Dynamic broadening alters triplet extinction coefficients in fluorene oligomers and polymers

Conjugated polymers are key components in “plastic electronics” due to their ability to act as both conductors for charges and excitons and their ability to absorb or emit photons. Understanding of these properties is key for improving the design of conjugated polymers used in applications such as organic photovoltaics (OPV),^1–3 organic light emitting diodes (OLEDs),^4–12 and molecular electronics.¹³ Derivatives of polyfluorenes have been applied in these areas, as well as other novel optoelectronic applications.^14–17 Excitons and excess charges on these materials are known to be delocalized over a number of repeat units as polarons, depending on the nature of the polymer.^18–22 The importance of delocalization has led several groups to examine how oligomer length affects the properties of charges and excitons in solution.^23–30 A particular interest in triplet excitons stems from their potential to support alternative visions of triplet-based OPV utilizing long distance triplet transport.^9,31

A valuable aid to understanding has been the oligomer extrapolation approximation,^27,32 which notes that many properties, for example, excited state absorption spectra, fluorescence, and phosphorescence spectra, shift consistently with oligomer length until they approach an effective conjugation length or delocalization length of the polaron.³³ Beyond this length, many properties become constant, taking on value characteristic of the corresponding polydisperse polymers. This dependence is widely fit in terms of models having roots in a particle in a 1D box,³⁴ using a functional form of 1/n or e⁻ⁿ. A modification to such universal scaling was recently proposed by Chen,³⁵ who showed that properties such as triplet free energies and redox potentials have a component due to entropic effects, which results in continued shifts past the delocalization length producing substantial differences between oligomers and polymers.

Triplet excited states of conjugated polymers are of particular current interest because they can efficiently carry excitation energy over long distances on conjugated chains within their lifetime.^36–40 By contrast, singlets typically have exciton diffusion lengths less than 10 nm^41–47 although recent work established that they can be as much as 34 nm in solution.⁴⁸ Triplets in conjugated polymer have been shown to be more localized than singlets in both liquid^27,49–52 and solid phases.^29,53 Their motion along and between chains is governed by electronic coupling through the Dexter exchange mechanism. The short range of Dexter couplings might be expected to make triplet motion very sensitive to the polymer structure and to defects. Wasserberg examined length-dependence of T_n ← T₁ absorption spectra of oligofluorenes and polyfluorenes in low temperature glasses,²⁹ but there are no reports of the corresponding extinction coefficients, necessary for quantitative probes of dynamics and transport of triplets. Triplet formation is an energy loss mechanism in OPV and OLEDs, and extinction coefficients are needed to know how many are formed. In solution, excitation of conjugated polymers in the presence of electron donors (D) or acceptors (A) or with D or A attached yields rich photophysics including electron transfer (ET) reactions that may produce triplets.^40,52 In polymer films, triplet formation often has a central role. Quantitative interpretation of all these experiments requires knowledge of the extinction coefficients of the triplets. Indeed, knowledge of triplet extinction coefficients might often be key to understanding such experiments. While spectra of many triplets have been examined, extinction coefficients are less available. For many conjugated polymers, low intersystem crossing yields make measurements of extinction coefficients of triplets difficult. This work reports room temperature measurements in solution of triplet extinction coefficients, triplet transfer rates, and T_n ← T₁ spectra for 9,9-dihexyl-fluorene oligomers (oF_n) and polymers (pF_n) with lengths of n = 2–84 repeat units. The method used is pulse radiolysis, which can produce triplets quickly and in large yield without the need for singlet precursors.⁵⁴ 

During these measurements, we found that triplet extinction coefficients have an unusual dependence on oligomer length and that spectral widths provide information on the motion of triplets on chains. The results also seek to understand a puzzle. Computation finds triplets to have short delocalization length in part because their optimized geometries have one flat (∼0°) dihedral angle. Movement of a triplet exciton would thus require movement of this substantial deformation which should be difficult, so transport might be slow. Triplet transport appears to be fast however. The results below will provide understanding of this puzzle.

