Understanding Photoinduced Charge Transfer Dynamics of Single

Article pubs.acs.org/JPCC

Understanding Photoinduced Charge Transfer Dynamics of Single Perylenediimide Dyes in a Polymer Matrix by Bin-Time Dependence of their Fluorescence Blinking Statistics Masaaki Mitsui,*,† Aki Unno,† and Syun Azechi‡ †

Department of Chemistry, College of Science, Rikkyo University, 3-34-1, Nishiikebukuro, Toshima-ku, Tokyo, 171-8501, Japan Department of Chemistry, Faculty of Science, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan

‡

S Supporting Information *

ABSTRACT: Fluorescence on−off blinking of single perylenediimide (DMP−PDI) dyes embedded in a disordered matrix of poly(methyl methacrylate) (PMMA) was investigated by single-molecule fluorescence spectroscopy. In particular, we examined the bin-time dependencies of the complementary cumulative distribution functions (cCDF) of the on-time and off-time durations. It appears that intersystem crossing (ISC) within DMP−PDI competes with charge transfer between DMP−PDI and PMMA. The single-molecule and ensemble cCDFs of the on-time and off-time durations (induced by charge transfer) were best described by Weibull functions and log-normal functions, respectively. The dispersive kinetics of the on-time duration were attributed to radical ion pair ISC from triplet charge separation (3CS) state to the corresponding singlet (1CS) state via charge hopping between charge trap sites in PMMA. This process should compete with triplet charge recombination (3CR), which restores the triplet state. On-to-off switching occurs only when 3CS → 1CS ISC overcomes the 3CR process. The resultant Weibull distribution (with A < 1) reflects that the probability of charge hopping rapidly decreases with time. On the other hand, the offtime log-normal kinetics of single-molecules are explained by a Gaussian distribution of activation energies for charge recombination (the so-called Albery model). A simulation study revealed that the ensemble off-time cCDF included power-law kinetics of charge hopping in the polymer matrix.

1. INTRODUCTION The fascinating phenomenon known as fluorescence blinking has attracted much attention over the past 20 years for its role in single-molecule/single-particle detection.1−3 Fluorescence blinking is characterized by discrete switching between the emitting (“on”) and nonemitting (“off”) periods over time, commonly denoted as on-time (ton) and off-time (toff), respectively. The blinking statistics (i.e., the on-time and offtime distributions) have been extensively utilized to understand the photophysics underlying the blinking behavior4−26 and the fluctuation dynamics of local environments encompassing single emitters.27−29 One well-known source of blinking in organic molecular systems is intersystem crossing (ISC) from an excited singlet state (S1) to a triplet state (T1), followed by ISC from T1 to the ground state (S0). The on-time and off-time durations of this process are exponentially distributed over microseconds to milliseconds. Such blinking is usually referred to as “triplet blinking” and is characterized by a single rate constant (i.e., first-order kinetics).4−12 Blinking also occurs by charge transfer between a photoexcited single-molecule and its surrounding environmental matrix,13−26 which is the main focus of our © XXXX American Chemical Society

present work. The off states generated by this process are longer-lived than triplet blinking states, with lifetimes up to several hundred seconds. In various environments such as glass surfaces,13 polymers,14−18,28 single crystals,3,19,20 biological systems,21,22 and semiconductor interfaces,23−26 the probability density functions (PDFs) of on-time and off-time durations typically follow an inverse power-law P(t) ∝ t −m, where P(t) denotes the PDF and the power-law exponent m usually ranges from 1 to 3.13−26 In power-law kinetics, the time constant is not fixed, but varies with time. The power-law off-time statistics of the charge-transfer-induced blinking have often been reported, whereas the on-time statistics appear to depend on the system under investigation. For example, the statistics of organic dyes in biological system,22 and at semiconductor interfaces25 clearly differ from power-law behavior (e.g., multiexponential kinetics). Although the power-law distribution of the on-time kinetics has been explained by several models,2,28,30 its mechanism is Received: April 24, 2016 Revised: June 26, 2016

