J. Phys. Chem. B 2001, 105, 9139-9149
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Dispersive Relaxation Dynamics of Photoexcitations in a Polyfluorene Film Involving Energy Transfer: Experiment and Monte Carlo Simulations Stefan C. J. Meskers,† Jens Hu1 bner,‡ Michael Oestreich,‡,§ and Heinz Ba1 ssler*,† Institute of Physical Chemistry, Nuclear Chemistry and Macromolecular Chemistry, Philipps-UniVersity of Marburg, Hans Meerweinstrasse, D-35032 Marburg, Germany, Institute of Physics, Philipps-UniVersity of Marburg, Renthof 5, D-35032 Marburg, Germany, and Institute of Solid State Physics, UniVersity HannoVer, Appelstrasse 2, D-30167 HannoVer, Germany ReceiVed: April 10, 2001
Time-resolved fluorescence spectroscopy is used to investigate relaxation of electronic excitations in films of π-conjugated polymer 1 in the ps time domain. The position of the fluorescence band and its width are measured as a function of time and excitation energy. Both low (15 K) and room-temperature behavior are investigated. For high energy excitation, the fluorescence band shows a continuous red shift with time. The energy associated with the maximum of the fluorescence band E is proportional to log(t), with t being the time after excitation. For excitation in the tail of the lowest absorption band, the fluorescence remains stationary and selective excitation of a subset of chromophoric chain segments is possible. At intermediate excitation energy the time required for the excitations to make their first jump depends on the excitation energy and is longer at lower energy. At low temperature and high energy excitation the fluorescence bands are found to narrow with time, while for low energy excitation a broadening with time is observed. The experimental data are consistent with dispersive relaxation dynamics for the photoexcitations by incoherent hopping between localized states. Monte Carlo simulations are performed to obtain the average energy and the width of the energy distribution for an ensemble of photoexcitations in an energetically disordered molecular solid assuming Fo¨rster type energy transfer. A Fo¨rster radius R0 ∼ 30 Å is found to give good agreement between experiment and simulations. In addition, the measurements indicate that for excitation energies >2.94 eV additional relaxation processes, ascribed to ultrafast intrachain vibrational relaxation, are operative.
1. Introduction Disorder in molecular solids consisting of identical chromophoric groups results in inhomogeneous broadening of the optical transitions of the chromophore. When a photoexcitation is created within the inhomogeneously broadened density of states (DOS), it can relax by energy transfer to a neighboring molecule with lower excitation energy. Once the excitation has moved to lower energy, absorption of a phonon is required for transfer to a site with higher energy. At very low temperature (kT, σ, with σ the width of the broadened DOS), thermal phonons are not available and the relaxation is a downhill process. As the energy of the excitation lowers through successive transfer steps, the probability for the next jump decreases because the concentration of sites with an even lower excitation energy gradually diminishes. The mobility of the quasiparticles is thus time dependent, and the associated dynamics of the particle has been termed “dispersive transport”. This type of transport has been observed experimentally for triplet excitations in disordered materials1,2,3 and singlet excitations of dye molecules inside polymeric solids.4,5 Outside the realm of organic molecules, dispersive transport has also been observed in (disordered) inorganic semiconductors6,7 and glasses containing rare earth ions.8,9 π-Conjugated polymers have become a major research subject in recent years, not in the least because of their promising * To whom correspondence should be addressed. Institute of Physical Chemistry, Philipps-University of Marburg, Hans Meerweinstrasse, D-35032 Marburg, Germany. Fax: +49-6421-28-28916, Phone: +49-642128-22190, e-mail:
[email protected]. † Institute of Physical Chemistry, Nuclear Chemistry and Macromolecular Chemistry, Philipps-University of Marburg. ‡ Institute of Physics, Philipps-University of Marburg. § University Hannover.
