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K. Kamada et al.
when two excitons locate neighboring sites along the A–B pair direction. It means
that the triplet energy transfer occurs one-dimensionally.
Judging only from the TTA reaction times, the performance of DPA in the TTA
process is superior to C7-sDPA, which contradicts with the experimental finding.
Compared to the TTA process, the difference in the minimum reaction time for
TTET between DPA and C7-sDPA is small, i.e., 0.2 μs for DPA and 0.3 μs for
C7-sDPA. However, this result does not also explain why DPA exhibits less efficient
TTA-UC than C7-sDPA. These results suggest that other mechanisms are needed
for considering TTA-UC efficiency. One possible reason is the difference in the
lifetime of triplet exciton, τ triplet . τ triplet for DPA and C7-sDPA are 54 and 140 μs,
respectively. TTET occurs at most n TTET = 335 (510) times, estimated from n TTET =
τ triplet /min(τ TTET ). This means that the triplet exciton possibly encounters more often
in C7-sDPA than in DPA. The encounter probability is directly related to the quantum
yield and emission intensity. Thus, in order to measure the TTA-UC performance
between DPA and C7-sDPA quantitatively, we analyze the encounter probability of
triplet excitons by referring both the differences in the dimensionality, the TTET
reaction times, and the experimentally measured triplet state lifetime.
9.3.3 Random Walk Models for the Encounter of Triplet
Excitations
Before formulating the encounter probability, we must consider the experimental
conditions. The diffusion length L D of triplet exciton is approximately estimated as
L D =
R 2 n TTET /2m, where R and m are the averaged distance between neighboring
emitters and the dimensionality. The diffusion length is estimated as 60.1Å for DPA
and 71.3Å for C7-sDPA, respectively. In this sense, C7-sDPA encounters more likely
than DPA. However, the situation is not so simple. Since the concentration of the
triplet sensitizer is 1/1000 of that of DPAs, the average distance between two sensitizers is approximately 10 times longer than the lattice constant of DPAs for each
direction, 91–211 Å for DPA and 149–154 Å for C7-sDPA, which clearly exceed
both L D values. This implies that it is likely that two excitons are generated from
the same sensitizer with a lag time, τ lag , which is defined as the mean recovery time
to the initial state of the sensitizer after TTET to the emitter. The normalized lag
time is defined as n lag = τ lag /min(τ TTET ). Hereafter, we first consider the encounter
probability of two excitons as a function of n TTET and n lag . Then, we go back to the
encounter probability of the triplet excitons, which are generated independently from
two different sensitizers.
Here, we consider a rough estimation of an encounter probability of two excitons
by using uniform random walk models, where we assume that the TTET between two
emitters is treated as a stochastic problem. For the one-dimensional (1D) systems,
the probability that one finds an exciton at the position i after n steps, P i (n), is given
by
K. Kamada et al.
when two excitons locate neighboring sites along the A–B pair direction. It means
that the triplet energy transfer occurs one-dimensionally.
Judging only from the TTA reaction times, the performance of DPA in the TTA
process is superior to C7-sDPA, which contradicts with the experimental finding.
Compared to the TTA process, the difference in the minimum reaction time for
TTET between DPA and C7-sDPA is small, i.e., 0.2 μs for DPA and 0.3 μs for
C7-sDPA. However, this result does not also explain why DPA exhibits less efficient
TTA-UC than C7-sDPA. These results suggest that other mechanisms are needed
for considering TTA-UC efficiency. One possible reason is the difference in the
lifetime of triplet exciton, τ triplet . τ triplet for DPA and C7-sDPA are 54 and 140 μs,
respectively. TTET occurs at most n TTET = 335 (510) times, estimated from n TTET =
τ triplet /min(τ TTET ). This means that the triplet exciton possibly encounters more often
in C7-sDPA than in DPA. The encounter probability is directly related to the quantum
yield and emission intensity. Thus, in order to measure the TTA-UC performance
between DPA and C7-sDPA quantitatively, we analyze the encounter probability of
triplet excitons by referring both the differences in the dimensionality, the TTET
reaction times, and the experimentally measured triplet state lifetime.
9.3.3 Random Walk Models for the Encounter of Triplet
Excitations
Before formulating the encounter probability, we must consider the experimental
conditions. The diffusion length L D of triplet exciton is approximately estimated as
L D =
R 2 n TTET /2m, where R and m are the averaged distance between neighboring
emitters and the dimensionality. The diffusion length is estimated as 60.1Å for DPA
and 71.3Å for C7-sDPA, respectively. In this sense, C7-sDPA encounters more likely
than DPA. However, the situation is not so simple. Since the concentration of the
triplet sensitizer is 1/1000 of that of DPAs, the average distance between two sensitizers is approximately 10 times longer than the lattice constant of DPAs for each
direction, 91–211 Å for DPA and 149–154 Å for C7-sDPA, which clearly exceed
both L D values. This implies that it is likely that two excitons are generated from
the same sensitizer with a lag time, τ lag , which is defined as the mean recovery time
to the initial state of the sensitizer after TTET to the emitter. The normalized lag
time is defined as n lag = τ lag /min(τ TTET ). Hereafter, we first consider the encounter
probability of two excitons as a function of n TTET and n lag . Then, we go back to the
encounter probability of the triplet excitons, which are generated independently from
two different sensitizers.
Here, we consider a rough estimation of an encounter probability of two excitons
by using uniform random walk models, where we assume that the TTET between two
emitters is treated as a stochastic problem. For the one-dimensional (1D) systems,
the probability that one finds an exciton at the position i after n steps, P i (n), is given
by