Article Cite This: J. Phys. Chem. B 2018, 122, 7185−7190
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Extraction of Radiative and Nonradiative Rate Constants of SuperRadiant J‑Aggregates from Emission Spectra Christopher M. Pochas*
J. Phys. Chem. B 2018.122:7185-7190. Downloaded from pubs.acs.org by UNIV OF WINNIPEG on 07/24/18. For personal use only.
Department of Chemistry and Biochemistry, University of Colorado Boulder, Boulder, Colorado 50309-0215, United States ABSTRACT: Equations relating the radiative and nonradiative decay rate constants of singlet excitons in superradiant aggregates to the relative intensities of the 0−0 and 0− 1 peaks in the emission spectrum were derived and tested against experimental data. In many instances, these equations eliminate the need for a time-dependent fluorescence decay measurement. They can also be useful in determining the fluorescence quantum yield.
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INTRODUCTION
Figure 1 depicts the spectral characteristics of J-aggregation and super-radiance as they appear in the steady-state spectra of pseudoisocyanine (PIC) aggregates. It shows the absorption and emission spectra of a PIC monomer10 along with the absorption and emission spectra of a J-aggregate of PIC.11 The spectra of Figure 1 display the characteristic red shift and enhancement of the 0−0 peak typically associated with Jaggregation. This red shift and enhancement occur in both the absorption and emission spectra of the J-aggregate. A property central to J-aggregates and excitons in general is the spatial coherence number Ncoh, which, in the case of organic aggregates, can be defined as the number of coherently coupled chromophores. Reference 1 shows that information about Ncoh is contained within the spectral line shape of the emission spectrum of a J-aggregate. Two other central properties of J-aggregates are the radiative and nonradiative singlet exciton decay rate constants, krJ and knrJ, respectively. These constants determine the rates at which singlet-state excitons in the J-aggregate decay by emitting light (krJ) or by any other process (knrJ). The purpose of this paper is to derive and test a new set of equations for krJ and knrJ that allow their calculation from the fluorescence spectra. Often these rate constants are determined by observing the fluorescence decay lifetime; however, other methods, such as the Strickler−Berg equation and energy gap law, are also utilized. The equations derived in this paper render the lifetime measurement unnecessary in many cases if at least the first two vibronic peaks in the emission spectrum of the super-radiant aggregate can be resolved. The equations of this paper also tend to be more accurate than the Strickler−Berg equation and are comparable in accuracy to the energy gap law for those systems that were studied.
Super-radiant molecular aggregates continue to be a topic of interest theoretically,1−5 experimentally,6,7 and commercially,8,9 primarily for use in organic light emitting diode (OLED) devices. The theory of J-aggregates has become quite relevant to commercial applications recently, after studies were published showing that a single π-conjugated polymer chain can be treated theoretically like a J-aggregate,4 and that there can be J-like intermolecular interactions in polymer films.5 These findings are especially significant because many commercially available OLED devices are made from polymer films.9
Figure 1. Steady-state spectra of PIC. Normalized spectral line shapes for the PIC monomer10 absorption (dashed blue line), monomer emission (dashed red line), PIC J-aggregate11 absorption (solid blue line), and J-aggregate emission (solid red line). © 2018 American Chemical Society
Received: May 6, 2018 Revised: June 12, 2018 Published: June 20, 2018 7185
DOI: 10.1021/acs.jpcb.8b04326 J. Phys. Chem. B 2018, 122, 7185−7190
Article
The Journal of Physical Chemistry B
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limit of no disorder and T = 0 K; however, it seems to remain quite accurate even in the presence of disorder and at room temperature. The 0−0 and 0−1 line strengths are related to the squares of the matrix elements of the transition dipole moment operator and so they do not include the cubic frequency dependence from the photon density of states (see the Einstein expression for spontaneous emission). As a result, to use eq 8, the cubic frequency dependence needs to be deconvoluted from the measured spectrum. That deconvolution was carried out for all of the experimental data studied in this article except for the data from ref 18 because those authors directly reported I0−0/I0−1. This deconvolution is carried out by dividing the intensity of the spectrum at each point by the cube of the photon frequency. Spano and Yamagata only considered Frenkel excitons and long-range Coulombic coupling; however, a more recent study by Hestand and Spano3 showed that eq 8 is still valid for aggregates that have large orbital overlap couplings and significant charge-transfer excitonic character. Combining eqs 5−8, eqs 9 and 10 are obtained.
