Fluorescence Lifetime Analysis of Graphene Quantum Dots

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Fluorescence Lifetime Analysis of Graphene Quantum Dots Magnus Röding, Siobhan Julie Bradley, Magnus Nyden, and Thomas Nann J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp510436r • Publication Date (Web): 25 Nov 2014 Downloaded from http://pubs.acs.org on November 29, 2014

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The Journal of Physical Chemistry

Fluorescence Lifetime Analysis of Graphene Quantum Dots Magnus Röding,∗ Siobhan J. Bradley, Magnus Nydén, and Thomas Nann Ian Wark Research Institute, University of South Australia, Adelaide, Australia. E-mail: [email protected] Phone: +61 (0) 8 8302 3290. Fax: +61 (0) 8 8302 3683

∗

To whom correspondence should be addressed

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Abstract Fitting of the fluorescence lifetimes for fluorophores can be achieved using various techniques. Problems arise when the fluorophore sample itself is heterogeneous and the fluorescence decay is far from exponential, as a single sample may have a distribution of lifetimes. We study the fitting of fluorescence lifetimes for two systems, a homogeneous and near-exponential fluorescein dye and a highly heterogeneous system of graphene quantum dots. We fit a variety of different models to experimental time-correlated single photon counting (TCSPC) fluorescence data from each system, evaluating the validity of each model for the two systems. We find that for the near-exponential system, there is little difference between the fit of each model. However, for strongly non-exponential behaviour, the models give quite different estimates of mean and standard deviation indicating that appropriate model choice and assessment is crucial in obtaining meaningful lifetimes from fluorescence lifetime data. Moreover, we perform simulation studies that strengthen these conclusions.

Keywords: time-correlated single photon counting, luminescence, model, maximum likelihood, fitting


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Introduction The understanding of luminescence (fluorescence) decay dynamics of excited states of any material is significant for the understanding of its electro-optic and catalytic properties. At the heart of many new discoveries in this area lies often fluorescence, time-resolved fluorescence spectroscopy and lifetime imaging to spatially map the fluorescence in complex materials. 1–3 In the simplest case, the system can be described by a single decay rate (or characteristic time scale), corresponding to an exponential distribution of fluorescence lifetimes. However, perfectly exponential decay appears quite uncommon, and more often 4–16 than not, 17–20 deviations from this simple model are observed, making modeling and interpretation of more sophisticated, non-exponential fluorescence lifetime distributions an important matter. Graphene quantum dots (GQDs) are a relatively unstudied graphitic material consisting of fragments of graphene, typically below 20 nm. 21 They are of interest not only due to the plethora of properties displayed by graphene itself but also due to their opto-electronic properties that are similar to conventional quantum dots. 22 The cheapest, simplest and most employed synthesis methods for producing GQDs are top-down approaches. 23–27 These approaches yield particles which vary in size, shape and chemical composition and therefore display complex fluorescence behaviour, caused by the high heterogeneity of the sample. 25,28–30 For example, the entire sample likely generates a distribution of lifetimes rather than one discrete lifetime. We emphasize that GQDs synthesized by bottom-up approaches will not show the same degree of heterogeneity. In this paper, we discuss fluorescence lifetime analysis of highly heterogeneous GQDs using time-correlated single photon counting (TCSPC), 31–33 together with a comparison to a standard, homogenous fluorescein dye. We explore four different models to interpret the TCSPC data: the mono-exponential, the stretched exponential, the lognormal, and the inverse gamma distribution models. By simulation studies, we compare computational speed and we study how well the methods perform asymptotically, i.e. for infinitely high photon 3 ACS Paragon Plus Environment


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count, when the data cannot be completely described by the assumed model beacuse the model is misspecified. The results of this study provide a heuristic for spectroscopists and practitioners to help in the meaningful interpretation of fluorescence lifetime data of highly heterogeneous samples, as frequently found in the nanosciences, e.g. quantum dots.

