Diffusion affected magnetic field effect in exciplex fluorescence Anatoly I. Burshtein and Anatoly I. Ivanov Citation: The Journal of Chemical Physics 141, 024508 (2014); doi: 10.1063/1.4886809 View online: http://dx.doi.org/10.1063/1.4886809 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Hyperfine interaction mechanism of magnetic field effects in sequential fluorophore and exciplex fluorescence J. Chem. Phys. 138, 124102 (2013); 10.1063/1.4795576 Magnetic field effect in fluorescence of excited fluorophore equilibrated with exciplex that reversibly dissociates into radical-ion pair undergoing the spin conversion J. Chem. Phys. 137, 024511 (2012); 10.1063/1.4734306 Rapid fluorescence quenching of S 2 -xanthione by 3,3-diethylpentane in perfluorohydrocarbons J. Chem. Phys. 120, 8166 (2004); 10.1063/1.1695325 Ultrafast charge separation and exciplex formation induced by strong interaction between electron donor and acceptor at short distances J. Chem. Phys. 112, 7111 (2000); 10.1063/1.481326 Magnetic-field effects on the recombination fluorescence of anthracene cation and perfluorocarbon anions J. Chem. Phys. 109, 7354 (1998); 10.1063/1.477341

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THE JOURNAL OF CHEMICAL PHYSICS 141, 024508 (2014)

Diffusion affected magnetic field effect in exciplex fluorescence Anatoly I. Burshtein1 and Anatoly I. Ivanov2,a) 1 2

Weizmann Institute of Science, Rehovot 76100, Israel Volgograd State University, University Avenue, 100, Volgograd 400062, Russia

(Received 12 April 2014; accepted 20 June 2014; published online 10 July 2014) The fluorescence of the exciplex, 1 [D+δ A−δ ], formed at contact of photoexcited acceptor 1 A∗ with an electron donor 1 D, is known to be very sensitive to an external magnetic field, reducing the spin conversion efficiency in the resulting geminate radical ion pair, 1, 3 [D+ . . . A− ]. The relative increase of the exciplex fluorescence in the highest magnetic field compared to the lowest one, known as the magnetic field effect, crucially depends on the viscosity of the solvent. This phenomenon first studied experimentally is at first reproduced here theoretically. The magnetic field effect is shown to vanish in both limits of high and low solvent diffusivity reaching a maximum in between. It is also very sensitive to the solvent dielectric constant and to the exciplex and radical-ion pair conversion rates. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4886809] I. INTRODUCTION

The Magnetic Field Effect (MFE) on chemical reactions with radical-forming stages has been studied for several decades.1–14 Much attention was paid to the MFEs on fluorescence intensity in electron transfer quenching.2, 5, 13, 14 In such reactions, an exciplex composed from a pair of partially charged donor, D+δ , and acceptor of electron, A−δ , can be produced. Exciplex formation can proceed in two ways known as the Weller’s Scheme I and Scheme II. In accordance with the Weller’s Scheme I exciplex formation occurs at the contact of the excited electron donor and acceptor, followed by reversible dissociation to the radical-ion pair (Scheme I):15, 16 hν

A+D − → A∗ + D

Diff usion

− →

A−δ D +δ ← → [A− + D + ].

The second way (Scheme II) can be thought as distant electron transfer from the donor to the acceptor producing directly the ions, D+ and A− , which can reversibly associate into exciplex at contact encounter. In what follows the Scheme I is exploited. Although it was shown in a recent Ref. 17 that upon increasing the solvent polarity the relative importance of the distant electron transfer quenching increases nonetheless the direct exciplex formation (Scheme I) dominates. Moreover, in a medium polarity solvent, like tetrahydrofuran, ion pairs are almost exclusively generated upon dissociation of the exciplex and that their recombination also occurs with the exciplex as intermediate.18 The mechanism of the MFE on exciplex fluorescence quantum yield is now well understood. The magnetic sensitive stage in these reactions is the spin conversion of a geminate radical-ion pair (RIP) when the ions are well separated and the singlet-triplet splitting becomes less than the hyperfine interaction (HFI) energy. The reversible singlet-triplet conversion of the RIP, 1