Synthesis and characterization of oligomeric and polymeric 2,7-(9,9-dihexylfluorenes) followed established methods^28,55,56 and were described previously.^20,57 Oligomer samples were synthesized stepwise to have chains with identical length without fractional separation for length, denoted oF_n, where n is the length in fluorene repeat units. Longer polymers, which are polydisperse, were separated into narrowed length distribution fractions with different average lengths, n, by preparative scale GPC, similarly denoted pF_n. Average lengths of each fraction were determined by multiangle light scattering (MALS) using the oligomers as molecular weight standards. Polydispersities of pF_n fractions ranged from 1.5 to 1.9. Samples were prepared in an inert argon atmosphere glovebox in 0.5 cm path length sealed Suprasil spectrophotometric cells. Solvents used were benzene, toluene, or p-xylene from Aldrich, dried over 3A molecular sieves. Benzophenone (Aldrich) was sublimed. Steady-state absorption spectra of oF_n have been previously reported in tetrahydrofuran²² and are given along with those for pF_n in toluene at 21 and −78 °C in the supplementary material (Fig. S0).

Polymer triplets were produced using pulse radiolysis at Brookhaven National Laboratory’s Laser-Electron Accelerator Facility (LEAF).⁵⁸ The experiments used <50 ps pulses of ∼9 MeV electrons giving an absorbed dose of about 25 (±30%) Gy in 0.5 cm cells, producing 3–4 μM solute triplets. Probe light from a pulsed xenon arc lamp, with wavelengths selected by 10 nm bandpass interference filters with 10 nm increments between filters, was detected with a silicon EG&G FND-100Q detector and digitized with either a LeCroy Waverunner HRO 640ZI or HRO 66Zi oscilloscope, giving approximately 1 ns time resolution. Data analysis used the Igor Pro program, with task specific fitting functions.

Electronic structure computations used the Gaussian09 and Gaussview5 programs.^59,60 All geometries were determined using density functional theory (DFT) with the B3LYP functional and 6-31(d) basis set. Triplet excited state transitions were modeled using time-dependent DFT (TD-DFT). No solvation models were used.

Pulse radiolysis can efficiently produce triplet excited states, even for molecules with low triplet quantum yields, and is widely used to determine their extinction coefficients.^54,61 The current experiments produced 3–4 μM concentrations of ³oF_n^* or ³pF_n^* by transfer from a high concentration solute, as shown in Scheme 1. Ionization (1) of solvent molecules was followed by rapid capture (2) of electrons primarily by 100 mM benzophenone (Bzph). Recombination (3) of Bzph^•− with Bz^•+ resulted in production of Bzph^*. About half of these ion-recombination events are expected to form triplets.⁶² The other half form Bzph singlets, ¹Bzph^*, which intersystem cross to triplets (4) in 30 ps with unit quantum yield.⁵⁴ The large benzophenone concentration prevents most recombination of e− with Bz^•+, which produces short-lived singlet and triplet Bz^*, but any Bz^* made by recombination or direct excitation will also transfer rapidly to make Bzph^*. Bimolecular energy transfer, equation (5), forms triplets of oligofluorenes or polyfluorenes. The concentrations of oF_n and pF_n used were kept low so that most electrons were captured by Bzph but high enough that capture of more than 1 triplet per molecule is unlikely, avoiding significant intrachain triplet-triplet annihilation.^38,39 In the longest chains, less than 1% may attach more than 1 triplet; for others, the fraction becomes vanishingly small. Initial experiments were performed in toluene or p-xylene. Triplets of benzophenone were found to have shortened lifetimes due to abstraction of H atoms from these solvent molecules, so benzene was a better alternative when low concentrations of oligomers or polymers were used. Similar results were obtained in any of the three aromatic solvents if corrections were applied for the lower yields of triplets transferred to oF’s and pF’s in toluene and p-xylene.

Figure 1 shows typical pulse radiolysis data. Each trace is an average of 4 accelerator shots, with per shot correction for electron pulse size fluctuations using a faraday cup after the sample. The concentration of ³Bzph^* produced was determined at its peak absorbance in benzene at 530 nm, using the widely accepted value of ε(530 nm, benzene) = 7220 M⁻¹ cm⁻¹.⁶³ oFn and pFn were measured in benzene for direct comparison with ³Bzph^* and to avoid losses due to H atom abstraction. Older reported values of ε(³Bzph^*) in benzene were generally larger^61,64,65 and would result in lower ³oF_n^* or ³pF_n^* determinations. The data were fit to yield the oF_n or pF_n extinction coefficient and the rate of triplet transfer from ³Bzph^* to the solute. Additional details of the fits are given in Sec. S1 of the supplementary material.

