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Diffusion Mediated Delayed Fluorescence by Singlet Fission and Geminate Fusion of Correlated Triplets Kazuhiko Seki, Yoriko Sonoda, and Ryuzi Katoh J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b02234 • Publication Date (Web): 03 May 2018 Downloaded from http://pubs.acs.org on May 5, 2018

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The Journal of Physical Chemistry

Diffusion Mediated Delayed Fluorescence by Singlet Fission and Geminate Fusion of Correlated Triplets Kazuhiko Seki,∗,† Yoriko Sonoda,‡ and Ryuzi Katoh¶ Nanomaterials Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 5, 1-1-1 Higashi, Tsukuba, Ibaraki, 305-8565, Japan, Electronics and Photonics Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 5, 1-1-1 Higashi, Tsukuba, Ibaraki, 305-8565, Japan, and Department of Chemical Biology and Applied Chemistry, College of Engineering, Nihon University, Koriyama, Fukushima, 963-8642, Japan E-mail: [email protected]

Abstract Singlet fission (SF) has been extensively studied through the use of magnetic field effects and analyzed by the Johnson–Merrifield model or its extended models. We extended the Johnson–Merrifield model to study delayed fluorescence originating from diffusion mediated reversible geminate fusion of triplets; the emission intensity was shown to decay by a A/tβ law, where β = 3/2 in one and three dimensions and β = 1 ∗

To whom correspondence should be addressed AIST1 ‡ AIST2 ¶ Nihon †

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ACS Paragon Plus Environment

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in two dimensions. Analytical expressions for the amplitude as well as the exponent of the asymptotic power law decay were obtained. We have shown that a wide range of SF materials from polyene-based molecular crystals to amorphous rubrene exhibit A/t3/2 asymptotic decay. The temperature dependence of the asymptotic emission decay supports a mechanism involving delayed fluorescence mediated by diffusion. Moreover, we show that the decay lines at different field strengths will not cross each other when fission occurs effectively. In the opposite limit when fusion occurs effectively the two decay lines will cross each other.

1

Introduction

Singlet fission (SF) is the process whereby an excited singlet exciton splits into a pair of triplet excitons (an associated triplet pair) and has potential to achieve effective conversion of photons into multiple electrons in organic photovoltaics by doubling the number of initially generated excitons. 1,2 When an associated triplet pair is formed by singlet fission, fusion of the triplet pair can be spin allowed. Triplet fusion (TF) induces delayed emission and is expected to increase the efficiency of organic light-emitting diodes. 3 In general, the long-life time of triplets, owing to a spin forbidden back-transition to the ground state, is advantageous for organic electronic devices. The triplet yields are related to the singlet fission and the triplet fusion rate constants, which can be estimated from kinetic studies; it is important to study the kinetics of singlet fission and triplet fusion as fundamental processes in organic electronic devices. 1,2,4,5 Materials that exhibit singlet fission and triplet fusion are limited, because of spin conservation, the life times of singlet excited states under favorable coupling conditions, and the requirement for energy conservation to match the energy of the singlet state and the sum of the energy of the two triplets interacting with their surroundings. 1,2 Despite these restrictions, the number of SF materials is increasing; singlet fission has been reported for crystals of polyene-based molecules in addition to previously reported polyacenes such as 2


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anthracene and tetracene. 1,2,6–9 SF has been extensively studied through the use of magnetic field effects and analyzed by the Johnson–Merrifield model or its extended models. 10–13 The Johnson–Merrifield model is a kinetic model which takes into account spin states and spin selectivity for SF and TF. More recently, time-resolved fluorescence measurements have been performed and analyzed by the Johnson–Merrifield model to describe the emission changes on up to 10 ns time scales. 14–17 Although SF and TF kinetics up to 10 ns have been widely investigated, kinetics at longer times are not yet well understood. Because the transition from an excited triplet state to the ground singlet state is spin forbidden, deactivation of triplet excitons requires a long-time, allowing them to dissociate and migrate over long distances. Dissociated triplets generated from the same singlet can occasionally form associated triplet pairs again. Some fraction of the associated triplet pairs could form singlets giving rise to fluorescence and others might separate again. The triplets escaping from the above processes through diffusion are denoted as free triplets. Some fraction of the dissociate triplet pairs can escape from pairwise recombination and become free if there are more than two spatial dimensions according to the recurrence theorem of random walks. 18 In three dimensions, dissociated triplet pairs either recombine or ultimately escape from pairwise recombination. The situation is different in one dimension; dissociated triplet pairs ultimately recombine provided a long enough time is allowed for diffusion. The free triplets may still undergo triplet-triplet fusion by bimolecular reactions. To date, the effects of diffusion on triplet fusion have been studied mainly for bimolecular reactions where fusion is assumed to occur between any triplet pair regardless of whether they are the same pair generated by singlet fission. 19–22 When delayed fluorescence originates from bimolecular triplet-triplet fusion, the intensity decays exponentially with the rate twice the decay rate for an isolated triplet and the intensity varies with the square of the initial triplet concentration. 19,20,22 At time regimes shortly after photoexcitation, geminate triplet fusion mainly occurs while bimolecular triplet fusion is dominant at longer time regimes. The time scale of geminate triplet fusion depends the density of singlets; most triplets recombine pair-

3



wise as long as the triplet pairs generated by singlet fission do not interfere among themselves by diffusion. Therefore, there are time regimes where geminate triplet fusion predominates and delayed fluorescence is emitted by pairwise regenerated singlets even under diffusion. We focus on geminate triplet fusion where triplets undergo diffusion but still recombine with the counter triplet generated by singlet fission. In general, the time distribution required to regenerate associated triplet pairs by random migration after dissociation is non-exponential. To study the long-time kinetics involving migration, the non-exponential features of the regeneration time distribution as well as spin conservation for SF and TF should be taken into account. In this work, we develop analytical expressions describing long-time emission decay obtained by taking the regeneration time distribution into the Johnson–Merrifield model. The analytical expressions are applied to analyze the emission decay curves of recently identified SF materials, polyene-based crystals and conventional polyacene films. We study the effects of temperature, magnetic field dependence and dimensionality dependence of triplet migration on the delayed emission. The density decay of separate triplet pairs as well as that of associated triplet pairs is also studied; the yield of free triplets is expressed in terms of the dissociation rate constant renormalized from the bare dissociation rate constant owing to reversible triplet fusion by diffusive migration. Recently, a formal expression for the Laplace transform of the excited singlet state density was obtained by taking into account the diffusive exciton migration in three dimensions using the stochastic Liouville equation. 23 After assuming equal reactivity and equal non-reactivity of spin states for SF and TF, the long-time power law decay was obtained by applying an inverse Laplace transformation. 23 In this paper, asymptotic decay of the emissive singlet density is explicitly expressed in a form similar to the TF rate constant known for the Johnson–Merrifield model. 21,24,25 Analytical expressions for the amplitude as well as the exponent of the asymptotic power law decay are shown and compared with the measured emission decay. Using the analytical expressions, the temperature dependence of the emission curves at times longer than 10 ns was analyzed.