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DOI: 10.1021/acs.jpcc.6b04114 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C incompletely understood, and a unified view appears to be still lacking. Importantly, the shapes of the on- and off-time distributions are affected by artifacts caused by the analytical procedure (e.g., binning and thresholding in the histogram method) and depend on the signal-to-background (S/B) ratios and the total fluorescence intensity.31−34 It has been demonstrated that the choice of the bin-time (i.e., the time resolution of the fluorescence intensity time trace) significantly influences the distributions, especially the on-time distribution.11,31,32 Short off-events that are observed at a particular bin-time are unresolved as the bin-time increases. Ideally, the bin-time should be as small as possible, but a smaller bin-time leads to a lower S/B ratio, which may easily generate artifacts in the analysis. As another important point, if plural photophysical processes (e.g., ISC and charge transfer) compete in the investigated system, the shapes of the on-time/off-time distributions may be distorted with bin-time because of the change of their relative contributions in the distribution. Thus, these facts suggest that a detailed investigation of the binning effect on the distribution is necessary to extract reliable kinetic information from the blinking statistics. Recently, Reid and co-workers analyzed blinking behavior by a combined maximum likelihood estimation (MLE) and the Kolomogorov−Smirnov (KS) test (hereafter denoted as MLEKS).35 This method derives two data sets of cumulative distribution functions (CDFs) of the on-time (or off-time) from the experimental data and the MLE. The PDF is hypothesized as power-law, Weibull, or log-normal. The twosample KS test is used to determine the likelihood that the two data sets arose from the same PDF. Remarkably, this approach has demonstrated that both the on-time and off-time kinetics of the photoinduced charge transfer of nile red dye in poly(vinylidene fluoride) polymer film deviate from the power law.36 Furthermore, the interfacial charge transfer dynamics in rhodamine dye-TiO2 systems exhibit a (partial) power-law distribution of the on-times and a log-normal distribution of the off-times.37,38 Therefore, a fundamental review of the blinking statistics is essential. Perylenemonoimide (PMI) and perylenediimide (PDI) based chromophores are among the most important classes of organic fluorophores used in single-molecule fluorescence spectroscopy (SMFS) studies. Due to their exceptional photochemical and photophysical stability, high extinction coefficients (∼105 M−1 cm−1) and high fluorescence quantum yields (Φf ∼ 1), they outperform other fluorescent dyes at the single-molecule level in terms of brightness and survival time.39,40 For these reasons, their blinking behaviors have been investigated in various environments. For example, the on- and off-times of PMI and PDI derivatives exhibit power-law behavior in polymers matrixes16−18 and on Al2O3 (0001),41 ITO (tin-doped indium oxide),23 TiO2,42 ZrO2.43 Motivated by these observations, we employed a PDI derivative (hereafter termed DMP−PDI, see Figure 1a) as a fluorescent probe to disclose the bin-time effect on the blinking statistics governed by its charge transfer reaction with an encompassing poly(methyl methacrylate) (PMMA) polymer matrix (Figure 1b). Under linear polarization excitation, the fluorescence intensity time traces of single DMP−PDI molecules exhibited extremely high signal-to-background (S/B) ratios, which allows us to neglect artifacts caused by binning and thresholding procedures.31−34 Consequently, we could change the bin-time by 5 orders of magnitude (from 0.08 ms to 1 s). By MLE-KS analysis

Figure 1. Chemical structures of (a) DMP-PDI and (b) PMMA polymer.

of the on-time and off-time distributions at properly selected bin-times, we propose the entire scheme of the photophysical processes underlying the blinking phenomenon and determine the associated time constants and quantum yields.