RESULTS According to Horng and Quitevis,11 in a J-aggregate, Ncoh and krJ are related by k rJ = k rMNcoh
(1)
where kr is the radiative rate constant of the monomer. 1 is only valid in the electric dipole approximation when the coherence length (the product of [Ncoh − 1] and the average intermolecular spacing) is much smaller than the optical wavelength. 1 is valid for J-aggregates with strong coupling (i.e., with a free-exciton bandwidth W that is much larger than the relaxation energy λ2ω0). However, it neglects to consider vibronic coupling, which becomes relevant in the weakcoupling limit. Vibronic coupling can be accounted for by considering another expression for super-radiance. For a Jaggregate where fluorescence is dominated by the 0−0 emission component, the radiative rate is given by 11
M
k rJ ≈ [μ−2 |⟨G|M̂ |Ψex⟩|2 ]k rM
(2)
where μ is the magnitude of the monomer transition dipole moment, |G⟩ is the vibrationless electronic ground state, M̂ is the transition dipole moment operator,1 and Ψex is the wave function of the emitting exciton. The term in brackets is the dimensionless 0−0 line strength I
0−0
−2
≡ [μ
2
|⟨G|M̂ |Ψex⟩| ]
k rJ =
and k nrJ =
(3)
Inserting eq 3 into 2 gives k rJ ≈ I 0 − 0k rM
and according to ref 1, I0−0 = FN, and so eq 5 is obtained. (5)
where F is the generalized Franck−Condon factor. It is equal 2 to e−λ in the weak-coupling limit and is near unity in the strong-coupling limit.1 It should be pointed out that eqs 4 and 5 are approximate in that they neglect the difference between aggregate and monomer transition frequencies in the Einstein spontaneous emission expression. 5 correctly accounts for vibronic coupling, whereas eq 111 did not. The parameters krJ and knrJ are often determined by measuring the fluorescence lifetime and quantum yield according to
k rJ =
τf
k nrJ =
and k nrJ =
(6)
τf J
where τf is the fluorescence lifetime and Φf is the fluorescence quantum yield.11 According to Spano and Yamagata,1 the spectral ratio I0−0/ 0−1 I is J
Ncoh(σ , T )
=
I 0−0 I 0−1
(8)
where I and I are the intensities of the 0−0 and 0−1 peaks, respectively, λ2 is the Huang−Rhys factor of the single molecule, T is the temperature, and σ is the Gaussian disorder width. 8 only applies to J-aggregates, and it is only exact in the 0−0
(10)
I 0−ν 2 M λ Fk r I 0−1
(11)
yz I 0 − ν 2 Mijj 1 λ Fk r jj J − 1zzz 0−1 jΦ z I k f {
(12)
A simple summation of intensities in eqs 11 and 12 only approximates the rate constants since the cubic frequency dependencies occurring in the Einstein spontaneous emission expression for each 0−ν emission component have been omitted. Since the optical transition frequency is typically much greater than the vibrational frequency defining the vibronic progression, the approximation is usually a good one. Carrying out the deconvolution of the cubic frequency dependence from the measured spectrum, as discussed above, is a remedy for this approximation. 9−12 were tested against published experimental data obtained on various super-radiant aggregates: a solution of PIC ionically bound to poly(vinylsulfonic acid) sodium salt (PVS),11 anthracene thin films,12 tetracene thin films,13−15 zwitterionic Meisenheimer complex (MHC) nanoparticles,16 a PIC film,17 a series of covalently linked perylene diimide (PDI)
(7)
J
λ
∑ ν=0
J
1 − ΦfJ
2
∑ ν=0
ΦfJ
k rJ =
yz I 0 − 0 2 Mijj 1 zz j λ − Fk 1 j zz r j ΦJ I 0−1 f k {
(9)
9 and 10 assume that the 0−0 peak is dominant in emission. If the 0−0 peak was not dominant, the rate constants would depend on the sum (I0−0 + I0−1 + I0−2 + ...). This becomes apparent when the process of emission is considered in terms of conservation of energy; the rate at which the aggregate radiates energy away must depend on the intensity of each spectral peak. 9 and 10 are limiting cases, applicable when the 0−0 peak dominates the emission spectrum. In the case of a spectrum where the 0−0 peak is comparable in intensity to other peaks, eqs 11 and 12 should be used.