[D + . . . A− ]   3 [D + . . . A− ] ,

a) Author to whom correspondence should be addressed. Electronic mail:

[email protected] 0021-9606/2014/141(2)/024508/5/$30.00

occurring due to HFI1 decreases the number of singlet RIPs that can recombine backward to reproduce the exciplex. The external magnetic field lifting S − T± degeneracy can significantly decrease the degree of singlet to triplet transformations in RIPs. This increases the number of singlet RIPs and, hence, leads to more efficient recombination of RIPs reproducing the exciplex and the quantum yield of the exciplex fluorescence increases. This simultaneously decreases the quantum yield of free ions. The duration of HFI-induced singlet-triplet transformations are limited by the lifetime of the geminate RIP. This lifetime depends on both the encounter diffusion of ions and the magnitude of a Coulomb interaction which is strongly affected by the dielectric permittivity of the solvent. Therefore, the efficiency of the singlet-triplet transformations should strongly depend on both the dielectric and diffusion constants. Indeed, the MFE on the exciplex fluorescence was found to crucially depend on dielectric constant.5, 13, 14 The effect is negligibly small in both regions of high and low dielectric permittivity, being considerable only in between. Such a behavior is in full accord with theoretical calculations.19, 20 Although the MFE on an exciplex fluorescence has been known for a long time its variation with the solvent viscosity was experimentally investigated quite recently.21 Pyrene (Py) is used as the electron acceptor and N,N-dimethylaniline (DMA) as the electron donor.21 In the experiments, the dielectric permittivity is kept inalterable, while the diffusion constant is varied. This condition strongly limits the accessible range of diffusion constant variation. Moreover, for different values of the dielectric constant the accessible ranges of the diffusion constant are not overlapped.21 Nevertheless, the results obtained clearly demonstrate a rather strong dependence of the MFE on the diffusion constant. To get a full understanding of this dependence a theoretical analysis of the effect should be done. In this paper, we use the kinetic mechanism of the reaction presented in Figure 1 where the rate constants of the corresponding reversible reactions are also indicated. The most appropriate approach for a description of such reactions is the

141, 024508-1

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024508-2

A. I. Burshtein and A. I. Ivanov

J. Chem. Phys. 141, 024508 (2014)

conventional Integral Encounter Theory (IET) well presented in a number of recent works.22–26 In Sec. II, this theory will be applied to the sequence of reactions presented in Figure 1. We will specify the kernels of the integral equations used here accounting for the RIP spinconversion proceeding in the Coulomb field, in the presence or absence of an external magnetic field. The main purpose of this paper is to comprehend the mechanism underlying the solvent diffusivity influence on the MFE on the exciplex fluorescence and to develop a theory that is capable of adequately describing this MFE.

II. FLUORESCENCE QUANTUM YIELDS AND MFE

The problem considered here has been already addressed in Ref. 19 though ignoring the singlet and triplet RIP recombination into the ground state and into the triplet excited state of the fluorophore. So, we have to only generalize the results by accounting for singlet and triplet RIPs annihilation. The

quantum yield of the fluorophore fluorescence, ηf , is written as19 1 ηf = , (2.1) 1 + cτA kg where kg is expressed via the Laplace transformations of the memory kernels R˜ ij (s),19, 26 kg = R˜ 11 (0) −

   R22 (0) = R12 (0) + kd 1 −

=

G0 (s) =

1 1 , √ kD 1 + sτd

R˜ 21 (0)(1 − ηf ) . R˜ 11 (0) + τe (R˜ 11 (0)R˜ 22 (0) − R˜ 12 (0)R˜ 21 (0))

(2.7)

(2.3)