2. EXPRIMENTAL DETAILS 2.1. Sample Preparation. N,N′-bis(2,6-dimethylphenyl)perylene-3,4,9,10-tetracarboxylicdiimide (DMP−PDI, ≥ 90%, Sigma-Aldrich; see Figure 1a), PMMA (glass-transition temperature Tg = 99 °C, average molecular weight Mw ∼ 1.2 × 105, Sigma-Aldrich; see Figure 1b), N,N′-bis(3-methylphenyl)N,N′-bis(phenyl)-benzidine (TPD, Sigma-Aldrich), and toluene (spectroscopic grade, Wako) were used as received. DMP−PDI-dispersed PMMA films were prepared by spincoating (2000 rpm) one drop of a toluene solution containing DMP−PDI (∼10−10 M) and PMMA (5 mg/mL) onto thoroughly cleaned cover glasses (Matsunami). Consequently, DMP−PDI molecules were dispersed on the PMMA thin film. By atomic force microscopy (AFM, SPM-9700, Shimadzu), the thickness of the PMMA film was determined as 50−100 nm. The sample-coated cover glass was set on an O-ring, forming the top face of a small vacuum chamber as described in ref 11. Prior to measurements, the sample was evacuated for 30 min under high vacuum conditions (10). Therefore, we hereafter analyze the 10 ms bintime distributions, which originate from only the CR process with no ISC contribution and can provide an accurate analysis result of the CR process. Figure 5 plots the cCDFs of the off-time in the MLE-KS analysis of 1 and 68 molecules at 10 ms bin-time. In Figure 5a, the single-molecule cCDF of the off-time is well reproduced by a log-normal function with μ = 0.98 and σ = 2.40 (p = 1.0). However, the fit of the Weibull function also yielded a high pvalue (p = 0.99). As pointed out by Reid and co-workers, less accurate p-value is obtained with decreasing number of data points.35 Then, the accuracy of the p-value for this data set was estimated to be 0.13. Thus, it is difficult to determine which is the best model function. The power-law function has a smaller p-value (0.56), thus is excluded as a model function candidate. Other single-molecule cCDFs were also well represented by both log-normal and Weibull distributions. In contrast, the composite off-time cCDF (Figure 5b) was best represented by a log-normal function (p = 0.51) with μ = 0.10 and σ = 2.84 (the average value: τoff = 61.8 s). The p-values of the Weibull and power-law functions are 0.24 and 0.00, respectively. Note that the differences of p-values between log-normal and Weibull functions are much larger than the accuracy of p-value for the data set (0.025). This result implies that the log-normal distribution best describes the off-time cCDFs of singlemolecules (i.e., single-molecule CR kinetics). As can be seen in Figure 5b, the large p-value (p = 0.51) was obtained for the log-normal function; nevertheless, the distribution remarkably deviates from the best fitting curve at toff ∼ 100 s. The number of off-events in this data set (i.e., 389 events) is not enough to produce a highly accurate p-value. In this case, the relatively large p-value is obtained from eq 2, but it does not necessarily mean the significant fitting of the distribution. Then, what is the cause of this deviation? It may result from the limited observation time of single molecules, which reduces the likelihood of observing long off-time events.14 Consequently, the probability of these relatively rare events will be underestimated. To test this supposition, the ensemble cCDFs were constructed after gradually excluding the long off-time events, as shown in Figure 6. However, the resulting distributions exhibit prominent bending tails, and the entire distributions cannot be reproduced by a single log-normal function. As exemplified in Figure 5a, the ensemble cCDF is the

are discussed in this and the subsequent subsection, respectively. Figure 3 displays the PDFs and cCDFs of the off-time, denoted by P(toff), obtained from a single molecule and 68 molecules. The PDFs and cCDFs could be fitted to two types of distribution. From 80 μs to 2 ms, the off-time PDF and cCDF of the single-molecule data were well reproduced by the single-exponential function exp(−toff/τoff) with τoff = 170 μs. Therefore, this period was assigned to the T1 → S0 ISC process (with a triplet lifetime τT of 170 μs in DMP−PDI molecules). In all panels of Figure 3, the two distributions exhibit a clear boundary around 2 ms. Assuming the exponential distribution with τoff = 170 μs, the estimated probability of triplet blinking at toff = 2 ms is of the order of 10−6. Therefore, ISC makes negligible contribution to the distribution when toff ≥ 2 ms. In this time frame, the PDFs and cCDFs can be explained by charge recombination (CR) dynamics between the ionized DMP−PDI and PMMA. Later, we will confirm this assignment by investigating the bin-time dependence of cCDFs. At first glance, the PDFs in the toff ≥ 2 ms region appear to obey power-law statistics; that is, P(toff) ∝ toff−moff (see panels a and c of Figure 3). However, the corresponding cCDFs are clearly nonlinear and show a clear curvature at long durations. In general, the CDF is obtained by integrating the PDF: CDF(t ) =

∫t

t

P(t )dt min

(5)

Because the integral of a power-law function is also a power-law function, the curved CDFs (toff ≥ 2 ms) unequivocally demonstrate that the CR kinetics deviate from a power law. To reveal the true off-time distribution reflecting the CR dynamics, we plot the bin-time dependence of the combined off-time cCDFs in Figure 4. Here, the bin time was varied as 0.08, 0.2, 0.5, 1, 10, and 100 ms. The distribution due to the T1 → S0 ISC process appears at toff < 2 ms and is especially prominent in the distributions of the 0.08 and 0.2 ms bin-times.