Figure 2 shows T_n ← T₁ absorption spectra in toluene for oF₁₀ and pF₂₈ along with those previously reported for oF₂-oF₆.³⁵ Spectra for pF₅₇ and pF₈₄ are identical in shape and position to that for pF₂₈ but have lower extinction coefficient and are not shown. All spectra are assumed to be identical in different aromatic solvents, confirmed with oF10 in benzene, toluene, and p-xylene. Each spectrum is scaled here to match the peak extinction coefficients determined in benzene below. The spectra were recorded following pulse radiolysis in toluene at 293 K by a similar method to that in scheme 1, using 30 mM acetophenone instead of benzophenone, and added 10 mM biphenyl or naphthalene to rapidly make an intermediate long-lived aromatic triplet, which subsequently transferred to the oligofluorenes and polyfluorenes with high yield. The spectra show a large red shift from oF₂ to pF₂₈, accompanied by noticeable broadening of the bands to oF₁₀, but a slight narrowing for pF₂₈. Wasserberg et al.²⁹ reported a similar trend for oF₃, oF₅, and pF in 80 K methyl-THF glass although the 80 K spectra peak at slightly longer wavelengths. The spectrum for pF₂₈ is consistent with previous reports for polyfluorene.^37,66 Longer polyfluorenes showed no significant additional red-shift or changes in width. Note that ³oF₂^*, peaking at 541 nm, is also red shifted compared to ³fluorene^*, which has been reported to peak at 376 nm in hexane.⁶⁷ Additional analysis of the spectral shift with polymer length is given in Sec. S2 (supplementary material).

Oscillator strengths, f_n, and transition dipole moments, μ_n, for the T_n ← T₁ transition were determined from the absorption spectra in Fig. 2. The oscillator strength gives a measure of the total transition probability, integrated over the absorption band in wavenumbers, $υ̃$⁠, determined using the following equation:⁶⁸ 

where n₀ is the sample index of refraction. Kedenburg⁶⁹ determined n₀ in toluene to be 1.501–1.487 at the peak wavelength of each spectrum. The magnitude of the transition dipole moment between the T₁ and T_n states is related to f_n by the following equation:⁶⁸ 

Figure 3 plots f_n and μ_n as functions of chain length, n. Results for pF₅₇ and pF₈₄ are not included to highlight changes in the oligomers. These long polymers have smaller f and μ due only to differences in extinction coefficient from pF₂₈ as they have identical spectra and are suggested in the discussion to be due to chain defects. For the oligomers, both f_n and μ_n grow rapidly with n from oF₂ and approach a maximum value for oligomers at 10 repeat units long, saturating at a similar length as the T_n ← T₁ absorption spectral shift. Strong absorption by ground-state polymer prevented measurement for wavelengths less than ∼400 nm truncating the triplet spectra on the high energy side. Neglecting any new peaks at wavelengths <400 nm, this truncation decreases the oscillator strengths by 10% at most. The truncation will affect results for the shortest oligomers, oF₂ the most; however, the trends seen in Fig. 3 would not be strongly perturbed. The values for pF₂₈ appear to be 3%–8% lower than that for oF₁₀. This decrease may be explainable by chain defects, as explored in Sec. IV. The magnitudes of the transition dipole moments rival those for S₁ ← S₀ absorptions^26,30 and are known to be oriented along the polymer chain.⁴⁹ 

Determination of T_n ← T₁ absorption extinction coefficients, ε, was made in benzene for fluorene oligomers, oF_n, of lengths n = 2–6, 10, polyfluorenes, pF_n, with average lengths of n = 28, 57, and 84 units, plus two reference molecules. Results are summarized in Table I. These determinations depend on electron pulse size, which can fluctuate by ±10%, corrected by the use of a faraday cup. To provide the most internally consistent and reliable set of extinction coefficients across all samples, individual determinations used an average of 4 shots each and were further averaged by cycling through all samples, including reference samples, 4–9 times on 1 or 2 different days. Because accuracy of the determinations is best with conditions described in Sec. II, the repeat unit concentrations of oF_n and pF_n used were mostly kept near 3 mM, giving triplet capture times within a factor of about 2 of each other for all lengths. Errors noted are standard deviations of all determinations per sample. Overall accuracy is estimated at ±10%, including the reported approximate 4% accuracy of ε for the reference, ³Bzph^*.⁶³ 

