Are super-exponential luminescence decays possible?

Chemical Physics 445 (2014) 14–20

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Chemical Physics journal homepage: www.elsevier.com/locate/chemphys

Are super-exponential luminescence decays possible? Tiago Palmeira, Mário N. Berberan-Santos ⇑ CQFM – Centro de Química-Física Molecular and IN – Institute of Nanoscience and Nanotechnology, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal

a r t i c l e

i n f o

Article history: Received 29 August 2014 In final form 7 October 2014 Available online 22 October 2014 Keywords: Luminescence decay Stretched exponential Relaxation kinetics Fluorescence Phosphorescence Triplet–triplet absorption

a b s t r a c t Luminescence decay functions describe the time dependence of radiation emitted by a sample after excitation. An overview of the mathematical aspects and systematics of luminescence decays is presented. In particular, super-exponential (faster-than-exponential) decays are defined and the possibility of their observation in single species physicochemical systems discussed. It is shown that this type of behavior can be both spontaneous and induced (by acting upon the system in real time). Spontaneous super-exponentiality is identified for the first time in experimental decays, these being phosphorescence decays affected by triplet–triplet absorption. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction

is always the case for completely monotonic functions [4], i.e., functions for which the nth-order derivatives I(n)(t) obey

A luminescence decay function, I(t), is the function describing the time dependence of the intensity of radiation spontaneously emitted by a previously excited sample at a given wavelength. This emission may result from a single species (containing a luminophore) or from several species. function is a dI A strict decay  monotonically decaying function dt < 0 for all t with a finite R1  light sum  ( 0 IðtÞdt, a time constant proportional to the total number of light quanta emitted) and results from a single species. For convenience, and without loss of generality, the decay function is normalized at t = 0, I(0) = 1. The decay function can in principle be related to a model describing the luminescence mechanism and respective dynamics, but remains a valuable tool if this model is not available. The luminescence decay function can be written in the following form, see Ref. [1, pp. 308–312]:

IðtÞ ¼

Z

1

gðkÞekt dk:

This relation is generally valid as I(t) always has an inverse Laplace transform, g(k). The function g(k), also called the eigenvalue specR1 trum, is normalized, as I(0) = 1 implies 0 gðkÞdk ¼ 1. In many situations the function g(k) is nonnegative for all k > 0, and g(k) can be regarded as a distribution of rate constants - strictly, a probability density function (pdf) [1-3]. It was previously shown [2] that this

E-mail address: [email protected] (M.N. Berberan-Santos). http://dx.doi.org/10.1016/j.chemphys.2014.10.011 0301-0104/Ó 2014 Elsevier B.V. All rights reserved.

ð2Þ

However, in a few cases the decay function does not comply with this definition and the respective g(k) also takes negative values [2,3]. For the discussion of the time behavior of decay functions a second formal approach is useful [2,3,5]. Let us consider the following defining equation

wðtÞ ¼ 

d ln IðtÞ ; dt

ð3Þ

where w(t) is the decay rate coefficient, with a possible time dependence. For a monotonic decay function, w(t) > 0 for all t. Eq. (3) implies that the decay can be written as

 Z t  IðtÞ ¼ exp  wðuÞdu :

ð4Þ

0

ð1Þ

0

⇑ Corresponding author.

ð1Þn IðnÞ ðtÞ > 0 ðn ¼ 0; 1; 2; . . .Þ:

Using Eq. (1), the time-dependent rate coefficient becomes

R1 Z 1 k gðkÞekt dk wðtÞ ¼ R0 1 ¼ khðk; tÞdk; gðkÞekt dk 0 0

ð5Þ

where h(k, t) is a new time-dependent distribution function that remains normalized for all times [2,3],

hðk; tÞ ¼ R 1 0

gðkÞekt : gðkÞekt dk

ð6Þ

T. Palmeira, M.N. Berberan-Santos / Chemical Physics 445 (2014) 14–20

15

Fig. 1. Classification of decay functions. The most general class (larger circle) corresponds to all continuous and positive functions that tend asymptotically to 0 when t ? 1. R1 Stricto sensu decay functions are monotonic, however only those with a finite «light sum» 0 IðtÞdt can have physical meaning.