Figure 4. Bin-time dependence of off-time cCDFs for an ensemble of 68 DMP−PDI molecules (identically to Figure 3c,d). The bin time is varied as 0.08, 0.2, 0.5, 1, 10, and 100 ms. E


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The Journal of Physical Chemistry C

Figure 5. Off-time cCDFs of (a) 1 molecule (14 events) and (b) 68 molecules (389 events), fitted to log-normal (solid lines), Weibull (dashed lines), and power-law (dotted lines) functions. See text for further details.

Figure 6. Composite off-time cCDFs (68 molecules), obtained at tmax ≤ 0.1, 1, 10, and 100 s. The solid lines are the best-fitted log-normal functions.

Figure 7. Single-molecule (a) PDF and (b) cCDF of on-time (6082 events; the accuracy of p-value is approximately 0.006), obtained at a bin-time of 0.08 ms. The solid lines represent the best-fitted singleexponential functions obtained by the MLE-KS analysis. The on-time lifetimes are τon = 11.9 ms (p = 0.015) for PDF and cCDF, respectively.

composite of many single-molecule distributions, each of which can be represented by a log-normal function with different parameters (i.e., μ and σ, see Figure S2). Hence, the log-normal fit of the ensemble cCDF deviates mainly by the static heterogeneity of single-molecule CR kinetics. However, this deviation is probably caused by not only the inhomogeneous CR kinetics but also the charge diffusion in PMMA, as discussed later. 4.3. On-Time Distribution. Figure 7 illustrates the PDF and cCDF of the on-time at a bin time of 0.08 ms. The short bin-time may lead to artificial separation of one on-state into several shorter on- and off-states. This segmentation of onstates is responsible for the overestimation of the number of short on-times.33 However, the overestimation of the number of short on-events was not observed in Figure 7b and the entire parts of these curves were well fitted to a single exponential function of the form exp(−ton/τon) with τon = 11.9 ms, although a statistically insignificant value of p = 0.015 was obtained. Corresponding MLE-KS analysis for log-normal and power-law distributions yielded completely statistically insignificant results (Figure S4). The analysis for Weibull distribution yielded the most significant p-value (i.e., p = 0.92), but the parameter A was almost unity (i.e., A = 0.94). In this case, the CDF of Weibull distribution coincides with that of single exponential distribution (see eqsS5 and S10 in the Supporting Information). Therefore, we conclude that the empirical distribution is best

represented by single exponential distribution and is assignable to the S1 → T1 ISC process in a PDI molecule. The quantum yield (ΦISC) of the S1 → T1 ISC was estimated as 1.1 × 10−4 from the relationship ΦISC = (τonk01)−1, where 1k01 is the excitation rate from S0 to S1. The histograms of τT, determined from the single-molecule off-time distributions (e.g., Figure 3b), and ΦISC are given in Figure S3. Their average values (i.e., 120 μs and 1.1 × 10−4, respectively) are consistent with the reported values of similar PDI derivatives.46 According to Reid and co-workers,35 the cCDF is a continuously decaying function requiring no data smoothing, thus facilitating the distribution analysis. As is evident in Figure 7, the cCDF much better resembles the single exponential distribution than the corresponding PDF. Hence, our subsequent distribution analysis is performed only on the cCDFs. To reveal the distribution generated from the charge transfer process alone, we examined the bin-time effect on the combined on-time cCDF. The results are presented in Figure 8. At 0.08 ms bin-time, the cCDF is attributable to S1 → T1 ISC. However, rather than a single-exponential function, this cCDF is fitted to a stretched-exponential function (see Supporting Information) with β = 0.68 and τKKW = 9.4 ms (i.e., a weighted average of ⟨τon⟩ = 12.3 ms). As the cCDFs of F


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MLE-KS analysis disputes any power-law behavior of the ontime kinetics. Figure 9 plots the single-molecule and ensemble on-time cCDFs at 10 ms bin-time. The fitting curves of the lognormal, Weibull, and power-law functions are also shown. The cCDFs of single-molecules are well reproduced by both lognormal and Weibull functions. Because the p-values of both fittings are comparable within the accuracy of the p-values (ca. 0.1), the most appropriate model function for single-molecule cCDFs is difficult to determine. In contrast, the ensemble cCFD is best fitted by the Weibull distribution (A = 0.56, β = 0.80, p = 0.57). The differences in p-values among the Weibull and other functions are much larger than the accuracy of pvalue (i.e., 0.02). Thus, we infer that the Weibull function also best describes the single-molecule on-time kinetics.

