Phasor Representation of Monomer–Excimer Kinetics: General

Article pubs.acs.org/JPCB

Phasor Representation of Monomer−Excimer Kinetics: General Results and Application to Pyrene Liliana Martelo, Alexander Fedorov, and 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 ABSTRACT: Phasor plots of the fluorescence intensity decay (plots of the Fourier sine transform versus the Fourier cosine transform, for one or several angular frequencies) are being increasingly used in studies of homogeneous and heterogeneous systems. In this work, the phasor approach is applied to monomer−excimer kinetics. The results obtained allow a clear visualization of the information contained in the decays. The monomer phasor falls inside the universal circle, whereas the excimer phasor lies outside it, but within the double-exponential outer boundary curve. The monomer and excimer phasors, along with those corresponding to the two exponential components of the decays, fall on a common straight line and obey the generalized lever rule. The clockwise trajectories described by both phasors upon monomer concentration increase are identified. The phasor approach allows discussing in a single graphic not only the effect of concentration but also that of rate constants, including the evolution from irreversible kinetics to fast excited-state equilibrium upon a temperature increase. The obtained results are applied to the fluorescence decays of pyrene monomer and excimer in methylcyclohexane at room temperature. A straightforward method of monomer−excimer lifetime data analysis based on linear plots is also introduced.

1. INTRODUCTION A luminescence decay f unction, 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 can be area normalized, E (t ) =

I (t ) ∞

∫0 I(t ) dt

(1)

This definition has the advantage of including “decay” functions that start from zero, as occurs with the emission of intermediates, e.g., excimers, and also products of photochemical processes. The function E(t) has the meaning of the emission probability of a photon between t and t + dt, given that the photon was emitted. The cosine and sine Fourier transforms of E(t), G(ω) and S(ω), respectively, are defined by1−4 G[E] = G(ω) = S[E] = S(ω) =

∫0

∫0

Figure 1. Phasor space (unit circle), universal semicircle, and outer boundary curve for two-exponential decays. Also shown are the (truly) universal points corresponding to all decay functions for zero and infinite frequencies. Monomer (M) phasors fall in the blue area whereas excimer (D) phasors fall in the pink area (see text).

∞

cos(ωu) E(u) du

(2)

∞

sin(ωu) E(u) du

(3)

different paths between the two extreme points (ω = 0 and ω = ∞), when going from zero to infinite frequency. The outer boundary for two-exponential decays (eqs 21 and 22),5 located

where ω is the angular frequency. Each decay function, for a given frequency, is mapped onto a point inside the unit circle defined by G2 + S2 = 1, called the phasor space, Figure 1. It also follows from eqs 2 and 3 that G(0) = 1 and S(0) = 0, whereas S(∞) = G(∞) = 0, Figure 1. The universal semicircle, located in the first quadrant, defines the loci of all exponential decays. Other decay functions follow © 2015 American Chemical Society

Received: September 11, 2015 Revised: October 28, 2015 Published: November 9, 2015 15023

DOI: 10.1021/acs.jpcb.5b08875 J. Phys. Chem. B 2015, 119, 15023−15029

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The Journal of Physical Chemistry B in the first and second quadrants, sets the limit for the phasors of two-exponential decays. For a given frequency, the (G, S) pair defines a point or, equivalently, a vector P(ω) = G(ω)e1 + S(ω)e2, called the phase vector or phasor. This vector is the basis of the phasor approach to time-resolved luminescence (mainly fluorescence),2−25 which provides a simple graphical and modelindependent (when using experimental data) portrait of the kinetic properties of a system. Processes such as quenching, solvent relaxation, energy transfer, and excimer formation are defined by characteristic trajectories in the plane (at a fixed frequency or using several frequencies). 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| = (G2 + S2)1/2.1−4 When the measurement technique used is time domain fluorimetry, the phasor must be computed (numerically or analytically) from the measured decay according to eqs 2 and 3, at a conveniently chosen frequency (or set of frequencies). This decay is often distorted by the instrument response function (IRF), and the effect can be significant in phasor plots.25 To remove its effect, the experimental decay can be fitted with an empirical decay law, e.g., a sum of exponentials, from which the sine and cosine transforms are then computed. An alternative correction procedure was described recently.25 The precise location of the decay in the plane, defined by its phasor, is a function of frequency and decay characteristics, as mentioned. Single-exponential decays lie on a so-called universal circle (in fact a semicircle, for nonnegative frequencies), defined by S = (G(1−G))1/2, with 1 ≥ G ≥ 0,2−4 Figure 1. Indeed, if E (t ) =

⎛ t⎞ 1 exp⎜ − ⎟ ⎝ τ⎠ τ

results concerning monomer−excimer data analysis will also be presented.