Here the Laplace transformation of the kernels, R˜ ij (0) can be written in the form19 kf 1 + gf kf

(2.4)

,

kb , 1 + gf kf

(2.5)

 ka [αg1 + g0 + (α + 1)g0 g1 kt ] , α + 1 + [αg1 + g0 + (α + 1)g0 g1 kt ](ka + kc ) + (αg0 + g1 )kt

where gf = G0 (s + 1/τ A ), g0 = Gc (s), g1 = Gc (s + ks (α + 1)) taken at s = 0, kc and kt are the singlet and triplet RIP recombination rate constants, correspondingly. The quantity G0 (s) is the Laplace transformation of the reaction-free Green function for diffusing neutral reactants

(2.2)

The quantum yield of the exciplex fluorescence, ηe , has the form cτA R˜ 21 (0)ηf ηe = 1 + R˜ 22 (0)τe

21 (0) = 11 (0) = R R 12 (0) = R

R˜ 12 (0)R˜ 21 (0) . R˜ 22 (0) + 1/τe

(2.6)

where kD = 4π σ D is the diffusion rate constant of the contact reaction, while τ d = (σ 2 /D) is an encounter time of neutral partners, D is an encounter diffusion constant considered to be the same for neutral and charged reactants. Gc (s) is the Laplace transformation of the reaction-free Green function for continuous diffusion of ions in the Coulomb potential. A close analytical approximation for this Green function was obtained in Ref. 27. It takes the form 1 c (s) = 1 · , G kD μ0 (x) + κ(x, s) where μ0 (x) =

1/x , exp(1/x) − 1

κ(x, s) =

(2.8)

√ sτd + [ν(x)/ (x)] sτd , √ sτd + ν(x)

(x) = x 2 exp(1/x)[1 − exp(−1/x)]2 . Here ν(x) = {x 3 [exp(1/x)+exp(−1/x)−2]−x}/{ 16 [Ei(1/x) − exp(1/x)(x + x 2 + 2x 3 )]+  1 2 + [E1 (1/x) − exp(−1/x)(x − x 2 + 2x 3 )] + x 3 + x , 6 3

FIG. 1. The Weller Scheme I. The exciplexes are created directly from an excited fluorophore and a donor. The exciplexes can reversibly dissociate into radical-ion pairs. The designations of the rate constants are presented.

Ei and E1 are the integral exponential functions, while x = σ /rc and rc = e2 /( kB T) is the Onsager radius and is the static dielectric permittivity of the solvent, kB is the Boltzmann constant, and T is the temperature.

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024508-3

A. I. Burshtein and A. I. Ivanov

J. Chem. Phys. 141, 024508 (2014)

The rate constants kf and kb , as well as ka and kd , relate to each other according to the detailed balance principle kf = kb v exp(−Gexc /T ),

(2.9)

kd = ka exp(−GI /T )/v, ,

(2.10)

where v = 4π σ r is the volume of the attached reaction layer and r is its thickness.28, 29 In Figure 1, free energies of exciplex creation from neutral precursors, Gexc , is negative and RIP production from exciplex, GI , is positive. It is assumed that in zero field all 4 spin states are degenerate, while the highest field splits three triplet components in such a way that only two states (S and T0 ) remain in resonance. Therefore, the spin conversion from a singlet state, (αks ), proceeds in a former case three time faster there in the latter30  3 H = 0, α= (2.11) 1 H = ∞. 2

In the upper case of weak external magnetic field, the spinconversion equilibrates all 4 RIP substate populations leaving 1/4 on the singlet one that exchanges with the exciplex. In the lower case of strong magnetic field, the singlet population increases twice leaving highly splitted triplet states, T±1 , empty. As a result, the backward electron transfer from singlet state of RIPs forming first exciplex and then fluorophore is more probable in a high magnetic field. In the weak external magnetic field, one should select the value α = 3 and designate the quantities as ηfw and ηew , correspondingly. In a strong magnetic field, we should set α = 1, designating the corresponding values as ηfs and ηes . Then we define the MFE for the fluorescence of the fluorophore, A∗ , as χf =