5. DISCUSSION 5.1. A Proposed Kinetic Scheme and Estimation of Related Parameters. As reported for perylene orange, T1 → Tn absorption occurs under excitation by 488 nm laser light (2.54 eV) owing to its large overlap with S0 → S1 absorption.18 Therefore, charge transfer is thought to occur from the higher excited triplet Tn states of perylene orange to PMMA. The present system should follow the same kinetics, for the following reasons: (1) The π-conjugated framework and electronic states of DMP−PDI are very similar to those of perylene orange; and (2) the driving forces of electron transfer/ hole transfer (i.e., ΔGCS) are positive from the S1 and T1 states, but sufficiently negative from the Tn states to PMMA, i.e., ΔGCS ≤ − 0.5 eV.47 Notably, the fluorescence of DMP−PDI singlemolecules embedded in TPD film was almost completely quenched, because the driving force of hole transfer from the S1 state of DMP−PDI to TPD is approximately −0.6 eV, allowing ultrafast charge transfer from the S1 state. This result further supports that the CS does not arise from the S1 state in the DMP-PDI/PMMA system (where the strong fluorescence was observed, as exemplified in Figure 2a). Based on these facts, we propose an entire kinetic scheme of the charge transfer dynamics in the present system (see Figure 10). The Tn states formed through T1 → Tn absorption (3k1n) undergo the CS reaction with the PMMA matrix (3kCS), forming a triplet charge

Figure 8. Bin-time dependence of the composite cCDF of on-time (68 molecules).

individual molecules exhibit single-exponential distributions, this stretched-exponential character reflects heterogeneity in the S1 → T1 ISC yield of each single molecule (Figure S3b). Remarkably, the cCDF gradually changes shape as the bin-time increases from 0.08 to 1 ms, whereas the bending tails of the cCDFs remain unchanged beyond 1 ms. This result suggests that the contribution of triplet blinking gradually disappears with increasing bin-time. Interestingly, the triplet and chargetransfer-induced blinking are merged in the on-time distributions, but are clearly discernible in the off-time distributions (see Figure 3). Such a mixture of competing processes obscures the true distribution. However, as revealed in Figure 8, the triplet blinking is negligible in the cCDFs at bin times of 10 and 100 ms; thus, the distribution should be generated only by the charge transfer process. As also seen in Figure 8, the on-time cCDFs generated by the charge transfer process deviate far from linearity on a log−log plot. To date, the on-time kinetics have been interpreted as power-law; however, the present

Figure 9. (a) Single-molecule on-time cCDF (26 events) and the best-fitted log-normal (solid line, μ = −0.58, σ = 1.36, p = 1.0), Weibull (dashed line, A = 0.83, β = 1.10, p = 0.99), and power-law (dotted line, m = 1.36, p = 0.32) functions. (b) Ensemble on-time cCDF (459 events, 68 molecules) and the best-fitted log-normal (solid line, μ = −1.19, σ = 1.99, p = 0.12), Weibull (dashed line, A = 0.56, β = 0.80, p = 0.57), and powerlaw (dotted line, m = 1.29, p = 0.00) functions. The best fit is obtained by the Weibull function with A = 0.56 and β = 0.80 (average value τon = 1.33 s). G


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k1n values at the excitation wavelength of λ = 488 nm were calculated as follows:

3

1/3

k ij =

ln(10)ελPex NAhc /λ

(7) −2

where Pex is the excitation laser intensity in W cm , ελ is the molar absorption coefficient, NA is the Avogadro constant, c is the light velocity, and h is the Planck constant. The ε488 values for S0 → S1 and T1 → Tn transitions are approximately 4 × 107 and 6 × 107 cm2/mol in toluene, respectively,18 affording 1k01 = 7.5 × 105 s−1 and 3k1n = 1.1 × 106 s−1 at Pex = 2 kW/cm2. The mean τT and ΦISC for 166 molecules in PMMA were determined as approximately 120 μs and 1.1 × 10−4, respectively (see Figure S3). From these results, 1k01ΦISC and 3 k1nτT were calculated as 82.5 s−1 and 132, respectively. The former value corresponds to the formation rate constant of T1 state and the latter is the average number of T1 → Tn transitions during the residence time in T1 (i.e., triplet lifetime τT), respectively. The formation rate constant of the Tn states, 1 k01ΦISC3k1nτT, was then calculated as 1.1 × 104 s−1. The average τon was determined as 1.33 s from the Weibull function reproducing the ensemble on-time cCDF (Figure 9b). Therefore, the average formation rate constant of the 1CS state (the long-lived off-state) is only 0.8 s−1. From eq 6, the ΦCSΦRIP‑ISC (formation yield of 1CS from Tn states) was thereby estimated as 1.5 × 10−4. The photophysical parameters obtained in this study are summarized in Table 1. 5.2. On-Time Kinetics: Origin of Weibull Distribution. As shown in Figure 10, we consider that the on-time distribution, which is described by the Weibull distribution, correlates with the RIP-ISC from 3CS to 1CS. Considering (1) the sufficiently negative ΔGCS (approximately −0.5 eV) of electron/hole transfer from Tn states to PMMA18,47 and (2) the close vicinity of an excited dye molecule with surrounding acceptor sites, the CS process should be very fast. Presumably, its rate is larger than (or comparable to) the Tn → T1 IC rate, i.e., 3kCS ≥ 3kn1. However, the intrinsic low formation yield of the T1 state (i.e., ΦISC ∼ 10−4) significantly limits the chance of obtaining CS from the Tn states, even when ΦCS ∼ 1. Furthermore, the formation yield of 1CS from the Tn states (ΦCSΦRIP‑ISC) is very small (approximately 10−4) because the reaction competes with the rapid triplet CR pathway (3kCR).48 Consequently, the blinking events in the present system rarely occur through the charge transfer process (i.e., koff = 0.8 s−1). Since the electron-exchange interaction between an ionized DMP-PDI and the counter charge transferred to a PMMA acceptor site is an exponentially decreasing function of the distance between them, the singlet and triplet radical ion pair mixing becomes possible at a long distance (e.g., > ∼ 1 nm).51 Therefore, the RIP-ISC from 3CS to 1CS probably accompanies charge hopping from the first acceptor site to nearby other acceptor sites in PMMA, because it can increase the radical ion pair distance. The RIP-ISC process (or the “initial” charge hopping process) best fitted the Weibull function with A = 0.56

Figure 10. Proposed entire kinetic scheme of DMP−PDI molecules in a PMMA matrix.

separation (3CS) state (or triplet radical ion pair). Subsequently, the 3CS should return to the T1 state through spinallowed triplet CR (3CR), or undergo radical ion pair intersystem crossing (RIP-ISC) to yield the singlet charge separation (1CS) state (or singlet radical ion pair). The energy gap for the 3CR process (i.e., the energy difference between 3 CS and T1 states) is considerably smaller than that for the 1CR process (i.e., the energy difference between 1CS and S0 states), resulting in a much faster rate for 3CR (3kCR) than for 1CR (1kCR).48 Thus, it is expected that the 3CS → T1 charge recombination pathway is efficient, with the result that the 3CS → 1CS RIP-ISC rarely occurs. The route of T1 → Tn → 3CS → T1 cannot be observed as the blinking event and would be buried under on-time periods at long bin-times (e.g., 100 ms). Consequently, the 3CS state does not likely to be the “longlived dark state” (i.e., long off-time). We infer that the 1CS state formed through the RIP-ISC corresponds to the “long-lived dark state” in the present system. At this moment, we have no direct evidence for this proposal, but many photochemical studies have reported such a spin selective charge transfer dynamics.48−50 Hence, we propose that the “log-normal” offtime kinetics is dominated by this singlet CR (1CR) pathway; that is, 1kCR = 1/τoff. The 1CR process may associate with charge hopping (or charge shift) in the PMMA, because it can form long-lived radical ion pair (i.e., the long-lived charge separated state). This will be discussed later. The formation rate constant of the off-state (koff) produced by charge transfer process is the reciprocal of the average ontime (τon), given by 1 = 1k 01ΦISC3k1nτTΦCSΦRIP ‐ ISC koff = τon (6) where 3k1n stands for the excitation rates from T1 to Tn, respectively. ΦCS and ΦRIP‑ISC are the quantum yields of the CS from the Tn states (i.e., the formation of 3CS from Tn states) and the RIP−ISC from 3CS to 1CS, respectively. The 1k01 and

Table 1. Excitation Rates from S0 to S1 and from T1 to Tn, Ensemble-Averaged ISC Yield, and Triplet Lifetime of Single DMPPDI Molecules Embedded in PMMA Film at Room Temperature (Determined by SMFS under Excitation at 488 nm) k01/s−1

1

DMP-PDI/PMMA

7.5 × 10

5a

k1n/s−1

ΦISC

1.1 × 10

−4b

3

6a

1.1 × 10

τT/μs (166)

b

120 (166)

ΦCSΦRIP‑ISC 1.5 × 10

−4c

(68)

koff/s−1 0.8d (68)

a Obtained at laser power Pex = 2 kW/cm2. bObtained by assuming a Gaussian distribution. cObtained by eq 6. dFormation rate constant of the charge-transfer-induced off-state.