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Moreover, we show that the decay lines at different field strengths do not cross each other when fission occurs effectively, i.e. when the majority of triplet pairs generated by SF dissociate in competition to TF. At the opposite limit, when fusion occurs effectively the two decay lines cross each other.

2

Theory: Reversible Geminate Fusion of Triplets (a)

(Associated triplet pair state) (Separate triplet pair state)

kSF |Cs(j)|2 TP k dis S1 kr kTF |Cs(j)|2

FT

Diffusion

S0 (b)

(Associated triplet pair state) (Separate triplet pair state)

kSF |Cs(j)|2 TP kdis S1 kr kTF |Cs(j)|2

FT

Diffusion

krelax

S0 Figure 1: Kinetic model describing dissociation of an associated triplet pair to a separate triplet pair and regeneration of an associated triplet pair by diffusion. (a) Relaxation in the separate triplet pair state is ignored. (b) Relaxation in the separate triplet pair state is considered. We denote the density of singlet exciton (S1 ) at time t after photo-excitation by S(t). The rate constant of singlet fission depends on the relative orientation and distance between chromophores. For simplicity, we assume that the SF and the reverse process (TF) occur between neighboring chromophores. When a pair of triplets occupy nearest neighboring sites, such a pair is termed an associated triplet pair. When the distance between a pair of triplets exceeds the nearest neighbor distance, the pair is denoted as a separate triplet pair. The density of associated triplet pairs is denoted by C(t). To take into account diffusion effect analytically, we expand the Johnson–Merrifield model, which is the simplest model 5



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that includes spin selectivity. 21,24,25 A pair of triplets can be combined into 9 spin states. The density of associated triplet pairs with the spin state (j) is denoted by C (j) (t) and the P total density of associated triplet pairs is given by C(t) = 9j=1 C (j) (t). The transition rate (j)

of fission to the j-th triplet pair state is given by kSF |Cs |2 and the transition rate of fusion (j)

(j)

from the j-th triplet pair state is given by kTF |Cs |2 , where |Cs |2 indicates the projection of the associated triplet pair states to the singlet state and represents the weight of the singlet P (j) character of the associated triplet pair states with a constraint given by 9j=1 |Cs |2 = 1. The density of separate triplet pairs with the spin state (j) is denoted by f (j) (t) and the P total density of separate triplet pairs is given by f (t) = 9j=1 f (j) (t). For simplicity, we first consider the model (a) in Fig. 1, where relaxation of spin states is ignored. Later, we take into account relaxation of spin states and will show that the exponent of the long-time power law decay of the fluorescence is not influenced by relaxation of separate triplet pair states. The kinetic equation for the model (a) can be expressed as 9

X ∂ S(t) = − (kSF + krad ) S(t) + kTF |Cs(j) |2 C (j) (t), ∂t j=1 ∂ (j) C (t) = kSF |Cs(j) |2 S(t) − kTF |Cs(j) |2 + kdis C (j) (t) + kg(j) (t), ∂t ∂ (j) f (t) = kdis C (j) (t) − kg(j) (t), ∂t

(1) (2) (3)

where krad , kSF , kTF , and kdis indicate the radiative rate constant, the singlet fission rate constant, the triplet fusion rate constant and the dissociation rate constant of associated triplet pairs. The associated triplet pair dissociates into a separate triplet pair. The original Johnson–Merrifield model has already been extended to include separate triplet pair states. 14–17 Separate triplets diffuse by a sequence of hopping transitions. In the original Johnson–Merrifield model and the above mentioned extended model, diffusive migration is (j)

not considered. 14–17,21,24,25 kg (t) represents the regeneration rate of the associated triplet pair from the separate triplet pair by diffusion. When a triplet pair occupies nearest neighbor sites as a result of hopping transition by chance, the associated triplet pair is regenerated 6


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from a separate triplet pair. The regeneration rate can be expressed as

kg(j) (t)

Z ≡

t

dt1 R(t − t1 )kdis C(t1 ),

(4)

0

where the regeneration kernel R(t) represents the time distribution of regeneration after dissociation of the associated triplet pair: the right-hand side of Eq. (4) indicates that the dissociation occurs at any time after the photoexcitation from time 0 up to time t with the rate given by the dissociation rate constant kdis multiplied by the density of associated triplet pairs. The regeneration kernel R(t) reflects the migration of separate triplets. The migration of triplets is described by relative diffusion with the relative diffusion constant between the separate triplet pairs denoted by D. We assume that when triplets occupy nearest neighbor sites, they immediately form an associated triplet pair. We denote the distance between the triplets of the associated pair by a. The initial separation distance between the separate triplet pair after the dissociation of the associated triplet pair is denoted by a + b, where b indicates the hopping distance. The regeneration kernel expresses the rate of forming an associated triplet pair from a separate triplet pair at time t when the separate triplet pair is initially generated at the mutual distance a + b. The regeneration kernel is obtained by solving the diffusion equation with the mutual diffusion constant D under the perfectly absorbing boundary condition at a: we need the time distribution of the first arrival of the separate triplet pair at the relative distance a when the initial distance between the separate triplet pair is a + b. The perfectly absorbing boundary condition is set just to derive the regeneration kernel; the subsequent processes of TF and dissociation to separate triplet pairs are taken into account in Eqs. (1)-(3). Because the regeneration kernel R(t) is equal to the first passage time distribution mentioned above, we use the known results for the Laplace transform of the first passage time distribution given by 26

ˆ R(s) =

h i p 1−(d/2) K (a + b) s/D 1−(d/2) a+b p , a K1−(d/2) a s/D 7


(5)


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where d denotes the dimensionality, Kν (z) is a modified Bessel function of the second kind. 27 We have three linear equations given by Eqs. (1)-(3) for three unknowns S(t), C (j) (t), and f (j) (t) corresponding to each j. By eliminating C (j) (t), and f (j) (t) in the Laplace domain, S(t) can be obtained. By a Laplace transform of the regeneration kernel, the Laplace transform of S(t)/S(0) can be expressed as  9 X ˆ S(s)  = s + krad + kSF − S(0) j=1 s + k

(j) kSF kTF |Cs |4

(j) 2 TF |Cs |

+ kdis

ˆ 1 − R(s)

−1 

.

(6)

The asymptotic decay of the singlets S(t)/S(0) can be obtained from Eq. (6) as a power law S(t)/S(0) ≈ 1/t3/2 for one and three dimensional diffusion and S(t)/S(0) ≈ 1/t with a logarithmic correction for two dimensional diffusion. Singlet emission intensity is obtained from krad S(t). As shown in Fig. 2, the experimental data for singlet emission exhibits a 1/t3/2 asymptotic time dependence. The singlet emission intensity is proportional to the fraction of surviving singlets and the decay is consistent with the theoretical results obtained from the three-dimensional diffusion equation. Very recently, Shushin derived the asymptotic power law decay S(t)/S(0) ≈ 1/t3/2 by assuming three-dimensional diffusion when a number of associated triplet states have the equal singlet character. 23 Our results are consistent with this result in the sense that the exponent in the asymptotic power law decay reflects the kinetics of reversible geminate recombination by three-dimensional diffusion studied for other systems under different boundary conditions. 28–32 Below, we derive analytical expressions for the amplitude as well as the exponent of the asymptotic power law decay and compare the results with the emission decay measured for some organic solids.