The extinction coefficients for p-terphenyl and tetrathiophene determined in this work are also estimated to have a small error, ±10%, and are well within the ranges previously published. Reports of ε for tetrathiophene range widely, from 39 000–75 000 M⁻¹ cm⁻¹ in 1,4-dioxane,^70,71 with the value determined in the present work in the middle. For p-terphenyl, values of ε are scattered, reported in both benzene (ε = 90 000 M⁻¹ cm⁻¹, 460 nm)⁶¹ and hexane(ε = 75 000 M⁻¹ cm⁻¹, 440 nm).⁶⁷ Triplet extinction coefficients can depend remarkably on solvent. Bensasson compared T_n ← T₁ absorption spectra of naphthalene and anthracene in cyclohexane and benzene, finding that broader and ∼10 nm red-shifted spectra in benzene accompanied by a 30%–50% decrease in the extinction coefficient.⁶¹ This decrease is apparently opposite to that reported for p-terphenyl. While the present experiments found the TT spectrum for terphenyl to red-shift in benzene compared to that in hexane, its relatively broad width was not found to be sensitive to solvent (see Fig. S3 of the supplementary material). A likely explanation is that the bandwidth is dominated by transient variations of dihedral angles between phenyl rings in solution which is similar in both solvents; thus, samples are likely to have a similar extinction coefficient in different solvents. The lack of solvent dependence is expected to be true for all of the molecules in this study, with the effect of dihedrals explored further in Sec. IV.

If, like other properties, T_n ← T₁ extinction coefficients were expected to follow a trend given by the oligomer extrapolation approximation described in the Introduction, they would be expected to increase with length to a maximum value and then remain constant for longer oligomers and polymers. Figure 4 plots ε verses chain length; lines are added to note trends in the data. We find the expected increase in ε for short oligomers, oF₁ to oF₃; however, for longer oligomers, ε decreases nearly linearly with length, up to 13% by oF₁₀. The shortest polymer tested, pF₂₈, breaks the trend set by oligomers, having only a slightly smaller ε than oF₁₀. After this, ε continues to drop with a different trend, up to an additional 30% for a polymer ∼3× longer. Such decreases in the triplet-triplet extinction coefficient with length are unexpected. The total ∼40% drop from oF₃ to pF₈₄ may result in incorrect conclusions in studies where more than 1 length of polymer are compared or those where polymers have broader length distributions.

These measurements also determined the rate constants for triplet transfer (k_TT) from ³Bzph^* to oF_n and pF_n shown in Table I and plotted in Fig. 5. k_TT increases approximately linearly with length, n, but with k_TT ∝ 0.44 * n. These data are similar within the experimental error to that previously reported for triplet transfer from ³biphenyl^* to a narrower range of polyfluorenes with average lengths from 20 to 80 units, also showing a linear trend.³⁶ The addition of short oligomers in the current work show changes in rate that are consistent with those for longer polymers and likely are occurring at less than diffusion controlled rates.

The oligomer extrapolation approximation might suggest that T_n ← T₁ extinction coefficients, ε, oscillator strengths, f_n, and transition dipole moments, μ_n, would all increase to a maximum value at some optimal length and then remain constant as chain length continued to increase. This optimal length might plausibly be the delocalization length, n_D, of the T₁ state, which is often small.^{27,29,49,52,53,72,73} For oligofluorene triplets, Chen³⁵ reported n_D = 3.0 repeat units, which is shorter than delocalization lengths for anions, cations, and singlets in the same materials.²² Computations give a similar n_D, with a central dihedral angle of ∼0° irrespective of oligomer length up to oF₁₀ [B3LYP/6-31g(d), Sec. S5, supplementary material]. Results in Table I show that extinction coefficients increase from oF₂ to oF₃ but surprisingly decrease at longer lengths. Figure 3 shows that the oscillator strengths, f_n, and transition dipole moments, μ_n, increase to a maximum near a chain length of 10 units. They appear to follow the oligomer extrapolation approximation, but with a length of 10, much larger than the triplet delocalization length. The increases in f_n and μ_n to n = 10 repeat units, not n = 3, occur because the length of the T_n state increases until n ∼ 10, as suggested in the computed results (see Sec. S5 of the supplementary material).