H


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The Journal of Physical Chemistry C and β = 0.80 (see Figure 9b). As described by Reid and coworkers,35 the Weibull distribution incorporates statistical aging through the parameter A, where A > 1 indicates that the probability of a molecule leaving a particular emissive state increases with time (A < 1 denotes the opposite case). According to the Weibull distribution obtained in the present case, the probability of the molecule leaving the 3CS state decreases over time. This is likely caused by competition with the triplet CR process because the 1CS state can be formed only when its formation rate is sufficiently fast to prevail against the triplet CR process. This kinetic restriction is considered to yield the similar Weibull parameters for the on-time distributions of the single-molecules (see Figure S5). Consequently, the entire ensemble cCDF is well fitted by a single Weibull function, as seen in Figure 9b. Very recently, Wilma and co-workers reported that the free electrons in polystyrene film induce fluorescence blinking of PDI derivatives.52 In this case, the on-time periods are terminated by electron capture by a neutral ground-state PDI molecule from the surrounding polymer matrix. In other words, the on-time durations are dominated by electron diffusion processes in the polymer. Trap-to-trap electron diffusion is known to follow power-law kinetics;53,54 therefore, the on-time durations should exhibit a power-law distribution.30 However, no power-law component appears in the single-molecule and ensemble distributions at any bin-time (see Figure 8), implying that this blinking mechanism is absent or minimal in the present system. 5.3. Off-Time Kinetics: Origin of Log-Normal Distribution. Recently, single-molecule blinking dynamics on a TiO2 surface have been modeled by Monte Carlo simulations of the Albery model, which assumes a Gaussian distribution of activation energies in the CR process.37,38 The simulations successfully reproduced the experimentally observed lognormal CR kinetics (i.e., the log-normal distribution of the off-times). Thus far, power-law behavior of the off-time distributions of single-molecules in polymers (and also on glass and semiconductor surfaces) has been widely accepted. However, recent single-molecule studies using the MLE-KS method have challenged this view.35−38,55 In the present study, the CR process from 1CS to S0, which governs the off-time durations (i.e., τoff = 1/1kCR), follows log-normal kinetics, not power-law kinetics. We therefore consider that this result is also consistent with the Albery model. Since the Albery model assumes a Gaussian distribution of activation barriers to forward and back charge transfer, the corresponding rate constants are log-normally distributed.37,38 In the present system, the forward charge transfer from DMP-PDI to PMMA corresponds to the CS process from Tn to 3CS state (Figure 10). According to the Albery model, this process should exhibit a log-normal kinetics. However, we emphasize that this process does not dominate the on-time duration, because the 3CS state would be quickly depleted via the 3CS → T1 3CR process48 and does not correspond to the long off-state, as aforementioned. Thus, it is difficult to directly obtain the quantitative information about the forward charge transfer process (i.e., Tn → 3CS process) from the analysis of the on-time distribution. The origin of the Gaussian-distribution of activation energies remains unclear, especially for the single-molecule kinetics. Thermal fluctuations are known to follow a Gaussian distribution; therefore, the origin might be ascribed to a thermal fluctuation effect (i.e., thermal fluctuations of local

polymer environments encompassing single molecules) on charge transfer. Since the 1CS state is located above the T1 state whose transition energy is approximately 1.2 eV,56 the driving force ΔG1CR should be lower than −1.2 eV. Consequently, the 1 CR process might occur in the Marcus inverted region and has activation barriers, showing the appreciably slow dynamics. Indeed, the average value of the composite off-time distribution (Figure 5b) is very long; τoff = 61.8 s. All of the single-molecule off-time cCDFs were well reproduced by a log-normal function, but their average offtimes τoff widely varied (spanning approximately 3 orders of magnitude, as shown in Figure S2a). This suggests the presence of distinctly different CR pathways for individual molecules in the disordered PMMA matrix. Accordingly, the corresponding ensemble cCDF should not be reproducible by a single lognormal function (see Figures 5b and 6). To further investigate this issue, we conducted a simple simulation study. The detail is given in the Supporting Information. Specifically, we determined whether the ensemble distribution can be constructed from a number of log-normal distributions with different parameter values (i.e., μ and σ). In our simulation, 100 synthetic data sets of log-normal distributions with different average values from 1 to 100 s were created by using a dedicated macro tool (VBA Excel macro). Some examples are presented in Figure S6. When we supposed that the 100 simulated data sets were observed with same probability, they were combined intact. As shown in Figure 11, however, such

