Extraction of Radiative and Nonradiative Rate Constants of Super

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Extraction of the Radiative and Non-Radiative Rate Constants of Super-Radiant J-Aggregates from Emission Spectra Christopher M. Pochas J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b04326 • Publication Date (Web): 20 Jun 2018 Downloaded from http://pubs.acs.org on June 21, 2018

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The Journal of Physical Chemistry

Extraction of the Radiative and Non-Radiative Rate Constants of Super-Radiant J-Aggregates from Emission Spectra

Christopher M. Pochas,* Department of Chemistry and Biochemistry, University of Colorado Boulder, CO 50309-0215, United States Email: [email protected]

Abstract Equations relating the radiative and non-radiative decay rate constants of singlet excitons in super-radiant 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.

Introduction 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

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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 Jaggregate of PIC.11 The spectra of Figure 1 display the characteristic redshift and enhancement of the 0-0 peak typically associated with J-aggregation. This redshift 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 lineshape 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). It is the purpose of this paper 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.


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Figure 1: Steady State Spectra of PIC. Normalized spectral lineshapes for the PIC monomer10 absorption (dashed blue), monomer emission (dashed red), PIC J-aggregate11 absorption (solid blue), and J-aggregate emission (solid red).

Results According to Horng and Quitevis,11 in a J-aggregate Ncoh and krJ are related by

𝑘𝑟𝐽 = 𝑘𝑟𝑀 𝑁𝑐𝑜ℎ


(1)


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where krM is the radiative rate constant of the monomer.11 Eq. 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. Eq. 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 weak coupling limit. Vibronic coupling can be accounted for by considering another expression for super-radiance. For a J-aggregate where fluorescence is dominated by the 0-0 emission component, the radiative rate is given by 2

̂ |𝛹𝑒𝑥 ⟩| ] 𝑘𝑟𝑀 𝑘𝑟𝐽 ≈ [𝜇 −2 |⟨𝐺|𝑀

(2)

where μ is the magnitude of the monomer transition dipole moment, |𝐺⟩ is the vibrationless ̂ is the transition dipole moment operator,1 and 𝛹𝑒𝑥 is the wavefunction electronic ground state, 𝑀 of the emitting exciton. The term in brackets is the dimensionless 0−0 line strength, ̂ |𝛹𝑒𝑥 ⟩|2 ] 𝐼 0−0 ≡ [𝜇 −2 |⟨𝐺|𝑀

(3)

Inserting Eq. 3 into Eq. 2 gives 𝑘𝐽𝑟 ≈ 𝐼0−0 𝑘𝑀 𝑟

(4)

and according to ref. 1, 𝐼 0−0 = 𝐹𝑁, and so Eq. 5 is obtained. 𝑘𝑟𝐽 ≈ 𝑘𝑟𝑀 𝐹𝑁𝑐𝑜ℎ

(5) 2

F is called the generalized Franck Condon factor. It is equal 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 4 ACS Paragon Plus Environment

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frequencies in the Einstein spontaneous emission expression. Eq. 5 correctly accounts for vibronic coupling, whereas Eq. 111 did not. krJ and knrJ are often determined by measuring the fluorescence lifetime and quantum yield according to

𝑘𝑟𝐽

𝐽 𝑘𝑛𝑟

=

=

𝛷𝑓𝐽 𝜏𝑓𝐽

1 − 𝛷𝑓𝐽 𝜏𝑓𝐽

(6)

(7)

where τfJ is the fluorescence lifetime and ΦfJ is the fluorescence quantum yield.11 According to Spano and Yamagata,1 the spectral ratio 𝐼 0−0 /𝐼 0−1 is 𝑁𝑐𝑜ℎ (𝜎, 𝑇) 𝐼 0−0 = 0−1 𝜆2 𝐼

(8)

where I0-0 and I0-1 are the intensities of the 0-0 and 0-1 peaks, λ2 is the Huang-Rhys factor of the single molecule, T is the temperature and σ is the Gaussian disorder width. Eq. 8 only applies to J-aggregates, and it is only exact in the 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 the experimental data studied in this article except for the 5 ACS Paragon Plus Environment


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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, 6, 7, and 8, Eqs. 9 and 10 are obtained. 𝑘𝑟𝐽 =

𝐼 0−0 2 𝑀 𝜆 𝐹𝑘𝑟 𝐼 0−1

(9)

and

𝐽 𝑘𝑛𝑟 =

𝐼 0−0 2 𝑀 1 𝜆 𝐹𝑘𝑟 ( 𝐽 − 1) 𝐼 0−1 𝛷𝑓

(10)