It is seen that if g(k) is a pdf, then h(k, t) is also a pdf. In some special cases, such as the stretched exponential decay function [5], the corresponding distribution g(k), which is a one-sided Lévy stable distribution Lb,1(k) [6], does not have finite moments of any order. Eq. (5) thus implies that w(0) is infinite (see Fig. 2), as also follows from Eq. (3) in those cases. This singularity occurs only for t = 0, owing to the exponential factor in Eq. (6) [5]. 2. Exponential, sub-exponential, super-exponential, and unimodal decays The time-dependent rate coefficient w(t) can in principle exhibit a complex time dependence, but for monotonic decays there are only three important cases [2,3,5]: exponential decay, when

w(t) is constant; sub-exponential decay, when w(t) decreases with time; and super-exponential decay, when w(t) increases with time [7]. Sub-exponential and super-exponential decays are also called slower-than-exponential and faster-than-exponential decays, respectively (see nevertheless Ref. [5]with respect to the stretched exponential ambiguity). The possible types of decay function are shown as a Euler diagram [8] in Fig. 1. It follows from Eqs. (5) and (6) that [2,3]

"Z Z 1 2 # 1 dw 2 ¼ r2 ðtÞ; ¼ k hðk; tÞdk  khðk; tÞdk dt 0 0

Fig. 2. Schematic loci of decay function Eq. (10), as a function of parameter k0 (with k1 = a = 1).

ð7Þ

16

T. Palmeira, M.N. Berberan-Santos / Chemical Physics 445 (2014) 14–20

Fig. 3. Schematic loci of decay function Eq. (12), as a function of parameter b (b > 0, with k0 = k1 = 1).

where r2(t) is the variance of h(k, t) if h(k, t) is a pdf (i.e., nonnegative for all k). In this way, a decay function represented by a distribution of rate constants implies either an exponential (g(k) = d(k-k0), which has zero variance) or a subexponential decay (g(k) with nonzero variance). However, not all sub-exponential decay functions correspond to a distribution of rate constants, only those that are completely monotonic [2]. On the other hand, a super-exponential decay never corresponds to a distribution of rate constants, i.e., its inverse Laplace transform g(k) always takes negative values at some points.

Physically relevant monotonic decay functions can be exponential, super-exponential or sub-exponential. In the sub-exponential case, if complete monotonicity is verified, then the decay function corresponds to a pdf of rate constants, g(k). However, not all R1 R1 pdf ensure a finite  light sum  0 IðtÞdt ¼ 0 gðkÞ dk, e.g. g(k) = k exp(k) leads to a divergent integral, hence the fat-tailed 1/(1 + t) is not a bona fide decay function. In Fig. 1, the boundary of the super-exponential set osculates the boundary of the broader monotonic decay function set, as in some cases the  decay function  can be made to go from super-exponential to non-monotonic (decay function with a

Fig. 4. Schematic loci of decay function Eq. (13), as a function of parameter a (with k > 1).

T. Palmeira, M.N. Berberan-Santos / Chemical Physics 445 (2014) 14–20

maximum) by the continuous variation of a single parameter, as shown in Fig. 2 for Eq. (10) (see also Fig. 4). As mentioned, for a few sub-exponential decay functions, such as the stretched exponential [5], the corresponding distribution g(k) does not have finite moments and w’(0) is equal to 1 (see Fig. 6). The question to be answered in this work is the following: Can a super-exponential luminescence decay be observed for a single luminescent species? To the best of the authors’ knowledge, no single species (or otherwise) super-exponential decays have been identified as such in the literature. Possible super-exponential decay functions are the compressed exponential [5],

h i b IðtÞ ¼ exp ðk0 tÞ ;

17

luminophores proceeds in parallel, the fastest decaying configurations disappear first, on the average. As mentioned, this implies a sub-exponential decay. On the other hand, super-exponentiality means that the overall decay proceeds at an increasingly faster pace. One can conceive several possible ways of increasing the decay rate, by changing the conditions during the decay. Suppose, for instance, that

ð8Þ

with k0 > 0 and b > 1, and exponential of exponential type functions, suitably normalized, such as [5]

IðtÞ ¼ exp ½1  exp ðk0 t Þ;

ð9Þ

and

IðtÞ ¼ expfk1 t þ a½1  expðk0 tÞg;

ð10Þ

with k0, k1 and a > 0. The decay function Eq. (10) is asymptotically exponential. For k0 6 k1 the decay is super-exponential. For k0 > k1 the function has a maximum and ceases to be monotonic, as depicted in Fig. 2. Note that

IðtÞ ¼ exp fk1 t  a½1  exp ðk0 tÞg;

ð11Þ

which differs from Eq. (10) only by the sign of coefficient a, is always sub-exponential. Decay functions of this type occur in several physicochemical systems, see e.g. Ref. [1, p. 155–157]. Similarly, many instances (e.g. FRET in 1D, 2D and 3D rigid media) are known for which Eq. (8) holds with b < 1 [2,3,5], again with a multiplicative exponential, rendering the whole sub-exponential decay function asymptotically exponential,

h i b IðtÞ ¼ exp k1 t  ðk0 t Þ :

Fig. 5. Time evolution of decay function Eq. (13) as a function of parameter a (a) and respective rate coefficients w (b). Values of a are shown next to each curve. Rate constant k = 1.5. The decay curves, all asymptotically exponential (or exponential), correspond to: sub-exponential decay (a = 0.2), exponential decay (a = 1), superexponential decay (a = 3), and non-monotonic decay (a = 10), compare Fig. 4.