Figure 11. Experimental off-time cCDF (tmax = 100 s) and the bestfitted curve to composite log-normal synthetic data weighted by the inverse power-law weight function with m = 2.0. These data were not reproducible by any other exponents or weight functions.

combinations of the 100 data sets could not reproduce the ensemble off-time cCDF. This result suggests that their appearance probabilities are not same and should be weighted by some distribution caused by another underlying process. Therefore, the probabilities of observing data sets (i.e., the appearance number of times of each data set) were weighted using single-exponential, log-normal, or power-law functions. Finally, we constructed the synthetic weighted ensemble distributions (Figures 11 and S7). Among these weight functions, the power-law alone yielded the experimental cCDF (see Figures 11 and S7). The best reproducibility was obtained at a power law exponent of −2.0. As charge hopping dynamics are known to obey a power law,53,54 this fact intriguingly implies that charge hopping diffusion competes with the CR process. The contribution of charge hopping (i.e., power law dynamics) is not easily detected I



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ACKNOWLEDGMENTS M.M. is grateful to Mr. Takeo Saito from Shimadzu Corporation for the experimental work with AFM. This work is supported by Grants-in-Aid for Young Scientists (A), No. 20685001, Scientific Research (C), No. 24550018, Challenging Exploratory Research, No.23655007, and Basic Science Research Projects from The Sumitomo Foundation, No.130550.

among the few events in single-molecule distributions, but becomes prominent in the ensemble distribution. Hoogenboom and co-workers reported that the charge diffusion in polymers causes correlation losses between successive offevents (i.e., losses in the off−off correlations).28 Supporting this idea, no off−off correlations were found in the present system (Figure S8). In previous studies based on the PDF analysis, the power-law distribution of off-time has been found for several dye−polymer systems and ascribed to a heterogeneity of CR rates over a wide range.2,16−18,28 However, the present study demonstrates that the CR process follows log-normal kinetics, and not power-law kinetics. Furthermore, numerical simulation suggests the coexistence of the power-law kinetics of charge hopping, which are embedded in the log-normal distribution generated by the CR processes.

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ASSOCIATED CONTENT

* Supporting Information S

This material is freely available via the Internet at The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b04114. Representation of model functions used in this study. Additional data, including histograms of average values and parameters obtained by log-normal fits or Weibull fits of single-molecule distributions, histograms of singlemolecule triplet lifetimes and ISC yields, some examples of single-molecule simulation data, the composite lognormal synthetic data weighted by the single-exponential functions, and correlation plots of adjacent off-times. (PDF)

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REFERENCES

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6. CONCLUSIONS Our results highlight that the experimental distributions and corresponding kinetic models of blinking fluorescence can be deduced from the bin-time dependence of on-time/off-time distributions (e.g., on-time/off-time cCDFs). Importantly, when several intra/intermolecular photophysical processes (such as ISC and charge transfer) compete in the system under investigation, the on-time/off-time distribution can drastically vary with bin-time. Hence, the bin-time dependence of the on-time/off-time distribution is deemed imperative for unraveling the true kinetics of the underlying photophysical processes. Indeed, from an MLE−KS analysis of the bin-time dependent on-time and off-time CDFs, we established an entire kinetic scheme in the PDI−PMMA system. The probability of CS from Tn states, which is probably ultrafast and forms 3CS, is originally restricted by the very small S1−T1 ISC yield (approximately 10−4). In addition, the formation of 1CS (i.e., the long-lived off-state) from 3CS is hampered by the triplet CR pathway from 3CS to T1. Consequently, the charge-transferinduced off-state is rarely formed in this system. The present study shows that the MLE-KS analysis for the on-time/off-time distributions considering the bin-time dependence sheds new light on the underlying charge transfer kinetics of single molecules under heterogeneous environments.

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