If T_n ← T₁ spectral widths were constant, Eq. (1) predicts f_n to increase proportionately to ε, and Eq. (2), with correction for the position, predicts μ_n to increase as the square root of f_n. Because the observed results do not follow these predictions, results instead point to the dominant role of spectral widths. Going from oF₃ to oF₁₀, ε in Fig. 4 decreases from 159 400 to 139 300 M⁻¹ cm⁻¹ (a factor of 1.15). Over the same range, Fig. S4 (supplementary material) estimates the widths of T_n ← T₁ bands to increase by a factor of 1.99, so the integral in Eq. (1) increases by a factor of ∼1.7. Thus, the increased widths can account for the rise of f_n and μ_n from n = 3 to n = 10. By maximizing at 10 units, they suggest that beyond this length, the nature of the T_n ← T₁ absorption is unchanged. The shift of the absorption energy seen in Fig. 2 stops near 10 units as well, supporting this suggestion. Structural disorder will be identified below as a reason for the increasing width with chain length as suggested by the partially resolved vibrational structure in oF₂ becoming blurred out in the longer oligomers. The measured value of the oscillator strength for the shortest polymer, pF₂₈, is 8% smaller than that for oF₁₀. A small amount of the motional line narrowing in long polymers compared to short oligomers has been reported to be the result of the exciton exploring a larger region.⁷⁴ If the nature of the T_n ← T₁ transition is unchanged, this would imply larger extinction coefficients, not smaller as we observe. By contrast, we suggest below that the large decreases in the T_n ← T₁ extinction coefficient for long polymers has its main root in defects, rather than increases in spectral width observed for oF₃ to oF₁₀.

The striking geometry of the fully relaxed T₁ state, in which just one dihedral angle is driven nearly to zero, might be expected to play an important role in the motion of the triplet on the chain and to affect spectra. Analysis below will support the idea that due to triplet motion, this structure rarely occurs for chains longer than oF₂. Previous work assessed this motion finding that triplets generated on pF chains up to 170 units in length moved to end traps with a diffusion coefficient D ≥ 3 × 10⁻⁴ cm²/s.³⁷ With this minimum value of D, a 1 dimensional hopping model⁷⁵ predicts that a triplet hops by one repeat unit in ≤12 ps. Note that this is an upper limit for the hopping time because this value of D was limited by the rate triplets were attached to pF; hopping may be faster than this. If the central dihedral of T₁ cannot change to ∼0° faster than it hops, the chains would adopt a distribution of dihedral angles. Dihedral relaxation times are not known in the current molecules, but insight may be gained based on information from small molecules. Photoexcitation of biphenyl and derivatives produce singlet excited states with flattened central dihedrals, akin to the changes in triplets of oligofluorenes. While Mank reported that dihedral flattening between bare biphenyl rings in cyclopentane may be subpicosecond,⁷⁶ Lui reported a much slower time constant for an axially substituted biphenyl, 13 ps, in hexane after formation of the singlet excited state, which was slowed further in more viscous solvents.⁷⁷ At the same time, Lui noted that subpicosecond decay was largely solvent invariant, concluding it was due to electronic relaxation from higher excited states to S₁. In the current oligomers and polymers, dihexyl side chains on the 9 position of each fluorene unit likely further slow dihedral flattening to 0° and recovery to ∼37.5° as the triplet moves along the chain. Other bond length and angle changes are small by comparison and are expected to occur much faster than triplet motion. Jones recently reported similar vibrational and rotational relaxation (flattening) in an octathiophene (T₈) singlet excited state with a lifetime of 11.7 ps.⁷⁸ Again, we expect that dihedral changes are likely slower in the current samples as the T₈ molecules have only 1 hexyl side chain and lack the steric hindrance present in fluorenes.

From the above, a plausible picture is that triplet motion is too fast for complete flattening and recovery of dihedrals, so the actual geometry in longer oligomers and polymers when a triplet is present may be better described as having more than one partially flattened dihedral angles, depending on the residence time of the triplet in each location. Computations surprisingly found that the delocalization length of the T₁ state, measured from orbitals such as those in Fig. S5 (supplementary material), was almost constant when all chain dihedral angles were fixed to average values and is the same even if all repeat units are held coplanar or at the ground state 37.5° (see Fig. S6 of the supplementary material). While dihedral angles larger than 0° only increase the triplet energy up to a few kT, they have a large impact on spectral parameters, increasing f and red-shifting the absorption.