2. MATERIALS AND METHODS Pyrene was from Koch-Light Laboratories (pure) and methylcyclohexane (MCH) (99%, spectrophotometric grade) was from Sigma-Aldrich. Time-resolved fluorescence intensity decays were obtained by the single-photon timing method with laser excitation and microchannel plate detection, with the setup described in ref 21. Excitation wavelength was 335 nm, and the emission wavelengths were 380 nm (monomer) and 540 nm (excimer). A front-face geometry was used except for the more dilute solutions. The time scale varied between 425 ps/channel for the fastest decays (2.2 mM solution) and 2 ns/ channel for the slowest decays (10−5 M). Degassing was achieved by bubbling argon previously saturated in MCH vapor. 3. RESULTS AND DISCUSSION 3.1. Monomer−Excimer Kinetics: General Results. Intermolecular excimer formation kinetics is described by the kinetic scheme shown in Scheme 1.26 Scheme 1. Monomer−Excimer Kinetics

In nonviscous solvents, where diffusional transient effects are negligible,27 the solution of the rate equations corresponding to Scheme 1 (“Birks’ kinetics”) is well-known.26 In terms of the area-normalized decays, one has

(4)

EM(t ) = f1M λ1e−λ1t + f2M λ 2e−λ2t

then eqs 2 and 3 give1−4 G(ω) = S(ω) =

1 1 + (ωτ )2 ωτ 1 + (ωτ )2

(7)

E D(t ) = f1D λ1e−λ1t + f2D λ 2e−λ2t = f1D λ1(e−λ1t − e−λ2t ) (8)

(5)

where the steady-state generalized fractional contributions of the exponential components obey f1α + f 2α = 1 (α = M, D) with (6)

Complex decays usually, but not always, fall inside the universal circle.5 In the common case of a two-exponential decay with positive amplitudes, the corresponding point falls on a straight line connecting the phasors of the two components (“tie line”).2−5 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).2−5,10−12,16 Although not widely recognized, when one of the amplitudes is negative, as is the case with excimer decays, the phasor lies outside the universal circle.5 The lever rule was recently generalized to include this case.5 The purpose of the present work is to examine monomer− excimer kinetics according to the phasor approach and to use the developed formalism in the discussion of a canonical experimental system, pyrene in homogeneous solution. New

f1M =

λ 2 (λ 2 − X ) Y (λ 2 − λ1)

f1D =

λ2 λ 2 − λ1

and where

(9)

(10)

26

1 τ1,2 1 = {X + Y ∓ [(Y − X )2 + 4kMDkDM[M]]1/2 } 2

λ1,2 =

(11)

X = kM + kDM[M]

(12)

Y = kD + kMD

(13)

The typical evolution with concentration of the decay constants λ1 and λ2 and of the monomer and excimer fractional intensities f1M and f1D is shown in Figures 2 and 3, respectively. 15024

DOI: 10.1021/acs.jpcb.5b08875 J. Phys. Chem. B 2015, 119, 15023−15029

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The Journal of Physical Chemistry B Results for the extreme situations [M] → 0 and [M] → ∞ are summarized in Table 1.

usual method of data analysis26 for the determination of the other three parameters consists in using the two decay constants λ1 and λ2 and the relative amplitude of the two exponentials in the monomer decay (eq 7) of a more concentrated solution, where excimer formation is important. This is a nonlinear system of equations, whose sequential resolution has been described,26 but which is nevertheless subject to significant propagation errors. Measurements for several concentrations are preferable. In such a case, the following linear plot is used:26 λ1 + λ 2 = Y + X = Y + kM + kDM[M]

(16)

from which kDM and Y are immediately obtained. Rate constants kD and kMD are then computed using eqs 11 and 13. A simpler procedure is nevertheless possible. Indeed28 λ1λ 2 = kMY + kDkDM[M]

Figure 2. Evolution with concentration of the decay constants λ1 and λ2 for kM = 1/383 ns−1, kD = 1/55 ns−1, kMD = 1/156 ns−1 and kDM = 7.6 M−1 ns−1. The decay constants are closest for [M] = 1.2 mM. The points are experimental results for pyrene in MCH at 25 °C (see Table 2).