ð12Þ

The loci of decay function Eq. (12), as a function of parameter b (with k0 = k1 = 1) are shown in Fig. 3. For b > 1 (no experimental cases being known) the decay function is super-exponential. Perhaps surprisingly, still another possibility for a superexponential decay function is just an algebraic sum of two exponentials,

IðtÞ ¼ aet þ ð1  aÞekt ;

ð13Þ

with a > 0 and k > 1. This decay function is super-exponential if 1 < a < k/(k1), see Fig. 4. For a > k/(k1) the function is unimodal (has a maximum) and displays what is known as a rise time. For a < 1 the decay function is an ordinary sum of two exponentials with positive amplitudes and is sub-exponential. A few representative cases are plotted in Figs. 5 and 6. A negative curvature and a decaying rate coefficient characterize sub-exponential decays, whereas a positive curvature and an increasing rate coefficient define super-exponential decays. For unimodal decay functions, the rate coefficient is negative before the maximum (Fig. 5). From a physical perspective, the decay of the luminescence emitted by a single species, when proportional to its concentration, is easily understood in terms of Eq. (1): even if the species has a characteristic lifetime, and the decay is ideally exponential, several conformations, microenvironments, or quencher spatial configurations (e.g. in a rigid medium) around each luminophore, will give rise to a distribution of decay rates. If all quenching processes or configurations exist from the beginning, and the decay of all

Fig. 6. Time evolution of several luminescence decay functions, with all constants equal to 1 unless otherwise noted (a), and respective rate coefficients (b); (1) Exponential decay; (2) Sub-exponential decay, asymptotically exponential, Eq. (11); (3) Super-exponential decay, asymptotically exponential, Eq. (10); (4) Compressed exponential, Eq. (8), with b = 2; (5) Stretched exponential, Eq. (8), with b = ½.

18

T. Palmeira, M.N. Berberan-Santos / Chemical Physics 445 (2014) 14–20

Residuals

0.006

(c)

0.003 0.000 -0.003 0

20

40

60

80

100

120

140

160

180

Residuals

0.0015

200

(b)

0.0000 -0.0015

Phosphorescence Intensity

0

20

40

60

80

100

120

140

160

180

1

200

(a)

0.1

0.01 0

20

40

60

80

100 time/s

120

140

160

180

200

Fig. 7. A super-exponential decay of the phosphorescence of perdeuterated coronene in an ethanol glass at 77 K (excitation and emission wavelengths were 340 nm and 560 nm, respectively) (a). Also shown are the residuals plots for fits according to Eq. (24) (b1 = 0.638, b1 = 1.81, b2 = 3.42, b2 = 1.37, b3 = 94.9, b3 = 0.073) (b), and to Eq. (22) (a = 0.74) (c). sP = 32 s.

Residuals

0.005

(c)

0.000 -0.005 0

20

40

60

80

100

120

140

160

180

Residuals

0.002

(b)

0.000 -0.002 0

Phosphorescence Intensity

200

20

40

60

80

100

120

140

160

180

1

200

(a)

0.1

0.01 0

20

40

60

80

100 time/s

120

140

160

180

200

Fig. 8. A super-exponential decay of the phosphorescence of perdeuterated coronene in a Zeonex polymer film (excitation and emission wavelengths were 340 nm and 560 nm, respectively), at 77 K (a). Also shown are the residuals plots for fits according to Eq. (24) (b1 = 0.734, b1 = 1.91, b2 = 4.08, b2 = 3.18, b3 = 12.3, b3 = 2.73) (b), and to Eq. (22) (a = 0.88) (c). sP = 30 s.

luminescence is quenched by a dynamic collisional process in a microscopically well-mixed system (no transient effects). A continuous addition of quenchers to the medium during the excited state lifetime, thus continuously increasing the quencher concentration, will also continuously increase the rate coefficient. A stream of gas

with an increasing quencher (e.g. O2) composition flowing through a cell containing a luminescent porous solid is another example. Using the Stern–Volmer relation [1],

1

s

¼

1

s0

þ kq ½Q ;

