The article information
 Yonglei Sun, Jing Zhao
 孙咏雷, 赵晶
 Average Arrival Time: an Alternative Approach for Studying Fluorescent Behavior of Single Quantum Dots^{†}
 平均探测时间:研究单个量子点荧光现象的另一种思路
 Chinese Journal of Chemical Physics, 2018, 31(4): 595598
 化学物理学报, 2018, 31(4): 595598
 http://dx.doi.org/10.1063/16740068/31/cjcp1806149

Article history
 Received on: June 21, 2018
 Accepted on: July 23, 2018
b. Department of Chemistry, University of Connecticut, Storrs, Connecticut 06269, USA
Fluorescence lifetime is a critical parameter to characterize quantum dots (QDs) as well as other fluorophores on the ensemble or singleemitter level. The value of lifetime comes from singleexponential fitting of the timedependent fluorescence decay curve. This ideal situation happens when QDs have singular emission state; however, in real experiments QDs have been reported to have multiple emission states, as revealed by single particle level experiments [1]. The bright and dark emission states are indicated by high and low fluorescence intensity segments in single QD's intensity trajectory over time, a phenomenon also known as fluorescence intermittency or "blinking" [2]. The fluorescence decay kinetics is different for the bright and dark emission states. As a result, the decay curve deviates from perfect single exponential form. In principle, all decay kinetics curves can be fitted multiexponentially within a certain margin of error. However, this pure mathematical treatment has two disadvantages: (ⅰ) physical meanings are usually difficult to be assigned to the lifetime components from multiexponential fitting; (ⅱ) no information about the change of emission states over time can be obtained.
Another approach to interpreting the decay kinetics that deviates from single exponentiality is the lifetime distribution analysis [3]. In this approach, instead of finding discrete lifetime components, a Laplace transform is involved to determine the distribution or spectrum of lifetime:
$ \begin{eqnarray} I(t)=\int_0^\infty \sigma(\tau) \textrm{e}^{t/\tau} \textrm{d}\tau \end{eqnarray} $  (1) 
where
To solve the aforementioned problem, in this work we used the average arrival time (AAT) of fluorescent photons in each bin time to characterize the emission behavior of colloidal QDs, instead of the traditional exponential fitting method. To obtain the AAT of fluorescent photons, timetagged timeresolved (TTTR) mode was used in the experiments, which is readily available in many of the timecorrelated single photon counting modules. Without the preassumption of exponential lifetime, we can construct the trajectory and distribution of the AAT easily. And though different by definitions, we show the AAT is compatible with the traditional exponential lifetime in analysis. Therefore albeit still being statistical, our method could possibly reveal subtle changes of the emission behavior of single QDs.
Ⅱ. METHOD A. TheoreticalSuppose the bin time is chosen so that the decay kinetics has a single exponential form
$ \begin{eqnarray} \tau'&=&\left(\sum\limits_i^\infty t_i N_i\right)\Big/\left(\sum\limits_i^\infty N_i \right)\nonumber\\ &=&\left(\sum\limits_i^\infty t_i A\textrm{e}^{t_i/\tau} \Delta t\right)\Big/\left(\sum\limits_i^\infty A\textrm{e}^{t_i/\tau} \Delta t\right)\nonumber \\ &=&\left(\sum\limits_i^\infty t_i \textrm{e}^{t_i/\tau} \Delta t\right)\Big/\left(\sum\limits_i^\infty \textrm{e}^{t_i/\tau} \Delta t\right) \end{eqnarray} $  (2) 
The time resolution
$ \begin{array}{l} \mathop {\lim }\limits_{\Delta t \to 0} \tau ' \to \left( {\int_0^\infty t {{\rm{e}}^{  t/\tau }}{\rm{d}}t} \right)/\left( {\int_0^\infty {{{\rm{e}}^{  t/\tau }}} {\rm{d}}t} \right)\\ = \frac{{{\tau ^2}}}{\tau } = \tau \end{array} $  (3) 
Therefore we prove the AAT is compatible with the traditional exponential lifetime in analysis as long as the bin time is much bigger than the time resolution.
B. ExperimentalTo test the AAT method, single colloidal CdSe/CdS QDs were studied. The QDs were synthesized using previously published method [7]. A highly diluted solution of the QDs in hexane was spun cast onto a glass coverslip and then mounted onto a homebuilt confocal fluorescence microscope. The QDs were excited by a pulsed laser (PDL 800B, PicoQuant) at 450 nm with a repetition rate of 2.5 MHz. The pumping power of the laser was kept low to avoid significant multiexciton emission of QDs. The fluorescence signal of single QDs was collected through a 100
For each individual photon, a photon counting system operated in TTTR mode records not only the photon's arrival time after the laser synchronized pulse, but also its macroscopic arrival time with respect to the beginning of measurement, hence offering more flexibility in analysis. FIG. 1 shows the intensity trajectory over time and fluorescence decay dynamics of a single QD recovered from TTTR data. The various high and low intensity states are clearly seen in FIG. 1(a), proving the signal was from a single QD. The switch between bright and dark emission states affected the decay kinetics, as the decay curve in FIG. 1(b) cannot be fitted by a single exponential decay indicated by the dashed curve. The solid curve in FIG. 1(b) denotes a double exponential fitting:
The deviation from single exponential decay curve of this single QD was further investigated by constructing the AAT trajectory and distribution shown in FIG. 2. The time resolution of PicoHarp 300 is 4 ps, much smaller than our choice of bin time 50 ms, therefore the requirement in Section Ⅱ is obviously satisfied. FIG. 2(a) shows that the AAT trajectory over time that has high and low value segments, is similar to the intensity trajectory. Actually the bins with higher intensities are likely to have larger AAT values. The correlation coefficient between the intensity and AAT corr(
FIG. 2(b) shows the AAT distribution which features a peak and its tail at the small end, corresponding to the high and low value segments of the AAT trajectory. The broad peak and its tail clearly demonstrate the single QD's multiple emission states, thus explain for the deviation from single exponential decay curve. The positions of the solid lines in FIG. 2(b) represent the values of the lifetime components
We have demonstrated that the AAT method is not only compatible with the traditional exponential lifetime analysis, but also able to present small changes in temporal and spectral domains while the latter cannot. The AAT method is thus a more exquisite approach to characterizing a single QD, suitable for many research topics in studying the QD fluorescence. Examples are identifying single QDs, detecting the changes of emission behaviors when changing excitation condition or chemical environment, and so on. Finally we would like to point out this method is general, as it can be applied to other fluorophores like dye molecules and fluorescent proteins.
Supplementary materials: Intensity distribution, intensityAAT correlation, additional AAT trajectory and distribution are available in the supplementary materials.
Ⅴ. ACKNOWLEDGMENTSThis work was supported by the National Science Foundation CAREER award (CHE1554800).
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b. 美国康涅狄格大学化学系, 康涅狄格州, 斯托斯 06269