- Email: [email protected]
Phasor plots of luminescence decay functions
Accepted Manuscript Phasor plots of luminescence decay functions Mário N. Berberan-Santos PII: DOI: Reference:
S0301-0104(15)00009-9 http://dx.doi.org/10.1016/j.chemphys.2015.01.007 CHEMPH 9245
To appear in:
Chemical Physics
Received Date: Accepted Date:
5 November 2014 7 January 2015
Please cite this article as: M.N. Berberan-Santos, Phasor plots of luminescence decay functions, Chemical Physics (2015), doi: http://dx.doi.org/10.1016/j.chemphys.2015.01.007
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Phasor plots of luminescence decay functions
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 Phone: 351-218419254, E-mail: [email protected]
Abstract Luminescence decay functions describe the time dependence of the intensity of radiation emitted by electronically excited species. Decay phasor plots (plots of the Fourier sine transform vs. the Fourier cosine transform, for one or several angular frequencies) are being increasingly used in fluorescence, namely in lifetime imaging microscopy (FLIM). In this work, a detailed study of the sum of two exponentials decay function is carried out revealing that sub-exponential, super-exponential and unimodal decays have different phasor signatures. A generalization of the lever rule is obtained, and the existence of an outermost phasor curve corresponding to intermediate-like decays is demonstrated. A study of the behavior of more complex decay functions (sum of three exponentials, stretched and compressed exponentials, phosphorescence with reabsorption and triplet-triplet annihilation, fluorescence with quantum beats) allows concluding that a rich diversity of phasor plot patterns exists. In particular, superexponential decays can present complex shapes, spiraling at high frequencies. The concept of virtual phasor is also introduced.
1
1. Introduction 1.1 Luminescence decay functions A luminescence decay function, I(t), is the function describing the time dependence of the intensity of radiation spontaneously emitted at a given wavelength, by a previously excited sample. For convenience, and without loss of generality, the decay function is usually normalized at t = 0, I(0) = 1, and only nonnegative times are considered (t ≥ 0). The decay function can in principle be related to a model describing the luminescence mechanism and respective dynamics, but remains a valuable tool even if this model is not available. The luminescence decay function can be written in the following form [1]:
∞
I (t ) = ∫ g (k ) e− kt dk .
(1)
0
This relation is generally valid as I(t) always has an inverse Laplace transform, g(k). The function g(k), also called the eigenvalue spectrum, is normalized, as I(0) = 1 ∞
implies ∫ g (k ) dk = 1 . In many situations the function g(k) is nonnegative for all k > 0, 0
and g(k) can be regarded as a distribution of rate constants - strictly, a probability density function (pdf) [1,2]. It was previously shown [2] that this is always the case for completely monotonic functions, i.e., functions for which the nth-order derivatives I(n)(t) obey:
(−1)n I ( n ) (t ) > 0
(n = 0, 1, 2,….).
(2)
However, in some cases the decay function does not comply with this definition and the respective g(k) also takes negative values [2]. For the discussion of the time behavior of decay functions the following defining equation is useful [2,3]
w( t ) = −
d ln I (t ) , dt
(3)
2
where w(t) is the decay rate coefficient, with a possible time dependence. For a monotonic decay function, w(t) > 0 for all t. 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]: exponential decay, when w(t) is constant; subexponential decay, when w(t) decreases with time; and super-exponential decay, when w(t) increases with time. Most of the experimental decays belong to the first two categories. Super-exponential decays were recently discussed and experimentally identified in phosphorescence affected by triplet-triplet absorption [3,4]. With respect to non-monotonic decay functions, an important class is that of unimodal decay functions, i.e. functions with a single maximum, whose intensity rises initially, followed by passage through a maximum before decay proper. They may correspond to a w(t) steadily increasing with time, as with super-exponential decays, but now starting from w(0) < 0 [3]. This type of behavior is typical of, but not limited to, the luminescence of states or species not directly generated by excitation, like delayed fluorescence and FRET acceptor emission. Finally, a second class of non-monotonic decays is that of multimodal decay functions, i.e., functions with several maxima. This is the situation of decays displaying quantum beats, observed for a few molecules under specific conditions, like gaseous biacetyl and crystalline tetracene, and is probably the most complex of all physically relevant cases [5,6]. A decay function represented by a distribution of rate constants implies either an exponential (g(k) = δ(k-k0), which has zero variance) or a sub-exponential decay (g(k) with nonzero variance). On the other hand, super-exponential and non-monotonic decay functions never correspond to a distribution of rate constants, i.e., their inverse Laplace transform g(k) always take negative values at some points. A distribution of lifetimes h(τ) is also used to represent the decay function,
∞
I (t ) = ∫ h(τ ) e− t / τ dτ .
(4)
0
This function is related to the inverse Laplace transform g(k) by
h(τ ) =
1 g , τ τ 1
2
(5)
3
and therefore it may also take negative values at some points, as happens with g(k).
1.2 The “phasor approach” The cosine and sine Fourier transforms of I(t), G(ω) and S(ω), respectively, are defined as [1,7]
∞
∫ I (u ) cos(ω u)du , G (ω ) = 0 ∞ ∫0 I (u ) du
(6)
∞
∫ I (u ) sin(ω u)du , S (ω ) = 0 ∞ ∫0 I (u ) du
(7)
where ω is the angular frequency. Following Weber [8], the letter G is used instead of C (G is a convenient choice, as it avoids the concentration symbol while retaining graphic similarity with C). The denominators in Eqs. (6) and (7), which are not part of the general definition [9,10], imply a specific normalization for the decay function (it becomes a density [9], i.e., a function analogous to a one-sided probability density function), and were adopted in the phase-modulation (frequency domain) technique for convenience. In this way, each decay function, for a given frequency, is mapped onto a point inside the unit circle defined by G2 + S2 = 1 [9], which may be said to be the “phasor space”, see Fig. 1. Indeed, Eqs. (6) and (7) immediately imply that both |G| and |S| cannot exceed 1. It also follows from these equations that G(0) = 1 and S(0) = 0, while S(∞) = G(∞) = 0, see Fig. 1. Insertion of Eq. (4) into Eqs. (6) and (7) gives
∞
G (ω ) = ∫ f (τ ) 0
∞
S (ω ) = ∫ f (τ ) 0
1 dτ , 1 + (ωτ ) 2
ωτ dτ , 1 + (ωτ ) 2
(8)
(9)
4
Figure 1. The phasor space (white area) and the «universal» semicircle. Also shown are the (truly) universal points corresponding to all decay functions for zero and infinite frequencies. The «universal» semicircle, located in the first quadrant, defines the loci of all exponential decays. Other decay functions follow different paths (which can have complex shapes, as will be shown) between the two extreme points (ω = 0 and ω = ∞), when going from zero to infinite frequency.
where f(τ) is a new distribution of lifetimes (intensity-weighted, see Eq. (12) below),
f (τ ) =
τ h(τ ) ∞
.
(10)
∫ τ h(τ )dτ 0
Like h(τ), f(τ) can take in general both positive and negative values. For a given frequency, the (G, S) pair defines a point or, equivalently, a vector P = G e1 + S e 2 , called the phase vector or phasor. This vector is the basis of the phasor
approach to time-resolved luminescence (up to now, fluorescence) [1,7,11-28], which provides a simple graphical and model-independent portrait of a luminescent system. Besides luminophore identification (“fingerprinting”) in complex systems, processes such as quenching, solvent relaxation and energy transfer can be identified by characteristic trajectories in the plane (at a fixed frequency or using several frequencies). Owing to the limited computational requirements, linearity and robustness, the phasor approach is especially useful in fluorescence lifetime imaging
5
microscopy (FLIM) [15-18,22,25,26]. Application to measurements in solution (“single pixel data”) is nevertheless also of interest [13,14,23,24,27]. The precise location of the decay in the plane, defined by its phasor, is a function of frequency and decay characteristics. Single exponential decays lie on a so-called universal circle (in fact a semi-circle, for nonnegative frequencies), defined by
S = G(1 − G) , with 1 ≥ G ≥ 0 [1,7,11,16-18,23], see Fig. 1. Complex decays usually, but not always, fall inside it. The point corresponding to a multiexponential decay is located at an average distance from those of the components. In the case of a twoexponential decay with positive amplitudes, for instance, the corresponding point falls on a straight line connecting the phasors of the two components (“tie line”) [1,7,11,1618,23]. Analogously to the lever rule of thermodynamic phase diagrams, the fractional contribution of each of the two components to the total intensity is given by the length of the segment connecting the decay point (“average lifetime”) to the opposite component, divided by the length of the full segment uniting the two extreme points (components) [1,7,11,16-18,23]. When there are three or more components, the corresponding points define a polygon, with vertices located at or inside the circle, with the decay point lying in turn inside the polygon [1,7,16-18,23]. When the measurement technique used is frequency domain fluorimetry, based on sinusoidally modulated excitation, G and S are directly related to the two parameters obtained for each frequency, which are the modulation ratio, M, and the phase shift, Φ, by G = M cos Φ and by S = M sin Φ , hence tan Φ = S / G and M = P = G 2 + S 2 [8]. When the measurement technique used is time domain fluorimetry, the phasor must be computed from the measured decay according to Eqs. (6) and (7), at a conveniently chosen frequency (or set of frequencies). The decay is usually previously fitted with an empirical decay law, e.g. a sum of exponentials, in order to remove the effect of the finite width of the excitation impulse (instrument response function). When plotting luminescence decay functions (and not experimental data), numerical evaluation of their transforms is required, in general. However, if the decay is given by a sum of exponentials,
I (t ) = ∑ ai e− t /τ i ,
(11)
i
The function f(τ), Eq. (10), becomes 6
f (τ ) = ∑ fi δ (τ − τ i ) ,
(12)
i
where the fi are the fractional steady-state intensities,
fi =
aiτ i , ∑ aiτ i
(13)
i
and Eqs. (8) and (9) become
G (ω ) = ∑ f i i
S (ω ) = ∑ f i i
1
,
(14)
ωτ i . 1 + (ωτ i ) 2
(15)
1 + (ωτ i ) 2
It is worth noting that G(ω) and S(ω) are related to the real and imaginary parts of the complex Fourier transform F(ω),
∞
∫ I (t )e F (ω ) =
− iωt
dt = G (ω ) − iS (ω ) .