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3

Results in Three Dimensions

3.1

Asymptotic Decay

In this subsection, we derive asymptotic kinetics of excited singlet density S(t) from Eq. (6). The emission kinetics can be obtained from krad S(t). The Laplace transform of the ˆ regeneration kernel R(s) is obtained from Eq. (5) as

ˆ R(s) =

p a exp −b s/D a+b

(7)

p ˆ in three dimensions. By substituting 1 − R(s) ≈ [b/(a + b)] 1 + a s/D obtained from Eq. (7) into the part including summation in Eq. (6) and taking the limit of small s , we obtain, 9 X j=1

9

(j)

(j)

X kSF kTF |Cs |4 kSF kTF |Cs |4 ≈ p (j) (j) 2 † ˆ s + kTF |Cs |2 + kdis 1 − R(s) s/D j=1 kTF |Cs | + kdis 1 + a ! p 9 (j) † X kdis a s/D kSF kTF |Cs |4 1− , ≈ (j) 2 (j) 2 † † k |C | + k k |C | + k s s TF TF j=1 dis dis

(8)

(9)

where the renormalized dissociation rate constant defined by † kdis = kdis b/(a + b),

(10)

is introduced. The dissociation rate constant is renormalized from the bare rate constant kdis owing to reversible geminate fusion by diffusive migration. By further substituting Eq. (9) into Eq. (6) and assuming that s is small because we are interested in long time kinetics, we find " #−1 X (j) (j) p ˆ S(s) , = krad + kfiss + Yfus kfiss a s/D S(0) j

9


(11)


where kfiss =

P

j

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(j)

(j)

kfiss denotes the overall rate constant for singlet fission and kfiss is the rate

constant for singlet fission via the associated triplet pair of the state, j given by † kSF |Cs |2 kdis (j)

(j)

kfiss =

† kTF |Cs |2 + kdis (j)

,

(12)

(j)

and Yfus represents the singlet regeneration yield from the associated triplet pair of the state j expressed as (j)

(j)

Yfus =

kTF |Cs |2 † kTF |Cs |2 + kdis (j)

.

(13)

To derive Eq. (11), we use

kSF −

9 (j) X kSF kTF |Cs |4 † kTF |Cs |2 + kdis (j)

j=1

= kSF

9 X

|Cs(j) |2 −

j=1

= kSF

9 (j) X kSF kTF |Cs |4

|Cs(j) |2 1 −

j=1

(14)

(j)

j=1

9 X

† kTF |Cs |2 + kdis ! (j) kTF |Cs |2 † kTF |Cs |2 + kdis (j)

= kfiss .

(15)

In the limit of s → 0, Eq. (11) can be expressed as P (j) (j) ˆ p S(s) j Yfus kfiss ≈− a s/D. S(0) (krad + kfiss )2 From the Tauberian theorem, the inverse Laplace transform of

(16)

√ √ s is given by −1/(2 πt3/2 ). 33

By applying the inverse Laplace transform to Eq. (16), the asymptotic decay of S(t) is obtained as P (j) (j) a S(t) 1 j Yfus kfiss √ ≈ . S(0) 2(krad + kfiss )2 πD t3/2

(17)

As shown in Fig. 1, non-radiative decay paths, except via the singlet fission, are not taken into account in Eq. (17). If other non-radiative decay paths are present, krad in Eq. (17) should be the sum of krad and the non-radiative rate constant. 10


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† = kdis /2. Ns denotes the number of states which Table 1: Results obtained for a = b, kdis (j) 2 † satisfies kTF |Cs | > kdis . (A) indicates a field dependence where the fluorescence initially decreases at low fields followed by an increase at high fields with increasing field strength. (B) indicates a field dependence where the fluorescence initially increases or decreases with † increasing field strength depending on the values of kSF , kTF , kdis and krad .

kfiss > krad kTF >

† kdis

† kTF < kdis

kfiss < krad † Ns kSF kdis

√

a

1

(B)

† 2kTF (krad + Ns kSF kdis /kTF )2 πD t3/2 P (j) 4 kSF kTF j |Cs | a 1 √ (A) P (j) 3/2 † (krad + j kSF |Cs |2 )2 πD t 2kdis P P (j) (j) kTF j |Cs |4 kSF kTF j |Cs |4 a a 1 √ (A) 2 √ P † 2 t3/2 (j) 2 † πD πD 2k k dis rad |C | 2kdis kSF s j

In Eq. (17), S(t) is inversely proportional to

√

1 t3/2

(A)

D. Therefore, the amplitude of the

asymptotic fluorescence decay decreases owing to the increase of the diffusion constant. It is not obvious if associated triplet pairs could be easily formed from separate triplet pairs by increasing the value of the diffusion constant or if the fraction of ultimately separated free triplet pairs could be increased by enhancing diffusion. The former would result in an increase of the amplitude and the latter in a decrease of the amplitude. Equation (17) shows that the latter process is dominant. In Fig. 2, we show the results of theoretical analysis of fluorescence decay curves of 1,6-diphenyl-1,3,5-hexatriene (DPH) and its fluorinated derivatives without an applied magnetic field. [In Fig. 2, fluorinated DPH derivatives are denoted by (E,E,E)-1,6-bis(4-fluorophenyl)-1,3,5-hexatriene (MF), (E,E,E)-1,6-bis(2,4-difluorophenyl)-1,3,5-hexatriene (DF), (E,E,E)-1,6-bis(2,4,6-trifluorophenyl)-1,3,5-hexatriene(TF), and (E,E,E)-1,6-bis(perfluorophenyl)-1,3,5-hexatriene (PF), respectively. The molecular structures of DPH and its derivatives are shown in ref. 9. Details of the preparation and experimental methods are also given in ref. 9.] The data are fitted to a power law decay function with the exponent −3/2 or with slightly larger exponents ranging from −1.52 to −1.8. The 11



long-time asymptotic decays obey the power law with an exponent close to −3/2. We also note a slight increase of the absolute values of the exponent for DPH and PF. The slight increase of the exponents could be caused by factors ignored in the theory, such as relaxation, homogeneous bulk triplet-triplet fusion by bimolecular reactions, and heterofusion. Relaxation of spin states will not change the exponent of the long time power law decay as we will show later. Relaxation will most likely affect the transient decay at a certain time region and could be influenced by temperature. All the curves at long times roughly obey the power laws after 5 ns. We could not see the characteristic time associated with relaxation for the time region exceeding 5 ns. If the normalized decay is unaffected by excitation intensity, triplet-triplet bulk fusion can be ignored. 15 Triplet-triplet bulk fusion also causes deviation from the power law decay; 34 if the exponent is slightly changed while the decay still obeys power law, triplet-triplet bulk fusion is unlikely. The difference could be caused by impurities leading to heterofusion. The rate of heterofusion could be influenced by the magnetic field strength as shown previously. 25 Additional theoretical and experimental studies are required to clarify the effect of heterofusion on the power law decay. As shown in Fig. 2, emission from orthorhombic DPH exhibits a power law decay with the exponent −1.8 and the exponent is slightly larger than −3/2. Although, a maximum deviation of 20% from −3/2 was observed, the asymptotic decay was well expressed by the power law decay with the exponent −3/2 for certain types of diphenylhexatriene, which could be a fingerprint of diffusion mediated reversible geminate fusion. In Fig. 3, fluorescence decay curves of rubrene film (amorphous) without an applied magnetic field are shown. 35 The solid lines indicate the asymptotic power law A/t3/2 . These results suggest that regeneration of singlets via diffusion of triplet pairs occurs in three dimensions. We now examine the temperature effect. Singlet fission could be efficient if the absolute emission quantum yield is low, φf 1. According to measurements of the absolute emission quantum yield, 9,36 φf 1 is indeed obtained in crystals of DPH. If φf 1 holds, we have