Here, we examine the hypothesis, suggested by the computed results, that spectral widths may increase with oligomer chain length due to dynamics. Triplets in oF₂ are immobile; they have a single geometry and therefore a narrow spectrum with a partially resolved vibrational structure. In oF₃ – oF₁₀, the triplets may move rapidly, encountering varied dihedrals, which broadens the spectra. Increasing inhomogeneous broadening may occur due to the varied solvent environment along longer chains, but this effect is expected to be small.⁷⁴ In polymers with lengths of 28 repeats and longer, the dihedral angles may approach those of chains without triplets also reducing the range of dihedrals and spectral widths. Computations with fully relaxed triplets, which have one dihedral angle near 0°, predict a decrease in oscillator strength, f, from 6 to 10 repeat units, which does not occur in the experiments. This failure of the prediction could be one signal that triplets indeed move too rapidly to allow complete relaxation of dihedrals. The dynamics of dihedral flattening may impact and possibly control triplet transport rates. This effect was tested by low temperature experiments and a simulation of triplet hopping, described below.

Given the computational prediction that the triplet energy only rises 1–3 kT for a central dihedral angle up to 37.5 compared to the fully optimized 0°, reduced temperature experiments in liquid toluene were expected to be effective at substantially reducing the triplet hopping motion and possibly locking the triplet into one location on a chain. Figure 6 shows the T_n ← T₁ absorption spectrum of oF₁₀ at both room temperature and −80 °C, shifted +650 cm⁻¹ to overlap the peaks. The low temperature spectrum is narrowed by about a factor of 1.8 or 960 cm⁻¹. This narrowing might arise due to a reduction in inhomogeneous broadening from solvent configurations and interactions with the solute. However, we see in Fig. S7 (supplementary material) that the narrowing over the same temperature range is less than half as large for oF₂. We suggest that the ∼400 cm⁻¹ reduction in broadening for oF₂ at low temperature gives a measure of the reduction in inhomogeneous broadening due to the solvent common to both molecules, while the additional decrease seen in oF₁₀ reflects a reduction and possibly elimination of triplet hopping. In oF₂, there is only a single dihedral, so the triplet is stationary, and the dihedral will be nearly flat regardless of temperature. The hypothesis above implies that triplets move in longer chains, resulting in a range of dihedral angles that have varied absorption energies, giving broad spectra. At low temperature, the oF₁₀ triplet is likely frozen in mostly a single location, lacking the thermal energy necessary to rapidly hop to less energetically favorable locations on the chain. This allows the dihedral where the triplet is located to become flat like those in oF₂ and results in a narrower absorption spectrum due to a lack of the different geometries present at room temperature.

To get a sense of how the competition between hopping and dihedral relaxation might alter T_n ← T₁ spectra, we constructed a simple simulation program aimed to simulate distributions of dihedral angles created by triplet hopping. The simulation used the ratio of rate constants for hopping and dihedral propagation, k_hop/k_dih, as the key parameter. The source C-code and detailed explanations are given in Sec. S8 (supplementary material). The resulting distributions of dihedral angles were summed over 1000 simulations to provide average values. The decision to hop was made without regard to the change in T₁ free energy for which we have no method to give an accurate estimate; however, it is expected to be small (<k_bT) based on idealized B3LYP/6-31G(d) calculations where dihedrals were changed by small amounts. Inclusion of this effect might tend to narrow distributions, depending on k_hop and k_dih, but in most results below, this was a small effect because the distribution of angles is broad. Figure 7 shows simulation results at different ratios, k_hop/k_dih for oF₆, displaying a general trend common to all lengths. Results for other lengths are given in Fig. S9 (supplementary material).