Methods Experimental Graphene quantum dots are a highly heterogeneous material. 21 As mentioned above, they are usually prepared by breaking larger graphene sheets into smaller fragments. The resulting particles vary in size, aspect ratio, number of layers, chemical composition of their edges and ’surfaces’, and several other parameters. Moreover, heterogeneity of this type is a typical feature of many advanced materials, including nanomaterials such as semiconductor quantum dots or carbon nanotubes. GQDs have been chosen for this study as an extreme case of heterogeneity in a nanomaterial. Graphene quantum dots were prepared using a top-down hydrothermal approach. 27 The only modification made was the type of modified Hummers method used. 34 Particle heights were obtained using a Bruker Nanoscope 8 AFM (Bruker, Germany) with tapping mode in air. In Fig. 1a-1c, the AFM characterization is shown. The height distribution was determined from n = 10 line profiles, an example of which is shown. The results indiciate that very little aggregation is present. Particle sizes (diameters) were obtained using a 200 kV JEOL 2100F (JEOL, Japan) transmission electron microscope. In Fig. 1d-1f, the TEM characterization is shown. The diameter distribution (d = 8.3 ± 2.9 nm) was determined from n = 368 particles. Furthermore, the high resultion TEM image inset shows lattice spacing of 0.21 nm in accordance with the structure of graphene. Steady-state emission spectra of the GQDs in the range 350-700 nm were acquired using an Edinburgh Photonics FLS 980 time-resolved photoluminescence spectrometer (Edinburgh Instruments, UK) and 4 ACS Paragon Plus Environment

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shown in Fig. 2a. At the emission intensity maximum at λ = 430 nm, fluorescence lifetime measurements using time-correlated single photon counting (TCSPC) were performed, see Fig. 2b. Excitation lifetimes were recorded in a 500 ns range using 8192 channels. Time zero was determined by finding the channel with the highest photon count. The first 60 ns of the recorded lifetime range was kept for analysis, constituting 984 channels and containing in excess of 5 × 106 photons. Another part of each data set was used solely to estimate the dark count proportion pB by

pˆB =

Average count per channel in noise , Average count per channel in signal plus noise

(1)

where ’signal plus noise’ constitutes the first 60 ns part of the recorded lifetime distribution with noticable decay, and ’noise’ constitutes another part of the same data set for very long lifetimes where the ’dark count’ dominates (for more information, see the section on modelling). For assessing intra-sample variation of the lifetime measurements, 100 independent parts of 5 × 104 photons each were randomly extracted from the complete lifetime data set. As a reference in the discussion, lifetime measurements were also performed in an identical manner for fluorescein (BDH Chemicals, UK), a standard organic dye with known nearexponential lifetime distribution, at emission wavelength λ = 517 nm.

General theory Suppose that the fluorescent sample is excited with a pulse at time t = 0, so that an excitedstate lifetime distribution with probability density function (pdf) f (t) can be observed for t > 0. In the simplest case, the probability of all fluorophores to leave the excited state is the same and constant over time, yielding that the lifetime can be represented by a random variable T with an exponential distribution having pdf (

f (t; τ0 ) =

)

1 t exp − , τ0 τ0


(2)


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where τ0 is the characteristic time scale for decay. However, due to aspects of photochemistry that are not entirely understood, populations of fluorophores and even some individual fluorophores demonstrate more complex decay behavior. Many authors discuss possible reasons for non-exponential decay such as local environment variations and spatially dependent interaction strength between fluorophores , 9,10,35,36 and whether it is an intrinsic effect or an ensemble effect. 15 This work being more exploratory than explanatory though, we will not elaborate further on the physical underpinnings of fluorescence. The most common extension to the exponential is probably the I-component multiexponential, with pdf

f (t; θ1 , ..., θI , τ1 , ...τI ) =

I ∑ i=1

(

θi

)

t 1 exp − . τi τi

(3)

However, it is natural to also consider a more general lifetime distribution model with pdf ∫

∞

f (t; a) = 0

(

)

1 t g(τ ; a) exp − dτ, τ τ

(4)

where a is a general parameter vector and g is a continuous ’distribution of time scales’ (there is some confusion in the literature on this point: sometimes g is denoted ’lifetime distribution’, whereas this is the correct term for f ), describing a general superposition of exponential distributions. 37 The distribution can be parametric with a specific functional form or semiparametric (multi-component) like a continuous analogue of the multi-exponential. We will cover some particularly common and relevant choices of f and g in coming sections.

