11532
J. Phys. Chem. C 2008, 112, 11532–11538
Trap Limited Exciton Transport in Conjugated Polymers Stavros Athanasopoulos,† Emmanuelle Hennebicq,‡ David Beljonne,‡ and Alison B. Walker*,‡ Department of Physics, UniVersity of Bath, Bath BA2 7AY, U.K., and Laboratory for Chemistry of NoVel Materials, Center for Research in Molecular Electronics and Photonics, UniVersity of Mons-Hainaut, Place du Parc 20, B-7000 Mons, Belgium ReceiVed: March 28, 2008; ReVised Manuscript ReceiVed: May 9, 2008
A Monte-Carlo approach based on hopping rates computed from quantum-chemical calculations is applied to model the energy diffusion dynamics in a polyindenofluorene conjugated polymer on a predetermined chain morphology. While the model predicts faster time-dependent energy evolution than that seen by site-selective experiments and yields a diffusion length that is an order of magnitude larger than typical experimental values, we show that these discrepancies can be corrected by introducing a low concentration of traps in the transport simulations. Implications for conjugated polymer based opto-electronic devices are discussed. Introduction Conjugated polymers (CP) have found a wide range of applications as charge and energy transporting hosts in organic devices such as light emitting diodes (LEDs) for displays1 and lighting,2 solar cells3,4 field effect transistors5 and chemical sensors.6 Absorption of light creates an exciton, a bound electron-hole pair, whose subsequent energy relaxation is an important phenomenon relevant to the optical and elecronic properties of these polymers.7 The exciton diffusion length LD, the mean distance traveled by an exciton before it recombines, plays a key role in organic devices, for example in LEDs it controls the color of the emitted light, in organic solar cells, LD determines whether excitons reach the interface between electron and hole transporting materials where charge separation can occur before recombining, and in sensors LD has to exceed the typical distance between the point where the exciton is created and the position of an analyte for an amplified response. As in proteins, in CP there is link between the chemical structure and the optical and electronic properties that are critical to the device operation. This link is highly complex since the chemical structure affects not only the electronic properties but also the physical conformation of these chains and the way in which they pack together, namely their morphology. X-ray and neutron scattering measurements have demonstrated correlations between morphologies on 10 nm scales and device characteristics.8–10 Below we describe a model that links morphology and chemical structure of conjugated polymer films to their optical and electronic properties by predicting LD for a film whose morphology is inspired by X-ray data and that contains energetic disorder deduced from absorption spectra. A conjugated polymer (CP) chain consists of many covalently bound repeat units (monomers). Electrons are coupled between monomers via through-space interactions, where the monomers are close enough spatially that there is an overlap of their electron density distributions, or through-bond interactions, where there is an indirect orbital overlap via an intervening σ bond framework.11,12 On excitation, configurations are mixed by exchange interactions to create an exciton state whose spatial * To whom correspondence should be addressed. E-mail: a.b.walker@ bath.ac.uk. † University of Bath. ‡ University of Mons-Hainaut.