ð14Þ

19

T. Palmeira, M.N. Berberan-Santos / Chemical Physics 445 (2014) 14–20

1.4

1.0 0.8

0.8

12500

0.6 0.4

10000

0.2 0.0 275

300

325

350

Wavelength/nm

375

7500

0.6

-1 -1

Normalized Intensity

1.2

εΤ/M cm

Normalised Absorption

15000 1.0

5000

0.4

2500

0.2 0.0 500

520

540

560

580

600

0 620

Wavelength/nm Fig. 9. Phosphorescence spectrum of perdeuterated coronene in a Zeonex film at 77 K. The T–T absorption spectrum in the same spectral region is also shown, displayed as a dotted line (absorption coefficients taken from Ref. [13]). The inset shows the absorption spectrum of perdeuterated coronene in the same film.

where s is the lifetime, s0 is the lifetime in the absence of quencher Q (s0 = 1/k0), and [Q] is the time-independent quencher concentration, the decay function is exponential,

3. Possibility of spontaneous (intrinsic) super-exponential decays

  t IðtÞ ¼ exp   kq ½Qt :

The above examples seem a bit far-fetched: in the first case the system is subject to changes, in real time, by an active procedure, whereas in the second case the intrinsic decay is still exponential, intensity changes occurring only outside the system. Finally, in the third example, there are two species, one of them being generated during the observation period. What about a genuine single component system that evolves without any external interference, after excitation? Is spontaneous (intrinsic) super-exponentiality possible? The answer is yes. The characteristic signature of super-exponential decays (positive curvature in a semi-log plot, see Eq. (3) and Fig. 6) in single species systems was recently observed by us [9,10]. Another super-exponential decay (identified as such with hindsight) was reported long ago [11]. In both cases, the super-exponential behavior occurs owing to a time-dependent internal filter (reabsorption) mechanism: Phosphorescence photons are emitted by molecules in the triplet state. If the phosphorescence spectrum overlaps the triplet–triplet absorption spectrum of the same substance, and if the triplet concentration is high, a significant part of these photons is reabsorbed [9-11]. However, the triplet concentration continuously decreases with time. In this way, the reabsorption probability also decreases continuously and approaches zero for sufficiently long times. As a consequence, the phosphorescence decay asymptotically approaches an exponential function governed by the intrinsic phosphorescence lifetime, sP. Qualitatively, the time evolution corresponds to Eq. (18), rewritten as [10],

s0

ð15Þ

If, however, the quencher concentration increases linearly with time, [Q] = at, then the decay becomes super-exponential,

  t 2 IðtÞ ¼ exp   bt ;

s0

ð16Þ

with b = a kq, compare Eq. (8). This may seem difficult to put in practice for fast fluorescence decays; however for long phosphorescence decays such an experiment poses no special problems. Another way of progressively increasing the measured decay rate is as follows. Suppose that a filter, whose transmission rises with time, is placed between the luminescent sample and the detector. The continuous transmission increase will compensate, at least in part and for some time, for the intensity decrease owing to the intrinsic decay. For instance, if the transmittance T of the filter evolves as

TðtÞ ¼ exp ½b expðatÞ;

ð17Þ

i.e., starting from an initial value exp(b) and rising asymptotically to 1, an exponential decay I0(t) with lifetime s0 becomes, after passing through the filter (and after normalization),

  t IðtÞ ¼ exp  þ b½1  expðatÞ ;

s0

ð18Þ

which is precisely the form of super-exponential decay Eq. (10). The trick here is to start from a low intensity value (not apparent in the decay function owing to normalization). The algebraic sum of two exponentials, Eq. (13), allows a simple photophysical interpretation in terms of two-species kinetics, like a FRET donor–acceptor pair (see e.g. Ref. [1, pp. 235 and 239]) or the monomer-excimer scheme (see e.g. Ref. [1, pp. 163–167]). If both species emit at a common wavelength, the measured decay will be a weighted sum of the respective contributions, hence Eq. (13) is obtained, and super-exponentiality is possible under specific conditions. Another possibility is the selective observation of the decay of the excited-state product (e.g. FRET acceptor, excimer), provided a fraction is initially present (e.g. directly excited FRET acceptor, pre-formed excimer), Eq. (13) being again obtained.

