutions at N*(0) ) 1, which obeys the reduced equations eqs 3.1 and 2.7, respectively:

dN* ) -ckc(t)N*-N*/τD dt

UT

(3.6)

Figure 4. Irreversible quenching of the excitation which is accomplished by the exponential decay in UT (solid line) and the slower (power-time) false IET asymptote (dotted line). The rest of the parameters are σ ) 6 Å, D ) 100 Å2/ns, c ) 0.1 M, ka ) 2500 Å3/ns, and τD ) 103 ns.

dN* ) -ckaN*+cka dt

∫0t dτ F (t-τ) N*(τ) N*/τD

IET (3.7)

The contact rate constant kc(t) ) kan(σ,t) is specified in eq 2.11 while F(t) has to be obtained by the inverse Laplace transfor~

mation of F (s) defined in eq 2.6. The comparison of these two kinetics in Figure 4 provides evidence that they well coincide at the very beginning, but at long times the IET curve decreases slower than the exponential asymptotic decay of the excitation in UT. This is due to a wellknown false asymptotic behavior of N*(t) in IET that decreases slower than its DET/UT analog: as t-3/2 (see Figure 3.53 in ref 6). This weakness of IET becomes even more significant at larger concentrations. Therefore, we will operate further on with only the UT equations. B. Reversible Formation of the Exciplex. When W(r) ) 0, the exciplex formation is the unique mechanism of excitation quenching and the set of eqs 3.1 and 3.2 take the simplest form:

dN* ) -ckc(t)N*+kdNE - kd dt

∫0t dτ F (t-τ) NE(τ) N*/τD

dNE ) ckc(t)N*-kdNE + kd dt

(3.8)

∫0t dτ F (t-τ) NE(τ) - NE/τE (3.9)

As usual, the exponential fluorescence of the excitation with a natural time decay τD is followed by the delayed fluorescence originating in such a case only by dissociation of the exciplex. The reversible accumulation of the latter is accomplished with its time decay τE. The numerical solution of these equations confirms this prediction (Figure 5). From the kinetics of the reactants evolution after instantaneous excitation, one can specify the quantum yields of excitation and exciplex fluorescence: η ) ∫0∞N*(t) dt/τD and ηE ) ∫0∞NE(t) dt/ τE. The analytical expressions for the Laplace transformations N˜*(0) and N˜E(0) available in IET, allowed us to reproduce the IET results for the yields given in eq 1.5.

11510

J. Phys. Chem. A, Vol. 114, No. 43, 2010

Khokhlova and Burshtein

Figure 5. Reversible excitation quenching (N* line) and exciplex formation and dissipation (NE line). Due to energy quenching by exciplex formation the former proceeds faster at the beginning than the irreversible (natural) decay of the excitation (dotted line). The situation is the opposite at the very end of the excitation decay, which is actually the delayed fluorescence, supported by the backward energy transfer (originating from the exciplex dissociation). The rest of the parameters are σ ) 6 Å, D ) 6 Å2/ns, c ) 0.1 M, ka ) 500 Å3/ns, kd ) 0.03 ns-1, τD ) 10 ns, and τE ) 15 ns.

The corresponding UT results can be obtained in the same way but only for the kinetic limit (D f ∞) when

lim kc(t) ) lim k0 ) ka ) const

Df∞

Df∞

and lim F (t) ) 0

Figure 6. Stern-Volmer constant κ as a function of the encounter diffusion coefficient D calculated in the frameworks of IET (solid line) and UT: at c ) 0.1 M (dashed line) and c ) 1 M (dotted line). The rest of the parameters are σ ) 6 Å, kd ) 0.03 ns-1, ka ) 500 Å3/ns, τD ) 10 ns, and τE ) 15 ns. The linearity with D at small D indicated in eq 1.6 and shown in the inset manifests the transient effect peculiar to the diffusional quenching, while the horizontal dotted line shows the upper (kinetic) limit of the Stern-Volmer constant, κ0.

as well as DET. Therefore, the difference between these approaches is more pronounced at larger concentrations: diffusional κ in UT increases with c as was numerously indicated previously and well confirmed experimentally for irreversible excitation quenching.17,18

Df∞

IV. Distant Screening of the Exciplex Formation and eqs 3.8 and 3.9 reduce to the following ones:

dN* ) -ckaN*+kdNE - N*/τD dt

(3.10)

dNE ) ckaN*-kdNE - NE /τE dt

(3.11)

The kinetic Stern-Volmer rate constant, κ0, deduced from the fluorescence quantum yield obtained from these equations, appears to be

κ0 )

ka 1 + kdτE

Sometimes the contact quenching of the excitation is accompanied by either remote electron (energy) transfer or internal conversion induced by the same partner. This is a doublechannel energy quenching by electron transfer having the following reaction scheme:

The rate of the remote irreversible electron transfer is well approximated by either the exponent or the bell-shaped curve shifted a bit from the contact:1,7

(3.12)