technological applications. It has been realized early on that a solid film of π-conjugated material can in many respects be regarded as a disordered molecular material. Chemical and conformational defects on the polymer chain limit the delocalization of the electrons along the chain. The more or less random position of defects results in a rather broad distribution of the effective conjugation length of the chain segments separated by the defects. As the energy of quasiparticles such as charges or photoexcitations depends strongly on the effective conjugation length of the chain segment on which they reside, the distribution of conjugation lengths translates into a strong inhomogeneously broadened density of states for these particles. And so, due to the disorder, the charges or excitations are (strongly) localized and their transport can generally be modeled as incoherent hopping between localized states.10 The photoexcitations themselves have been modeled as relatively strongly bound electron-hole pairs with a binding energy of ∼0.4 eV. This is, of course, in strong contrast to the well-known behavior of inorganic semiconductors such as crystalline Si or GaAS. The exact magnitude of the binding energy in organic semiconductors is still a matter of ongoing debate. Nevertheless, the model featuring the photoexcitations as relatively strongly localized, bound electron-hole pairs has been very successful in explaining certain photophysical properties of π-conjugated polymers. For π-conjugated polymers, site-selective fluorescence spectroscopy, in which only chromophores in resonance with a narrow laser line are excited,11 has given strong evidence for dispersive transport of photoexcitations in films of π-conjugated polymer at low temperature.12,13 In these experiments the excitation energy is scanned through the absorption band and the fluorescence is monitored. For excitation at high energies,
10.1021/jp0113331 CCC: $20.00 © 2001 American Chemical Society Published on Web 08/31/2001
9140 J. Phys. Chem. B, Vol. 105, No. 38, 2001 the frequency associated with the maximum fluorescence intensity is independent of the excitation energy. This indicates that the photoexcitations diffuse through the film and find sites with lower excitation energy. When the excitation energy is below a certain value (often referred to as “localization energy”) in the low energy tail of the absorption band, the energy of maximum fluorescence intensity shifts linearly with the excitation energy. The linear shift shows that under these conditions the photoexcitations remain stationary after their creation and can no longer relax by energy transfer processes. This behavior is expected for disordered solids and energy transfer processes that depend strongly on distance. For a chromophore with energy , the average distance to a site with equal or lower energy will depend on . For low values of , the average distance will be large so that due to the distance dependence of the energy transfer probability, the rate for transfer is very small. The energy for which the (average) time required for the first energy transfer step is equal to the lifetime of the excitation serves a a demarcation point. Below this localization energy the excitations do practically not relax by energy transfer during their lifetime, while above this energy relaxation occurs. Time-resolved fluorescence measurements (with fixed excitation wavelength) have indeed revealed a time-dependent red shift of the fluorescence spectrum in accordance with a dispersive relaxation process.14-17 For poly-(para-phenylenevinylene) derivatives at low temperature, it has been observed that the maximum of the fluorescence spectrum shifts over 80 meV in the time window from 0 to 100 ps.15,18 The monotonic shift with time extends well into the nanosecond time domain,16,19 and so it seems that at low temperature the photoexcitations do not reach a stationary energy during their lifetime. The bathochromic shift of the fluorescence has been interpreted in terms of migration of the photoexcitations to chain segments with longer effective conjugation length and thus lower excitation energy through transport of energy by, for example, the Fo¨rster mechanism. Fluorescence decay curves of π-conjugated polymers recorded for various detection wavelengths upon excitation with a fixed photon energy could be interpreted in terms of dispersive transport.20,21 Relaxation by energy transfer between localized sites could also be inferred from steady-state fluorescence measurements on films of π-conjugated polymer doped with energy accepting molecules.22 In this study we use time-resolved fluorescence line narrowing to investigate the dispersive relaxation dynamics in a film of the π-conjugated polyfluorene 1.23 By recording the fluorescence spectrum as a function of time and excitation energy (above and below the localization energy) we obtain direct evidence for the dispersive nature of the relaxation. Such a systematic study has not, to our knowledge, been performed for films of π-conjugated polymers.