ηfs − ηfw ηfw

,

(2.12)

while for the exciplex MFE we get accordingly χe =

ηes − ηew . ηew

(2.13)

The last two equations give a rather simple analytical solution of the problem in the contact approximation. III. THE MECHANISM OF THE VISCOSITY DEPENDENCE OF THE MAGNETIC FIELD EFFECT

In the calculations the parameters typical for experiments with pyrene and DMA are used: concentration of the quencher c = 3 × 10−5 molecules/Å3 = 0.05 M, encounter radius (sum of donor and acceptor radii) σ = 6.85 Å, exciplex and fluorophore lifetimes τ e = 500 ns and τ A = 190 ns, singlet-triplet transition rate in RIP ks = 0.01 ns−1 , kf = 8000 Å3 /ns. The dependence of the exciplex fluorescence quantum yield on diffusion constant, D, is shown in Figure 2 for a few values of the dielectric constant used in the experiment.21 All curves demonstrate sharp increase in the area of small diffusion constant, D, however reaching the maximum the curves go down the faster the larger ε. This results in a decrease of the maximum magnitude of ηe with increasing ε. Such a behavior of ηe is similar to that obtained experimentally and

FIG. 2. The dependence of the exciplex fluorescence quantum yield, ηe , on the diffusion constant, D. The parameters are ks = 0.01 ns−1 , τ A = 190 ns, τ e = 500 ns, ka = 104 Å3 /ns, kf = 8000 Å3 /ns, Gexc = −0.2 eV, GI = 0.1 eV, kc = kt = 0 – solid lines, kt = 103 kc , kc = 10 Å3 /ns – dashed lines, (1) = 8.5 (red lines); (2) = 24.5 (blue lines); (3) = 36.7 (green lines). The yields ηe are calculated for strong magnetic field (α = 1).

presented in Figure 2 in Ref. 21. At the lowest D → 0, the diffusion is so slow that the fluorophore fluorescence decays before first encounter with a donor. In this extreme, the fluorophore fluorescence quantum yield approaches to unity (see Figure 4), while the exciplex fluorescence quantum yield – to zero. In opposite limit of large D, the exciplexes and RIP are produced with large rate besides the RIP production becomes nearly irreversible so that the total fluorescence yield η + ηe → 0 and the free ion yield approaches to unity. This is the reason of decreasing ηe when D becomes large. The increase of the maximal ηe with decreasing ε is a direct consequence of suppression of the free ion formation with decreasing ε due to increase of the Coulomb interaction. Figure 2 also shows the dependence of the exciplex fluorescence quantum yield on the recombination rate of triplet RIPs, kt , (see Figure 1). In the considered donor acceptor pair (Py/DMA), the triplet recombination rate constant, kt , is known to be much larger than that of singlet recombination rate, kc .1 This is the reason why the relation kt = 103 kc is used. As one can expect the increase of kt from 0 to 104 Å3 /ns results in considerable decrease of ηe for low and moderate dielectric permittivity (ε = 8.5 and 24.5), while for high ε = 36.7 the magnitude of ηe becomes weakly dependent on kt (there is a noticeable difference in the area of small D < 10−6 cm2 /s that is well seen in Figure 3). In the solvents with not large ε the probability of RIP recombination into exciplex is rather large and appearance of competing ions recombination to the ground and triplet excited states suppresses the exciplex production and, hence, its fluorescence yield. In the solvents with large ε practically all RIP transforms into free ions and probability of RIP recombination becomes small. In this extreme an additional recombination channel plays a minor role in the RIP fate. The fluorophore fluorescence quantum yield (see Figure 4) decreases monotonously when diffusion constant, D, increases. This decrease of ηf is caused by shortening of the fluorophore lifetime. The similar behavior was obtained experimentally and presented in Figure 1 of Ref. 21. In the

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024508-4

A. I. Burshtein and A. I. Ivanov

J. Chem. Phys. 141, 024508 (2014)

FIG. 3. The same as in Figure 2 but for small values of the diffusion constant, D.