For slow hopping, k_hop ≪ k_dih, the simulations find that only one dihedral is flat, while all of the others are near 37.5°. This intuitively expected result is the same as predicted by computations of fully relaxed triplets on oligofluroene chains discussed above, giving very short triplets. As the hopping rate increases, we see the development of a range of dihedral angles, having a maximum width when k_hop is 30 times k_dih for oF₆. At higher hopping rates, the range of dihedral angles decreases, with a peak value for each oF_n equal to 37.5 * (n − 1)/n. For oF₆, this most probable dihedral angle becomes 31° at high k_hop/k_dih. This behavior is expected for very fast transport where the triplet visits every site on the chain many times and dihedral rotation is slow by comparison, resulting in most dihedrals averaged to the same angle. In long chains such as oF₂₀ (see Fig. S9 of the supplementary material), the distribution of angles for k_hop/k_dih ≥ 1 is truncated at close to 37.5°, resulting in narrower distributions, consistent with the suggestion above that long chains may have little geometry change from neutral polymers.

The results establish that for fast hop rates, triplets on polyfluorene chains produce a much wider range of dihedral angles than predicted for a fully relaxed and stationary T₁ state and that the range of angles depends on oligomer length. Broader dihedral distributions will lead to broader T_n ← T₁ absorption spectra. Spectral widths from broadened distributions were estimated using sums of computed absorptions for different dihedral angles in Sec. 10 of the supplementary material. The present simple model qualitatively reproduces the experimental width dependence of triplet spectra with chain length best when the ratio k_hop/k_dih is between 30 and 100. While the model is not quantitative, it is of interest to note what this ratio might imply about k_hop and k_dih. Using the upper limit for the triplet hopping time of 12 ps derived above from the published minimum value of D,³⁷ one might predict a dihedral relaxation time of 0.4–1.2 ns. This is very slow compared to the 12–13 ps relaxation time reported by Jones or Liu discussed above.^77,78 The current molecules are much larger and have bulky side-groups that will make dihedral torsions slower, but it might be surprising if they slowed to this extent implying that hopping might be faster than reported D ≥ 3 × 10⁻⁴ cm²/s.³⁷ In any case, results of the simulations paint a picture in which triplets transport faster than dihedrals can fully respond, likely fast enough that the idealized picture of the T₁ state with one flat dihedral is rare. Instead, dihedral angles have broad distributions in oligomers and are nearly unchanged in longer polymers molecules from the starting ground state geometry.

The discussion above concluded that decreases in the extinction coefficient with increasing length for oligomers and short polymers is due to increases in spectral width that overcome increasing oscillator strength, f, to 10 units long. These factors cannot explain the apparent decreases in extinction coefficients in Fig. 4 observed in longer pF₂₈ to pF₈₄ chains. These decreases can be explained by defects in polymer chains and could provide estimates of the prevalence of defects. Triplets are expected to be very sensitive to defects, which may act as simple barriers to triplet motion, sites where they become immobilized or sites where triplets are either destroyed or changed into new species that absorb at different wavelengths. Defects are expected to occur randomly, with a likelihood proportional to chain length. Keto defects particularly for mono-alkylated polyfluorenes have been reported to trap charges and excitons, resulting in green emission.^79,80 Green emission has not been detected in the current materials,⁴⁸ so any defects are likely of a different type. The fast triplet diffusion in polyfluorenes reported in previous work, with D ≥ 3 × 10⁻⁴ cm²/s,³⁷ predicts that triplets explore the entire chain many times within the triplet lifetime and thus will find a defect if present. For defects to affect measured extinction coefficients, they must result in loss of polyfluorene triplets.

To seek evidence of such a loss of triplets due to defects, we transferred pF triplets to 0.3 mM anthracene (An) in solutions containing 200 mM benzophenone with 0.21 mM pF₅₇ (12 mM in repeat units). Most triplets are captured first by pF₅₇ and then transferred to An. Detailed analysis and figures are shown in Sec. S11 (supplementary material). The ratio of ³An^* made with and without pF₅₇ is 0.864, indicating that 13.6% of polymer chains experience loss of the triplet, plausibly due to the presence of one or more defect. Because only this ratio is needed, interpretation of this experiment required no knowledge of extinction coefficients. With 13.6% of the triplets lost to defects, the ³pF_n^* extinction coefficients in Table I are too small.