12


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10 10

Counts

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10 10 10 10 10 10

4

DPH 3 2 1

MF DF TF PF

0

-1 -2 -3

0.1

1

10

Time/ns Figure 2: The dots from top to bottom represent fluorescence decay curves of (orthorhombic) DPH, MF, DF, TF, and PF without an applied magnetic field, respectively. [See the main text for details.] Raw data are multiplied by the factor 1/5 to shift the data downward; the multiplication factor is 1/5 for MF and 1/25 for DF. The other red solid lines indicate power law fits with the exponents, -1.80(DPH), -1.60(MF),-1.55(DF), -1.52(TF), and -1.69 (PF). The black dashed lines represent the asymptotic power law with the exponent −3/2. † kfiss > krad . We may also assume kTF < kdis . Expressions obtained from Eq. (17) by

taking various limits are summarized in Table 1. According to the expression in Table 1 † corresponding to the limit of kfiss > krad and kTF < kdis , we find

S(t) ≈ S(0)

(j)

|Cs |4 a 1 . 2 √ P 3/2 t (j) 2 † πD 2kdis kSF j |Cs | kTF

P

j

(18)

√ The asymptotic decay amplitude is proportional to kTF /(kSF kdis D). The diffusion constant is proportional to kh b2 , where kh indicates the hopping transition rate constant. We also assume that the dissociation rate constant is given by the hopping transition rate constant. 37 If all transition rates result from thermal activation, we have kTF ∝ exp [−ETF /(kB T )], kSF ∝ exp [−ESF /(kB T )], kdis ∝ exp [−Eh /(kB T )] and D ∝ kh ∝ exp [−Eh /(kB T )]. The amplitude is proportional to exp [− (ETF − ESF − 3Eh /2) /(kB T )], where kB and T denote the Boltzmann constant and the temperature, respectively. ESF , ETF , and Eh indicate the activation energy of the rate denoted by kSF , kTF , and kh , respectively. Below, we

13



assume that SF is an exothermic process. 9 If ETF − ESF > 3Eh /2, the amplitude decreases as the temperature decreases and if ETF − ESF < 3Eh /2, the amplitude increases as the temperature decreases. From this temperature dependence we can study whether the ratio of (ETF − ESF )/Eh exceeds 3/2 or not. According to Fig. 3, the amplitude increases as the temperature decreases. The temperature dependence is consistent with the condition ETF − ESF < 3Eh /2. If ESF < 3Eh /2 holds in addition to ETF − ESF < 3Eh /2, the temperature dependence is governed by that of the hopping transition rate leading to diffusion. The activation energy extracted from the Arrhenius plot in Fig. 3 was 0.04 eV. This value is consistent with the activation energy of the thermal hopping transition.

(a)

(b) 4

10

150K

3

10

4

Amplitude

Counts

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2

10

200K 250K 300K

100K 150K 200K 300K

1

10

3

10

0

10

10

1

10

100

30

40

50

60

70

80

1/(kBT )[1/eV]

Time/ns

Figure 3: (a) Fluorescence decay curves of rubrene film (amorphous) without an applied magnetic field. 35 Solid red lines indicate the asymptotic power law A/t3/2 . (b) Amplitude A as a function of 1/(kB T ). Solid line indicates an Arrhenius plot with an activation energy of 0.04 eV. The asymptotic decay is obtained by taking the long-time limit. The lowest correction to the asymptotic power law decay is obtained from the inverse Laplace transform of Eq. (11) as 38 S(t) z(t) 1 2 √ − z(t) exp z (t) erfc (z(t)) , ≈ S(0) (krad + kfiss )t π

14


(19)

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where z(t) is defined by √ z(t) =

Dt krad + kfiss P (j) (j) . a Yfus kfiss

(20)

j

√ √ By applying an asymptotic expansion, given by z(t) exp (z 2 (t)) erfc (z(t)) ≈ 1/ π−1/[2 π z 2 (t)], Eq. (17) is recovered from Eq. (19). Within the simplification, the asymptotic power law dependence appears after the transit time t∗ defined by z(t∗ ) = 1, a2 t∗ = D

P

(j) (j)

Yfus kfiss krad + kfiss j

!2 .

(21)

The transit time to the asymptotic power law is inversely proportional to D. By decreasing temperature, D tends to decrease and the transit time increases. The increase of transit time with decreasing temperature could be another signature of diffusion controlled asymptotic power law decay. Figure 3 (a) shows such trends.

3.2

Transient Kinetics

Many studies focus on the initial decay for time scales before diffusion sets in. 8,9,14–17 For later use, we reproduce here the initial decay rates from Eq. (6) by taking the limit of ˆ s → ∞. We obtain lims→∞ R(s) → 0 and Eq. (6) is simplified to " #−1 9 (j) X ˆ kSF kTF |Cs |4 S(s) = s + krad + kSF − . (j) 2 S(0) j=1 s + kTF |Cs | + kdis

(22)

The initial exponential decay rate can be obtained by taking s → ∞ limit of Eq. (22) as

kini = krad + kSF .

(23)

As noted previously, 9,15 the initial decay rate is independent of the magnetic field because kini (j)

is independent of the singlet character given by |Cs |2 . When the transient decay component 15



Page 16 of 39

is characterized by the single rate constant, the next order correction to Eq. (23) is obtained from Eq. (22) as (t)

ktr = krad + kSF + kfiss ,

(24)

(t)

where kfiss denotes the transient fission rate constant without diffusional influence given by 1,39 (t) kfiss

=

9 (j) X kSF |Cs |2 kdis (j)

j=1

where kSF

P

j

kTF |Cs |2 + kdis

,

(25)

(j)

|Cs |2 was substituted from kSF in Eq. (22) to find Eq. (24). These well-known

results will be used to associate the transient decay and the amplitude of the long time asymptotic in the next section.