extent is determined by the degree of coherent sharing of the excitation among the monomers.7 Interactions between excitons and nuclear motions introduce time-dependent confinement effects, where the exciton is confined to a region of the chain of typically 5 monomers termed a chromophore. This selftrapping process occurs on timescales much shorter than 1 ps and gives a strongly bound (∼0.1 eV) polaron-exciton state.7,13 Self-trapping has been studied in conjugated polymers and small molecules by three-pulse stimulated echo peak shift measurements.13 In supramolecular systems, disorder and electronvibration coupling combine to localize the exciton to a few neighboring molecules.14 The measurements of ref 13 separated homogeneous and inhomogeneous line broadening in the emission spectra. Line broadening comes from fluctuations in the optical bandgaps due to the coupling of vibrational modes to the exciton states. Homogeneous line broadening results from fluctuations that occur on a short time scale compared to the rate at which the exciton transfers to other chromophores; inhomogeneous broadening denotes an effectively static frequency distribution. The change in geometry associated with photoexcitation of CPs gives rise to homogeneous broadening and contributes to fluctuations of the absorption spectra, whereas inhomogeneous broadening results from the distribution of chromophores. According to this picture, CPs are thus best described as a collection of weakly coupled chromophores with various conjugation lengths.15 The energy associated with a chromophore is related to its lengthsthe shorter the chromophore, the higher the energy. Energetic disorder also arises owing to the screening or polarizing influences of the medium, with chromophores embedded in different local dielectric environments displaying distinct excitation energies. This picture has been corroborated by the presence of multiple emission lines observed in ladder-type poly(p-phenylene)s using single molecule spectroscopy at cryogenic temperature.16 In reference,16 it was found that CPs whose chemical structure and hence electron-phonon interactions differ strongly adhere to this view which hence is likely to be valid for all CPs. As excitons migrate around the film, some of them decay radiatively, emitting photons whose energies are determined by the site reached on decaying. Site selective spectroscopy, in which transient photoluminescence spectra are measured fol-
10.1021/jp802704z CCC: $40.75 2008 American Chemical Society Published on Web 07/03/2008
Exciton Transport in Conjugated Polymers lowing excitation with monochromatic radiation, allows experimental investigation of exciton diffusion and maps out the distribution of site energies.15 Both intrachain (i.e., hopping of the excitation along the polymer backbone) and interchain (i.e., hopping of the excitation between two chromophores residing on different chains) transfer can take place17,18 or as well as delocalization over several chains.19 We have previously shown17,18 that energy transfer along chains in solution, where there is negligible possibility of interchain transfer, occurs on the same time scale, 500 ps, as radiative decay. A far more efficient energy migration process was found for films as excitons can use a combination of intra and interchain hops, and interchain hops are much faster than intrachain hops. Since intermolecular transfer is strongly affected by relative orientation of the two chromophores involved in the process, it is sensitive to chain packing. LD is determined by the competing processes of hopping and recombination so it is important to have a model of exciton migration that contains an explicit description of chain morphology. On the basis of detailed geometric and electronic structures of chromophores and a film morphology where chains are positioned on a hexagonal lattice with a chain separation determined from X-ray data, we model exciton migration in rigid-rod CP films with a random walk Monte Carlo (MC) simulation on an assembly of chromophores treated as rigid rods linked to form chains. Diagonal, i.e. energetic, disorder comes in through a chromophore length distribution deduced from the experimental absorption spectrum and off-diagonal, i.e., positional, disorder through a uniform distribution of interchromophore torsional angles. Transfer rates are calculated using the distributed monopole model17,18 for the electronic coupling between donor and acceptor, an improvement over the point dipole approximation of Fo¨rster20 in the case of closely spaced interacting chromophores. The time-dependent populations of the excited chromophores are monitored and averaged over the MC trajectories. The 0-1 photoluminescence (PL) peak energy relaxation predicted on the basis of this model is compared with time-resolved measurements of luminescence decay in poly(6,6′,12,12′-tetraalkyl-2,8-indenofluorene) (PIF) at 7 and 294 K.21 PIF is a stepladder blue emitting polymer promising for optoelectronic device applications. With the same simulations, we can establish the contributions of intra and interchain hopping to motion along and orthogonal to the chains before recombination and determine LD. We find that ∼1% traps are required to obtain results consistent with experiment, especially the values of LD in the range 5-20 nm that are typically found for CPs. This sensitivity to the presence of defects is problematic for solar cells, but is an advantage for sensors if traps are mostly the sites at which the molecules to be sensed are bound. If polymer LEDs can be made of material that is relatively free of intrinsic traps, this sensitivity will also be an advantage as extrinsic traps, whose location and electronic structure could be controlled, would be the emissive sites. Energy relaxation due to exciton migration in organic films have previously been modeled with transfer rates from Fo¨rster theory, coupled with the Miller-Abrahams model where energetic disorder is specified with a Gaussian distribution.22,23 Meskers et al.22 assume that excitons hop between sites on a simple cubic lattice, and use a fitted value for the Fo¨rster radius RF whereas in reference23 excitons hop between randomly spaced sites and RF is calculated from the measured spectra. Qualitatively, we observe the same behavior as references22,23 but, in our model, input parameters are linked to the physical properties of the polymer.