Modeling The time-correlated single photon counting (TCSPC) technique uses a series of excitation pulses with subsequent photon detection to obtain a distribution of excited-state lifetimes as a histogram of K channels with values Nk , k = 1, ..., K, representing the photon count in the intervals between tk−1 and tk with 0 = t0 < t1 < ... < tK . In modeling, the lifetime 6 ACS Paragon Plus Environment

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distribution f (which is continuous, whether g is continuous or not) is converted to a discrete counterpart with ’channel probabilities’

Pk (a) = P (tk−1 < T ≤ tk |T ≤ tK ) = P (tk−1 < T ≤ tk ) F (tk ; a) − F (tk−1 ; a) = , P (T ≤ tK ) F (tK ; a)

(5)

where F (t; a) is the cumulative distribution function (cdf) of f (t; a), i.e. ∫

t

f (u; a)du.

sF (t; a) =

(6)

0

The expression in Eq. 5 takes into account both the ’binning’ into discrete channels and the fact that lifetimes exceeding tK are not recorded which is accounted for by normalization. Also, a baseline must be included in the model to account for ’dark count’ photons. Assuming that a proportion pB of recorded photons are ’dark’ and that their occurrence is uniformly distributed in time over all K channels, the corrected ’channel probabilities’ are

Pk⋆ (a, pB ) = (1 − pB )Pk (a) + pB

1 . K

(7)

Although pB is intrinsic to the experimental setup and the sample but independent on the choice of model, it was found that when estimated jointly with the parameter a, the estimate of pB indeed varied with the choice of model. Therefore, pB was estimated prior to estimating a by pˆB =

Average count per channel in noise , Average count per channel in signal plus noise

(8)

where ’signal plus noise’ constitutes the first part of the recorded lifetime distribution with noticable decay, and ’noise’ constitutes another part of the same data set for very long lifetimes where the ’dark count’ dominates. In this manner, all models are used with one common estimate of the ’dark count’ which provides a more fair comparison.



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Estimation Since the 1970s, plenty of parameter estimation methods in fluorescence lifetime analysis have been proposed, as well as comparisons and analyses of accuracy, including nonlinear least squares, the method of moments, phase plane methods, Laplace transform methods, Fourier transform methods, maximum entropy methods, maximum likelihood, Bayesian methods, and variations and combinations thereof. 3,38–53 It appears the community has for the most part settled for weighted least squares. 54 However, using least squares implies the incorrect assumption of Gaussian noise making maximum likelihood 40,44,45,55–59 and maximizing the loglikelihood l(a) =

K ∑

Nk log Pk⋆ (a)

(9)

k=1

an appealing alternative. It has been argued that maximum likelihood outperforms least squares for realistic data set sizes, but asymptotically (for infinitely many photons) they are the same. 44,60 This latter claim is indeed correct if the model is correctly specified, i.e. the true ’data generating model’ is included in the assumed family of models. 61 Nevertheless, although mostly overlooked, misspecified models ought to be the norm and not the exception in science. Maximum likelihood estimation leads to minimization of the Kullback-Leibler divergence KL(a) =

K ∑

(

Pktrue

k=1

)

Pktrue , log Pk⋆ (a)

(10)

where Pktrue are the true probabilities, 62 a procedure which does typically not provide asymptotically unbiased parameter estimates if the model is incorrect. Neither can it be assumed that least squares will provide exactly the same estimates. We note (study not shown) that maximum likelihood generally produced less biased results when the model was misspecified. This further strengthens the case for maximum likelihood and it is safe to say that it is preferable over least squares. When comparing different models, we use estimated mean and standard deviation for the time scale distributions g (not for the lifetime distributions), computed as M = ⟨τ ⟩ and 8 ACS Paragon Plus Environment

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S = (⟨τ 2 ⟩ − ⟨τ ⟩2 )1/2 (the notation is chosen to keep it distinct from the parameter notation for the different models). These parameters M and S are derived from the parameter vector a, and due to the invariance property they too are maximum likelihood estimates. 63 Modeling and estimation are implemented in Matlab R2013b (version 8.2.0.701, Mathworks, Natick, MA, US). In the coming sections, we present the four models used in the study.