0 ∞
(16)
∫ I (t )dt 0
As follows from Eqs. (6) and (7), G(ω) and S(ω) are not independent. They are explicitly related by Hilbert transforms [9],
G (ω ) = −
2
∞
uS (u ) du , 2 − u2
π ∫ω 0
(17)
7
S (ω ) =
2ω
π
∞
G (u ) du . 2 − u2
∫ω 0
(18)
The purpose of the present work is to discuss the characteristic curves of a few representative luminescence decay functions, obtained by plotting the respective Fourier cosine and sine transform pairs (phasors) vs. frequency in the so-called polar or phasor plot. According to Eq. (16), these curves can also be viewed as plots of the conjugate of F(ω), F*(ω), in the complex plane [9,10] (in the context of control systems theory, this type of plot is known as the Nyquist plot [13,14], whereas in the dielectric relaxation field a similar plot is called the Cole-Cole plot [12,17].)
2. Results and Discussion 2.1 Sum of two exponentials The normalized decay function corresponding to a sum of two exponentials is
I (t ) = α e−t /τ1 + (1 − α )e−t /τ 2 ,
(19)
Without loss of generality, let us assume that τ1 > τ2. Then I(t) > 0 for all times implies α > 0. Eq. (19) can be rewritten as
I (θ ) = α e−θ + (1 − α )e−kθ ,
(20)
where θ = t/τ1 is a dimensionless (or reduced) time and k = τ1/τ2 is the lifetime ratio, with k > 1. Eq. (20) encompasses all types of time-dependent behavior defined above, excluding the multimodal: sub-exponential, exponential, super-exponential and unimodal, see Scheme 1 and Fig. 2 [3]. The decay is sub-exponential for 0 < α < l and both terms have positive pre-exponential factors. The transition from sub-exponential to super-exponential occurs at α = l (for α = l the decay is exponential), as one of the exponentials gains a negative pre-exponential factor. The transition from superexponential to unimodal, with the appearance of a maximum, occurs at α = k/(k-1).
8
Scheme 1
Figure 2. Decay curves corresponding to Eq. (20) with k = 1.5, all asymptotically exponential (or exponential): sub-exponential decay (α = 0.2), exponential decay (α = 1), super-exponential decay ( α = 3), and unimodal (α = 10), compare Scheme 1.
For a given frequency, the corresponding phasor P is related to the phasors of the two components (real or fictitious), P1 and P2, which lie in the universal circle. Using Eqs. (13) and (20), the generalized fractional steady-state intensities of these
components are:
f1 =
αk , 1 + α ( k − 1)
(21)
f2 =
1−α . 1 + α ( k − 1)
(22)
If α < 1 (sub-exponential decay) the lever rule applies, as mentioned above, and the phasor falls inside the circle, on the straight line (tie line) connecting τ1 and τ2, whose 9
slope is u(u2-k)/[u(k+1)] [15]. When α > 1 (super-exponential or unimodal decay) f1 > 1 and f2 < 0. In this case, the phasor also occurs on the same straight line, but now outside the circle, to the left of both τ1 and τ2, see Fig. 3. It can be shown that the ratio of the lengths of the segments |P-P2| and |P1-P2| is always f1, irrespective of α:
f1 =
P − P2 P1 − P2
.
(23)
The phasor plots of Eq. (20) shown in Figs. 3-8 are obtained from the respective cosine and sine Fourier transforms [15] that read, in the present notation:
G (u ) =
1 1−α 1 αk , + 2 1 + α (k − 1) 1 + u 1 + α ( k − 1) 1 + ( u / k )2
(24)
S (u ) =
1−α αk u u/k , + 2 1 + α ( k − 1) 1 + u 1 + α (k − 1) 1 + ( u / k )2
(25)
where u is a dimensionless frequency, u = ωτ1. From these relations it can be shown that the length of the segment uniting the loci of the two exponentials is
P1 (u ) − P2 (u ) =
u ( k − 1) (1 + u 2 )( k 2 + u 2 )
.
(26)
It follows from Eq. (26) that the maximum distance between the two phasors occurs for u = k and is equal to (k-1)/(k+1). In this situation S1 = S2 and the tie line is horizontal.
It also follows that
P( k ) − P2 ( k ) =
α k ( k − 1) . 1 + α ( k − 1) ( k + 1)
(27)
10
Figure 3. Extension of the «lever rule» to the situation where the phasor falls outside the universal circle (black semi-circle). The lifetime ratio is k = 2. A common and large α was used in both cases, α = 1000. For this reason, the blue line is practically equal to the outermost boundary of the phasor space, corresponding to phasors with α → ∞, P∞(u). The reduced angular frequency is (a) u = 0.75 and (b) u = 2.5. It is also seen that G∞ takes negative values at high frequencies, thus entering the second quadrant.
The maximum length is obtained for α → ∞, and is then equal to k/(k+1). In this situation, the phasor P( k ) = P∞ ( k ) has always G∞ = 0, see Fig. 4. The corresponding S∞ is
k / (1 + k ) .
11
Figure 4. Loci of the phasor P (blue dots) when u =
k and α → ∞, P∞
( k ) , for the values of k given
in the figure near the respective blue dots. The corresponding decay functions are given by Eq. (30). The phasors of the components (with lifetimes τ1 and τ2) are shown as black dots.
It can also be seen in Fig. 4 that the outermost boundary (blue line) shrinks and approaches the universal circle as k → ∞. From Eqs. (24) and (25) one obtains, for α → ∞ (blue lines), the parametric equations
G∞ (u ) =
k (k − u 2 ) k (k − u 2 ) = G1 (u ) , (1 + u 2 )(k 2 + u 2 ) k 2 + u2
(28)
S∞ (u ) =
k (k + 1) u k (k + 1) = 2 S1 (u ) . 2 2 2 (1 + u )(k + u ) k + u 2
(29)
For a given k, the decay function corresponding to the outermost boundary (α → ∞) ∞
can be obtained from Eq. (20) (after normalization, ∫ I (θ )dθ = 1 ) by finding the limit 0
when α → ∞. One gets
I ∞ (θ ) =
k e −θ − e − kθ ) , ( k −1
(30)
12
which is the characteristic time evolution of an «intermediate» A (unstable species whose initial concentration is zero, e.g. an excimer or a FRET acceptor), as depicted in kinetic Scheme 2:
1 k D → A → or k 1 D → A →
Schemes 2a and 2b
Species D can also decay by two or more competing channels, of which only one leads to species A, and the result will still be the same, provided the sum of the rate constants for the n channels is equal to 1 for Scheme 2a and equal to k for Scheme 2b. The most extreme case (k → 1) is obtained from Eq. (30) as I∞(θ) = θ e-θ.
(31)
The minimum value (α →∞) of the Fourier cosine and sine transforms can be shown to be
min ∞
G
1 k− k (k ) = − k + 1 k − 1
S∞min (k ) =
(
) ( k + 1) ( k + 1)
2
,
(32)
,
(33)
k k + k +1
2
the corresponding dimensionless frequency being
u∞min (k ) = − k
S∞min (k ) = G∞min (k )
k (k + 1) + k .
(34)
13
The most extreme value of G∞min ( k ) occurs for k → 1 and is G∞min (1) = -1/8 = -0.125, while S∞min (1) = 3 / 8 = 0.2165... and u∞min (1) = 3 . This extreme case corresponds to a phase of 120° and a modulation ratio of 1/4. Elimination of u from parametric equations (28) and (29) is possible when k → 1, allowing to obtain explicit relations for the upper ( u < 3 ) and lower ( u > 3 ) branches of the most extreme outermost curve (see Figs. 4 and 5),
S∞+ (G∞ ) =
− ∞
1 3 − 1 + 8G∞ 8
S (G∞ ) = 4
(
1/2
) (1 +
1 + 8G∞
(
−2G∞3 1 + 1 + 8G∞ + 2G∞
(
1 + 1 + 8G∞
)
2
)
3/2
,
).
(35)
(36)
For G∞ close to 0, the lower branch reduces to
S∞− (G∞ ) = 2 ( −G∞ )
3/2
.
(37)
Expansion of Eqs. (22) and (23) in power series of 1/u2, allows obtaining a general relation (valid for all k) for the lower branch, when G∞ is close to 0,
S∞− (G∞ ) =
k +1 3/2 ( −G∞ ) . k
(38)
The four types of decay (Scheme 1) have a clear graphic correspondence in the phasor plots, displayed in Figures 5-8 for selected values of k. Sub-exponential decays (0 < α < l) occur inside the universal circle. On the other hand, super-exponential decays (1 < α < k/(k-l)) occur outside the universal circle, up to the green line. Finally, unimodal decays (α > k/(k-l)) are exterior to the green line, and are delimited from above by the already discussed blue line (α →∞) that corresponds to Eq. (30). It is also seen that for unimodal decays (but not for super-exponential decays) G(u) takes negative values at high frequencies, i.e. the phase can be higher than 90°, as mentioned above. This was (marginally) observed experimentally by Itagaki and coworkers [14] in their study of the pyrene monomer-excimer system, where the fast
14
Fourier transform (FFT) method was used to produce phasor plots of fluorescence decays containing contributions from both monomer and excimer.
Figure 5. Phasor plot for Eq. (20) with k = 1.1. Black and red and curves, nearly perfectly superimposed, correspond, respectively, to exponential decays (universal circle) and to Eq. (20) with a fraction of ½ for each component (α = 0.476). The two component lifetimes, τ1 and τ2, are marked for a dimensionless frequency u = 1.817. The green line corresponds to the upper limit for super-exponential behavior (α = 11), and the blue line to the upper limit for a unimodal decay (α →∞). The point marked ∞ has the lowest
G value for the curve,
G∞min , and corresponds to the dimensionless frequency u∞min = 1.817 . The dashed
blue line corresponds to Eq. (38).