3.3

Field Dependence (t)

The transient decay rate constant given by Eq. (24) is field dependent because kfiss can be (t)

(t)

expressed as kfiss = kSF Yfiss with the use of the transient fission yield given by (t)

Yfiss =

9 X

|Cs |2

j=1

1 + kTd |Cs |2

(j)

(j)

,

(26)

(j)

where kTd = kTF /kdis and the singlet character of pair states characterized by |Cs |2 can be (t)

affected by the field strength. The field dependence of Yfiss has been already elucidated. 21,24,25 Here, we study the field dependence of the amplitude An of the asymptotic power law decay given by Eq. (17); Eq. (17) can be expressed as

S(t) ≈

kTF † kSF kdis

16

a An √ , 2 πD t3/2


(27)

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where the field dependence is included in (j)

An =

X j

|Cs |4 2 , 2 (j) 2 † † krS + Yfiss 1 + kTd |Cs |

(28)

† † † and kTd and krS indicate kTd = kTF /kdis and krS = krad /kSF , respectively. In Eq. (28), the

fission yield is defined by (j)

† Yfiss

=

X

|Cs |2

j

† |Cs |2 1 + kTd (j)

,

(29)

† which differs from Eq. (26) by the rate constant given by kdis = kdis b/(a + b). For simplicity † we ignore the difference between Yfiss and Yfiss in the discussion of field dependence. (t)

(t)

When Yfiss is close to unity, the majority of triplet pairs generated by SF dissociate in (t)

competition to TF; fission occurs effectively. When Yfiss is close to zero, the majority of triplet pairs generated by SF undergo TF in competition to dissociation; TF occurs effectively. As a result of TF, excited singlet states are regenerated followed by delayed fluorescence. We consider two decay curves corresponding to different field strengths. When the transient decay rate is increased and the asymptotic amplitude becomes smaller owing to changes of the field strength, the two decay curves will not cross each other. On the contrary, when both the transient decay rate and asymptotic amplitude become greater owing to the changing field strength, the two decay curves must cross each other. (See Fig. 4 as an example.) Decay curves do not cross for DPH crystals 8,17 but they cross for tetracene crystals. 14 We study the conditions that lead to the two decay curves crossing each other. First, we discuss the effects qualitatively and later we study particular cases quantitatively. When the Hamiltonian is symmetric with respect to the exchange of triplets, three out of nine states carry singlet character in the zero field limit. 21,24,25 By increasing the field strength, the other states also acquired singlet character owing to mixing of the zero field states. As a result, kfiss increases owing to increasing field strength as previously no-

17



Page 18 of 39

† ticed, 21,24,25,40 whereas An decreases with increasing field strength when kTF < kdis , because P (j) 4 j |Cs | decreases as the singlet character is spread uniformly over all states. Therefore,

the amplitude of S(t)/S(0) decreases with increasing field strength at low fields except in the limit of large kTF . In the limit of high fields, the representation of the states along the fields is suitable. It was shown that the number of states with singlet character was smaller than that in the zero P (j) field limit. 21,24,25 In this case, j |Cs |4 in the expression of An increases. Therefore, the amplitude of S(t)/S(0) increases with increasing field strength in the high field limit. These trends are consistent with the field dependence of the previously reported fluorescence decay curves of DPH. 8,9 Now, we study the simple case, where Ns of the associated triplet pair states have the (j)

same weight of singlet character expressed as |Cs |2 = 1/Ns and zero singlet character for the rest. It is known that the field dependence of the transient decay rate is given by 21,24,25 (t)

Yfiss (Ns ) =

1 . 1 + kTd /Ns

(30)

(t)

When 1 > kTd /Ns , Yfiss is close to unity and fission occurs effectively. In the opposite limit (t)

[1 < kTd /Ns ], Yfiss is close to zero and TF occurs effectively. The transient decay rate increases with increasing Ns . We obtain the amplitude of the asymptotic decay in Eq. (27) as

An (Ns ) =

1 Ns

1 1 + krS +

† krS kTd /Ns

2 .

(31)

† When singlet fission occurs effectively, we have (kdis /kTF )(kSF /krad + 1) > 1/Ns and

An (Ns ) ≈ 1/Ns is obtained; namely, the amplitude of the asymptotic decay decreases with increasing Ns . When the transient decay rate increases and the amplitude of the asymptotic decay decreases, the decay lines will not cross each other. Decay lines cross each other when

18


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the amplitude An (Ns ) increases with increasing Ns , i.e., the denominator of An (Ns ) decreases † with increasing Ns . The condition is obtained as 9 ≤ kTd krS /(1+krS ), which is satisfied when

fusion occurs effectively or the radiative decay rate is larger than the singlet fission rate. As stated above, it is reasonable to assume that Ns = 3 in the zero field limit. 21,24,25,40 We denote the field dependence of the transient decay rate and the field dependence of the (t,0)

(t)

(0)

amplitude of asymptotic decay as Yfiss = Yfiss (Ns = 3) and An = An (Ns = 3), respectively. When other states are mixed with zero-field states having the singlet character by increasing field strength, the situation can be approximated as Ns = 9. 21,24,25,40 When the fission occurs † effectively [(kdis /kTF )(kSF /krad + 1) > 1/Ns ], the transient decay rate increases and the

amplitude of the asymptotic decay decreases with increasing field strength. As a result, the decay lines do not cross each other. In the opposite limit when the fusion occurs effectively † [9 ≤ kTd krS /(1 + krS )], both the amplitude of the asymptotic decay and the transient decay

rate increase with increasing field strength. In this case, two decay lines cross each other. In the high field limit, we consider the two states as having singlet character, where (1)

(2)

(j)

|Cs |2 = 1/3 and |Cs |2 = 2/3. 24,25,40 The other coefficients expressed as |Cs |2 are set zero. The transient decay rate is known for this case and the field dependence can be expressed as 24 (t,∞)

Yfiss

2/3 1/3 + 1 + kTd /3 1 + 2kTd /3 1 + 4kTd /9 = . (1 + kTd /3) (1 + 2kTd /3)

(32)

=

(33)

We obtain the field dependence of the amplitude of the asymptotic decay as  A(∞) = n

1  9

 1

† /9 + 4kTd

4 1  2 + 2  krS + † † † 1 + kTd /3 1 + kTd /3 1 + 2kTd /3

1+

−2 

† 2kTd /3

. (34)

By comparing the transient decay rate in the high field limit and that in the zero field limit, 19



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*!('+*)('

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 39

!

!

!

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Figure 4: (a) Fluorescence decay curves showing effective singlet fission. The black thick solid line and red thin solid line indicate the decay curve in the absence of magnetic field and that in the limit of a strong magnetic field, respectively. The two lines do not cross each other. The thick black dashed line and thin red dashed line indicate the corresponding asymptotic power law A/t3/2 of Eq. (27). The parameter values are kTd = 1, krS = 0.1, kSF /kdis = 1, D/(a2 kSF ) = 1, and a = b. (b) Fluorescence decay curves showing effective triplet fusion. The black thick solid line and red thin solid line indicate the decay curve in the absence of a magnetic field and that in the limit of a strong magnetic field, respectively. The two lines cross each other. The thick black dashed line and the thin red dashed line indicate the corresponding asymptotic power law A/t3/2 of Eq. (27). The parameter value is kTd = 100. The other parameter values are the same as in (a).