A binomial distribution defect rate of 1 in 586 polymer repeat units was estimated considering the actual length distribution of chains in the pF₅₇ sample, as shown in Fig. S12 (supplementary material). Note that in pF_n samples, there are more chains longer than the average length than chains that are shorter. Longer chains are more likely to have defects and will also capture triplets at higher rates as seen in Fig. 5. This defect rate is consistent with a rough estimate given in previous work of less than 1 destructive defect in 500 repeat units based on triplet transport to end-cap trap groups; however, this estimate did not consider length distributions.³⁶ Armed with defect rates, Table II gives fractions of polymer chains in pF₂₈-pF₈₄ with defects, f_d, and makes corrections to the pF_n extinction coefficients in Table I, ε_d. The corrected extinction coefficients are 7%–20% higher than those in Table I. We assumed that the oligomers, which were synthesized by a different route, have no substantial defect rate.

Extinction coefficients for ³pF_n^* corrected for defects, ε_d, are plotted in Fig. 4. The corrected value for pF₅₇ is about 5% smaller than that for oF₁₀, within the overall 10% experimental error described in Sec. III. The corrected ε_d, based on triplet transfer to anthracene, still decreases with chain length for pF₂₈ to pF₈₄, with a slope nearly the same as for uncorrected values. The reason for this is not clear nor is the nature of the traps. The discussion above assumes that traps kill triplets or move them to a defect with a low enough triplet energy that they cannot transfer to anthracene. One possible issue may be the latter type of defect if they have a triplet absorption at the same wavelength as ³pF^*. These would result in extinction coefficients that are too low; however, these would not change the slope appreciably. If a chain attaches more than 1 triplet, triplet-triplet annihilation would lead to underestimations of T_n ← T₁ extinction coefficients. This effect would be most pronounced in the longest polymers, producing smaller apparent ε with increasing length. At the low concentration of triplets made, less than 1% of chains attached more than 1 triplet, so annihilation is insignificant. If we assume that ε_d should be constant for chains longer than 10 units, a likely explanation for pF_n results is that the defect rate per repeat unit is not constant but rather is proportional to chain length. This would require the defect rates, r_d, shown in Table II to match ε for oF₁₀. These values predict a linear growth of defect rate per repeat unit with polymer length. This might be a result of the polymer synthesis, where longer chains have vulnerable active growth sites for longer times than short ones.

Extinction coefficients, ε, are reported for T_n ← T₁ absorptions in a set of oligofluorenes and polyfluorenes. Together with measured spectra, these give oscillator strengths and transition moments that level off after rising smoothly from 2 to 10 fluorene repeat units, thus following the oligomer extrapolation approximation. The extinction coefficients themselves do not follow this trend. Their fall from a maximum at oF₃ to oF₁₀ is due to spectral broadening, much of which arises from a distribution of dihedral angles because triplet hopping along chains competes with dihedral relaxation. Application of an elementary model of this competition yields the conclusion that triplets diffuse so fast along chains that dihedral relaxation does not keep up in accord with earlier observations of the triplet diffusion coefficient, D ≥ 3 × 10⁻⁴ cm²/s.³⁷ 

At low temperature, the triplet is expected to move very slowly, so the T₁ state is delocalized over ∼3 units independent of chain length, with a 0° central dihedral angle in accord with the structure optimized by computation. That is, 0° dihedral would be rare in chains at room temperature. At low temperatures, spectral widths might not increase with oligomer length. For triplets in long polymers, dihedral angles may not differ significantly from the ∼37.5° in ground state as the triplet does not spend enough time in any one location for the low temperature geometry to develop, as suggested by the simple computer simulation described here.

The supplementary material contains the following: Steady-state absorption spectra; details for pulse radiolysis data fitting; analysis of the empirical 1/n fit of data in Fig. 2; comparison of T_n ← T₁ (TT) absorption spectra in benzene and hexane; TT absorption spectra widths on an energy axis; orbitals for TT transitions for optimized triplets on oF₂-oF₁₀; impact of central dihedral angle on T₁ delocalization length; −80 °C TT spectrum for oF₂; source C code and explanation for hopping simulation; distribution of simulated dihedral angles for oF₃-pF₂₀; connection of dihedral angle to TT absorption energy; modeling of raw data for defect rate determination; chain length distribution in the pF₅₇ sample.

This material is based on the work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences through Grant No. DE-AC02-98-CH10886, including use of the LEAF and Van de Graaff facilities of the BNL Accelerator Center for Energy Research.