Figure 6. Phasor plot for Eq. (20) with k = 3. Black and red and curves correspond to exponential decays (universal circle) and to Eq. (20) with a fraction of ½ for each component (α = 0.25), respectively. The two component lifetimes, τ1 and τ2, are marked for a dimensionless frequency u = 3.151. The green line corresponds to the upper limit for super-exponential behavior (α = 1.125), and the blue line to the upper min
limit for a unimodal decay (α →∞). The point marked ∞ has the lowest G value for the curve, G∞ , and corresponds to the dimensionless frequency u∞ = 3.151 . min
15
Figure 7. Phasor plot for Eq. (20) with k = 10. Black and red and curves correspond to exponential decays (universal circle) and to Eq. (20) with a fraction of ½ for each component (α = 0.091), respectively. The two component lifetimes, τ1 and τ2, are marked for a dimensionless frequency u = 6.692. The green line corresponds to the upper limit for super-exponential behavior (α = 1.010), and the blue line to the upper limit for a unimodal decay (α →∞). The point marked ∞ has the lowest G value for
the curve, G∞ , and corresponds to the dimensionless frequency u∞ = 6.692 . min
min
Figure 8. Phasor plot for Eq. (20) with k = 30. Black and red and curves correspond to exponential decays (universal circle) and to Eq. (20) with a fraction of ½ for each component (α = 0.033), respectively. The two component lifetimes, τ1 and τ2, are marked for a dimensionless frequency u = 14.135. The green line corresponds to the upper limit for super-exponential behavior (α = 1.001), and the blue line to the upper limit for a unimodal decay (α →∞). The point marked ∞ has the lowest G value for
the curve, G∞ , and corresponds to the dimensionless frequency u∞ = 14.135 . min
min
16
2.2 Sum of three exponentials
When compared with the sum of two exponentials, the existence of a third exponential term considerably complicates the analysis of the decay function, as there are now four independent parameters, instead of two:
I (θ ) = α0 e−θ + α1 e−k1θ + α2 e−k2θ ,
(39)
with α 0 + α1 + α 2 = 1 and k2 > k1 > 1. No systematic study is attempted here, and only a few selected cases and aspects are examined. The phasor plots of Eq. (39) are obtained using Eqs. (14) and (15) with three terms. It is interesting to note that any two of the three terms can be paired to yield a virtual phasor, for instance
G01 (ω ) =
f 0G0 (ω ) + f1G1 (ω ) , f 0 + f1
(40)
S01 (ω ) =
f 0 S0 (ω ) + f1S1 (ω ) , f 0 + f1
(41)
hence
G (ω ) = ( f 0 + f1 )G01 (ω ) + f 2G2 (ω ) ,
(42)
S (ω ) = ( f 0 + f1 ) S 01 (ω ) + f 2 S 2 (ω ) .
(43)
The position of the virtual phasor with respect to the phasors of the two component exponentials (which may also not have direct physical meaning) obeys the generalized lever rule, Eq. (23). However, unlike what was observed in the previous section, it may now also appear to the right of these phasors. This happens when f0 + f1 < 0. The virtual phasor may even fall outside the phasor space depicted in Fig. 1, as the coefficients in Eqs. (40) and (41) can be arbitrarily large when f0 >> 1 and f1 ≈ -f0. The overall phasor, expressed as a linear combination of the virtual phasor with the third phasor according to Eqs. (42) and (43), again following the lever rule, is always placed within the phasor space, as could be expected. 17
The types of decay function that can result from a sum of three exponentials obviously include all possibilities already observed for a sum of two exponentials: subexponential, exponential, super-exponential and unimodal. Nevertheless, a specific type of unimodal time evolution exists: the “double decay”, corresponding to a fast initial decay that may nearly reach zero, followed by a rise and a second decay. The time evolution of the luminescence of intermediate A2 present in kinetic Scheme 3, although not covering the full parameter space of Eq. (39), allows exploring all types of decay mentioned.
k1 k2 1 D → A1 → A2 →
Scheme 3
The (area normalized) decay function for intermediate A2 is
I (θ ) =
α − k2θ k1 e −θ e − k1θ 1 − + + e , β ( k1 − 1)( k2 − 1) ( k1 − 1)( k2 − k1 ) k1 ( k2 − k1 )( k2 − 1)
(44)
where β is a constant,
β=
k1
( k1 − 1)( k2 − 1)
−
1 1 k1 + α + , ( k1 − 1)( k2 − k1 ) k2 ( k2 − k1 )( k2 − 1)
(45)
and α = A2(0)/D(0). It is assumed that A1(0) = 0. The decay (and the phasor) are thus a function of only three independent parameters: k1, k2 and α. For large α the decay is nearly exponential. For α = 0 it is unimodal. For intermediate values of α and appropriate combinations of k1 and k2 the decay, although still unimodal, has the “double-decay” character mentioned. The phasor plot of Eq. (44) with k1 = 2, k2 = 2.65, and α = 0 (unimodal decay function) is shown in Fig. 9. The corresponding curve (red line) is exterior to the universal circle (black semicircle). The curve occurs between two violet curves, the inner one corresponding to Eq. (31) (attainable in this case for a very large k2), and the outer one to the case where k1 = k2 = 1 and α = 0, whose decay function is
18
I (θ ) =
θ2 2
e −θ .
(46)
The cosine and sine transforms corresponding to this decay function, defining the outer violet line, are
G (u ) =
S (u ) =
1 − 3u 2
(1 + u 2 )
3
u (3 − u 2 )
(1 + u 2 )
3
,
(47)
,
(48)
compare Eqs. (28) and (29) with k = 1. The outer violet curve is nevertheless not the outermost curve in the case of a sum of three exponentials, but seems to be the limit for Eq. (44). Eqs. (46)-(48) can be easily generalized for a sum of n exponentials.
Figure 9. Phasor plot (red line) corresponding to a sum of three exponentials, Eq. (44), with k1 = 2, k2 =
2.65, and α = 0 (f0 = 3.212, f1 = -4.077 and f2 = 1.865). The decay function is unimodal, with I(0) = 0. The points correspond to a reduced frequency u = 2. The virtual phasor (V) resulting from the pairing of the first two exponentials (dashed blue line) occurs outside the phasor space, and to the right of the phasors. The global phasor (P) can be viewed as the combination of the virtual phasor and of the third exponential (dashed red line). The positions of the other two possible virtual phasors can be found geometrically as the intersection of the respective straight lines ( P 0 with 12 , and P1 with 02 ). The two violet lines are discussed in the text.
19
The phasor plot of Eq. (44) with k1 = 1.01, k2 = 4, and α = 0.5 (“double-decay” function) is shown in Fig. 10.
Figure 10. Phasor plot (red line) corresponding to a sum of three exponentials, Eq. (44), with k1 = 1.01, k2
= 4, and α = 0.5 (f0 = 89.78, f1 = -89.19 and f2 = 0.408). The decay function is unimodal, of the “doubledecay” type, with I(0) > 0. The points correspond to a reduced frequency u = 1.5. The virtual phasor (V) resulting from the pairing of the first two exponentials (dashed blue line) occurs inside the phasor space, and to the left of the phasors. The global phasor (P) can be viewed as the combination of the virtual phasor and of the third exponential (dashed red line). The two violet lines are discussed in the text.
2.3 Stretched and compressed exponentials
A well-known and versatile decay function is I (θ ) = exp ( −θ β ) ,
(49)
where θ = t/τ is a dimensionless time and β > 0. The kind of decay behavior according to the value of β is summarized in Scheme 4: When β < 1 the decay function is a stretched exponential [30] and the decay is sub-exponential [3]; when β = 1 the decay function is exponential; and finally when β > 1 the decay function is a compressed exponential and the decay is super-exponential [3].
20
Scheme 4
There are many examples of application of the stretched exponential to luminescence decays, either as a result of physical models, or as an empirical decay function
[1,3,30,31]. The compressed exponential, on the contrary, remains to be observed or used. A situation where it should hold was discussed in [3]. The cosine and sine Fourier transforms of Eq. (49) are:
∞
G (u ) =
1 cos ( u x ) exp ( − x β ) dx , ∫ Γ (1 + 1 / β ) 0
(50)
∞
1 S (u ) = sin ( u x ) exp ( − x β ) dx . ∫ Γ (1 + 11// β ) 0
(51)
Numerical integration is required for the computation of these transforms as a function of the dimensionless frequency u = ωτ. A suitable transformation is necessary for the computation of reliable values when parameter β is large.
Phasor plots of Eq. (49) are shown in Fig. 11 for selected values of parameter β. It is seen that for β < 1 (sub-exponential decays) the phasor remains inside the universal semicircle, as expected, whereas for β > 1 (super-exponential decays) the phasor may occur both outside and inside the universal semicircle. This example shows that super-
exponential decays do not always lie outside the universal circle, unlike what was observed for a sum of two exponentials (see 2.1). For large β, G(u) can take negative values (for high frequencies), as observed for a sum of two exponentials (see 2.1).
However, in this case the S(G) curve has a more complex, nearly spiral-like form, before reaching the (0,0) point (corresponding to a hypothetical infinite frequency) according to a nearly vertical path. The spiraling is the more complex, the larger the
value of β.
21
Figure 11. Phasor plots for Eq. (49), including a zoom of the high frequency region. The values of
parameter β are: 0.5 (red), 1 (black) and 10 (blue). For β = 10, coordinate G crosses the zero line for u = 3.3, 6.6, 9.8, 12.9, 15.9, 18.9, …
22
2.4 A versatile phosphorescence decay function A simple decay function that describes with good accuracy experimental phosphorescence decays in the presence of excited-state absorption was recently
proposed [3,4]
I (θ ) =
1 , a + (1 − a ) eθ
(52)
where θ = t/τP is a dimensionless (or reduced) time, τP is the intrinsic phosphorescence lifetime and a is a dimensionless parameter accounting for reabsorption, 1 > a ≥ 0. For very small a, i.e. negligible reabsorption, Eq. (52) 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 [2 [29] and is
super-exponential, as shown in Scheme 5.
Scheme 5
The phosphorescence decay of a species subject to triplet-triplet annihilation in fluid medium is very similar to Eq. (52), see e.g. [30],
I (θ ) =
1 , −a + (1 + a ) eθ
(53)
where a = kTT C0 τP 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. In fact, Eq. (53) always corresponds to a sub-exponential decay, see Scheme 5.
23
In the present phasor plots, Eq. (52) will be used in all cases, with parameter a taking both negative and positive values (a < 1). In this way, both sub-exponential and superexponential situations will be covered. The cosine and sine Fourier transforms of Eq. (52) are:
G (u ) = −
∞ cos ( u x ) a dx , ∫ ln(1 − a ) 0 a + (1 − a)e x
(54)
S (u ) = −
∞ sin ( u x ) a dx . ∫ ln(1 − a) 0 a + (1 − a)e x
(55)
Numerical integration is required for the computation of these transforms as a function of the dimensionless frequency u. A suitable transformation is necessary for the computation of reliable values when parameter a is close to 1. Phasor plots of Eq. (52) are shown in Fig. 12 for several values of parameter a. It is seen that for a < 0 (sub-exponential decays) the phasor remains inside the universal semicircle, whereas for a > 0 (super-exponential decays) the phasor occurs, in general, both outside and inside the universal semicircle. For a close to 1, G(u) can take negative values at high frequencies, as observed previously (see 2.1, 2.2 and 2.3). As in the compressed exponential case, the S(G) curve takes a complex, spiral-like form, before reaching the (0,0) point (corresponding to a hypothetical infinite frequency) again according to a nearly vertical path, see Fig. 12. The spiraling is the more intrincate, the closer parameter a is to unity, and nearly-periodic, arabesque-like patterns can be observed at high frequencies (Fig. 12), whose analysis is beyond the scope of this work.