20


Page 21 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


we reproduce the known result that the transient decay rate decreases in the high field limit 24 (t,∞)

Yfiss

(t,0)

=

Yfiss

1 + 4kTd /9 . 1 + 2kTd /3

(35)

It is not trivial to determine the ratio between the amplitude of asymptotic decay in the † high field limit and that in the zero field limit. When krad /kSF < (2/3)/(1 + (1/3)kTF /kdis ), † † where (1 + 4kTd /9)/(1 + 2kTd /3) > 2/3 is used, we obtain

(∞) An (0) An

2 2 † † 1 + 2k /3 + 4 1 + k /3 Td Td 1 = , 2 3 † 1 + 4kTd /9

(36)

† which decreases with increasing kTd and the value varies from 5/3 to 3/2. In this case, the

amplitude of the asymptotic decay in the high field limit is larger than that in the zero field limit. Because the transient decay rate decreases and the amplitude of asymptotic decay increases in the high field limit, the decay line obtained in the high field limit does not cross that obtained in the zero field limit. The situation is shown in Fig. 4 based on the assumed † values of |Cs |2 . In the opposite situation expressed as krad /kSF > 1/(1 + (1/3)kTF /kdis ), we (j)

obtain (∞)

An

(0)

An

1 4 = + 3 3

† 1 + kTd /3

!2

† 1 + 2kTd /3

,

(37)

† which decreases by increasing kTd and the value varies from 5/3 to 2/3. We find An /An > √ √ (∞) (0) † † 1 for kTd < 3 2/2 and An /An < 1 for kTd > 3 2/2. When krad /kSF > 1/(1 + (∞)

(0)

† (1/3)kTF /kdis ), the decay line obtained in the high field limit does not cross with that √ † obtained in the zero field limit for kTd < 3 2/2; however, the lines cross each other for √ † > 3 2/2. The situation exhibiting crossing is shown in Fig. 4 based on the assumed kTd (j)

values of |Cs |2 .

21



3.4

Page 22 of 39

Density of Triplet Pairs

Here, we study the asymptotic kinetics of the density of associated triplet pairs and that of separate triplet pairs. The ultimate yield of free triplets that escape from the geminate triplet fusion is also determined. As stated in Introduction, some fraction of the triplet pairs ultimately escape from pairwise recombination in more than two spatial dimensions. 18 In three dimensions, dissociated triplet pairs either recombine or ultimately escape from pairwise recombination. In the Laplace domain, the density of the associated triplet pairs and that of separate triplet pairs can be expressed as ˆ S0 − (s + krad ) S(s) ˆ C(s) = ˆ s + kdis 1 − R(s)

(38)

  h i 1 1 ˆ  −  , fˆ(s) = S0 − (s + krad ) S(s) s s+k ˆ dis 1 − R(s)

(39)

and

respectively. In three dimensions, the asymptotic kinetics of the density of separate triplet pairs and that of the associated triplet pairs are determined to be P (j) (j) f (t) a j Yfus kfiss √ ≈ 1 − φf + φf S(0) (krad + kfiss ) πDt

(40)

and C(t) ≈ S(0)

(j) (j)

P

j Yfus kfiss 1 − φf − φf (krad + kfiss )

22

!

a √

† 2kdis


1 πD

t3/2

,

(41)

Page 23 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


where the fluorescence quantum yield is given by

φf =

krad , krad + kfiss

(42)

and kfiss denotes the overall rate constant for singlet fission defined below Eq. (11). We note from Eq. (40) that 1 − φf represents the ultimate yield of separate free triplet pairs. kfiss in 1 − φf differs from the conventional definition of the overall rate constant for (t)

singlet fission, which is denoted by kfiss and given by Eq. (25). In contrast to the dissociation † rate constant kdis in kfiss , the dissociation rate constant in kfiss is given by kdis = kdis b/(a + b), (t)

which is renormalized by the factor b/(a + b) owing to reversible triplet fusion by diffusive migration. Under the effect of migration, the ultimate yield of separate free triplet pairs decreases from the value estimated by the conventional definition of the overall rate constant for singlet fission. Both the density of associated triplet pairs and that of separate triplet pairs exhibit power law decays. However, the exponents are different. The exponent of the asymptotic density decay for associated triplet pairs is −3/2, which is the same as that of the excited singlet state. The exponent of the separate triplet pairs is −1/2. The density of associated triplet pairs decays faster than that of separate triplet pairs. In terms of asymptotic kinetics, most of the triplet pairs are separated and the relationship with the density of singlet excited Rt states is given by f (t) ≈ S(0) − krad 0 dt1 S(t1 ). The absorption spectrum of associated triplet pairs is similar to that of separate triplet pairs, though they can be distinguished by careful studies. 41 We have shown that their kinetics might be different at long times. The theoretical results could be helpful to differentiate the similar spectra between associated and separate triplet pairs.

23



3.5

Page 24 of 39

Relaxation Effect

Here, we consider the effects of the relaxation of spin states for separate triplet pairs. Relaxation of associated triplet pairs can be ignored for organic crystals. 14,15,21,24,25 We analyze the kinetic model (b) in Fig. 1 in the high temperature limit. In the high temperature limit, relaxation can be characterized by a single rate constant denoted by krelax . In the presence of relaxation, Eq. (3) changes into X1 ∂ (j) f (t) = kdis C(t)(j) − kg(j) (t) − krelax f (j) + krelax f (`) . ∂t 8 `6=j

(43)

The last two terms represent spin lattice relaxation introduced previously to describe the fluorescence decay of rubrene and tetracene films. 14,15 The key quantity in this manuscript is (j)

kg (t), which represents the diffusional regeneration of associated triplet pairs from separate triplet pairs as explained below Eq. (3). Under relaxation, the regeneration rate is different from Eq. (4), as shown in Eq. (61) in the Appendix. After some calculations, shown in the Appendix, the asymptotic decay is obtained as † † 1 S(t) Yfus θr (1 + θr ) kfiss a √ ≈ , † † 2 πD t3/2 S(0) krad 1 + Yfus θr + kfiss

(44)

† † where kfiss denotes the overall rate constant for singlet fission; kfiss can be explicitly written

as † kfiss =

9 (j) ‡ X kSF |Cs |2 kdis ‡ kTF |Cs |2 + kdis (j)

j=1

,

(45)

† ‡ ˆ relax /8) . Y † represents the singlet where kdis in Eq. (12) is replaced by kdis = kdis 1 − R(9k fus regeneration yield from the associated triplet pair defined by † Yfus

=

9 X

kTF |Cs |2

j=1

‡ kTF |Cs |2 + kdis

(j)

(j)

24


,

(46)

Page 25 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and θr indicates the effect of relaxation given by

θr =

ˆ relax /8) θrd R(9k , ˆ relax /8) (1 + θrd ) 1 − R(9k

(47)

where the ratio of the relaxation rate and the diffusional association rate for the separation distance b is defined by bkrelax . θrd = p 8 9krelax D/8

(48)

† † † Yfus is a short notation for Yfus (0), where Yfus (s) is defined by Eq. (67) in the Appendix.

The asymptotic decay of the fluorescence is given by krad S(t). From Eq. (44), the exponent of the power law decay is not altered from the result without relaxation given by Eq. (17), although the amplitude is changed.