(4)
k rJ ≈ k rMFNcoh
I 0−0 2 M λ Fk r I 0−1
0−1
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DOI: 10.1021/acs.jpcb.8b04326 J. Phys. Chem. B 2018, 122, 7185−7190
Article
The Journal of Physical Chemistry B Table 1. Monomer and Aggregate Parameters krM/107 s−1
chromophore 12
anthracene tetracene 114 tetracene 213 PIC/PVS11,23 PIC film17,23 MHC16 BODIPY20 BIC21,22 PDI dimer18,19 a
Ncoh, krJ,
5 3.69 3.69 27 27 5.2 1.6 78 25.7
λ2
krJ/s−1 (exp)
krJ/s−1 (calcd)a
1.1 1 1 0.605 0.605 0.7 0.39 0.45 1.12
× × × × × × × × ×
× × × × × × × × ×
2.5 9.5 1.2 1.3 1.1 2.7 1.5 6.7 3.7
8
10 107 108 109 109 108 108 109 108
1.1 7.8 1.2 1.2 1.1 1.2 1.4 5.1 3.8
8
10 107 108 109 109 108 108 109 108
knrJ/s−1 (exp) N/A 1.2 × 1.5 × 5.6 × 2.1 × 6.5 × 2.4 × N/A 2.8 ×
1010 1010 1010 1010 109 109 107
knrJ/s−1 (calcd)a N/A 9.6 × 1.5 × 5.2 × 2.1 × 2.9 × 2.2 × N/A 2.9 ×
109 1010 1010 1010 109 109 107
Ncoha
Φf
1.2 2.1 3.2 4.3 4.1 1.5 8.6 6.6 1.5
N/A 0.008 0.008 0.022 0.05 0.04 0.06 N/A 0.93
and knr are calculated from eqs 8−12. J
dimers,18,19 a BODIPY aggregate,20 and a 1,1′-diethyl-3,3′-bis(3-sulfopropyl)-5,5′,6,6′-tetrachlorobenzimidazolocarbocyanine (BIC) aggregate.21,22 The anthracene and MHC aggregates had 0−1 peaks comparable in intensity to their 0−0 peaks, so eqs 11 and 12 were used for those two aggregates only, truncating at ν = 1. In some cases, peak intensities were reported by the authors, otherwise the intensity of each peak was determined by fitting the Gaussian functions to the emission spectra and then integrating them. All of the aggregates presented in Table 1 fall into the regime of a large free-exciton bandwidth (W ≫ λ2ω0), where the generalized Franck−Condon factor F is near unity. 1 Bandwidths were extracted from the absorption spectra of the aggregates. This method yields an underestimate of W because super-radiant J-aggregates have dark states at the highenergy end of the band, which contribute to W but are not visible in the absorption spectra. Table 1 gives the values of krM, which were either reported or calculated using an equation similar to eq 6, the values of λ2, which were either reported or calculated from the monomer absorption spectrum, the calculated values of Ncoh, and the experimental and calculated values of krJ and knrJ. The measured values of krJ and knrJ are compared to the values calculated from eqs 9−12 in Figure 2. The quantum yields in Table 1 were measured. In the anthracene experiment, different fluorescence spectra were detected at different times: a short-lived spectrum that exhibits super-radiance and a long-lived spectrum that does not. The super-radiant spectrum comes from excitons, which decay radiatively while in the crystalline phase, whereas the long-lived spectrum comes from excitons, which have become trapped in defect sites that are bright in emission. For the purposes of this paper, only the short-lived spectrum was used. Similarly, in the tetracene experiment,13 different decay rates and spectra were detected from the same sample, and a biexponential function had to be used to fit the decay curve. There were a short-lived super-radiant emission spectrum and a long-lived non-super-radiant spectrum. The biexponential function again had short- and long-lived components. However, the short-lived exponential function was major, with a much larger coefficient, and was the only component considered. As with anthracene, these effects are attributed to exciton migration to non-super-radiant defect sites. Comparison with Nonradiative Rate Constants Calculated from the Energy Gap Law. Here, knrJ is calculated using the energy gap law24 for the series of PDI dimers of ref 18. These rate constants are then compared to those calculated from eq 10, to test it against another method of determining knrJ. The dimers are named “DX” in refs 18 and 19, where X is the number of benzene linkers between the two
Figure 2. Radiative and nonradiative decay rate constants. Measured and calculated log(krJ) (circles) and log(knrJ) (triangles) at room temperature: anthracene film (orange circle solid), tetracene film 1 (blue circle solid/blue triangle up solid), tetracene film 2 (●/▲), MC nanoparticles (gray circle solid/gray triangle up solid), PIC film (red circle solid/red triangle up solid), PIC/PVS solution (purple circle solid/purple triangle up solid), BODIPY aggregate (pink circle solid/ pink triangle up solid), BIC aggregate (sky blue circle solid), and “D0” PDI dimer (green circle solid/green triangle up solid). krJ and knrJ are reported in units of inverse seconds. Quantum yields were not reported for the anthracene film and the BIC aggregate, so knrJ is unknown for those systems. The rate constants are reported in units of s−1.