24
Figure 12. Phasor plots for Eq. (52), including two zooms of the high frequency region. The values of
parameter a are: -20 (red), 0 (black) and 0.99999 (blue). For a = 0.99999, coordinate G crosses the zero line for u = 0.27, 0.55, 0.82, 1.09, 1.36, 1.64, …
25
A remark worth making in the present context is the following: It is widely recognized that the two main luminescence lifetime techniques, “frequency domain” and “time domain”, are theoretically equivalent, although in practice each has its own advantages and limitations [1,7]. However, it is not always remembered that this equivalence only exists as long as a linearity principle applies, i.e. that response of the system to successive excitations is always the same, and thus application of the convolution integral is valid. This holds only when a small fraction of the luminophore population is excited, and ground-state depletion is negligible. If a large fraction of the population goes into the excited state, then the response will be a function of this fraction, which is precisely the case for Eq. (52) [4]. The phosphorescence decay under triplet-triplet absorption results from a substantial triplet population at time zero and the response of the system to a continuous excitation is not a convolution of Eq. (52) with the excitation [4]. In this way, the phasor plot for Eq. (52) is a mathematical plot, characterizing the decay function, but will not necessarily correspond to the actual plot if a modulated excitation source is used. In such a case the actual plot will be a function of several technical parameters such as the (maximum) intensity of the source.
2.5 A decay function leading to phasors in all quadrants
The theoretical decay function for luminescence affected by excited-state absorption and with a single path length is [29]
I (θ ) = exp[−θ − B0 exp(−θ )] ,
(56)
where B0 is the initial optical thickness of the medium at the emission wavelength. This decay function is super-exponential for 0 < B0 < 1 and unimodal for B0 > 1. The maximum of the function occurs at θ = ln B0. For high values of B0 the function has negligible values at times significantly shorter than ln B0, i.e., corresponds to a “delayed” emission. The cosine and sine Fourier transforms of Eq. (56) are:
∞
B0 G (u ) = cos(ux) exp[ − x − B0 exp( − x)] dx , 1 − e− B0 ∫0
(57)
∞
S (u ) =
B0 sin(ux) exp[ − x − B0 exp(− x)] dx . 1 − e− B0 ∫0
(58) 26
The computation of these transforms as a function of the dimensionless frequency u is best done using series expansions of the integrals. Phasor plots of Eq. (56) are shown in Figs. 13-16 for several values of parameter B0. It is seen that for B0 ≤ 1 (super-exponential decay function) the phasor is outside the universal semicircle but remains in the first quadrant, whereas for B0 > 1 (unimodal decay function) the phasor acquires a spiral form and occurs in all quadrants.
Figure 13. Phasor plot for Eq. (56) with B0 = 1. Also shown is the universal circle (black line).
27
Figure 14. Phasor plot for Eq. (56) with B0 = 10.
Figure 15. Phasor plot for Eq. (56) with B0 = 50.
28
Figure 16. Phasor plot for Eq. (56) with B0 = 5000.
2.6 A multimodal decay function: Fluorescence with quantum beats
A decay function with several maxima results from the existence of quantum beats [5,6]. In order to investigate the corresponding phasor plot, Eq. (59) will be used:
I (θ ) =
1 + cos(αθ + ϕ ) −θ e , 1 + cos ϕ
(59)
where α = ω0τ, ω0 being the beat angular frequency, θ = t/τ and ϕ is the phase. This decay function applies to a four-level system when all transition moments and decay rates are identical, leading to full modulation [5]. Here, it will also be assumed, for simplicity, that the phase is zero, hence the decay function becomes
I (θ ) =
1 [1 + cos(αθ )] e−θ . 2
(60)
The cosine and sine Fourier transforms of Eq. (60) are:
29
1 u2 +1+ α 2 + , 2 4 2 2 2 2 1 + u u + 2(1 − α )u + (1 + α )
G (u ) =
1+ α 2 2 +α 2
S (u) =
1+ α 2 1 u2 +1− α 2 + u. 2 2 4 2 2 2 2 2 + α 1 + u u + 2(1 − α )u + (1 + α )
(61)
(62)
Phasor plots are shown in Figs. 17-22 for selected values of α.
Figure 17. Phasor plot for Eq. (60), with α = 1. Also shown is the universal circle.
Figure 18. Phasor plot for Eq. (60), with α = 2.
30
Figure 19. Phasor plot for Eq. (60), with α = 3.
Figure 20. Phasor plot for Eq. (60), with α = 5.
31
Figure 21. Phasor plot for Eq. (60), with α = 10.
Figure 22. Phasor plot for Eq. (60), with α = 50.
It is seen that for this multimodal decay law, S can take negative values. On the other hand, G is always nonnegative, hence in this case the phasor can be observed not only in the first but also in the fourth quadrant. A fixed pattern with a circular loop (Fig. 22) occurs for sufficiently large α. 32
3. Conclusion Luminescence decay functions show a rich diversity of patterns in their phasor plots. It was shown that the phasors of luminescence decays may fall on all four quadrants of the circular phasor space. In the case of a sum of two exponentials, a detailed study revealed that sub-exponential, super-exponential and unimodal decays have clearly different phasor signatures, in location and/or in shape. The existence of a limiting curve (outer boundary) was demonstrated and a generalization of the lever rule obtained. In the case of the sum of three exponentials, the concept of virtual phasor arose. Study of more complex decay functions, such as the stretched and compressed exponentials, and those for phosphorescence with reabsorption and triplet-triplet annihilation, showed that super-exponential decays can present complex shapes, spiraling at high frequencies while entering and leaving the sub-exponential region. A decay law of the exponential of exponential type gives rise to spirals and may display phasors in all four quadrants. A model multimodal decay function of fluorescence with quantum beats again revealed new patterns. The distinction between the phasor plot as a mathematical procedure to graphically characterize a family of decay functions, and the phasor plot as a result of “frequency domain” measurement, and their possible divergence in cases of nonlinear response, was also discussed.
Acknowledgements This work was carried out within projects PTDC/QUI-QUI/123162/2010 and RECI/CTM-POL/0342/2012 (FCT, Portugal).
33
References [1] B. Valeur and M.N. Berberan-Santos, Molecular Fluorescence. Principles and Applications, Wiley-VCH, Weinheim, 2nd ed., 2012. [2] M.N. Berberan-Santos, B. Valeur, J. Lumin. 126 (2007) 263. [3] T. Palmeira, M.N. Berberan-Santos, Chem. Phys. 445 (2014) 14. [4] T. Palmeira, M.N. Berberan-Santos, J. Lumin. 158 (2015) 510. [5] R.T. Carter, J.R. Huber, Chem. Soc. Rev. 29 (2000) 305. [6] J.J. Burdett, C.J. Bardeen, J. Am. Chem. Soc. 134 (2012) 8597. [7] D.M. Jameson, Introduction to Fluorescence, CRC Press, Boca Raton, 2014. [8] G. Weber, J. Phys. Chem. 85 (1981) 949.
[9] A. Papoulis, The Fourier Integral and its Applications, McGraw-Hill, New York, 1962. [10] R.N. Bracewell, The Fourier Transform and its Applications, 3rd. ed., McGrawHill, Singapore, 2000. [11] D.M. Jameson, E. Gratton, R.D. Hall, Appl. Spectrosc. Rev. 20 (1984) 55. [12] M. Berberan-Santos, J. Lumin. 50 (1991) 83. [13] M. Itagaki, K. Watanabe, Bunseki Kagaku 43 (1994) 1143. [14] M. Itagaki, M. Hosono, K. Watanabe, Anal. Sci. 13 (1997) 991. [15] P.J. Verveer, P.I.H. Bastiaens, J. Microsc. 209 (2003) 1. [16] A.H.A. Clayton, Q.S. Hanley, P.J. Verveer, J. Microsc. 213 (2004) 1. [17] G.I. Redford, R.M. Clegg, J. Fluoresc. 15 (2005) 805. [18] M.A. Digman, V.R. Caiolfa, M. Zamai, E. Gratton, Biophys. J. 94 (2008) L14. [19] A.H.A. Clayton, J. Microsc. 232 (2008) 306. [20] Y.-C. Chen, R.M. Clegg, Photosynth. Res. 102 (2009) 143.
34
[21] Y.-C. Chen, B.Q. Spring, C. Buranachai, G. Malachowski, R.M. Clegg, Proc. SPIE 7183 (2009) 718302. [22] C. Stringari, A. Cinquin, O. Cinquin, M.A. Digman, P.J. Donovan, E. Gratton, Proc. Natl. Acad. Sci. U.S.A. 108 (2011) 13582. [23] M. Stefl, N.G. James, J.A. Ross, D.M. Jameson, Anal. Biochem. 410 (2011) 62. [24] N.G. James, J.A. Ross, M. Stefl, D.M. Jameson, Anal. Biochem. 410 (2011) 70. [25] E. Hinde, M.A. Digman, C. Welch, K.M. Hahn, E. Gratton, Microsc. Res. Tech. 75 (2012) 271. [26] M.A. Digman, E. Gratton, in Fluorescence Lifetime Spectroscopy and Imaging: Principles and Applications in Biomedical Diagnostics (ed. L. Marcu, P.M.W. French, and D.S. Elson) CRC Press, Boca Raton, 2012. [27] F. Menezes, A. Fedorov, C. Baleizao, B. Valeur, M.N. Berberan-Santos, Methods Appl. Fluoresc. 1 (2013) 015002. [28] E. Hinde, M.A. Digman, K.M. Hahn, E. Gratton, Proc. Natl. Acad. Sci. U.S.A. 110 (2013) 135. [29] T. Palmeira, A. Fedorov, M.N. Berberan-Santos, ChemPhysChem, in press (2015). DOI: 10.1002/cphc.201402695 [30] M.N. Berberan-Santos, E.N. Bodunov, B. Valeur, Chem. Phys. 317 (2005) 57. [31] M. Berberan-Santos, E.N. Bodunov, B. Valeur, Ann. Phys. (Berlin) 17 (2008) 460.
35
36
Highlights
•
First complete study of the phasor plots of a sum of two exponentials (2E).
•
The existence of a 2E limiting curve (outer boundary) is demonstrated.
•
A generalization of the lever rule is presented.