J. Phys. Chem. C, Vol. 112, No. 30, 2008 11533 TABLE 1: 0-1 Emission Peak Energy E01em(l), Calculated Radiative Lifetime τdl, and Probability pl for Chromophore Lengths l Considered in the Simulation l (monomers)
E01em(l) (eV)
pl (%)
τdl (ps)
3 4 5 6 7 8
2.789 2.744 2.717 2.699 2.686 2.676
7.5 47.0 30.5 9.5 3.5 2.0
655 591 470 457 387 404
The layout of the paper is as follows. Section 2 covers the theoretical methodology, specifying the morphology, and providing a brief description of the quantum-chemical calculations and the MC scheme. Section 3 presents our results and comparisons with the experimental data of Herz et al.21 and Section 4 our conclusions. Theoretical Methodology Setting up the Model. An energy distribution for the chromophores was extracted by fitting a Gaussian with a mean of 3.00 eV and standard deviation of 0.05 eV to the absorption spectrum of PIF.21 The chromophore energy distribution can be transformed to a chromophore length distribution by fitting the form24
Eabs 00 (l) ) E∞ + (E1 - E∞)exp[-a(l - 1)]
(1)
to experimental 0-0 absorption maxima of PIF oligomers in solution25 where l is the number of monomers in the chromophore. We obtain E∞ ) 2.8719 eV, E1 ) 3.439 eV and a ) 0.43771. The derived chromophore length distribution pl is given in Table 1. Based on this distribution we randomly assign the length of each chromophore site. Furthermore, we introduce torsional disorder between successive chromophore units from a normal distribution with a mean of 20° and standard deviation 5°. Emission 0-0 peak energies were obtained by fitting to25 0.54203 Eem ⁄l 00 (l) ) 2.7685 +
(2)
To correct for the difference between 0-0 and 0-1 peaks a constant energy of 0.16 eV (i.e., one quantum of the dominant stretch vibrational mode) was subtracted from the fitted 0-0 emission maxima to obtain the 0-1 peak energies shown in Table 1. Each polymer chain consists of a group of 11 chromophores which are treated as linked rigid rods. This number of chromophores comes from the measured number averaged molecular weight for PIF of 3.3 × 104 g/mol25 and the average number of monomers per chromophore of 4.6 obtained from the chromophore length distribution in Table 1 (given that there are 56 carbon atoms per monomer and neglecting the contribution of hydrogen atoms to the molecular weight). The chains are aligned parallel to the z axis and form a hexagonal lattice with a lattice constant of 1.2 nm taken from X-ray data for poly(9,9) dioctyl fluorene,26 a polymer with side chains resembling those of PIF. Three stacks of chains were considered in the simulation cell, each stack containing 13 × 13 chains, and periodic boundary conditions assumed. Evaluation of the Transfer Rates by Quantum-Chemical Methods. In these systems, the magnitude of the electronic coupling (∼1-100 cm-1) is less than the reorganization energy (∼1000 cm-1). Thus we are in the weak coupling regime so a Fo¨rster-like approach has been adopted. The transfer rate
11534 J. Phys. Chem. C, Vol. 112, No. 30, 2008
Athanasopoulos et al.
Figure 1. Mean 0-1 emission peak energy E01 vs time t. Each curve corresponds to a given starting site length, in descending order 3mers, 4mers, 5mers, 6mers, 7mers, and 8mers. Experimental results21 at excitation energies in eV of 3.062 (filled circles), 2.962 (empty circles), 2.884 (diamonds), 2.851 (triangles), are also shown. Left panel: temperature T ) 294K, solid lines: homogeneous line width Γ ) 0.04 eV, dashed lines: Γ ) 0.08 eV. The dotted line shows the thermodynamic limit (eq 8). Right panel: T ) 7K, solid lines: Γ ) 0.02 eV.