(17)

From this linear plot, kD and hence kMD are obtained, but subject to significant error propagation. A third linear plot offering advantages over eq 17 is proposed here: elimination of kDM[M] from eqs 16 and 17 yields λ1 + λ 2 =

⎛ k ⎞ 1 λ1λ 2 + kD + ⎜1 − M ⎟kMD kD kD ⎠ ⎝

(18)

allowing a determination of kD and kMD with minimal error propagation, by plotting λ1 + λ2 versus λ1 λ2. 3.2. Pyrene Monomer−Excimer Kinetics in Methylcyclohexane. Molecular excimers were discovered by Förster and Kasper (1954) in pyrene solutions,29 which display a characteristic structureless emission at energies lower than that of a very dilute solution, and whose relative intensity increases with concentration, Figure 4, as expected from the mechanism

Figure 3. Evolution with concentration of the monomer and excimer fractional intensities f1M and f1D for kM = 1/383 ns−1, kD = 1/55 ns−1, kMD = 1/156 ns−1, and kDM = 7.6 M−1 ns−1. The excimer fractional contribution of the longest exponential (corresponding to λ1) has a maximum for [M] = 2.6 mM. The points are experimental results for pyrene in MCH at 25 °C (see Table 2).

Table 1. Decay Constants λ1 and λ2 (ref 26) and Monomer and Excimer Fractional Intensities f1M and f1D in the Limiting Cases of Infinite Dilution and High Concentration [M] → 0 [M] → ∞

λ1

λ2

f1M

f1D

kM kD

Y ∼ kDM [M]

1 kMD/Y

Y/(Y − kM) 1

Figure 4. Absorption (black) and fluorescence of a dilute solution (10−5 M, violet, λmax = 384 nm) and of a concentrated solution (0.019 M, blue-green, λmax = 476 nm) of pyrene in MCH at 25 °C.

The absolute difference between the decay constants λ1 and λ2 has a minimum for [M]min =

kD − (kM + kMD) kDM

(14)

shown in Scheme 1. The large separation between monomer and excimer emission, along with the forbidden nature of the S1 ← S0 transition, precluding significant reabsorption,3,30 make pyrene one of the best molecules for monomer−excimer data analysis. Pyrene monomer and excimer fluorescence decays in MCH at 25 °C were measured as a function of concentration, in degassed solutions. Two-exponential fits were satisfactory in all

whereas the excimer generalized fractional contribution f1D attains a maximum at [M]max =

(kD − kM)Y − kMkMD kDkDM

(15)

Scheme 1 contains four rate constants. One of them (kM = 1/ τM) is obtained from the decay of a very dilute solution. The 15025

DOI: 10.1021/acs.jpcb.5b08875 J. Phys. Chem. B 2015, 119, 15023−15029

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The Journal of Physical Chemistry B cases (white-noise residuals and reduced chi-squared lower than 1.1). The results are summarized in Table 2. Table 2. Two-Exponential Analysis of Fluorescence Decays of Pyrene in MCH at 25 °C monomer (380 nm) C/M 1.0 × 6.8 × 5.2 × 1.2 × 2.5 × 4.8 × 0.010 0.015 0.020 0.022

−5

10 10−5 10−4 10−3 10−3 10−3

excimer (540 nm)

τ1 (ns) (f1M )

τ2 (ns)

τ1 (ns) ( f1Da)

τ2 (ns)

383 (1.00) 302 (1.00) 190 (0.970) 118 (0.929) 74.4 (0.741) 66.5 (0.572) 59.7 (0.420) 58.1 (0.371) 56.4 (0.373) 56.6 (0.304)

− − 46.4 36.2 25.0 20.2 12.4 8.8 6.4 5.4

383(1.12) 320 (1.15) 197 (1.25) 116 (1.44) 74.7 (1.55) 67.9 (1.44) 61.1 (1.26) 58.8 (1.16) 57.7 (1.12) 56.5 (1.10)

41.6 42.4 39.1 35.7 26.4 20.6 12.6 8.2 6.2 5.3

a

Figure 6. Plot of the product of decay parameters, λ1λ2, versus concentration.

a Computed from the experimental pre-exponential factors a1 and a2, f1α = a1τ1/(a1τ1 + a2τ2). For the excimer, a1 = −a2.