•
Virtual phasor concept.
•
Study of several decay laws displaying a diversity of patterns.
37
S0301-0104(15)00009-9 http://dx.doi.org/10.1016/j.chemphys.2015.01.007 CHEMPH 9245
To appear in:
Chemical Physics
Received Date: Accepted Date:
5 November 2014 7 January 2015
Please cite this article as: M.N. Berberan-Santos, Phasor plots of luminescence decay functions, Chemical Physics (2015), doi: http://dx.doi.org/10.1016/j.chemphys.2015.01.007
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Phasor plots of luminescence decay functions
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 Phone: 351-218419254, E-mail: [email protected]
Abstract Luminescence decay functions describe the time dependence of the intensity of radiation emitted by electronically excited species. Decay phasor plots (plots of the Fourier sine transform vs. the Fourier cosine transform, for one or several angular frequencies) are being increasingly used in fluorescence, namely in lifetime imaging microscopy (FLIM). In this work, a detailed study of the sum of two exponentials decay function is carried out revealing that sub-exponential, super-exponential and unimodal decays have different phasor signatures. A generalization of the lever rule is obtained, and the existence of an outermost phasor curve corresponding to intermediate-like decays is demonstrated. A study of the behavior of more complex decay functions (sum of three exponentials, stretched and compressed exponentials, phosphorescence with reabsorption and triplet-triplet annihilation, fluorescence with quantum beats) allows concluding that a rich diversity of phasor plot patterns exists. In particular, superexponential decays can present complex shapes, spiraling at high frequencies. The concept of virtual phasor is also introduced.
1
1. Introduction 1.1 Luminescence decay functions A luminescence decay function, I(t), is the function describing the time dependence of the intensity of radiation spontaneously emitted at a given wavelength, by a previously excited sample. For convenience, and without loss of generality, the decay function is usually normalized at t = 0, I(0) = 1, and only nonnegative times are considered (t ≥ 0). The decay function can in principle be related to a model describing the luminescence mechanism and respective dynamics, but remains a valuable tool even if this model is not available. The luminescence decay function can be written in the following form [1]:
∞
I (t ) = ∫ g (k ) e− kt dk .
(1)
0
This relation is generally valid as I(t) always has an inverse Laplace transform, g(k). The function g(k), also called the eigenvalue spectrum, is normalized, as I(0) = 1 ∞
implies ∫ g (k ) dk = 1 . In many situations the function g(k) is nonnegative for all k > 0, 0
and g(k) can be regarded as a distribution of rate constants - strictly, a probability density function (pdf) [1,2]. It was previously shown [2] that this is always the case for completely monotonic functions, i.e., functions for which the nth-order derivatives I(n)(t) obey:
(−1)n I ( n ) (t ) > 0
(n = 0, 1, 2,….).
(2)
However, in some cases the decay function does not comply with this definition and the respective g(k) also takes negative values [2]. For the discussion of the time behavior of decay functions the following defining equation is useful [2,3]
w( t ) = −
d ln I (t ) , dt
(3)
2
where w(t) is the decay rate coefficient, with a possible time dependence. For a monotonic decay function, w(t) > 0 for all t. 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]: exponential decay, when w(t) is constant; subexponential decay, when w(t) decreases with time; and super-exponential decay, when w(t) increases with time. Most of the experimental decays belong to the first two categories. Super-exponential decays were recently discussed and experimentally identified in phosphorescence affected by triplet-triplet absorption [3,4]. With respect to non-monotonic decay functions, an important class is that of unimodal decay functions, i.e. functions with a single maximum, whose intensity rises initially, followed by passage through a maximum before decay proper. They may correspond to a w(t) steadily increasing with time, as with super-exponential decays, but now starting from w(0) < 0 [3]. This type of behavior is typical of, but not limited to, the luminescence of states or species not directly generated by excitation, like delayed fluorescence and FRET acceptor emission. Finally, a second class of non-monotonic decays is that of multimodal decay functions, i.e., functions with several maxima. This is the situation of decays displaying quantum beats, observed for a few molecules under specific conditions, like gaseous biacetyl and crystalline tetracene, and is probably the most complex of all physically relevant cases [5,6]. A decay function represented by a distribution of rate constants implies either an exponential (g(k) = δ(k-k0), which has zero variance) or a sub-exponential decay (g(k) with nonzero variance). On the other hand, super-exponential and non-monotonic decay functions never correspond to a distribution of rate constants, i.e., their inverse Laplace transform g(k) always take negative values at some points. A distribution of lifetimes h(τ) is also used to represent the decay function,
∞
I (t ) = ∫ h(τ ) e− t / τ dτ .
(4)
0
This function is related to the inverse Laplace transform g(k) by
h(τ ) =
1 g , τ τ 1
2
(5)
3
and therefore it may also take negative values at some points, as happens with g(k).
1.2 The “phasor approach” The cosine and sine Fourier transforms of I(t), G(ω) and S(ω), respectively, are defined as [1,7]
∞
∫ I (u ) cos(ω u)du , G (ω ) = 0 ∞ ∫0 I (u ) du
(6)
∞
∫ I (u ) sin(ω u)du , S (ω ) = 0 ∞ ∫0 I (u ) du
(7)
where ω is the angular frequency. Following Weber [8], the letter G is used instead of C (G is a convenient choice, as it avoids the concentration symbol while retaining graphic similarity with C). The denominators in Eqs. (6) and (7), which are not part of the general definition [9,10], imply a specific normalization for the decay function (it becomes a density [9], i.e., a function analogous to a one-sided probability density function), and were adopted in the phase-modulation (frequency domain) technique for convenience. In this way, each decay function, for a given frequency, is mapped onto a point inside the unit circle defined by G2 + S2 = 1 [9], which may be said to be the “phasor space”, see Fig. 1. Indeed, Eqs. (6) and (7) immediately imply that both |G| and |S| cannot exceed 1. It also follows from these equations that G(0) = 1 and S(0) = 0, while S(∞) = G(∞) = 0, see Fig. 1. Insertion of Eq. (4) into Eqs. (6) and (7) gives
∞
G (ω ) = ∫ f (τ ) 0
∞
S (ω ) = ∫ f (τ ) 0
1 dτ , 1 + (ωτ ) 2
ωτ dτ , 1 + (ωτ ) 2
(8)
(9)
4
Figure 1. The phasor space (white area) and the «universal» semicircle. Also shown are the (truly) universal points corresponding to all decay functions for zero and infinite frequencies. The «universal» semicircle, located in the first quadrant, defines the loci of all exponential decays. Other decay functions follow different paths (which can have complex shapes, as will be shown) between the two extreme points (ω = 0 and ω = ∞), when going from zero to infinite frequency.
where f(τ) is a new distribution of lifetimes (intensity-weighted, see Eq. (12) below),
f (τ ) =
τ h(τ ) ∞
.
(10)
∫ τ h(τ )dτ 0
Like h(τ), f(τ) can take in general both positive and negative values. For a given frequency, the (G, S) pair defines a point or, equivalently, a vector P = G e1 + S e 2 , called the phase vector or phasor. This vector is the basis of the phasor
approach to time-resolved luminescence (up to now, fluorescence) [1,7,11-28], which provides a simple graphical and model-independent portrait of a luminescent system. Besides luminophore identification (“fingerprinting”) in complex systems, processes such as quenching, solvent relaxation and energy transfer can be identified by characteristic trajectories in the plane (at a fixed frequency or using several frequencies). Owing to the limited computational requirements, linearity and robustness, the phasor approach is especially useful in fluorescence lifetime imaging
5
microscopy (FLIM) [15-18,22,25,26]. Application to measurements in solution (“single pixel data”) is nevertheless also of interest [13,14,23,24,27]. The precise location of the decay in the plane, defined by its phasor, is a function of frequency and decay characteristics. Single exponential decays lie on a so-called universal circle (in fact a semi-circle, for nonnegative frequencies), defined by
S = G(1 − G) , with 1 ≥ G ≥ 0 [1,7,11,16-18,23], see Fig. 1. Complex decays usually, but not always, fall inside it. The point corresponding to a multiexponential decay is located at an average distance from those of the components. In the case of a twoexponential decay with positive amplitudes, for instance, the corresponding point falls on a straight line connecting the phasors of the two components (“tie line”) [1,7,11,1618,23]. Analogously to the lever rule of thermodynamic phase diagrams, the fractional contribution of each of the two components to the total intensity is given by the length of the segment connecting the decay point (“average lifetime”) to the opposite component, divided by the length of the full segment uniting the two extreme points (components) [1,7,11,16-18,23]. When there are three or more components, the corresponding points define a polygon, with vertices located at or inside the circle, with the decay point lying in turn inside the polygon [1,7,16-18,23]. When the measurement technique used is frequency domain fluorimetry, based on sinusoidally modulated excitation, G and S are directly related to the two parameters obtained for each frequency, which are the modulation ratio, M, and the phase shift, Φ, by G = M cos Φ and by S = M sin Φ , hence tan Φ = S / G and M = P = G 2 + S 2 [8]. When the measurement technique used is time domain fluorimetry, the phasor must be computed from the measured decay according to Eqs. (6) and (7), at a conveniently chosen frequency (or set of frequencies). The decay is usually previously fitted with an empirical decay law, e.g. a sum of exponentials, in order to remove the effect of the finite width of the excitation impulse (instrument response function). When plotting luminescence decay functions (and not experimental data), numerical evaluation of their transforms is required, in general. However, if the decay is given by a sum of exponentials,
I (t ) = ∑ ai e− t /τ i ,
(11)
i
The function f(τ), Eq. (10), becomes 6
f (τ ) = ∑ fi δ (τ − τ i ) ,
(12)
i
where the fi are the fractional steady-state intensities,
fi =
aiτ i , ∑ aiτ i
(13)
i
and Eqs. (8) and (9) become
G (ω ) = ∑ f i i
S (ω ) = ∑ f i i
1
,
(14)
ωτ i . 1 + (ωτ i ) 2
(15)
1 + (ωτ i ) 2
It is worth noting that G(ω) and S(ω) are related to the real and imaginary parts of the complex Fourier transform F(ω),
∞
∫ I (t )e F (ω ) =
− iωt
dt = G (ω ) − iS (ω ) .