The mono-exponential model The mono-exponential model has already been introduced in Eq. (2) and has pdf (

)

1 t , fME (t; τ0 ) = exp − τ0 τ0

(11)

where τ0 is the (only) characteristic time scale. Note that this corresponds to the time scale distribution gME (τ ; τ0 ) = δ(τ − τ0 ) (a Dirac delta). The cdf is (

)

t FME (t; τ0 ) = 1 − exp − . τ0

(12)

Of course, mean and standard deviation are MME = τ0 and SME = 0.

The stretched exponential model The stretched exponential model is frequently used in physics on both theoretical and empirical grounds, e.g. for modeling capacitor discharge, 64 dielectric relaxation, 65 optical Kerr effects, 66 and intermolecular energy transfer. 67 Although its interpretation is debated, one possible motivation for the model would be that of random multiplicative processes. 68 In the probabilistic form of the stretched exponential law (the original decay curve attributed to R.



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Kohlrausch looks somewhat different), the lifetimes follow a Weibull distribution with pdf β fSE (t; τ0 , β) = τ0

(

t τ0

)β−1

(

(

t exp − τ0

)β )

.

(13)

For 0 < β ≤ 1 this is a monotonically decreasing function of t, and by a theorem attributed to Bernstein, 69 all monotonically decreasing functions can be written as superpositions of exponentials. Consequently, the stretched exponential can be interpreted as an integral of the type in Eq. (4), albeit that the corresponding time scale distribution gSE (τ ; τ0 , β) is generally not available in closed form. However, the first moments can be acquired from the first moments of fSE (t; τ0 , β) by the moment relations ⟨t⟩ = ⟨τ ⟩ and ⟨t2 ⟩ = 2⟨τ 2 ⟩, and one obtains MSE = τ (1/β)Γ(1/β) and SSE = τ0 ((1/β)Γ(2/β) − (1/β 2 )Γ(1/β)2 )1/2 . The cdf of the lifetimes is simply

(

(

t FSE (t; τ0 , β) = 1 − exp − τ0

)β )

.

(14)

The lognormal model The lognormal model has been used repeatedly in fluorescence lifetime analysis. 4,6,7 The lognormal distribution no doubt has an appeal due to its wide applicability across the sciences 70 and due to it being the multiplicative analogue of the normal distribution. It is based on directly modeling the distribution of time scales by (

)

(log τ − µ)2 exp − , gLN (τ ; µ, σ) = 2σ 2 τ σ 2π 1 √

(15)

for µ and σ being the mean and standard deviation of the distribution of log(τ ). The corresponding lifetime pdf is numerically computed as ∫

fLN (t; µ, σ) =

0

∞

1 t gLN (τ ; µ, σ) e− τ dτ. τ


(16)

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By integration with respect to t under the integral sign, the cdf is numerically computed as

FLN (t; µ, σ) = 1 −

∫

∞ 0

(

gLN (τ ; µ, σ) exp −

)

t dτ. τ

(17)

Mean and standard deviation are computed by MLN = exp (µ + σ 2 /2) and SLN = (exp (σ 2 ) − 1)1/2 exp (µ + σ 2 /2). As σ approaches zero, the model collapses to the exponential distribution.

The inverse gamma model The inverse gamma model has been used in various flavours in fluorescence lifetime analysis 71–74 and other fields as well, e.g. for modeling signal decay in nuclear magnetic resonance. 75 The appeal of using the inverse gamma pdf (

)

β β α −α−1 gIG (τ ; α, β) = τ exp − , Γ(α) τ

(18)

as the distribution of time scales (equivalent to assuming a gamma pdf for the rate constants) is that the integral in Eq. (4) becomes analytically tractable and the corresponding lifetime distribution is a Pareto Type II or Lomax distribution 76 with pdf α fIG (t; α, β) = β and cdf

(

β β+t

(

)α+1

β FIG (t; α, β) = 1 − β+t

(19)

)α

.