It is seen that for a given concentration nearly identical lifetime components are recovered from monomer and excimer decays. The concentration dependence of the decay constants and of the fractional intensities is plotted in Figures 2 and 3, respectively. Analysis of excimer data according to eqs 16-18 is displayed in Figures 5−7. Monomer data gives very similar results. Equation 16, as shown in Figure 5, gives a good fit, with kDM = 7.6 ± 0.4 M−1 ns−1 and Y = 0.026 ± 0.010 ns−1 (uncertainties always refer to a 95% confidence level).

Figure 7. Plot of the sum of decay parameters versus the product of the same decay parameters.

For the purpose of phasor space analysis, carried out in section 3.3, the selected set of rate constants, consistent with experimental results, is kM = 1/383 ns−1, kD = 1/55 ns−1, kMD = 1/156 ns−1, and kDM = 7.6 M−1 ns−1. This set is in very good agreement with the published values for pyrene in MCH at 25 °C.31 3.3. Monomer−Excimer Kinetics in the Phasor Space. The monomer and excimer phasors (α = M, D) are given by 1 1 Gα(ω) = f1α + f2α 2 1 + (ωτ1) 1 + (ωτ2)2 (19) Sα(ω) = f1α

ωτ1 2

1 + (ωτ1)

+ f2α

ωτ2 1 + (ωτ2)2

(20)

As mentioned, the fact that the excimer decay contains an exponential term with negative amplitude implies that the respective phasor falls outside the universal circle.5 Starting from zero, and thus going always through a maximum, the excimer “decay” is an example of a unimodal decay function.5,32 In this way, when the two excimer decay constants are very close, the phasor trajectory displays an extreme behavior, nearing the outer boundary curve, S = B(G), for twoexponential decay functions,5 whose upper (S+) and lower (S−) branches are (eq 22 is equivalent to eq 36 in ref 5)

Figure 5. Plot of the sum of decay parameters, λ1 + λ2, versus concentration.

As shown in Figure 6, eq 17 also gives a good fit. From the slope, kD = 0.018 ± 0.003 ns−1, or τD = 1/kD = 55 ± 9 ns. The intercept, (0.5 ± 1.0) × 10−4, gives Y = 0.02 ns−1, close to the value obtained with eq 16, but with an uncertainty larger than the value itself. Using from the fit with eq 16, Y = 0.026 ns−1 (together with kD = 0.018 ns−1), it is obtained that kMD = (8 ± 8) × 10−3 ns−1, again with a very large uncertainty. Equation 18, as shown in Figure 7, gives an excellent fit. From the slope, τD = 55.2 ± 0.5 ns, and from the intercept, kMD = (6 ± 1) × 10−3 ns−1. The estimated uncertainties (computed assuming no correlation between the errors of λ1 + λ2 and λ1 λ2) are considerably lower than those resulting from the use of eq 17, and the best pair of equations for monomer−excimer linear data analysis appears to be formed by eqs 16 (giving kDM) and 18 (giving kD and kMD).

S +(G) =

1 (3 − 8

1 + 8G )1/2 (1 +

1 + 8G )3/2

⎛ 1 ⎞ ⎜− < G < 1⎟ ⎝ 8 ⎠

(21) S −(G) =

1 (3 + 8

1 + 8G )1/2 (1 −

1 + 8G )3/2

⎛ 1 ⎞ ⎜− < G < 0⎟ ⎝ 8 ⎠

(22)

See also Figure 1. A representative plot (2.5 mM pyrene) for monomer−excimer kinetics is shown in Figure 8. 15026

DOI: 10.1021/acs.jpcb.5b08875 J. Phys. Chem. B 2015, 119, 15023−15029

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The Journal of Physical Chemistry B

nonoverlapping intervals of variation with monomer concentration, as was already shown for their reciprocals in Figure 2. As the monomer concentration is increased, the lifetime τ1 decreases from τM to τD, whereas τ2 goes from τY to 0. Experimental results are shown in Figure 9 also for 20 MHz. It is seen that the red and blue curves computed with the set of

Figure 8. Phasor plot for a monomer−excimer system corresponding to the kinetic constants of pyrene in MCH at 25 °C with a 2.5 mM concentration. The dashed black line is the outer boundary curve for two-exponential decays (eqs 21 and 22). The angular frequency is 20 MHz.