FIG. 5. The dependence of the MFE magnitude on exciplex, χ e , on the diffusion constant, D. The parameters are ks = 0.01 ns−1 , τ A = 190 ns, τ e = 500 ns, ka = 104 Å3 /ns, kf = 8000 Å3 /ns, Gexc = −0.2 eV,

extreme D → 0 the diffusion is so slow that the fluorophore fluorescence approaches to unity (the area of very small values of D is not shown in Figure 4 therefore the curves start from the values of approximately 0.3). The sensitivity of ηf to dielectric constant is considerably weaker than that of ηe . The dielectric permittivity affects only the RIP motion and, hence, the RIP to exciplex transformation, but the dependence of ηf on ε is due to the reversibility of the exciplex formation from the fluorophore. For accepted value Gexc = −0.2 eV, the reversibility is rather weak and the effect is not as large. The influence of kt on ηf is much weaker than that on ηe (not shown in Figure 4). This is also a consequence of small probability of backward transformation of an exciplex to neutral reagents. The MFE diffusional dependencies, ηe (D), for a few large values of are shown in Figure 5. All curves demonstrate a maximum that shifts to smaller values of D with increasing . For small values of D the probability of RIP association into exciplex is much larger than the probability of the free ion formation. In other words, the RIP cannot avoid the association into the exciplex in the limit D → 0. In this case, a conversion of an exciplex into a RIP means only a termination of its fluo-

GI = 0.1 eV, kc = kt = 0 – solid lines, kt = 103 kc , kc = 10 Å3 /ns – dashed lines, (1) ε = 24.5 (red lines); (2) ε = 36.7 (blue lines); (3) ε = 46 (green lines).

FIG. 4. The dependence of the fluorophore fluorescence quantum yield, ηf , on the diffusion constant, D. The parameters are ks = 0.01 ns−1 , τ A = 190 ns, τ e = 500 ns, ka = 104 Å3 /ns, kf = 8000 Å3 /ns, Gexc = −0.2 eV, GI = 0.1 eV, kc = kt = 0, (1) ε = 8.5 (red lines); (2) ε = 24.5 (blue lines); (3) ε = 36.7 (green lines). The yields ηf are calculated for strong magnetic field (α = 1).

rescence and transformation into a fluorophore. Some delay in the exciplex evolution caused by its temporary sojourn in the RIP state cannot change the exciplex fluorescence quantum yields. This means that the MFE should vanish in the limit D → 0. In the opposite limit of large values of D, the MFE is also negligible since the reaction 1 [D+δ A−δ ] → 1 [D+ + A− ] becomes irreversible and, hence, the singlet-triplet evolution of the RIP cannot affect the fluorescence quantum yields of the exciplex. In this limit the probability of the free ion formation is much larger than the probability of RIP association into exciplex. So, the maximum of the MFE should be expected when the probabilities of free ion formation and the RIP association into exciplex are equal. Obviously, this condition is met when the equation ηe = (1 − ηf )/2 is fulfilled. From this equation and Eq. (2.3) for kt = kc = ks = 0 we obtain kD =

ka . μ0 (τe kd − 1)

(3.1)

This equation provides a rough estimation of the position of the MFE maximum. It should be noted that the bell-shaped curve corresponding to the largest ε = 46 is qualitatively similar to that obtained experimentally at the same (see Figure 5 in Ref. 21). The variation of kt from 0 to 104 Å3 /ns (see Figure 5) leads to considerable increase of MFE, χ e , and to a shift of its maximum to the region of smaller values of D. To clarify the mechanism of this effect we note that for the parameters considered in this paper the probability of an exciplex dissociation into RIP is rather large and the probability of RIP association to produce exciplex at kt = kc = 0 varies nearly from unity to zero with variation of both D and . Obviously, the RIPs recombination with the rates kt and kc decreasing the RIPs number decreases the probability of the RIP association to produce exciplex and, hence, decreases almost equally the quantum yields ηes and ηew . The difference ηes − ηew (the nominator in Eq. (2.13)) is practically independent of kt , while ηew (the denominator in Eq. (2.13)) decreases with kt increase.