Equations 9 and 10 assume that the 0-0 peak is dominant in emission. If the 0-0 peak were 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. Eqs. 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. 𝑘𝑟𝐽 = ∑ 𝜈=0

𝐼 0−𝜈 2 𝑀 𝜆 𝐹𝑘𝑟 𝐼 0−1

and


(11)

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𝐽 𝑘𝑛𝑟 =∑ 𝜈=0

𝐼 0−𝜈 2 𝑀 1 𝜆 𝐹𝑘𝑟 ( 𝐽 − 1) 𝐼 0−1 𝛷𝑓

(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-v 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. Eqs. 9-12 were tested against published experimental data obtained on various superradiant aggregates: a solution of PIC ionically bound to poly(viny1sulfonic 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) dimers,18,19 a BODIPY aggregate,20 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 Gaussian functions to the emission spectra and then integrating them. All 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 high-energy 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 7 ACS Paragon Plus Environment


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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 experimental and calculated values for krJ and knrJ. The measured values of krJ and knrJ are compared to values calculated from Eqs. 9-12 in Figure 2. The quantum yields in Table 1 were measured.

Table 1: Monomer and Aggregate Parameters Chromophore

krM/107 s-1

λ2

krJ/s-1(exp)

krJ/s-1(calcd)a knrJ/s-1(exp) knrJ/s-1(calcd)a Ncoha

Φf

_____________________________________________________________________________________ Anthracene12 Tetracene 114 Tetracene 213 PIC/PVS11,23 PIC Film17,23 MHC16 BODIPY20 BIC21,22 PDI Dimer18,19

5 3.69 3.69 27 27 5.2 1.6 78 25.7

1.1 1 1 0.605 0.605 0.7 0.39 0.45 1.12

2.5 x 108 9.5 x 107 1.2 x 108 1.3 x 109 1.1 x 109 2.7 x 108 1.5 x 108 6.7 x 109 3.7 x 108

1.1 x 108 7.8 x 107 1.2 x 108 1.2 x 109 1.1 x 109 1.2 x 108 1.4 x 108 5.1 x 109 3.8 x 108

N/A 1.2 x 1010 1.5 x 1010 5.6 x 1010 2.1 x 1010 6.5 x 109 2.4 x 109 N/A 2.8 x 107

N/A 9.6 x 109 1.5 x 1010 5.2 x 1010 2.1 x 1010 2.9 x 109 2.2 x 109 N/A 2.9 x 107

N/A 1.2 2.1 0.008 3.2 0.008 4.3 0.022 4.1 0.05 1.5 0.04 8.6 0.06 N/A 6.6 1.5 0.93

______________________________________________________________________________ a

Ncoh, krJ, and knrJ are calculated from Eqs. 8-12.


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Figure 2: Radiative and Non-Radiative Decay Rate Constants. Measured and calculated log(krJ) (circles) and log(knrJ) (triangles) at room temperature: anthracene film (●), tetracene film 1 (●/▲), tetracene film 2 (●/▲), MC nanoparticles (●/▲), PIC film (●/▲), PIC/PVS solution (●/▲), BODIPY aggregate (●/▲), BIC aggregate (●) and “D0” PDI dimer (●/▲). 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.

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 which 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 9 ACS Paragon Plus Environment


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trapped in defect sites that are bright in emission. For the purposes of this paper, only the shortlived 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 was a short-lived super-radiant emission spectrum and a long-lived non-super-radiant spectrum. The biexponential function again had short-lived 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 Non-Radiative 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 references 18 and 19, where X is the number of benzene linkers between the two 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 2

regime. In this regime, for PDI the generalized Frank Condon factor F = 𝑒 −𝜆 = 0.33. The energy gap law24 for the limit of weak coupling is given by 𝐽 𝑘𝑛𝑟 ∝ 𝑒 −𝛾𝐸𝑚

(13)

Here 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


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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 𝐽 𝑙𝑜𝑔(𝑘𝑛𝑟 ) = 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 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.

Table 2: Comparison of knrJ for PDI Dimers Dimer logknrJ (exp)

logknrJ(eq. 10)

logknrJ(eq. 14)

Em

Φf

λ2

k rM

_____________________________________________________________________________ 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

18695 18828 18835 18843

0.93 0.96 0.96 0.97

1.12 1.12 1.12 1.12

2.57 2.57 2.57 2.57

______________________________________________________________________________

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 𝑘𝑟 = 2.880 × 10−9 𝑛2 〈𝜈̃𝑓−3 〉−1 𝐴𝑣

𝑔𝑡 ∫ 𝜀 𝑑 𝑙𝑛𝜈̃ 𝑔𝑢


(15)


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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 𝑙𝑛𝜈̃, the natural logarithm of the frequency of the absorption spectrum. This analysis was carried out on the absorption spectra of a PIC J-aggregate11, 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.