between a donor at site ri and an acceptor chromophore at site rj in ps-1 is18
kij ) 1.18|Vij|2Jij
(3)
where Vij is the electronic coupling in cm-1. Jij, the overlap between the donor fluorescence Fi(ω) and acceptor absorption spectrum Aj(ω) normalized on an energy scale in cm-1, is given by27
Jij )
∫0∞ Fi(ω)Aj(ω)dω
(4)
Spectra for Jij were obtained from the Franck-Condon approximation and a displaced harmonic oscillator model. Here, following reference,18 two modes were taken into account in the calculation of Jij: (i) a high-frequency mode (at 1300 cm-1) representative of the dominant carbon-carbon stretching and ring breathing vibrational motions in phenylene-based CPs; (ii) a low-frequency mode (at 80 cm-1) that accounts for the libration motion between indenofluorene units but also effectively encompasses modes associated with the bath (i.e., surroundings). The Huang-Rhys factors associated with the high-frequency vibrational mode have been extracted from the calculated relaxation energies (at the INDO/SCI level) while constraining the molecule to a planar conformation; the Huang-Rhys factors for the low-frequency mode have been adjusted to fit the Stokes shifts measured in oligomer solutions at room temperature, see reference.18 Chromophore absorption and emission spectra have been convoluted by means of Lorentzian functions with a full width at half-maximum Γ to account for homogeneous broadening; a value of 0.08 eV (full width at half-maximum) was inferred by comparison with the oligomer experimental spectra.18 Vij is the sum of the product of the transition densities F using the distributed monopole model:18
Vij )
Fi(rd)Fj(ra) 1 4πε0 a*d |rd - ra|
∑
(5)
where the sum runs over the donor (d) and acceptor (a) sites. The transition densities F have been computed at the INDO/ SCI28 level on the basis of the excited-state relaxed geometries, hence assuming full thermalization of the donor prior to hopping.18 The rate kij varies with temperature T through the Boltzmann weighted Franck-Condon integrals that determine the probability that a given vibrational mode will contribute to the spectrum. However, because of the simplicity of the model used
to compute the spectral line shapes and widths, the rates deviate from detailed balance conditions at low temperatures. To correct for this, uphill transition rates were computed by multiplying the downhill transition rates by a Boltzmann factor, viz exp[-(Ej - Ei)/(kBT)] where Ei is E00em for the ith chromophore and kB the Boltzmann constant. For each site i, values for kij were computed for its 50 nearest neighbors. Jumps outside this range were ignored as they would be infrequent and could exceed half the cell width, which would be inconsistent with the periodic boundary conditions (minimum image convention). The radiative lifetimes for an exciton on a chromophore of given length l were estimated as the inverse of the Einstein coefficient for spontaneous emission
A(l) )
8π2νfi3 µfi2 3ε0pc3
(6)
where µfi is the (INDO/SCI) transition dipole moment and νfi the corresponding absorption frequency for the transition f f i. MC Simulations. An exciton of given energy is created on a chromophore of the corresponding length chosen at random. At each MC step the exciton either moves to a neighboring chromophore or decays radiatively. For an exciton at site i, a waiting time τij is calculated for a hop to the jth neighbor:
1 τij ) - ln(X), kij
(7)
where X is a random number from a uniform distribution between 0 and 1. The same expression is used to find the waiting time for emission τdl in terms of the decay rate for a chromophore of length l, kdl ) 1/τdl. The event with the smallest waiting time is executed and the simulation time advances by that waiting time. If the chosen event is emission then we stop the MC run and create a new exciton. 105 trajectories were sufficient to generate results that are not noisy except at times exceeding 300 ps. A record is kept of the position, and therefore energy, and the dwell time at each site for all the excitons, enabling us to construct time dependent averages. Results and Discussion Figure 1 compares the time-dependent experimental and theoretical photoluminescence average energies (0-1 peak) at 294 and 7 K for site-selective excitation of the sample. As can be seen in the left panel of Figure 1, at 294 K, high energy chromophores relax predominantly by moving downhill and low
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J. Phys. Chem. C, Vol. 112, No. 30, 2008 11535
energy chromophores through moving uphill in energy space until a stationary state is reached, which occurs after approximately 20 ps according to the calculations for Γ ) 0.08 eV. It is interesting to note that photoexcitation of the longest chromophores is followed by an increase in the average PL emission energy suggesting that energy transfer pathways reshuffling the populations toward higher-energy chromophores occur at room temperature. This increase in energy arises because the available thermal energy exceeds the energy barrier from the site energy mismatch due to the different conjugation lengths. In the long time limit, the predicted mean excitation energy of the 0-1 peak converges to the experimental value (2.72 eV), which is only slightly higher than the thermodynamic j th of 2.71 eV where limit E