0 ∞
(16)
∫ I (t )dt 0
As follows from Eqs. (6) and (7), G(ω) and S(ω) are not independent. They are explicitly related by Hilbert transforms [9],
G (ω ) = −
2
∞
uS (u ) du , 2 − u2
π ∫ω 0
(17)
7
S (ω ) =
2ω
π
∞
G (u ) du . 2 − u2
∫ω 0
(18)
The purpose of the present work is to discuss the characteristic curves of a few representative luminescence decay functions, obtained by plotting the respective Fourier cosine and sine transform pairs (phasors) vs. frequency in the so-called polar or phasor plot. According to Eq. (16), these curves can also be viewed as plots of the conjugate of F(ω), F*(ω), in the complex plane [9,10] (in the context of control systems theory, this type of plot is known as the Nyquist plot [13,14], whereas in the dielectric relaxation field a similar plot is called the Cole-Cole plot [12,17].)
2. Results and Discussion 2.1 Sum of two exponentials The normalized decay function corresponding to a sum of two exponentials is
I (t ) = α e−t /τ1 + (1 − α )e−t /τ 2 ,
(19)
Without loss of generality, let us assume that τ1 > τ2. Then I(t) > 0 for all times implies α > 0. Eq. (19) can be rewritten as
I (θ ) = α e−θ + (1 − α )e−kθ ,
(20)
where θ = t/τ1 is a dimensionless (or reduced) time and k = τ1/τ2 is the lifetime ratio, with k > 1. Eq. (20) encompasses all types of time-dependent behavior defined above, excluding the multimodal: sub-exponential, exponential, super-exponential and unimodal, see Scheme 1 and Fig. 2 [3]. The decay is sub-exponential for 0 < α < l and both terms have positive pre-exponential factors. The transition from sub-exponential to super-exponential occurs at α = l (for α = l the decay is exponential), as one of the exponentials gains a negative pre-exponential factor. The transition from superexponential to unimodal, with the appearance of a maximum, occurs at α = k/(k-1).
8
Scheme 1
Figure 2. Decay curves corresponding to Eq. (20) with k = 1.5, all asymptotically exponential (or exponential): sub-exponential decay (α = 0.2), exponential decay (α = 1), super-exponential decay ( α = 3), and unimodal (α = 10), compare Scheme 1.
For a given frequency, the corresponding phasor P is related to the phasors of the two components (real or fictitious), P1 and P2, which lie in the universal circle. Using Eqs. (13) and (20), the generalized fractional steady-state intensities of these
components are:
f1 =
αk , 1 + α ( k − 1)
(21)
f2 =
1−α . 1 + α ( k − 1)
(22)
If α < 1 (sub-exponential decay) the lever rule applies, as mentioned above, and the phasor falls inside the circle, on the straight line (tie line) connecting τ1 and τ2, whose 9
slope is u(u2-k)/[u(k+1)] [15]. When α > 1 (super-exponential or unimodal decay) f1 > 1 and f2 < 0. In this case, the phasor also occurs on the same straight line, but now outside the circle, to the left of both τ1 and τ2, see Fig. 3. It can be shown that the ratio of the lengths of the segments |P-P2| and |P1-P2| is always f1, irrespective of α:
f1 =
P − P2 P1 − P2
.
(23)
The phasor plots of Eq. (20) shown in Figs. 3-8 are obtained from the respective cosine and sine Fourier transforms [15] that read, in the present notation:
G (u ) =
1 1−α 1 αk , + 2 1 + α (k − 1) 1 + u 1 + α ( k − 1) 1 + ( u / k )2
(24)
S (u ) =
1−α αk u u/k , + 2 1 + α ( k − 1) 1 + u 1 + α (k − 1) 1 + ( u / k )2
(25)
where u is a dimensionless frequency, u = ωτ1. From these relations it can be shown that the length of the segment uniting the loci of the two exponentials is
P1 (u ) − P2 (u ) =
u ( k − 1) (1 + u 2 )( k 2 + u 2 )
.
(26)
It follows from Eq. (26) that the maximum distance between the two phasors occurs for u = k and is equal to (k-1)/(k+1). In this situation S1 = S2 and the tie line is horizontal.
It also follows that
P( k ) − P2 ( k ) =
α k ( k − 1) . 1 + α ( k − 1) ( k + 1)
(27)
10
Figure 3. Extension of the «lever rule» to the situation where the phasor falls outside the universal circle (black semi-circle). The lifetime ratio is k = 2. A common and large α was used in both cases, α = 1000. For this reason, the blue line is practically equal to the outermost boundary of the phasor space, corresponding to phasors with α → ∞, P∞(u). The reduced angular frequency is (a) u = 0.75 and (b) u = 2.5. It is also seen that G∞ takes negative values at high frequencies, thus entering the second quadrant.
The maximum length is obtained for α → ∞, and is then equal to k/(k+1). In this situation, the phasor P( k ) = P∞ ( k ) has always G∞ = 0, see Fig. 4. The corresponding S∞ is
k / (1 + k ) .
11
Figure 4. Loci of the phasor P (blue dots) when u =
k and α → ∞, P∞
( k ) , for the values of k given
in the figure near the respective blue dots. The corresponding decay functions are given by Eq. (30). The phasors of the components (with lifetimes τ1 and τ2) are shown as black dots.
It can also be seen in Fig. 4 that the outermost boundary (blue line) shrinks and approaches the universal circle as k → ∞. From Eqs. (24) and (25) one obtains, for α → ∞ (blue lines), the parametric equations
G∞ (u ) =
k (k − u 2 ) k (k − u 2 ) = G1 (u ) , (1 + u 2 )(k 2 + u 2 ) k 2 + u2
(28)
S∞ (u ) =
k (k + 1) u k (k + 1) = 2 S1 (u ) . 2 2 2 (1 + u )(k + u ) k + u 2
(29)
For a given k, the decay function corresponding to the outermost boundary (α → ∞) ∞
can be obtained from Eq. (20) (after normalization, ∫ I (θ )dθ = 1 ) by finding the limit 0
when α → ∞. One gets
I ∞ (θ ) =
k e −θ − e − kθ ) , ( k −1
(30)
12
which is the characteristic time evolution of an «intermediate» A (unstable species whose initial concentration is zero, e.g. an excimer or a FRET acceptor), as depicted in kinetic Scheme 2:
1 k D → A → or k 1 D → A →
Schemes 2a and 2b
Species D can also decay by two or more competing channels, of which only one leads to species A, and the result will still be the same, provided the sum of the rate constants for the n channels is equal to 1 for Scheme 2a and equal to k for Scheme 2b. The most extreme case (k → 1) is obtained from Eq. (30) as I∞(θ) = θ e-θ.
(31)
The minimum value (α →∞) of the Fourier cosine and sine transforms can be shown to be
min ∞
G
1 k− k (k ) = − k + 1 k − 1
S∞min (k ) =
(
) ( k + 1) ( k + 1)
2
,
(32)
,
(33)
k k + k +1
2
the corresponding dimensionless frequency being
u∞min (k ) = − k
S∞min (k ) = G∞min (k )
k (k + 1) + k .
(34)
13
The most extreme value of G∞min ( k ) occurs for k → 1 and is G∞min (1) = -1/8 = -0.125, while S∞min (1) = 3 / 8 = 0.2165... and u∞min (1) = 3 . This extreme case corresponds to a phase of 120° and a modulation ratio of 1/4. Elimination of u from parametric equations (28) and (29) is possible when k → 1, allowing to obtain explicit relations for the upper ( u < 3 ) and lower ( u > 3 ) branches of the most extreme outermost curve (see Figs. 4 and 5),
S∞+ (G∞ ) =
− ∞
1 3 − 1 + 8G∞ 8
S (G∞ ) = 4
(
1/2
) (1 +
1 + 8G∞
(
−2G∞3 1 + 1 + 8G∞ + 2G∞
(
1 + 1 + 8G∞
)
2
)
3/2
,
).
(35)
(36)
For G∞ close to 0, the lower branch reduces to
S∞− (G∞ ) = 2 ( −G∞ )
3/2
.
(37)
Expansion of Eqs. (22) and (23) in power series of 1/u2, allows obtaining a general relation (valid for all k) for the lower branch, when G∞ is close to 0,
S∞− (G∞ ) =
k +1 3/2 ( −G∞ ) . k
(38)
The four types of decay (Scheme 1) have a clear graphic correspondence in the phasor plots, displayed in Figures 5-8 for selected values of k. Sub-exponential decays (0 < α < l) occur inside the universal circle. On the other hand, super-exponential decays (1 < α < k/(k-l)) occur outside the universal circle, up to the green line. Finally, unimodal decays (α > k/(k-l)) are exterior to the green line, and are delimited from above by the already discussed blue line (α →∞) that corresponds to Eq. (30). It is also seen that for unimodal decays (but not for super-exponential decays) G(u) takes negative values at high frequencies, i.e. the phase can be higher than 90°, as mentioned above. This was (marginally) observed experimentally by Itagaki and coworkers [14] in their study of the pyrene monomer-excimer system, where the fast
14
Fourier transform (FFT) method was used to produce phasor plots of fluorescence decays containing contributions from both monomer and excimer.
Figure 5. Phasor plot for Eq. (20) with k = 1.1. Black and red and curves, nearly perfectly superimposed, correspond, respectively, to exponential decays (universal circle) and to Eq. (20) with a fraction of ½ for each component (α = 0.476). The two component lifetimes, τ1 and τ2, are marked for a dimensionless frequency u = 1.817. The green line corresponds to the upper limit for super-exponential behavior (α = 11), and the blue line to the upper limit for a unimodal decay (α →∞). The point marked ∞ has the lowest
G value for the curve,
G∞min , and corresponds to the dimensionless frequency u∞min = 1.817 . The dashed
blue line corresponds to Eq. (38).