4

Asymptotic Kinetics in One and Two Dimensions

The time distribution of regenerating associated triplet pairs depends on the dimensionality of diffusive migration. In this section, we study the asymptotic decay of the excited singlet state when diffusion occurs in one and two dimensions in the absence of relaxation. The ˆ Laplace transform of the regeneration kernel R(s) is obtained from Eq. (5) as p ˆ R(s) = exp −b s/D

(49)

in one dimension. By substituting Eq. (49) into Eq. (6), the asymptotic decay of S(t) is obtained as S(t) 9kSF kdis b 1 √ ≈ . 2 3/2 S(0) 2kTF krad πD t

25


(50)


Page 26 of 39

In one dimensions, the fluorescence decays by the power law with the exponent −3/2 as in three dimensions. The result, however, indicates the absence of a field dependence if (j)

additional relaxation does not present because |Cs |2 cancels out in Eq. (50); the field (j)

dependence originates from |Cs |2 in the Johnson–Merrifield model. 21,24,25 We also note that the ultimate yield of separate free triplet pairs is zero in one dimension. This result reflects the recursive nature of the random walk in one dimension. 18 The measured fluorescence yield in DPH is low. 9 If isotropic diffusion is assumed, the low fluorescence yield is consistent with the three-dimensional result, although the same exponent −3/2 of the asymptotic power law decay can be obtained for delayed fluorescence in one dimension. In this respect, three-dimensional diffusive migration could occur in DPH, although one dimensional diffusion could be possible to some extent judging from the molecular structures and the anisotropic crystalline structures. To complete the list of theoretical results, we summarize the results obtained for two ˆ dimensions. The Laplace transform of the regeneration kernel R(s) in the limit of s → 0 is obtained from Eq. (5) as i p ln (a + b) s/D ˆ p R(s) ≈ ln a s/D h

(51)

in two dimensions. By substituting Eq. (51) into Eq. (6), the asymptotic decay of S(t) is obtained as S(t) kdis kSF ln [(a + b)/a] ≈ 18 2 . S(0) krad kTF t [ln (Dt/a2 )]2 The asymptotic decay is different from that in one and three dimensions.

26


(52)

Page 27 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5

Conclusion

When delayed fluorescence originates from diffusion mediated reversible geminate fusion of triplets, the emission intensity is shown to decay by a A/tβ law, where β = 3/2 in one and three dimensions, and β = 1 in two dimensions with a logarithmic weak time dependence in two dimensions. Our results share the same time dependence obtained recently for three dimensions. 23 By extending the Johnson–Merrifield model to include diffusive migration, we obtained analytical expressions for the amplitude A in addition to the exponent β. The exponent of the asymptotic power law decay is the same in one and three dimensions. The difference lies in the amplitude A, which can be distinguished by applying magnetic field. Using the Johnson–Merrifield approach, we show that the field dependence is absent in one dimension when relaxation of spin states can be ignored. These results are in sharp contrast to field dependence of emission decay derived for three dimensions. The ultimate yield of separate free triplet pairs can be non-zero in three dimensions but is zero in one dimension because of the recursive nature of random walk in one dimension. 18 In other words, a low fluorescence yield cannot be obtained in one dimension if non-radiative deactivation of excitons can be ignored. We have shown that the wide range of SF materials ranging from polyene-based molecular crystals to amorphous rubrene exhibit a A/t3/2 asymptotic decay. For polyene-based molecules, one dimensional diffusive migration might occur. Judging from the low fluorescence yield in DPH, and clear the magnetic field effect, 9 diffusive migration could occur in three dimensional in DPH. To scrutinize the effects of diffusive migration, the temperature dependences of the emission decay curves were studied for rubrene films. By assuming an Arrhenius temperature dependence for elementary rate constants, the temperature dependence of long-time emission decay can be interpreted as reversible geminate triplet fusion controlled by diffusive migration. The transit time to the asymptotic power law is increased by decreasing temperature, which is shown to be consistent with the consequence of diffusion controlled delayed 27



emission. The amplitude of asymptotic decay A is given by Eq. (17) and Eq. (44) in the absence and presence of relaxation for separate triplet pairs, respectively. Although the expressions are more complicated, they are given in terms of the singlet character distributed over the triplet pair states as the Johnson–Merrifield definition of the triplet fusion rate is given by Eq. (25). 21,24,25 As the Johnson–Merrifield definition of the triplet fusion rate is used to study field dependence, the magnetic field effect on the delayed emission under diffusive migration can be studied from the amplitude of the power law decay with the use of Eqs. (17) and (44). Unlike the early decay kinetics before the influence of diffusive migration takes place, magnetic field effects on the amplitude of asymptotic decay depend on the fission efficiency. As a result, the two decay curves corresponding to different field strengths cross each other when triplet fusion occurs effectively and singlet fission occurs ineffectively; however, they do not cross each other when singlet fission occurs effectively and triplet fusion occurs ineffectively; fission efficiency can be qualitatively estimated from the magnetic field effects on asymptotic emission decay. We ignored lattice structures and molecular orientation to obtain analytical expressions. The molecularity and anisotropy in hopping transitions are particularly important for studying early time kinetics. The formalism developed in this work could allow us to study the effects if the lattice Green function can be calculated. We need to solve matrix equations to obtain the lattice Green function to take into account formation of associated triplet pairs. These results could only be evaluated numerically and depend on the structure of crystals. Singlet fission together with triplet fusion processes were mainly confirmed by studying magnetic field effects. Our study suggests that occurrence of singlet fission followed by triplet fusion processes can be corroborated by studying the kinetics of delayed fluorescence; if delayed fluorescence decays by a power law with an exponent of 3/2, the delayed component of fluorescence could be attributed to geminate fusion of triplet pairs generated by singlet fission for three dimensional crystals. We have also shown the magnetic field effects on the

28


Page 28 of 39

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kinetics of delayed fluorescence when singlet fission is followed by geminate triplet fusion. By combining the static and kinetic aspects of the magnetic field effects on the fluorescence, a considerable amount of information on singlet fission and triplet fusion kinetics can be obtained. In organic solids, a triplet exciton can be also generated by intersystem crossing (ISC) from a singlet state. The delayed fluorescence of E-type can be generated by reverse ISC. 42 Recently, considerable attention has been paid to E-type delayed fluorescence for harvesting triplets in organic light emitting diodes (OLEDs). 43 Our results indicate that the triplets generated by SF lead to the characteristic power law decay of delayed fluorescence by reversible geminate recombination. Detailed study of fluorescence decay kinetics might be used to prove triplets generated by SF.

Acknowledgement This work was supported by JSPS KAKENHI under Grant No. 15K05406. We would like to thank Prof. M. Wakasa and Dr. T. Yago for their helpful comments. We would also like to thank Prof. Zuhong Xiong and Dr. Yong Zhang for their correspondence on ref. 33.