PDI chromophores, and that naming convention is used here. According to the authors of ref 18, the dimers D2 and D3 fall into the weak-coupling regime. In this regime, for PDI, the 2 generalized Frank−Condon factor F = e−λ = 0.33. The energy gap law24 for the limit of weak coupling is given by k nrJ ∝ e−γEm
(13)
where Em is the energy at which the maximum of the emission spectrum occurs and γ is an empirically determined constant. In this case, because the rate constants were previously determined by observing the time-dependent decay of emission intensity, a constant of proportionality C can be found, and eq 13 can be rewritten as log(k nrJ) = C − γEm
(14)
By fitting a curve described by eq 14 to the data from ref 18, it is determined that C = 57.8 and γ = 0.00269 cm when Em has 7187
DOI: 10.1021/acs.jpcb.8b04326 J. Phys. Chem. B 2018, 122, 7185−7190
Article
The Journal of Physical Chemistry B Table 2. Comparison of knrJ for PDI Dimers dimer
log knrJ (exp)
log knrJ (eq 10)
log knrJ (eq 14)
Em/cm−1
Φf
λ2
krM/s−1
D0 D1 D2 D3
7.44 7.14 7.09 6.96
7.46 7.16 6.62 6.48
7.44 7.08 7.06 7.04
18 695 18 828 18 835 18 843
0.93 0.96 0.96 0.97
1.12 1.12 1.12 1.12
2.57 2.57 2.57 2.57
× × × ×
108 108 108 108
Table 3. Comparison of krJ from Time-Dependent Decay, eq 5, and Strickler−Berg Equation chromophore 11,23
PIC BODIPY20 BIC21,22
krJ/s−1 (exp)
krJ/s−1 (eq 9)
krJ/s−1 (eq 15)
ν̃f/cm−1
∫ ε d ln ν̃
λ2
krM/s−1
1.3 × 10 1.5 × 108 6.7 × 109
1.3 × 10 1.4 × 108 5.1 × 109
3.1 × 10 8.8 × 107 1.2 × 109
17 575 16 025 16 830
11 205 4175 50 500
0.605 0.39 0.45
2.7 × 108 1.6 × 107 7.8 × 108
9
9
8
units of cm−1. Rate constants are calculated using eq 14 and compared in Table 2 to those calculated from eq 10 and those obtained from measuring the fluorescence decay lifetime. Comparison with Radiative Rate Constants Calculated from the Strickler−Berg Equation. Here, krJ is calculated using the Strickler−Berg equation25 for various aggregates. These rate constants are then compared to those calculated from eq 9, to test it against another method of determining krJ. The Strickler−Berg equation can be written as g k r = 2.880 × 10−9n2⟨νf−̃ 3⟩−Av1 t ε d ln ν ̃ gu (15)
and long-lived decay components in the fluorescence, so it is likely that the results of eq 11 are being confounded by a second non-super-radiant crystal phase or defect sites in the MHC aggregates. The crystalline aggregates discussed in this article (anthracene, tetracene, and the MHC aggregate) are not strictly J-aggregates; however, they are super-radiant and geometrically J-like enough that eqs 9−12 are still quite accurate when applied to these systems. For the PIC film experiment,17 a large uncertainty was reported for krJ due to a large experimental error in measuring ΦfJ. This scenario illustrates another use for the equations presented in this writing. In situations like this, where τfJ can be determined with high accuracy (τfJ = 40 ± 10 ps), but Φf cannot (ΦfJ = 0.005 + 0.05), eqs 6 and 9 can be used to determine ΦfJ. Using this method yields a result ΦfJ ≈ 0.05 for the layer-by-layer PIC aggregate of Figure 2 in ref 17, and that value was used here. Figure 2 contains the relevant data for one of the PDI dimers used in spectroscopic studies of a series of PDI dimers with covalent linkers systematically varied in length.18,19 9 and 10 predict krJ and knrJ for all of the dimers studied in refs 18 and 19, although only the data for the shortest dimer are shown in Figure 2 for simplicity. Table 2 shows a good agreement between experimentally obtained rate constants and eq 10 for the strongly coupled dimers, whereas rate constants obtained from the energy gap law (eq 14) appear to be comparably accurate. In the weakcoupling limit (D2 and D3), the energy gap law is a better predictor of the rate constants. However, to determine the constant of proportionality C in eq 14, another separate method of determining knrJ is required, which is not necessary for eq 10. The range of Em presented in Table 2 might be considered small if one were comparing the spectra of different monomers; however, in this case, the comparison is between aggregates of the same monomer. Because these aggregates are disorder-free and their spacing is tightly controlled, small variations in the intermolecular distance naturally lead to small variations in the intermolecular coupling, Em, and both krJ and knrJ. With regard to the comparison of the equations of this paper to the Strickler−Berg equation, Table 3 shows that eq 9 tends to give more accurate results than the Strickler−Berg equation compared to experimentally obtained rate constants. This might be expected, considering that eq 15 was derived with single molecules in mind, not aggregates. This contrasts with eqs 8 and 9, which were found by considering J-aggregates specifically.