Figure 6. Phasor plot for Eq. (20) with k = 3. Black and red and curves correspond to exponential decays (universal circle) and to Eq. (20) with a fraction of ½ for each component (α = 0.25), respectively. The two component lifetimes, τ1 and τ2, are marked for a dimensionless frequency u = 3.151. The green line corresponds to the upper limit for super-exponential behavior (α = 1.125), and the blue line to the upper min
limit for a unimodal decay (α →∞). The point marked ∞ has the lowest G value for the curve, G∞ , and corresponds to the dimensionless frequency u∞ = 3.151 . min
15
Figure 7. Phasor plot for Eq. (20) with k = 10. Black and red and curves correspond to exponential decays (universal circle) and to Eq. (20) with a fraction of ½ for each component (α = 0.091), respectively. The two component lifetimes, τ1 and τ2, are marked for a dimensionless frequency u = 6.692. The green line corresponds to the upper limit for super-exponential behavior (α = 1.010), and the blue line to the upper limit for a unimodal decay (α →∞). The point marked ∞ has the lowest G value for
the curve, G∞ , and corresponds to the dimensionless frequency u∞ = 6.692 . min
min
Figure 8. Phasor plot for Eq. (20) with k = 30. Black and red and curves correspond to exponential decays (universal circle) and to Eq. (20) with a fraction of ½ for each component (α = 0.033), respectively. The two component lifetimes, τ1 and τ2, are marked for a dimensionless frequency u = 14.135. The green line corresponds to the upper limit for super-exponential behavior (α = 1.001), and the blue line to the upper limit for a unimodal decay (α →∞). The point marked ∞ has the lowest G value for
the curve, G∞ , and corresponds to the dimensionless frequency u∞ = 14.135 . min
min
16
2.2 Sum of three exponentials
When compared with the sum of two exponentials, the existence of a third exponential term considerably complicates the analysis of the decay function, as there are now four independent parameters, instead of two:
I (θ ) = α0 e−θ + α1 e−k1θ + α2 e−k2θ ,
(39)
with α 0 + α1 + α 2 = 1 and k2 > k1 > 1. No systematic study is attempted here, and only a few selected cases and aspects are examined. The phasor plots of Eq. (39) are obtained using Eqs. (14) and (15) with three terms. It is interesting to note that any two of the three terms can be paired to yield a virtual phasor, for instance
G01 (ω ) =
f 0G0 (ω ) + f1G1 (ω ) , f 0 + f1
(40)
S01 (ω ) =
f 0 S0 (ω ) + f1S1 (ω ) , f 0 + f1
(41)
hence
G (ω ) = ( f 0 + f1 )G01 (ω ) + f 2G2 (ω ) ,
(42)
S (ω ) = ( f 0 + f1 ) S 01 (ω ) + f 2 S 2 (ω ) .
(43)
The position of the virtual phasor with respect to the phasors of the two component exponentials (which may also not have direct physical meaning) obeys the generalized lever rule, Eq. (23). However, unlike what was observed in the previous section, it may now also appear to the right of these phasors. This happens when f0 + f1 < 0. The virtual phasor may even fall outside the phasor space depicted in Fig. 1, as the coefficients in Eqs. (40) and (41) can be arbitrarily large when f0 >> 1 and f1 ≈ -f0. The overall phasor, expressed as a linear combination of the virtual phasor with the third phasor according to Eqs. (42) and (43), again following the lever rule, is always placed within the phasor space, as could be expected. 17
The types of decay function that can result from a sum of three exponentials obviously include all possibilities already observed for a sum of two exponentials: subexponential, exponential, super-exponential and unimodal. Nevertheless, a specific type of unimodal time evolution exists: the “double decay”, corresponding to a fast initial decay that may nearly reach zero, followed by a rise and a second decay. The time evolution of the luminescence of intermediate A2 present in kinetic Scheme 3, although not covering the full parameter space of Eq. (39), allows exploring all types of decay mentioned.
k1 k2 1 D → A1 → A2 →
Scheme 3
The (area normalized) decay function for intermediate A2 is
I (θ ) =
α − k2θ k1 e −θ e − k1θ 1 − + + e , β ( k1 − 1)( k2 − 1) ( k1 − 1)( k2 − k1 ) k1 ( k2 − k1 )( k2 − 1)
(44)
where β is a constant,
β=
k1
( k1 − 1)( k2 − 1)
−
1 1 k1 + α + , ( k1 − 1)( k2 − k1 ) k2 ( k2 − k1 )( k2 − 1)
(45)
and α = A2(0)/D(0). It is assumed that A1(0) = 0. The decay (and the phasor) are thus a function of only three independent parameters: k1, k2 and α. For large α the decay is nearly exponential. For α = 0 it is unimodal. For intermediate values of α and appropriate combinations of k1 and k2 the decay, although still unimodal, has the “double-decay” character mentioned. The phasor plot of Eq. (44) with k1 = 2, k2 = 2.65, and α = 0 (unimodal decay function) is shown in Fig. 9. The corresponding curve (red line) is exterior to the universal circle (black semicircle). The curve occurs between two violet curves, the inner one corresponding to Eq. (31) (attainable in this case for a very large k2), and the outer one to the case where k1 = k2 = 1 and α = 0, whose decay function is
18
I (θ ) =
θ2 2
e −θ .
(46)
The cosine and sine transforms corresponding to this decay function, defining the outer violet line, are
G (u ) =
S (u ) =
1 − 3u 2
(1 + u 2 )
3
u (3 − u 2 )
(1 + u 2 )
3
,
(47)
,
(48)
compare Eqs. (28) and (29) with k = 1. The outer violet curve is nevertheless not the outermost curve in the case of a sum of three exponentials, but seems to be the limit for Eq. (44). Eqs. (46)-(48) can be easily generalized for a sum of n exponentials.
Figure 9. Phasor plot (red line) corresponding to a sum of three exponentials, Eq. (44), with k1 = 2, k2 =
2.65, and α = 0 (f0 = 3.212, f1 = -4.077 and f2 = 1.865). The decay function is unimodal, with I(0) = 0. The points correspond to a reduced frequency u = 2. The virtual phasor (V) resulting from the pairing of the first two exponentials (dashed blue line) occurs outside the phasor space, and to the right of the phasors. The global phasor (P) can be viewed as the combination of the virtual phasor and of the third exponential (dashed red line). The positions of the other two possible virtual phasors can be found geometrically as the intersection of the respective straight lines ( P 0 with 12 , and P1 with 02 ). The two violet lines are discussed in the text.
19
The phasor plot of Eq. (44) with k1 = 1.01, k2 = 4, and α = 0.5 (“double-decay” function) is shown in Fig. 10.
Figure 10. Phasor plot (red line) corresponding to a sum of three exponentials, Eq. (44), with k1 = 1.01, k2
= 4, and α = 0.5 (f0 = 89.78, f1 = -89.19 and f2 = 0.408). The decay function is unimodal, of the “doubledecay” type, with I(0) > 0. The points correspond to a reduced frequency u = 1.5. The virtual phasor (V) resulting from the pairing of the first two exponentials (dashed blue line) occurs inside the phasor space, and to the left of the phasors. The global phasor (P) can be viewed as the combination of the virtual phasor and of the third exponential (dashed red line). The two violet lines are discussed in the text.
2.3 Stretched and compressed exponentials
A well-known and versatile decay function is I (θ ) = exp ( −θ β ) ,
(49)
where θ = t/τ is a dimensionless time and β > 0. The kind of decay behavior according to the value of β is summarized in Scheme 4: When β < 1 the decay function is a stretched exponential [30] and the decay is sub-exponential [3]; when β = 1 the decay function is exponential; and finally when β > 1 the decay function is a compressed exponential and the decay is super-exponential [3].
20
Scheme 4
There are many examples of application of the stretched exponential to luminescence decays, either as a result of physical models, or as an empirical decay function
[1,3,30,31]. The compressed exponential, on the contrary, remains to be observed or used. A situation where it should hold was discussed in [3]. The cosine and sine Fourier transforms of Eq. (49) are:
∞
G (u ) =
1 cos ( u x ) exp ( − x β ) dx , ∫ Γ (1 + 1 / β ) 0
(50)
∞
1 S (u ) = sin ( u x ) exp ( − x β ) dx . ∫ Γ (1 + 11// β ) 0
(51)
Numerical integration is required for the computation of these transforms as a function of the dimensionless frequency u = ωτ. A suitable transformation is necessary for the computation of reliable values when parameter β is large.
Phasor plots of Eq. (49) are shown in Fig. 11 for selected values of parameter β. It is seen that for β < 1 (sub-exponential decays) the phasor remains inside the universal semicircle, as expected, whereas for β > 1 (super-exponential decays) the phasor may occur both outside and inside the universal semicircle. This example shows that super-
exponential decays do not always lie outside the universal circle, unlike what was observed for a sum of two exponentials (see 2.1). For large β, G(u) can take negative values (for high frequencies), as observed for a sum of two exponentials (see 2.1).
However, in this case the S(G) curve has a more complex, nearly spiral-like form, before reaching the (0,0) point (corresponding to a hypothetical infinite frequency) according to a nearly vertical path. The spiraling is the more complex, the larger the
value of β.
21
Figure 11. Phasor plots for Eq. (49), including a zoom of the high frequency region. The values of
parameter β are: 0.5 (red), 1 (black) and 10 (blue). For β = 10, coordinate G crosses the zero line for u = 3.3, 6.6, 9.8, 12.9, 15.9, 18.9, …
22
2.4 A versatile phosphorescence decay function A simple decay function that describes with good accuracy experimental phosphorescence decays in the presence of excited-state absorption was recently
proposed [3,4]
I (θ ) =
1 , a + (1 − a ) eθ
(52)
where θ = t/τP is a dimensionless (or reduced) time, τP is the intrinsic phosphorescence lifetime and a is a dimensionless parameter accounting for reabsorption, 1 > a ≥ 0. For very small a, i.e. negligible reabsorption, Eq. (52) 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 [2 [29] and is
super-exponential, as shown in Scheme 5.
Scheme 5
The phosphorescence decay of a species subject to triplet-triplet annihilation in fluid medium is very similar to Eq. (52), see e.g. [30],
I (θ ) =
1 , −a + (1 + a ) eθ
(53)
where a = kTT C0 τP 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. In fact, Eq. (53) always corresponds to a sub-exponential decay, see Scheme 5.
23
In the present phasor plots, Eq. (52) will be used in all cases, with parameter a taking both negative and positive values (a < 1). In this way, both sub-exponential and superexponential situations will be covered. The cosine and sine Fourier transforms of Eq. (52) are:
G (u ) = −
∞ cos ( u x ) a dx , ∫ ln(1 − a ) 0 a + (1 − a)e x
(54)
S (u ) = −
∞ sin ( u x ) a dx . ∫ ln(1 − a) 0 a + (1 − a)e x
(55)
Numerical integration is required for the computation of these transforms as a function of the dimensionless frequency u. A suitable transformation is necessary for the computation of reliable values when parameter a is close to 1. Phasor plots of Eq. (52) are shown in Fig. 12 for several values of parameter a. It is seen that for a < 0 (sub-exponential decays) the phasor remains inside the universal semicircle, whereas for a > 0 (super-exponential decays) the phasor occurs, in general, both outside and inside the universal semicircle. For a close to 1, G(u) can take negative values at high frequencies, as observed previously (see 2.1, 2.2 and 2.3). As in the compressed exponential case, the S(G) curve takes a complex, spiral-like form, before reaching the (0,0) point (corresponding to a hypothetical infinite frequency) again according to a nearly vertical path, see Fig. 12. The spiraling is the more intrincate, the closer parameter a is to unity, and nearly-periodic, arabesque-like patterns can be observed at high frequencies (Fig. 12), whose analysis is beyond the scope of this work.