(20)

Some authors have elaborated on possible physical underpinnings of this model in fluorescence lifetimes 71 and otherwise. 77,78 A peculiarity of the inverse gamma distribution is that the m:th moment ⟨τ m ⟩ is only defined, i.e. non-divergent, for α > m. Hence, mean and standard deviation are computed by MIG = β/(α − 1) (but finite only for α > 1) and



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√ SIG = β/((α − 1) α − 2) (but finite only for α > 2). As a consequence of the moment relations ⟨t⟩ = ⟨τ ⟩ and ⟨t2 ⟩ = 2⟨τ 2 ⟩, diverging moments of the time scale distribution will cause the corresponding moments of the lifetime distribution to diverge as well. This is a well-known feature of power-law type decays in general, with infinite means in empirical data having been reported 79–81 although we will not elaborate on their interpretation.

Results and discussion Simulation study Simulated data sets were generated in Matlab R2013b (version 8.2.0.701, Mathworks, Natick, MA, US) using all four models for data generation. In all cases, virtual photon counting was performed using 1000 channels in the range 0-100 ns, replicating conditions similar to those of the experimental data studied further on. No baseline (’dark count’) was used in the simulation study. First, a comparison of computational workload between the models were performed. The estimated parameter values are immaterial here; the purpose is only to assess the algorithmic speed under varied and realistic conditions. Random samples from the continuous lifetime distribution models are generated and the data are binned into histograms. Combinations of stretched exponential, lognormal, and inverse gamma model components, forming 1-, 2-, or 3-component mixtures, are randomly selected, with mean and standard deviation selected randomly in the range 0-10 ns. Moreover, the number of collected photons is randomly selected in the range 103 − 107 . We argue that the randomization of all parameters in the simulation in this fashion provides for a fair comparison of computational speed across models. Maximum likelihood estimation is performed using all four models and additionally with the multi-exponential, since that is a more natural competitor to the continuous models than the mono-exponential. The average execution time is calculated for 1000 random data sets, see Fig. 3. Naturally, the (mono- and multi-)exponential is the overall fastest 12 ACS Paragon Plus Environment

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model by virtue of its simple form and small parameter set. The stretched exponential and the inverse gamma models are comparable in terms of speed, and note in particular that switching from a multi-exponential to anyone of these models can reduce processing time. Not surprisingly, the lognormal model that requires numerical integration is by far the slowest, and more than two orders of magnitude slower than its competitors. It is worth noting that the increase in execution time as the number of components of the multiexponential model increases is underestimated, because the measured times are based on a single random initialization of parameter values in the likelihood maximization. In practice the number of random initializations would have to increase as model complexity increases to ensure reliable convergence. Second, an asymptotic study, i.e. with infinitely many photons, was performed of model fit and the obtained parameter estimates. With infinitely many collected photons, the theoretical channel probabilities Pk for each of the models is used directly as data (the changed scaling when replacing counts by normalized probabilities does not affect maximum likelihood estimation). It should be noted that already for data set sizes of about 105 photons, the histograms rather closely resemble these ideal curves, so whereas this part of the study is indeed relevant also for realistic data set sizes, finite-sample bias is eliminated, focusing on the asymptotic bias of the estimated values of the parameters that is an effect of model misspecification alone. A small-sample study, while valuable in its own right, would be less informative in this regard. Using mean value M = 10 ns in all cases, and standard deviation S varying in the range 0, 0.25, 0.5, ..., 9.75, 10 (S = 0 corresponding to the exponential distribution, regardless of model), three series of data sets from the stretched exponential, lognormal, and inverse gamma models were generated. Parameter estimation was then performed, using all models on all data sets to systematically investigate how the difference between models influences parameter estimation. The results are shown for data generated from the stretched exponential, see Fig. 4, the lognormal, see Fig. 5, and the inverse gamma, see Fig. 6. If the model is correctly specified, the asymptotic bias is zero and the parameter



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estimates are correct. However, there is an exception in practice for S = 0, since the lognormal and the inverse gamma are not numerically well-behaved for very small S which leads to overestimation of that parameter. The stretched exponential does not suffer from the same problem. When the model is not correctly specified though, estimated parameter values start to deviate increasingly from the true values as S increases (after all, this is natural since all models approach the mono-exponential for small S and thus are quite alike in that regime). A general trend is that with respect to both M and S, the inverse gamma model gives the largest parameter estimates, followed by the lognormal, and last the stretched exponential (the place of the mono-exponential in this order varies a bit). One important point is that when the true model is the stretched exponential, a large enough true S can lead to diverging estimated S and even diverging estimated M for the inverse gamma model. Thus, if the model is misspecified, a finite moment can be mistaken for an infinite one. One should therefore probably be quite careful in interpreting an infinite moment in physical terms.