IðhÞ ¼ expfh þ b½1  expðhÞg;

ð19Þ

with h = t/sP. For a quantitative description, a distribution f(b) of optical thicknesses, b, must be considered [10], and Eq. (19) is replaced by

IðhÞ ¼

Z

1

gðbÞ exp ½h  b exp ðhÞdb;

ð20Þ

0

with g(b) given by

gðbÞ ¼ R 1 0

f ðbÞ : f ðbÞeb db

ð21Þ

For the purpose of decay data analysis, specific distributions of optical thickness can be used. A simple super-exponential decay function that describes with very good accuracy experimental phosphorescence decays in the presence of excited-state absorption is [9,10]

20

IðhÞ ¼

T. Palmeira, M.N. Berberan-Santos / Chemical Physics 445 (2014) 14–20

1 ; a þ ð1  aÞeh

ð22Þ

where a is a dimensionless parameter accounting for reabsorption, 1 > a P 0. For very small a, i.e. negligible reabsorption, Eq. (22) reduces to an exponential decay. The same occurs for sufficiently long times irrespective of the value of a. This decay function results from an exponential distribution of optical thicknesses [10]. It is interesting to note that the phosphorescence decay of a species subject to triplet–triplet annihilation in fluid medium is very similar to Eq. (22), see e.g. [12],

IðhÞ ¼

1 ; a þ ð1 þ aÞeh

ð23Þ

where a = kTT C0 sP is an intrinsically nonnegative dimensionless constant, kTT being the effective bimolecular triplet–triplet annihilation rate constant and C0 the initial triplet concentration. Owing to the fact that parameter a is nonnegative in both equations, they cannot be interconverted, and Eq. (23) corresponds to a subexponential decay. Analysis of phosphorescence decays in the presence of excitedstate absorption can also be made using the discretized form of Eq. (20) [10],

IðhÞ ¼

X bi exp ½h  bi exp ðhÞ;

ð24Þ

i

P where the bi are numerical coefficients such that i bi ebi ¼ 1. Fits of Eqs. (22) and (24) to a representative experimental phosphorescence decay with strong reabsorption are shown in Figs. 7 and 8. The corresponding absorption and emission spectra are shown in Fig. 9. It is clear that the fit is quite satisfactory for both decay models, since the residuals are very small. However, Eq. (24) is somewhat better as the respective residuals plots are closer to white noise. The same conclusion is reached after analyzing a large set of decays [10]. 4. Conclusions An overview of the mathematical aspects and systematics of luminescence decays was presented. Super-exponential (faster-

than-exponential) decays were defined and the possibility of their observation in physicochemical systems discussed. It was shown that this type of behavior could be induced by acting upon the system in real time (i.e., during the decay). Spontaneous (intrinsic) super-exponentiality was identified for the first time in experimental decays. These were phosphorescence decays affected by triplet–triplet absorption (time-dependent inner filter effect). Conflict of interest There is no conflict of interest among authors. Acknowledgements This work was carried out within projects PTDC/QUI–QUI/ 123162/2010 and RECI/CTM-POL/0342/2012 (FCT, Portugal). TP was supported by a research grant from project PTDC/QUI–QUI/ 123162/2010 (FCT, Portugal). References [1] B. Valeur, M.N. Berberan-Santos, Molecular Fluorescence. Principles and Applications, second ed., Wiley-VCH, Weinheim, 2012. [2] M.N. Berberan-Santos, B. Valeur, J. Lumin. 126 (2007) 263. [3] M.N. Berberan-Santos, E.N. Bodunov, B. Valeur, in: M.N. Berberan-Santos (Ed.), Fluorescence of Supermolecules, Polymers and Nanosystems, Springer, Berlin, 2008, pp. 67–103. [4] D.V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946, p. 145. [5] M.N. Berberan-Santos, E.N. Bodunov, B. Valeur, Chem. Phys. 315 (2005) 171. [6] J.-P. Bouchaud, A. Georges, Phys. Rep. 195 (1990) 127. [7] The same classification was used for the decay of the intensity of radiation with distance owing to scattering, see R.A. Shaw, A.B. Kostinski, D.D. Lanterman, J. Quant. Spectrosc. Radiat. Transfer 75 (2002) 13. [8] L. Euler, Lettres à une princesse d’Allemagne sur divers sujets de physique et de philosophie, Academie Impériale des Sciences, vol. 2, Saint-Petersburg, 1768. pp. 95-126 (available at archive.org). [9] T. Palmeira, M.N. Berberan-Santos, J. Math. Chem. 52 (2014) 2271. [10] T. Palmeira, M.N. Berberan-Santos, J. Lumin. (2014), http://dx.doi.org/10.1016/ j.jlumin.2014.10.054. in press. [11] Yu.V. Naboikin, L.A. Ogurtsova, I.D. Fil, Opt. Spectrosc. 20 (1966) 27. [12] M.N. Berberan-Santos, E.N. Bodunov, B. Valeur, Chem. Phys. 317 (2005) 57. [13] W.R. Dawson, J.L. Kropp, J. Phys. Chem. 73 (1969) 693.