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024508-5

A. I. Burshtein and A. I. Ivanov

J. Chem. Phys. 141, 024508 (2014)

conversion and recombination to either an exciplex or the neutral products D + A∗   1 [D +δ A−δ ]   1 [D + + A− ]   3 [D + + A− ].

FIG. 6. The dependence of the MFE magnitude on exciplex, χ e , on the diffusion constant, D. The parameters are ks = 0.01 ns−1 , τ A = 190 ns, τ e = 500 ns, ka = 104 Å3 /ns, kf = 8000 Å3 /ns, Gexc = −0.2 eV, GI = 0.1 eV, kc = kt = 0, (1) ε = 8.5 (red lines); (2) ε = 10.5 (blue lines); (3) = 12.5 (green lines).

This is the reason why the magnitude of the MFE increases with kt . Therefore, all the dashed lines representing MFE at strong charge recombination appear above the MFE curves at zero recombination rates (solid lines). Figure 6 demonstrates the dependence of the MFE magnitude on exciplex, χ e , on the diffusion constant, D, for a few values of in the area of relatively low dielectric permittivity. The position of the maximum is very sensitive to variation of

in this area. When increases from 8.5 to 12.5 the maximum position shifts from D  10−5 cm2 /s to D  8 × 10−5 cm2 /s and its magnitude noticeable decreases. In this paper, we do not differentiate the diffusion constants of the neutral and charged particles. In fact, the diffusion constant for neutral particles A∗ and D can considerably differ from that of the ions A− and D+ due to much stronger solvation of the charged particles by the polar molecules of the solvent. The theory developed can be easily generalized assuming different diffusion constants for the neutral particles, Dn , and for the charged particles, D. The calculations show that MFE is nearly independent of Dn . Thus, the diffusional dependencies of the MFE presented in Figures 5 and 6 reflect the MFE dependencies on the diffusion constant of the radical-ions. In real solvents the diffusion constant Dn can considerably exceed the constant D. Taking into account that in experimental studies an estimation of D for neutral particles is typically exploited, one should expect a considerable shift of the position of the experimental maximum of χ e (D) to the region of larger D relative to that predicted by Eq (2.13). Indeed, for ε = 46 in Figure 4 the maximum of χ e (D) is placed at D  0.03 × 10−5 cm2 /s, while for the same experimental curve has a maximum at D  0.1 × 10−5 cm2 /s (see Figure 5 in Ref. 21). IV. CONCLUDING REMARKS

We developed the theory of the reversible fluorescence quenching by exciplex formation following by reversible charge separation forming RIPs. The last are subjected to spin

The IET suggested in this paper explains the almost full suppression of MFE in the regions of low and high dielectric permittivity of the solvent and its non-monotonous dependence on the diffusion constant. The role of the charge recombination is also elucidated. The qualitative comparison of the theory with experimental studies performed in Ref. 21 is rather hard because the experimental technique used there is different than simple detection of fluorescence after instantaneous excitation presumed here. Nonetheless the qualitative comparison of some results was possible and showed a reasonable agreement. ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research (Grant No. 14-03-00261). 1 H.-J.

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Diffusion affected magnetic field effect in exciplex fluorescence.

The fluorescence of the exciplex, (1)[D(+δ)A(-δ)], formed at contact of photoexcited acceptor (1)A(*) with an electron donor (1)D, is known to be very...
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