For best displaying the results, the angular frequency chosen was 20 MHz. The phasors of the two exponential components of both monomer and excimer decays, τ1 and τ2, fall on the universal circle. A straight line segment is in this way defined (dashed green line), where the monomer (M) and excimer (D) phasors also fall, one inside the universal circle (M) and the outer outside it (D), according to the generalized lever rule.5 The excimer phasor is nevertheless inside the outer boundary line (dashed black curve). Starting with a very dilute solution (D0 and τM phasors, see Figure 8), and upon increasing the monomer concentration, the monomer and excimer phasors describe the blue and red paths, respectively. The monomer starts from the universal circle, because the decay of a dilute solution is single-exponential, but ends inside the universal circle (M∞), because the decay of a very concentrated solution is double-exponential, although one of the components, τ2, has a very short lifetime (phasor 0). The other component, phasor τD, has lifetime τD. Using the results given in Table 1 and eqs 19 and 20, it can be shown that the monomer phasor for a very concentrated solution (M∞) has components ∞ GM (ω) =

⎤ kMD 1⎡ ⎢k D + ⎥ 2 Y⎣ 1 + (ω/kD) ⎦

(23)

∞ SM (ω) =

ω / kD kMD Y 1 + (ω/kD)2

(24)

Figure 9. Pyrene monomer and excimer phasors in MCH at 25 °C (for concentrations, see Table 2). The angular frequency is 20 MHz.

rate constants already mentioned give a good description of the behavior of the system, for which previous results8 in terms of Fourier analysis were only exploratory and did not differentiate between monomer and excimer emissions. The phasor plot also provides insight into the effect of individual rate constants on the overall kinetics. When kMD is changed, the extreme situations of irreversible kinetics26 (kMD = 0), Figure 10, and fast excited-state equilibrium26,33 (large kMD), Figure 11, are clearly depicted. These situations correspond to low- and high-temperature behavior, respectively.26

In the case of the excimer phasor, it starts outside the universal circle (D0), with phasor components τM and τY (τY = 1/Y), and ends on the circle, because the decay, although double-exponential, becomes overwhelmingly dominated by the τ1 component (with lifetime τD) for large [M] (see Figure 3). Using again the results given in Table 1 and eqs 19 and 20, it can be shown that the excimer phasor for a very dilute solution (D0) has components G D0(ω) = SD0(ω) =

Figure 10. Phasor plot for a monomer−excimer system in the irreversible case. Monomer (blue) and excimer (red) curves were computed with the kinetic constants of pyrene in MCH at 25 °C, except that kMD = 0. The angular frequency is 20 MHz.

1 − ω 2 /(YkM) 2

2

[1 + (ω/kM) ][1 + (ω/Y ) ]

The relation between the monomer and excimer phasors can also be understood using convolution kinetics.27 The excimer time evolution is given by27

(25)

(1/kM + 1/Y )ω 2

2

[1 + (ω/kM) ][1 + (ω/Y ) ]

[D](t ) = kDM[M][M*](t ) ⊗ e−Yt (26)

(27)

Hence, the excimer decay is the convolution of the monomer decay with the intrinsic excimer decay (decay constant Y = 1/ τY )

It is also noted that the two exponential components of both monomer and excimer decays, τ1 and τ2, have different and 15027

DOI: 10.1021/acs.jpcb.5b08875 J. Phys. Chem. B 2015, 119, 15023−15029

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The Journal of Physical Chemistry B

phasor approach allows discussing in a single graphic not only the effect of concentration but also that of rate constants, including the evolution from irreversible kinetics to fast excitedstate equilibrium, e.g., upon a temperature increase. The obtained results were applied to the fluorescence decays of pyrene monomer and excimer in methylcyclohexane at room temperature. A straightforward method of monomer−excimer lifetime data analysis based on linear plots was introduced and applied to the same data. The obtained results are also transposable to kinetic schemes isomorphic to the monomer− excimer one, e.g., excited-state proton transfer and thermally activated delayed fluorescence.

■

Figure 11. Phasor plot for a monomer−excimer system in fast excitedstate equilibrium. Monomer (blue) and excimer (red) curves were computed with the kinetic constants of pyrene in MCH at 25 °C, except that kMD = 0.4 ns−1. The angular frequency is 20 MHz.

E D(t ) = EM(t ) ⊗ (Ye−Yt )

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

(28)

ACKNOWLEDGMENTS This work was carried out within projects PTDC/QUI-QUI/ 123162/2010 and RECI/CTM-POL/0342/2012 (FCT, Portugal). L.M. was supported by doctoral Grant SFRH/BD/78032/ 2011 (FCT, Portugal).