Eth )
∑
[ ] [ ]
El p E exp l l l kBT El p exp l l kBT
∑
(8)
Excitons transfer between sites at rates that are faster than the decay rate so their final distribution is close to thermal equilibrium but with a slightly higher average energy because of the finite excited-state lifetime. Meskers et al.22 and Hennebicq et al.18 have also reported convergence to a final energy j th. that slightly exceeds E Direct comparison between the measured and calculated data should be made with care, because our representation of polymer chains into a discrete energy set of chromophores is oversimplified and it is likely that chromophores of different lengths are simultaneously excited by the (broad) laser pulse used in the experiments of reference.21 Yet, it is clear from Figure 1 that the theoretical data yield significantly higher relaxation rates compared to experiment. As the spectral overlaps Jij and therefore the hopping rates kij depend on the homogeneous line widths Γ, one could expect a slower relaxation rate for smaller Γ. To test this idea, we have repeated the same calculations but using Γ ) 0.04 eV, i.e., half the value extracted by fitting the experimental spectra. In this case, the relaxation time for the site selective simulations increases slightly to ≈30 ps but is still much faster than the experimental result. As Γ is not expected to be significantly outside the range tested, we conclude that the uncertainty in Γ is not the main culprit for the discrepancy between experiment and theory. At low temperatures, as shown in the right-hand panel of Figure 1, hops upward in energy are unlikely and there is a demarcation energy below which the PL peak energy remains fixed as the time to make a hop exceeds the decay lifetime.22 Here, for theory and experiment, E01(t) is still decreasing when most of the excitons have decayed and so also never reaches j th which is close to the lowest chromophore energy, i.e., 2.676 E eV. For excitation energies of 3.062 and 2.962 eV, a similar time variation is seen in experiment as in our theoretical predictions for excited 4mers and 5mers. We have applied the same model with Γ ) 0.08 eV to calculate the exciton diffusion length LD assuming that excitons start on any chromophore site with equal probability. LD at room temperature does not depend on the initial distribution of excited chromophores as by the time most of the excitons decay, the distribution has changed significantly. Our simulations predict LD ≈ 62 nm, which is about 1 order of magnitude larger than the experimental values reported so far for CPs, see Table 229–34 Note, however, that they are close to diffusion lengths reported
TABLE 2: Experimentally Determined Exciton Diffusion Lengths polymer/molecule
LD (nm)
29,30
5-7 Poly(phenylenevinylene) PPV Poly(2-methoxy,5-(2′ethylhexoxy)-1,4- phenylenevinylene) 5-8 MEH-PPV31 Alkoxy-Poly(phenylenevinylene) MDMO-PPV32 6 Polyfluorenes33 5 Ladder-type polyparaphenylene LPPP34 6-14 Sexithiophene 6T 60 Pentacene36 65 88 Perylenetetracarboxylic dianhydride PTCDA37 Carbon Nanotubes38 100 166 Pyronine39
for small molecules35–37 and semiconducting carbon nanotubes.38 In addition, MC simulations on amorphous Al tris(8hydroxyquinoline) (Alq3) films predict LD ) 26nm40 and simulations of intrachain exciton migration on isolated poly(3-(2,5-dioctylphenyl)thiophene) (PDOPT) chains41 and MEH-PPV chains42 show that in both cases LD ≈ 15 nm. Reducing Γ by a factor of 2, from 0.08 to 0.04 eV, only reduces LD from ≈ 62 nm to ≈ 45 nm as shown in Figure 2. LD is also sensitive to the conjugation length distribution. We have isolated the effect of energetic disorder by considering a morphology where all chromophores are hexamers. As expected, the diffusion length is increased to 42-83 nm depending on the value of Γ. Part of the observed effect is due to the increase in the average conjugation length, from 4.6 monomers in the energetically disordered samples to 6 monomers in the homogeneous system. In addition, energetic disorder reduces LD because uphill migration slows down diffusion. Nevertheless, the simulations systematically yield an overestimated LD value. The diffusion length is sensitive to the width of the emission energy distribution, which at sigma∼0.025 eV, is narrower than the absorption energy distribution. A broader emission energy distribution will result in smaller values of LD as the uphill rates are reduced by the Boltzmann factors. To test this hypothesis we repeated the Monte Carlo simulations using an emission energy distribution as broad as the absorption spectra, with a standard deviation σ ) 0.05 eV, and found a small reduction in LD, from 45 to 31 nm for Γ) 0.04 eV. In a CP film, diffusion of electronic excitations may be affected by the presence of low-lying sites acting as traps. Although chemical defects (such as fluorenones found in the polyfluorenes with chemical structures closely related to PIF) can not be ruled out, we speculate that in rigid rod polymers
Figure 2. Exciton diffusion length LD vs homogeneous line width Γ. Solid line: chromophore length distribution from Table 1, dashed line 6mers only.