Experimental data For each of the GQD and fluorescein lifetime data sets, all four models, the mono-exponential, the stretched exponential, the lognormal distribution, and the inverse gamma distribution models were fitted, and the estimated values of mean and standard deviation were studied. In Fig. 7, the results for fluorescein are presented. In this case, the near-exponential lifetime distribution makes it relatively unimportant what model is chosen. The average value of the ˆ = 4.14 ns for all four models) are very close to each other. 100 estimated mean lifetimes (⟨M⟩ The stretched exponential consistently collapses into the mono-exponential by yielding βˆ = 1 and hence Sˆ = 0. This is as such not proof of a perfectly exponential distribution though, especially not considering the results of the lognormal distribution and the inverse gamma distribution models, where the standard deviation is estimated to non-zero in most cases. However, there are two natural explanations for this behavior. First, the lognormal and the inverse gamma models are not numerically well-behaved for very small S which leads 14 ACS Paragon Plus Environment

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to overestimation of that parameter, as discussed in the simulation study. The stretched exponential does not suffer from the same problem. Second, small random deviations in the data sets will yield small deviations in the estimated parameters, which will yield an apparent spread (S > 0) even if this does not reflect the truth. Nevertheless, it can be of some importance to avoid overinterpretation even in this simple case by assessing different models. In Fig. 8, the results for GQDs are presented and the impact of model choice is now substantially more pronounced. Noticeably, this is so even though the goodness of fit (as measured by the loglikelihood value) is virtually identical for all models except for the monoexponential (although the lognormal provided the best fit). Both in terms of mean (average ˆ ME ⟩ = 6.58 ns, ⟨M ˆ SE ⟩ = 6.71 ns, ⟨M ˆ LN ⟩ = 6.88 ns, and ⟨M ˆ IG ⟩ = 6.92 estimated values ⟨M ˆSE ⟩ = 2.07 ns, ⟨S ˆLN ⟩ = 3.06 ns, and ns) and standard deviation (average estimated values ⟨S ÎG ⟩ = 3.30 ns), the estimated values differ quite noticeably, and by more than 50 % for ⟨S the standard deviation. Interestingly, we see that the estimated values for both mean and standard deviation follow the same pattern as in the simulation study, namely that among the three continuous models, the stretched exponential always yields the smallest mean and smallest standard deviation, and the inverse gamma always yields the largest mean and largest standard deviation. In Fig. 9, a GQD lifetime data set and the corresponding model fits are shown.

Conclusion We have studied four different models for fluorescence lifetime analysis using time-correlated single photon counting. In a simulation study, we compared computational speed and studied asymptotic bias in estimated parameters when the model was misspecified, i.e. when the true data generating model was not in the family of assumed models. It was found that the difference in estimated values of mean and standard deviation for different models could vary considerably and more so for strongly non-exponential data. Also in terms of computational



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speed, both the stretched exponential and the inverse gamma distribution model are worthy competitors of the mono- and multi-exponential model. Hence they may be preferable depending on the context and intended interpretation of the model. In an experimental study, we compare different models for analysis of graphene quantum dots and a standard fluorescein dye. We conclude that for near-exponential lifetime distributions, the models yield closely resembling results, even though some care in model choice must still be taken. Noticeably, this is so even though the goodness of fit (as measured by the loglikelihood value) is virtually identical for all models except for the mono-exponential (although the lognormal provided the best fit). However, for strongly non-exponential behavior as in the case of the graphene quantum dots, the models yield quite different estimates of mean and standard deviation. Consequently, model choice becomes crucial, and it should be preferable to try different models on the same data set in order to more thoroughly assess the impact of model choice. In summary, all models come with pros and cons. The mono-exponential is simple and fast, and provides surprisingly accurate estimates of the mean time scale even when it is far from the correct model. The stretched exponential is, as opposed to the lognormal and inverse gamma, numerically well-behaved in the limit as it collapses to a mono-exponential, and can therefore yield more reliable estimates of extremely small deviations from exponentiality. A preferential property of its competitors, on the other hand, is that the distribution of time scales can be expressed explicitly and in closed form. From an interpretation standpoint, the inverse gamma and the lognormal seems like the preferred choices. From a computational speed and ease of implementation standpoint, the inverse gamma comes out as the preferred of the two. However, ultimately, it is the goodness of fit as well as the physical underpinnings and the interpretation of the fit that should influence model choice.