This UT result coincides identically with that following from the general IET expression (1.6) as D f ∞. However, the UT analog of the general κ(D) dependence can be obtained from η and ηE by only numerical calculations of N˜*(0) and N˜E(0) from the general UT equations (3.8) and (3.9). The diffusional dependencies of the general Stern-Volmer constants calculated with IET and UT are compared in Figure 6. The difference between the IET and UT results is seen only at small D (under diffusional control) and disappears with D (in the kinetic limit) where κ0 is the same for both. The results differ under diffusional control due to the false IET asymptotic decay of N*(t) (Figure 4), though this effect appearing at very long times is rather negligible. More important is that the IET used here is the lowest order concentration approximation of the integral theory,6 unlike UT, which is concentration unlimited

W(r) ) W0 exp

-2(r - σ) l in the normal Marcus region (4.2)

W(r) ) W0/cosh2

( r -∆ R )

in the inverted Marcus region

(4.3) Here l is approximately the tunneling length, while ∆ is related to the reorganization energy of electron transfer in polar media and R > σ. There is a pronounced difference between the rate constants of a single channel energy quenching, when it is contact, kc(t), or distant, kf(t) ) ∫W(r) n(r,t) d3r. The latter is obtained with the reflecting boundary condition for either the exponential (solid line) or bell-shaped W(r) (dotted line), using eqs 4.2 and 4.3,

Exciplex Formation/Dissipation

Figure 7. (a) Exciplex formation rate constant of a single-channel quenching reaction, kc(t) (red), in comparison with the contact component, kan(σ,t), of the double-channel reaction, having the electron transfer component, W(r), of either exponential (solid blue line) or bell shape (dotted blue line), in the normal and inverted Marcus regions, respectively. (b) Energy quenching rate constants of the single-channel distant reaction, kf(t), in either the normal (solid red line) or inverted regions (dotted red line), in comparison with the corresponding total (double-channel) rate constants, k(t) (solid or dotted blue lines, respectively). The rest of the parameters are σ ) 6 Å, ka ) 500 Å3/ns, W0 ) 10 ns-1, l ) 1.5 Å, R ) 10 Å, and ∆ ) 0.2 Å. The components of the bimolecular reaction rate constant k(t) (3.3) were calculated using the SSDP2 program (http://www.fh.huji.ac.il/krissinel/software.html).

respectively. All of them are shown by the red curves: the contact one in Figure 7a and two alternative distant constants in Figure 7b. The total rate constants of the double channel reaction k(t) (blue curves in Figure 7b are, of course, larger than those of distant quenching alone (red curves) in either the normal or inverted regions. However, the difference between them is rather small because the contact quenching contribution is greatly reduced in comparison with the same quenching, acting alone, Figure 7a. This is an effect of screening the contact reaction by a distant one. The latter prevents the excitations from approaching the contact by quenching them on their way. This effect is larger in the inverted Marcus region, although at the very beginning it is insignificant, until distant electron transfer weakly affects the excitation distribution near the contact. The screening effect discovers itself in the diffusional dependence of the quantum yields shown in Figure 8. Diffusion facilitates both the distant and contact quenching reducing the luminescence yield (η) and increasing the exciplex production (ηE). However, in the double-channel reaction the exciplex production is hindered due to the screening effect. It can also

J. Phys. Chem. A, Vol. 114, No. 43, 2010 11511

Figure 8. Quantum yields of luminescence η and reversible exciplex formation ηE as a function of the encounter diffusion D in the case of a single-channel reversible exciplex formation/dissociaton (red lines) and in the case of reversible exciplex formation/dissociaton accompanied by remote electron transfer (blue lines). The latter proceeds in either the normal (a) or inverted (b) regions. The dotted black lines in (a) indicate two limits: (upper line) static quenching and (lower line) kinetic quenching. The inset in (b) demonstrates the slow diffusional behavior of the exciplex quantum yield. The rest of the parameters are c ) 0.1 M, σ ) 6 Å, kd ) 0.03 ns-1, ka ) 500 Å3/ns, τD ) 10 ns, τE ) 15 ns, W0 ) 10 ns-1, l ) 1.5 Å, R ) 10 Å, and ∆ ) 1.5 Å.

be seen from the insert in Figure 8b that there are two stages: at very slow diffusion the exciplexes are formed from only those excitations that are between the contact and the top of W(r) inherent to it in the inverted Marcus region. At faster diffusion even distant excitations partly cross the remote reaction layer, reaching contact and turning into an exciplex. This is a qualitatively different picture than that obtained in ref 14, where both competing reactions were assumed to be contact. To make the screening effect even clearer, let us appeal to the rate constants of the stationary reactions:

k(∞) ) 4πRQD

and

kan(σ,∞) ) 4πrQD

(4.4) The effective radii of total and contact reactions decrease similarly in the kinetic regime (at growing D) but oppositely when D decreases, i.e., under diffusional control (Figure 9). In this limit RQ grows as usual, thus enlarging the sphere around the contact where most of excitations do not penetrate. Only a few of them survive crossing the remote reaction sphere and reach the contact where they can be transformed into exciplexes. As a result, the effective radius of the contact reaction rQ f 0.