11536 J. Phys. Chem. C, Vol. 112, No. 30, 2008
Figure 3. Mean 0-1 emission peak energy E01 vs time t. Symbols as for Figure 1. Here, T ) 294 K and homogeneous line width Γ ) 0.04 eV, solid lines: 0.5% traps, dashed lines: 1% traps, dashed-dotted lines: 1.5% traps.
like polyindenofluorenes the traps most likely result from a higher local packing density of the chains, i.e., formation of aggregates in more ordered regions of the films.19 Such traps are attractive, since, for forward transfer, the aggregates are in the acceptor configuration and the electronic coupling for energy transfer is expected to be enhanced by the in-phase combination of molecular transition dipoles in the optically allowed excitedstate of the aggregate. In contrast, backward transfer occurs after fast relaxation to the bottom of the aggregate exciton band, which is dark for H-aggregates, making it hard for the exciton to leave the aggregate. To take into account such aggregates, we have introduced into our polymer morphology a small percentage of chromophore sites assigned to be active traps. At these sites, the transfer rate into the site is a hundred times larger and the transfer rate out of the site a hundred times smaller than the transfer rates had it not been a trap site.43 First, we look at the influence of traps on energy relaxation in Figure 3 (Γ ) 0.04 eV). By including just 0.5% traps, the time-dependent average emission energy shows a noticeable improvement over the trap-free simulations and is now much closer to the experimental results. The presence of traps does not affect strongly the early time dynamics (first 10 ps) but leads to a slower decrease of the average energy toward the thermodynamic limit at longer time delays. This result can be interpreted in the following way. Within the first ps, the probability for finding around the excited chromophore a nontrap site with a significant transition rate is high, so that the diffusion proceeds as in absence of the traps. At longer times, the electronic excitations that are attracted by the trap centers with larger Fo¨rster radii compared to nontrap sites stay in those traps, slowing down the relaxation process. Next, we have explored how LD varies with trap concentration c for Γ ) 0.04 eV, see Figure 4. For LD to decrease from 45 to 15 nm requires c as little as 0.5%. Our results are expected from a simple statistical analysis. Montroll44 predicted that the number of hops performed by a carrier in a 1-dimensional (1D) stack containing a small fraction of randomly positioned defects scales as c-2. This relation becomes 1.51638/c for 3D diffusion on a 3D cubic lattice. LD ) r0nhop1/2, with r0 the average hopping length and nhop the average number of hops so that LD ) r0/c (1D), LD ) 1.23r0/(c1/2) (3D). Assuming r0 ) 1 nm and c ) 1%, LD ) 100 nm for 1D but only 12.3 nm for 3D motion. A reduced diffusion length is thus predicted for 3D systems. At low concentrations of traps where the analytical results in ref 44 apply, an electronic excitation created far away from the nearest defect in an isolated chain is unlikely to reach the trapping site prior to recombination while the activation of
Athanasopoulos et al.
Figure 4. LD variation with the percentage of sites occupied by traps, c. Solid lines: T ) 294 K, dashed lines T ) 7 K. Filled circles intraand interchain hops, empty circles intrachain hops only. The chained line is the prediction of ref 44.