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Acknowledgements This work was performed in part at the UniSA node of the Australian National Fabrication Facility, a company established under the National Collaborative Research Infrastructure Strategy to provide nano- and micro-fabrication facilities for Australias researchers.

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Figure 1: Sample characterization using AFM and TEM. AFM has been used to characterize height. In (a), an example of raw AFM data is shown, together with an example of a line along which a height profile was extracted. In (b), the height distribution as estimated from 10 line profiles is shown. The skew towards low value suggest that aggregation is limited. In (c), the height profile following the line in (a) is shown. TEM has been used to characterize lateral size and diameter. In (d), an example of raw TEM data is shown, together with a high resolution inset of the fringe pattern arising from the crystal structure. In (e), the diameter distribution as manually estimated from n = 368 particles is shown (d = 8.3 ± 2.9 nm). Also, fits using a lognormal (blue) and an inverse gamma (yellow) distribution model are shown (this distribution being correlated to the distribution of lifetimes, a topic of future work). In (f), the intensity profile of the fringe pattern along the line indicated in (d) is shown. The periods are 0.21 nm, in accordance with what is expected from graphene.


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Figure 4: Estimation results when the true model is stretched exponential. In (a), mean M is shown and in (b), standard deviation S is shown. The best attainable values for mono-exponential (red, only M), stretched exponential (green), lognormal (blue), and inverse gamma (yellow) are shown. The true values, i.e. the lines M = 10 ns and S = S, are shown in black. Note that in this case, the inverse gamma model gives an extremely large bias in parameter estimates up to the point when the first and second moment diverges and hence, values of M and S are infinite.


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Figure 5: Estimation results when the true model is lognormal. In (a), mean M is shown and in (b), standard deviation S is shown. The best attainable values for mono-exponential (red, only M, partially occluded), stretched exponential (green), lognormal (blue), and inverse gamma (yellow) are shown. The true values, i.e. the lines M = 10 ns and S = S, are shown in black.

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Figure 6: Estimation results when the true model is inverse gamma. In (a), mean M is shown and in (b), standard deviation S is shown. The best attainable values for mono-exponential (red, only M), stretched exponential (green), lognormal (blue), and inverse gamma (yellow) are shown. The true values, i.e. the lines M = 10 ns and S = S, are shown in black. 29 ACS Paragon Plus Environment


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Figure 7: Experimental study of fluorescein. In (a), estimated means and in (b), estimated standard deviations are shown for 100 repeated lifetime measurements of fluorescein at emission wavelength λ = 517 nm. Results are shown for the mono-exponential (red, occluded but coinciding perfectly with the stretched exponential), stretched exponential (green), lognormal (blue), and inverse gamma (yellow).


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Figure 8: Experimental study of GQD. In (a), estimated means and in (b), estimated standard deviations are shown for 100 repeated lifetime measurements of GQD at emission wavelength λ = 430 nm. Results are shown for the mono-exponential (red), stretched exponential (green), lognormal (blue), and inverse gamma (yellow).



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Figure 9: Examples of data analysis results in-depth. In (a), lifetime data sets (black) with the corresponding model fits for mono-exponential (red), stretched exponential (green), lognormal (blue, occluded by inverse gamma), and inverse gamma (yellow) for fluorescein (lower) and GQD (upper). In (b), the estimated mean lifetimes for mono-exponential and stretched exponential and the estimated time scale distributions for lognormal and inverse gamma are shown for only GQD.


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