24
Figure 12. Phasor plots for Eq. (52), including two zooms of the high frequency region. The values of
parameter a are: -20 (red), 0 (black) and 0.99999 (blue). For a = 0.99999, coordinate G crosses the zero line for u = 0.27, 0.55, 0.82, 1.09, 1.36, 1.64, …
25
A remark worth making in the present context is the following: It is widely recognized that the two main luminescence lifetime techniques, “frequency domain” and “time domain”, are theoretically equivalent, although in practice each has its own advantages and limitations [1,7]. However, it is not always remembered that this equivalence only exists as long as a linearity principle applies, i.e. that response of the system to successive excitations is always the same, and thus application of the convolution integral is valid. This holds only when a small fraction of the luminophore population is excited, and ground-state depletion is negligible. If a large fraction of the population goes into the excited state, then the response will be a function of this fraction, which is precisely the case for Eq. (52) [4]. The phosphorescence decay under triplet-triplet absorption results from a substantial triplet population at time zero and the response of the system to a continuous excitation is not a convolution of Eq. (52) with the excitation [4]. In this way, the phasor plot for Eq. (52) is a mathematical plot, characterizing the decay function, but will not necessarily correspond to the actual plot if a modulated excitation source is used. In such a case the actual plot will be a function of several technical parameters such as the (maximum) intensity of the source.
2.5 A decay function leading to phasors in all quadrants
The theoretical decay function for luminescence affected by excited-state absorption and with a single path length is [29]
I (θ ) = exp[−θ − B0 exp(−θ )] ,
(56)
where B0 is the initial optical thickness of the medium at the emission wavelength. This decay function is super-exponential for 0 < B0 < 1 and unimodal for B0 > 1. The maximum of the function occurs at θ = ln B0. For high values of B0 the function has negligible values at times significantly shorter than ln B0, i.e., corresponds to a “delayed” emission. The cosine and sine Fourier transforms of Eq. (56) are:
∞
B0 G (u ) = cos(ux) exp[ − x − B0 exp( − x)] dx , 1 − e− B0 ∫0
(57)
∞
S (u ) =
B0 sin(ux) exp[ − x − B0 exp(− x)] dx . 1 − e− B0 ∫0
(58) 26
The computation of these transforms as a function of the dimensionless frequency u is best done using series expansions of the integrals. Phasor plots of Eq. (56) are shown in Figs. 13-16 for several values of parameter B0. It is seen that for B0 ≤ 1 (super-exponential decay function) the phasor is outside the universal semicircle but remains in the first quadrant, whereas for B0 > 1 (unimodal decay function) the phasor acquires a spiral form and occurs in all quadrants.
Figure 13. Phasor plot for Eq. (56) with B0 = 1. Also shown is the universal circle (black line).
27
Figure 14. Phasor plot for Eq. (56) with B0 = 10.
Figure 15. Phasor plot for Eq. (56) with B0 = 50.
28
Figure 16. Phasor plot for Eq. (56) with B0 = 5000.
2.6 A multimodal decay function: Fluorescence with quantum beats
A decay function with several maxima results from the existence of quantum beats [5,6]. In order to investigate the corresponding phasor plot, Eq. (59) will be used:
I (θ ) =
1 + cos(αθ + ϕ ) −θ e , 1 + cos ϕ
(59)
where α = ω0τ, ω0 being the beat angular frequency, θ = t/τ and ϕ is the phase. This decay function applies to a four-level system when all transition moments and decay rates are identical, leading to full modulation [5]. Here, it will also be assumed, for simplicity, that the phase is zero, hence the decay function becomes
I (θ ) =
1 [1 + cos(αθ )] e−θ . 2
(60)
The cosine and sine Fourier transforms of Eq. (60) are:
29
1 u2 +1+ α 2 + , 2 4 2 2 2 2 1 + u u + 2(1 − α )u + (1 + α )
G (u ) =
1+ α 2 2 +α 2
S (u) =
1+ α 2 1 u2 +1− α 2 + u. 2 2 4 2 2 2 2 2 + α 1 + u u + 2(1 − α )u + (1 + α )
(61)
(62)
Phasor plots are shown in Figs. 17-22 for selected values of α.
Figure 17. Phasor plot for Eq. (60), with α = 1. Also shown is the universal circle.
Figure 18. Phasor plot for Eq. (60), with α = 2.
30
Figure 19. Phasor plot for Eq. (60), with α = 3.
Figure 20. Phasor plot for Eq. (60), with α = 5.
31
Figure 21. Phasor plot for Eq. (60), with α = 10.
Figure 22. Phasor plot for Eq. (60), with α = 50.
It is seen that for this multimodal decay law, S can take negative values. On the other hand, G is always nonnegative, hence in this case the phasor can be observed not only in the first but also in the fourth quadrant. A fixed pattern with a circular loop (Fig. 22) occurs for sufficiently large α. 32
3. Conclusion Luminescence decay functions show a rich diversity of patterns in their phasor plots. It was shown that the phasors of luminescence decays may fall on all four quadrants of the circular phasor space. In the case of a sum of two exponentials, a detailed study revealed that sub-exponential, super-exponential and unimodal decays have clearly different phasor signatures, in location and/or in shape. The existence of a limiting curve (outer boundary) was demonstrated and a generalization of the lever rule obtained. In the case of the sum of three exponentials, the concept of virtual phasor arose. Study of more complex decay functions, such as the stretched and compressed exponentials, and those for phosphorescence with reabsorption and triplet-triplet annihilation, showed that super-exponential decays can present complex shapes, spiraling at high frequencies while entering and leaving the sub-exponential region. A decay law of the exponential of exponential type gives rise to spirals and may display phasors in all four quadrants. A model multimodal decay function of fluorescence with quantum beats again revealed new patterns. The distinction between the phasor plot as a mathematical procedure to graphically characterize a family of decay functions, and the phasor plot as a result of “frequency domain” measurement, and their possible divergence in cases of nonlinear response, was also discussed.
Acknowledgements This work was carried out within projects PTDC/QUI-QUI/123162/2010 and RECI/CTM-POL/0342/2012 (FCT, Portugal).
33
References [1] B. Valeur and M.N. Berberan-Santos, Molecular Fluorescence. Principles and Applications, Wiley-VCH, Weinheim, 2nd ed., 2012. [2] M.N. Berberan-Santos, B. Valeur, J. Lumin. 126 (2007) 263. [3] T. Palmeira, M.N. Berberan-Santos, Chem. Phys. 445 (2014) 14. [4] T. Palmeira, M.N. Berberan-Santos, J. Lumin. 158 (2015) 510. [5] R.T. Carter, J.R. Huber, Chem. Soc. Rev. 29 (2000) 305. [6] J.J. Burdett, C.J. Bardeen, J. Am. Chem. Soc. 134 (2012) 8597. [7] D.M. Jameson, Introduction to Fluorescence, CRC Press, Boca Raton, 2014. [8] G. Weber, J. Phys. Chem. 85 (1981) 949.
[9] A. Papoulis, The Fourier Integral and its Applications, McGraw-Hill, New York, 1962. [10] R.N. Bracewell, The Fourier Transform and its Applications, 3rd. ed., McGrawHill, Singapore, 2000. [11] D.M. Jameson, E. Gratton, R.D. Hall, Appl. Spectrosc. Rev. 20 (1984) 55. [12] M. Berberan-Santos, J. Lumin. 50 (1991) 83. [13] M. Itagaki, K. Watanabe, Bunseki Kagaku 43 (1994) 1143. [14] M. Itagaki, M. Hosono, K. Watanabe, Anal. Sci. 13 (1997) 991. [15] P.J. Verveer, P.I.H. Bastiaens, J. Microsc. 209 (2003) 1. [16] A.H.A. Clayton, Q.S. Hanley, P.J. Verveer, J. Microsc. 213 (2004) 1. [17] G.I. Redford, R.M. Clegg, J. Fluoresc. 15 (2005) 805. [18] M.A. Digman, V.R. Caiolfa, M. Zamai, E. Gratton, Biophys. J. 94 (2008) L14. [19] A.H.A. Clayton, J. Microsc. 232 (2008) 306. [20] Y.-C. Chen, R.M. Clegg, Photosynth. Res. 102 (2009) 143.
34
[21] Y.-C. Chen, B.Q. Spring, C. Buranachai, G. Malachowski, R.M. Clegg, Proc. SPIE 7183 (2009) 718302. [22] C. Stringari, A. Cinquin, O. Cinquin, M.A. Digman, P.J. Donovan, E. Gratton, Proc. Natl. Acad. Sci. U.S.A. 108 (2011) 13582. [23] M. Stefl, N.G. James, J.A. Ross, D.M. Jameson, Anal. Biochem. 410 (2011) 62. [24] N.G. James, J.A. Ross, M. Stefl, D.M. Jameson, Anal. Biochem. 410 (2011) 70. [25] E. Hinde, M.A. Digman, C. Welch, K.M. Hahn, E. Gratton, Microsc. Res. Tech. 75 (2012) 271. [26] M.A. Digman, E. Gratton, in Fluorescence Lifetime Spectroscopy and Imaging: Principles and Applications in Biomedical Diagnostics (ed. L. Marcu, P.M.W. French, and D.S. Elson) CRC Press, Boca Raton, 2012. [27] F. Menezes, A. Fedorov, C. Baleizao, B. Valeur, M.N. Berberan-Santos, Methods Appl. Fluoresc. 1 (2013) 015002. [28] E. Hinde, M.A. Digman, K.M. Hahn, E. Gratton, Proc. Natl. Acad. Sci. U.S.A. 110 (2013) 135. [29] T. Palmeira, A. Fedorov, M.N. Berberan-Santos, ChemPhysChem, in press (2015). DOI: 10.1002/cphc.201402695 [30] M.N. Berberan-Santos, E.N. Bodunov, B. Valeur, Chem. Phys. 317 (2005) 57. [31] M. Berberan-Santos, E.N. Bodunov, B. Valeur, Ann. Phys. (Berlin) 17 (2008) 460.
35
36
Highlights
•
First complete study of the phasor plots of a sum of two exponentials (2E).
•
The existence of a 2E limiting curve (outer boundary) is demonstrated.
•
A generalization of the lever rule is presented.
•
Virtual phasor concept.
•
Study of several decay laws displaying a diversity of patterns.
37