A

Detailed Derivation of Eq. (44)

By taking the summation of Eq. (43), we obtain an equation for f (t) = X ∂ f (t) = kdis C(t) − kg(j) (t). ∂t j

P

j

f (j) (t) as

(53)

When diffusion is taken into account by assuming three dimensional isotropic systems, the spatial distribution of f (t) denoted by f (r, t) satisfies, ∂ δ(r − a − b) f (r, t) = D∇2 f (r, t) + kdis C(t) , ∂t 4π(a + b)2

29


(54)


Page 30 of 39

where the perfectly absorbing boundary condition expressed as f (a, t) = 0 is set because associated triplet pairs are assumed to be formed immediately when the triplet pair separation distance is a. The solution in the Laplace domain can be expressed as h i ˆ a + b|s) − G(r, ˆ a|s)R(s) ˆ ˆ fˆ(r, s) = G(r, kdis C(s),

(55)

ˆ where R(s) is given by Eq. (5) and G(r, r0 |t) represents the Green function of the diffusion equation, δ(r − r0 ) ∂ G(r, r0 |t) = D∇2 G(r, r0 |t) + δ(t) , ∂t 4πr02

(56)

ˆ r0 |s) can be explicitly under the perfectly reflecting boundary condition set at r = a. G(r, written as 44 1 ˆ r0 |s) = G(r, 8πDrr0

r

" p r r # 1 − a s/D D s s p exp −|r − r0 | . − exp −(r + r0 − 2a) s D D 1 + a s/D (57)

ˆ r0 |s) fˆ(r, s) is obtained under the perfectly absorbing boundary condition at r = a, while G(r, represents the Green function for the perfectly reflecting boundary condition. The first term on the right-hand side of Eq. (55) can be the formal solution of fˆ(r, s) expressed by the Green function if the Green function is obtained for the perfectly absorbing boundary condition at r = a. Here, the Green function is obtained for the reflecting boundary condition. The second term on the right-hand side of Eq. (55) is needed to take into account the difference in the boundary condition for the Green function. The second term represents the fraction that would have reached a at some earlier time, subsequently reaching the distance r. To obtain the density under the perfectly absorbing boundary condition at a, the latter fraction should be subtracted from the total density obtained by assuming that a perfectly reflecting boundary presents at r = a.

30


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We have used the known result for the regeneration rate R(t) when the initial separation distance is a + b. Under relaxation of triplet pair spin states, we need the regeneration rate when the initial separation distance is r, which is denoted by R(r, t). Obviously, the Laplace transform of R(r, t) is obtained from Eq. (5) by substituting r for a + b. Under relaxation, (j)

kg (t) should also be generalized from Eq. (4). For the generalization, we first note that the spatial distribution of f (j) (t) denoted by f (j) (r, t) satisfies, δ(r − a − b) 9 1 ∂ (j) f (r, t) = D∇2 f (j) (r, t) + kdis C (j) (t) − krelax f (j) (r, t) + krelax f (r, t), 2 ∂t 4π(a + b) 8 8 (58) (j)

where the perfectly absorbing boundary condition expressed as f (j) (a, t) = 0 is set. kg (t) can be obtained by adding −kreact f (j) (r, t)δ(r − a)/(4πa2 ) on the right-hand side of Eq. (58), solving for f (j) (a, t) using the Laplace transformation and taking the limit of kreact → ∞ for (j)

the Laplace transform of kreact f (j) (a, t). In the Laplace domain, kg (t) can be expressed as, ˆ (z) kdis C (s) + 1 krelax =R 8

kˆg(j) (s)

ˆ (j)

Z

∞

ˆ (r|z) fˆ(r, s), dr4πr2 R

(59)

a+b

(j)

where z = s + 9krelax /8. kg (t) represents the rate of the first arrival at the association distance a when spin relaxation is taken into account. We ignored spin modulation owing to hopping transitions when associated triplet pairs dissociate and separate triplets execute hopping leading to diffusion. By substituting Eq. (55) into Eq. (59) and evaluating the spatial integration, we obtain h i ˆ rxs (s)kdis Cˆ (j) (s) + R ˆ rxc (s)kdis C(s) ˆ . kˆg(j) (s) = R

(60)

The regeneration rate is obtained from the inverse Laplace transform of Eq. (60) as

kg(j) (t)

Z ≡

t

dt1 Rrxs (t − t1 )kdis C (j) (t1 ) + Rrxc (t − t1 )kdis C(t1 ) ,

0

31


(61)


Page 32 of 39

where the Laplace transform of Rrxs (t) can be expressed using Eq. (5) as ˆ rxs (s) = R(s ˆ + 9krelax /8) R

(62)

h i 2 ˆ ˆ a + b|s) 1 − R(s) ˆ ˆ rxc (s) = π p (a + b) krelax Rrxs (s) p G(a, . R 2 (s + 9krelax /8)/D + s/D

(63)

and

By substituting the Laplace transform of Eq. (61) into Eq. (2), we obtain (j) ˆ ˆ rxc (s)kdis C(s) ˆ +R kSF |Cs |2 S(s) . Cˆ (j) (s) = (j) ˆ rxs (s) s + kTF |Cs |2 + kdis 1 − R

(64)

Substituting Eq. (64) into Eq. (1) yields † ˆ ˆ + Y † (s)R ˆ rxc (s)kdis C(s), ˆ (s + krad + kSF ) S(s) = S0 + krs (s)S(s) fus

(65)

† (s) denotes the regeneration rate of the singlet excited state by triplet fusion without where krs

generating separate triplet pairs, (j)

† krs (s) =

X j

kSF kTF |Cs |4 , (j) ˆ rxs (s) s + kTF |Cs |2 + kdis 1 − R

(66)

† and Yfus (s) denotes the triplet fusion yield from the associated triplet pairs (j)

† Yfus (s) =

X j

kTF |Cs |2 . (j) ˆ rxs (s) s + kTF |Cs |2 + kdis 1 − R

(67)

By applying the Laplace transform and taking the sum of Eq. (2) to find the expression for

32


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P ˆ C(s) = j Cˆ (j) (s), we obtain from Eq. (1) h i ˆ − S0 + sC(s) ˆ ˆ − kdis C(s) ˆ ˆ rxs (s) + R ˆ rxc (s) kdis C(s). ˆ sS(s) = −krad S(s) + R

(68)

Equation (68) can be rearranged into ˆ 0) ˆ 1 − (krad + s) (S/S C(s) . = S0 ˆ rxs (s) − R ˆ rxc (s) s + kdis 1 − R

(69)

By substituting Eq. (69) into Eq. (65), we obtain the Laplace transform of S(t)/S(0) as † ˆ 1 + Yfus S(s) (s)θ† (s) = , † † S0 s + krad + kSF − krs (s) + (s + krad ) Yfus (s)θ† (s)

(70)

where θ† (s) expresses relaxation effects

θ† (s) = s + kdis

ˆ rxc (s)kdis R . ˆ ˆ 1 − Rrxs (s) − Rrxc (s)

(71)

By taking the small s-limit of Eq. (70) and using the Tauberian theorem for the inverse Laplace transformation we obtain Eq. (44) as we derived Eq. (17) from Eq. (16).

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Graphical TOC Entry

kSF |Cs(j)|2 TP kdis S1 kr kTF |Cs(j)|2

Diffusion

FT krelax

S0

39


Diffusion Mediated Delayed Fluorescence by ... - ACS Publications

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