11512

J. Phys. Chem. A, Vol. 114, No. 43, 2010

Khokhlova and Burshtein In contrast, the ratio ηE/(1 - η) decreases and turns to zero as D f 0 if one of the reactions proceeds distantly, screening the contact one. V. Conclusions

Figure 9. Total RQ and contact rQ effective radii as a function of the encounter diffusion D, provided that the distant quenching is exponential. The value of RQ was multiplied for convenience by the coefficient 0.1. The rest of the parameters are σ ) 6 Å, ka ) 500 Å3/ ns, W0 ) 10 ns-1, and l ) 1.5 Å.

Figure 10. Fraction of the exciplexes in the total amount of the quenching products (curved line), in comparison with that obtained earlier,14 when both quenching reactions were assumed to be contact (straight line). The other parameters are: c ) 0.1 M, σ ) 6 Å, ka ) 500 Å3/ns, kd ) 0.03 ns-1, W0 ) 10 ns-1, l ) 1.5 Å, kF ) 4347.2 Å3/ns, τE ) 15 ns, and τD ) 10 ns.

Additional evidence of the screening effect and its location is given in Figure 10, where the yield of the exciplexes, ηE, is related to the total products yield of the double-channel quenching, 1 - η, assuming the distant quenching is exponential (normal). When the reactions in both channels are contact, this ratio appears to be constant at any diffusion as follows from eq 4.17 of ref 14.

ηE kE ) const ) 1-η kF + kE

(4.5)

where kF ) ∫W(r) d3r is the kinetic rate constant of the distant reaction, while kE ) ka/(1 + kdτE) is that of the contact reaction.14

The common IET was used to describe well a single contact reaction of the reversible exciplex formation. The Laplace transformation of the IET integral equations allows solving them algebraically, getting the kinetics of the reactants evolution by the inverse Laplace transformation of the solutions obtained. The results confirm the long-time asymptotic solution of the problem obtained earlier14 and describe well the equilibration of the system when the reactants are stable. The same results were obtained by a numerical solution of the initial IET equations so that the algorithm developed proves itself to be a good tool for solving UT problems as well. The newly reformulated UT is in fact preferable: it is valid for larger quenching concentrations than IET, thus accounting for the increase of the Stern-Volmer constant with concentrations. In the case of irreversible quenching, UT eliminates the false IET asymptotic behavior. The kinetics of reversible quenching includes the delayed fluorescence supported by exciplex dissociation. However, the main advantage of our UT is the inclusion of distant electron transfer preceding the exciplex formation at a contact distance. Such a double-channel reaction facilitates the excitation quenching but hinders the exciplex formation especially at slow diffusion. This is the screening effect first indicated in ref 19, but for a fully absorbing boundary condition at contact (ka ) ∞), assuming product formation to be irreversible. Since the exciplex production is reversible in principle, we had to develop a new approach that allowed us to confirm that the screening effect continues to exist in spite of the reversibility of the contact reaction. Acknowledgment. We gratefully acknowledge Profs. Y. Prior, S. Vega, and G. Kurizky for their personal assistance in getting a one year postdoctoral position in the Chemical Physics Department of the Weizmann Institute of Science for Dr. S. Khokhlova. References and Notes (1) Burshtein, A. I. AdV. Chem. Phys. 2000, 114, 419. (2) Smoluchowski, M. V. Z. Phys. Chem. 1917, 92, 129. (3) Collins, F. C.; Kimball, G. E. J. Colloid. Sci. 1949, 4, 425. (4) Gladkikh, V.; Angulo, G.; Burshtein, A. I. J. Phys. Chem. A 2007, 111, 3458. (5) Ivanov, A. I.; Burshtein, A. I. J. Phys. Chem. A 2008, 112, 6392. (6) Burshtein, A. I. AdV. Chem. Phys. 2004, 129, 196. (7) Burshtein A. I. AdV. Phys. Chem., doi: 10.1155/2009/214219. (8) Lee, S.; Karplus, M. J. Chem. Phys. 1987, 86, 1883. (9) Szabo, A. J. Chem. Phys. 1999, 95, 2481. (10) Berg, O. G. Chem. Phys. 1978, 31, 47. (11) Burshtein, A. I. J. Chem. Phys. 2002, 117, 7640. (12) Gopich, I. V.; Szabo, A. J. Chem. Phys. 2002, 117, 507. (13) Popov, A. V.; Burshtein, A. I. J. Phys. Chem. A 2003, 107, 9688. (14) Fedorenko, S. G.; Burshtein, A. I. J. Phys. Chem. A 2010, 114, 4558. (15) Doctorov, A. B.; Kipriyanov, A. A. Physica A 2003, 317, 41. (16) Kipriyanov, A. A.; Doctorov, A. B. Physica A 2003, 317, 63. (17) Gladkikh, V.; Burshtein, A. I.; Angulo, G.; Pages, S.; Lang, B.; Vauthey, E. J. Phys. Chem. A 2004, 108, 6667. (18) Feskov, S. V.; Burshtein, A. I. J. Phys. Chem. A 2009, 113, 13528. (19) Khokhlova S. S.; Burshtein A. I. Chem. Phys., in press.

JP1047216