This convolution relation implies25 that the D, M and Y ≡ τY phasors (defined in Figure 8) obey G[D] = G[Y]G[M] − S[Y]S[M]

(29)

S[D] = G[Y]S[M] + G[M]S[Y]

AUTHOR INFORMATION

■

(30) 25

or, in terms of modulation ratio and phase shift, MD = MYMM and ΦD = ΦM + ΦY, which can be viewed as a counterclockwise rotation of the Y phasor by ΦM, followed by multiplication of its modulus by MM, Figure 12.

REFERENCES

(1) Weber, G. Resolution of the Fluorescence Lifetimes in a Heterogeneous System by Phase and Modulation Measurements. J. Phys. Chem. 1981, 85, 949−953. (2) Jameson, D. M.; Gratton, E.; Hall, R. D. The Measurement and Analysis of Heterogeneous Emissions by Multifrequency Phase and Modulation Fluorometry. Appl. Spectrosc. Rev. 1984, 20, 55−106. (3) Valeur, B.; Berberan-Santos, M. N. Molecular Fluorescence. Principles and Applications, 2nd ed.; Wiley-VCH: Weinheim, 2012. (4) Jameson, D. M. Introduction to Fluorescence; CRC Press: Boca Raton, FL, 2014. (5) Berberan-Santos, M. N. Phasor Plots of Luminescence Decay Functions. Chem. Phys. 2015, 449, 23−33. (6) Berberan-Santos, M. The Time Dependence of Rate Coefficients and Fluorescence Anisotropy for Non-delta Production. J. Lumin. 1991, 50, 83−87. (7) Itagaki, M.; Watanabe, K. Determination of Fluorescence Lifetime with Transfer Function Processed by Fast Fourier Transformation. Bunseki Kagaku 1994, 43, 1143−1148. (8) Itagaki, M.; Hosono, M.; Watanabe, K. Analysis of Pyrene Fluorescence Emission by Fast Fourier Transformation. Anal. Sci. 1997, 13, 991−996. (9) Verveer, P. J.; Bastiaens, P. I. H. Evaluation of Global Analysis Algorithms for Single Frequency Fluorescence Lifetime Imaging Microscopy Data. J. Microsc. 2003, 209, 1−7. (10) Clayton, A. H. A.; Hanley, Q. S.; Verveer, P. J. Graphical Representation and Multicomponent Analysis of Single-Frequency Fluorescence Lifetime Imaging Microscopy Data. J. Microsc. 2004, 213, 1−5. (11) Redford, G. I.; Clegg, R. M. Polar Plot Representation for Frequency-Domain Analysis of Fluorescence Lifetimes. J. Fluoresc. 2005, 15, 805−815. (12) Digman, M. A.; Caiolfa, V. R.; Zamai, M.; Gratton, E. The Phasor Approach to Fluorescence Lifetime Imaging Analysis. Biophys. J. 2008, 94, L14−L16. (13) Clayton, A. H. A. The Polarized AB Plot for the FrequencyDomain Analysis and Representation of Fluorophore Rotation and Resonance Energy Homotransfer. J. Microsc. 2008, 232, 306−312. (14) Chen, Y.-C.; Clegg, R. M. Fluorescence Lifetime-resolved Imaging. Photosynth. Res. 2009, 102, 143−155.

Figure 12. Geometrical interpretation in the phasor space of eqs 27−30.

4. CONCLUSIONS The phasor approach was applied to monomer−excimer kinetics. The results obtained allow a clear visualization of the information contained in the decays. The monomer phasor falls inside the universal circle, whereas the excimer phasor lies outside it, but within the double-exponential outer boundary curve. The monomer and excimer phasors, along with those corresponding to the two exponential components of the decays, fall on a common straight line and obey the generalized lever rule. The geometric relation between monomer and excimer phasors, formulated in terms of modulation ratios and phase shifts, was obtained from a convolution approach. The clockwise trajectories described by both phasors upon monomer concentration increase were also identified. The 15028

DOI: 10.1021/acs.jpcb.5b08875 J. Phys. Chem. B 2015, 119, 15023−15029

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DOI: 10.1021/acs.jpcb.5b08875 J. Phys. Chem. B 2015, 119, 15023−15029