Figure 5. Evolution of the occupation probability pl (%) of chromophores of length l, for the case where 3mers are excited initially, in the absence of traps. Left panel: T ) 294 K. Right panel: T ) 7 K.
interchain transfer opens up additional and faster pathways that lead to efficient funneling of the excitation to the defect. Figure 4 shows that our simulation results for LD in 3D vary as c-1/2 in line with the prediction of reference.44 However for 1D diffusion LD is almost independent of c as intrachain hops occur with hopping times comparable to the lifetime of the excitation. At low temperatures, in 1D and 3D, LD is independent of c since in both cases, excitons decay before finding a trap. When starting on a short chromophore ie 3mers, 4mers and 5mers they diffuse further than if starting on a longer chromophore. For a more detailed understanding of the energy relaxation processes, we have shown how the distribution of excited chromophore lengths evolves in Figure 5, starting from excitation of 3mers. At 294 K, the 3mers become rapidly depopulated as the system relaxes, since there are only a few of them and because a transition involving a decrease in energy is more likely than a transition without an energy change, in part due to the detailed balance requirement. At long times, the distribution tends toward the thermally weighted probability distribution (eq 8) but never reaches it as the excitons decay before that can happen. At 7 K, the 3mer population evolves on a similar time scale as at 294 K, but the decay in the populations of the higher energy chromophores is accentuated by a much greater probability of transfer to lower energy chromophores compared with transfers in the opposite direction. We also investigate the directionality of the exciton transport. Figure 6 shows contour plots of the recombination probability
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J. Phys. Chem. C, Vol. 112, No. 30, 2008 11537
Figure 6. Left panel: Contour plots of recombination probability vs distance F ) (∆x2 + ∆y2)1/2 moved in the x-y plane, transverse to chains, and distance ∆z moved parallel to the z axis. Left panel T ) 294 K, right panel: T ) 7 K.
along the z axis and the x-y plane at different temperatures. The plots demonstrate that excitons travel further along the chain axis even though an exciton performs more interchain than intrachain hops, since an intrachain hop covers a larger distance than an interchain hop, and each interchain hop has a component along the z axis. Figure 6 also shows that the excitons diffuse further at higher temperatures, consistently with the temperature variation of LD discussed above.
energetic disorder only comes in via the length distribution. An alternative explanation to trapping for the slower energy relaxation times and shorter LD values compared to our predictions is inhomogeneous broadening of the chromophore spectra due to the random environment. Our model can take this broadening into account using relevant experimental input, and we find it does have a significant effect on LD. This result will be addressed in future work.
Conclusions
Acknowledgment. The work is supported by the UK Engineering and Physical Sciences Research Council (Bath), the Royal Society (Bath), Cambridge Display Technology (Bath) and the European Commission STREP project MODECOM (NMP-CT-2006-016434) (Bath,Mons). The work in Mons is partly supported by the Interuniversity Attraction Pole program of the Belgian Federal Science Policy Office (PAI 6/27) and by FNRS-FRFC. E.H. and D.B. are, respectively, postdoctoral researcher and research director of FNRS.
We have investigated how the dynamics of exciton migration are controlled by the following variables: the dimensionality of the motion (via the ratio of inter- to intrachain transfer probability), temperature, energetic disorder, homogeneous line broadening and the concentration of attractive trap sites. Our results have been achieved with an atomistic model whose predictions are specific to the material of interest, by employing realistic microscopic structures inspired by experimental morphologies and transfer rates computed by the distributed monopole model that allows for the topology of the excited states. By monitoring the time dependent energy distribution of excited states we compared the time for the emission energy to relax to its near equilibrium value with experimental timeresolved photoluminescence data. Our results showed that relaxation was too fast on the ordered chain arrangement considered, however by introducing attractive trap sites at a concentration of just 0.5%, we could obtain an equilibration time much closer to experiment. We also showed that LD in our ordered morphology greatly exceeds typical measured values for conjugated polymers and that the discrepancy can be remedied with the inclusion of a low concentration of attractive trap sites. Thus, we have confirmed that exciton migration in conjugated polymers is trap limited rather than transfer rate limited. Our model shows that it is possible to obtain this characteristic behavior, which is exploited in organic sensors, with a model that is readily capable of generalization to more realistic spatial morphologies. Our model is a first step toward identifying trends in for materials with varying local packing densities within the polymer film and toward comparing different materials on a length scale which is comparable with the measured LD values in molecular solids. To reduce the number of input parameters in our model, we assume that there is a single energy for each chromophore, thus
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