Table 3: Comparison of krJ from Time-Dependent Decay, Equation 5, and Strickler-Berg

Chromophore krJ/s-1(exp)

krJ/s-1(eq 9)

krJ/s-1(eq 15)

𝜈̃𝑓 /cm-1

∫ 𝜀 𝑑 𝑙𝑛𝜈̃

λ2

krM/s-1

______________________________________________________________________________ PIC11,23 1.3 x 109 1.3 x 109 3.1 x 108 17575 11205 0.605 2.7 x 108 BODIPY20 1.5 x 108 1.4 x 108 8.8 x 107 16025 4175 0.39 1.6 x 107 BIC21,22 6.7 x 109 5.1 x 109 1.2 x 109 16830 50500 0.45 7.8 x 108 ______________________________________________________________________________

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 smaller than what was measured. The authors posit that this is because their measured rate constant k1 may be the sum of the 12 ACS Paragon Plus Environment

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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 notice shortlived 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 Eqs. 9



and 10 predict krJ and knrJ for all the dimers studied in references 18 and 19, though only the data for the shortest dimer are shown in Figure 2 for simplicity. Table 2 shows good agreement between experimentally obtained rate constants and Eq. 10 for the strongly coupled dimers, while rate constants obtained from the energy gap law (Eq. 14) appear to be comparably accurate. In the weak coupling 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 when compared with experimentally obtained rate constants. This might be expected, considering that Eq. 15 was derived with single molecules in mind, not aggregates. This contrasts with equations 8 and 9, which were found by considering J-aggregates specifically. Eq. 9 also reveals a simple relationship; the radiative rate of a J-aggregate 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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Conclusion 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 timedependent 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 strong coupling 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 Acknowledgement I thank Professor Josef Michl for all his support and guidance. This work was supported by the U.S. D.O.E. Division of Chemical Sciences, Biosciences, and Geosciences (DESC00077004).

References 1) Spano, F.C.; Yamagata, H. Vibronic Coupling in J-aggregates and Beyond: A Direct Means of Determining the Exciton Coherence Length from Photoluminescence Spectrum, J. Phys. Chem. B 2010, 11, 5133-5143. 15 ACS Paragon Plus Environment


2) Hestand, N.J.; Spano, F.C. Molecular Aggregate Photophysics beyond the Kasha Model: Novel Design Principles for Organic Materials, Acc. Chem. Res. 2017, 50, 341-350. 3) Hestand, N.J.; Spano, F.C. Determining the Spatial Coherence of Excitons from the Photoluminescence Spectrum in Charge-Transfer J-aggregates, J. Chem. Phys. 2016, 481, 262271. 4) Yamagata, H.; Spano, F.C. Strong Photophysical Similarities between Conjugated Polymers and J-Aggregates, J. Phys. Chem. Lett. 2014, 5, 622-632. 5) Yamagata, H.; Pochas, C.M.; Spano, F.C. Designing J- and H-Aggregates Through Wave Function Overlap Engineering: Applications to Poly(3-hexylthiophene), J. Phys Chem. B 2012, 120, 11394-11406. 6) Thomas, A.K.; Garcia, J.A.; Ulibarri-Sanchez, J.; Gao, J.; Grey, J.K. High Intrachain Order Promotes Triplet Formation from Recombination of Long-Lived Polarons in Poly(3hexylthiophene) J-Aggregate Nanofibers, ACS Nano 2014, 8, 10559-10568. 7) Sloniec-Myszk, J.; Resch-Genger, U.; Hennig, A. Chiral, J-Aggregate-Forming Dyes for Alternative Signal Modulation Mechanisms in Self-Immolative Enzyme-Activatable Optical Probes, J. Phys. Chem. B. 2016, 120, 877-885. 8) Akselrod, G.M.; Bawendi, M.G.; Bulovic, V.; Tischler, J.R.; Tisdale, W.A.; Walker,B.J. Device and Method for Luminescence Enhancement by Resonant Energy Transfer from an Absorptive Thin Film, U.S. Patent Application 20150153493 A1 2014.


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Extraction of Radiative and Nonradiative Rate Constants of Super

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