∫
where n is the index of refraction (n = 1.33 in each case presented here), ν̃f is the fluorescence frequency (taken to be the frequency of the 0−0 emission peak), gt and gu are the degeneracies of the lower and upper states, respectively, and ε is the molar absorptivity coefficient, which is integrated with respect to ln ν̃, the natural logarithm of the frequency of the absorption spectrum. This analysis was carried out on the absorption spectra of a PIC J-aggregate,11 as well as mesotrifluoromethyl BODIPY20 and BIC aggregates.21,22 In each of these references, the investigators included radiative rate constants obtained from time-dependent decay curves of the fluorescence intensity. Rate constants calculated from eqs 9 and 15 are compared to those experimentally measured in Table 3.
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DISCUSSION The rate constants obtained from fluorescence decay curves are in good agreement with the rate constants calculated from eqs 9−12, except in the cases of the anthracene aggregate, and to a lesser extent, the MHC aggregate (Figure 2). The investigators who examined anthracene12 also had a theoretical model, which produced a rate constant lower than what was measured. The authors posit that this is because their measured rate constant k1 may be the sum of the radiative rate constant and another rate constant which represents exciton migration to defect sites, which do not emit. This contribution from dark sites was not confirmed experimentally, but the authors’ theoretical model supports this explanation for the disagreement between the theoretical and measured rate constants. The rate constants calculated for the MHC aggregate appear to be slightly less accurate than those calculated for other aggregates. This could be a result of a complex crystal structure in the MHC nanoparticles. The authors of ref 16 state that the complex crystal structure makes it difficult to predict whether a J or H aggregate will form. Additionally, they also note short7188
DOI: 10.1021/acs.jpcb.8b04326 J. Phys. Chem. B 2018, 122, 7185−7190
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9 also reveals a simple relationship; the radiative rate of a Jaggregate is proportional to λ2. This is useful for guiding the selection of new materials for OLED devices, where a high radiative rate is desirable.
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CONCLUSIONS Equations relating krJ and knrJ to the steady-state fluorescence spectrum were derived and allow a calculation of these rate constants from the emission spectrum. The equations were tested against experimental data from a variety of systems and found to be accurate. They are useful for guiding experiment, because in many cases, they eliminate the need to perform time-dependent measurements. Furthermore, they are more accurate than the Strickler−Berg equation for the systems studied in this article. The equations of this paper are comparable in accuracy to the energy gap law in the strongcoupling limit. They can also be used to accurately determine ΦfJ in cases where τrJ is known to high accuracy, but ΦfJ is not.17 Additionally, because krM can be calculated accurately,26 as can the spectral ratio, I0−0/I0−1,1,3 these equations provide a straightforward method of calculating the radiative rate constants of J-aggregates purely theoretically. Finally, this paper and the published experimental results11−23 together provide experimental verification for eq 8.1
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Christopher M. Pochas: 0000-0002-2747-9803 Notes
The author declares no competing financial interest.
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ACKNOWLEDGMENTS The author thanks Professor Josef Michl for all his support and guidance. This work was supported by the U.S. DOE Division of Chemical Sciences, Biosciences, and Geosciences (DESC00077004).
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REFERENCES
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