Theoretical Background. Dispersive transport has been investigated theoretically.24-27 Energy transport within an energetically dispersed manifold of donor and acceptor states
Meskers et al. is modeled as an incoherent, stochastic random walk process. It is characterized by a loss of phase memory after each step and can be described by a generalized master equation28
dni )dt
ni
∑Wijni(t) + ∑Wjinj(t) - τ0
(1)
where ni is the occupational density of site i characterized by their position Rj and energy i. The last term applies to optical excitations that decay monomolecularly after an intrinsic lifetime τ0. The simplest ansatz for the hopping rate W is that due to Miller and Abrahams
Wij )
{
[
k0χij(R) exp k0χij(R)
]
j - i if j > i kT if j < i
(2)
It is a product of a frequency factor, a term that accounts for the distance dependence of the interaction strength and a Boltzmann factor for jumps up in energy. For downward jumps the Boltzmann factor is set equal to unity. For exchange interaction that applies to charge carriers and triplet excitons
χij(R) ) exp(-2γRij)
(3)
while for Fo¨rster-type energy transfer via dipole-dipole coupling
χij(R) ) (R0/R)6
(4)
In writing eq 4, the dependence of the transfer rate on the relative orientation of the dipoles involved has been neglected. The asymmetry of the rates is a major obstacle for any analytic treatment, in particular if the density of states distribution is of the Gaussian shape. One way to solve this problem is to apply the effective medium approach as used by Movaghar and coworkers.26,27 In this approach the motion of the quasiparticle in the aVeraged medium is described rather then the average behavior of a set of (noninteracting) particles each placed a different point in the disorder medium with random site energies. For Fo¨rster type transport (eq 4) and intermediate temperatures (σ/kT ) 4.5), the theory predicts that after an initial phase with rapid decay, the average energy E of excitations with high initial energy relaxes according to E(t) ∝ log(t). Interestingly, for excitations with low initial energy, the theory predicts that the relaxation processes are “delayed”. By this we mean the time needed for the excitation to make its first jump becomes increasingly longer, the lower the initial energy of the excitation is. This can be interpreted as follows. The average distance between a particular site i and the closest site j with lower or equal energy depends on the energy of site i (i). Clearly, when i is low and at the lower end of the Gaussian distribution, sites with even lower energy will be scarce and thus on average far away. Since at low temperature only jumps to sites with lower energy are allowed, the time needed for the excitation to make its first jump is longer because the jump rate depends with an inverse sixth power on the distance. As pointed out by Movaghar et al., the effective medium approximation breaks down at very low temperatures.27 To avoid the problems in theoretical analysis referred to above, one can use an alternative approach to study relaxation at low temperature, namely Monte Carlo simulations. These simulations can be considered as idealized experiments with an adjustable degree of complexity premised upon prescribed transfer rates and a DOS distribution. In the present work, these
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simulations will be used to clarify two questions. The first is the evolution of the average energy with time and excitation energy at low but finite temperatures. For this cases it is known that effective medium theories are incorrect. The second question regards the time dependence of the standard deviation of the energy in the ensemble of photoexcitations as a function of time and energy. At very low temperatures (kT , σ), the transport of photoexcitations takes place under nonequilibrium conditions. The time required to reach the thermal equilibrium is normally much longer than the lifetime of the photoexcitations. Moreover, the relaxation time tends to infinity as the temperature approaches the absolute zero. The term “frustration” may be used to describe this situation. It implies that thermal equilibrium values will not be reached within a finite period of time. Given a Gaussian density of states
( )
g() ) (2πσ)-1/2 exp -
2 2σ2
(5)
it can be shown easily that the equilibrium value for the average energy of the excitations is
〈∞〉 )
∫-∞+∞ g() exp(- kT ) d ∫-∞+∞ g() exp(- kT ) d
)-
σ2 kT
(6)
The associated variance of the energies of the ensemble of quasiparticles is given by
Var∞ ) 〈∞2 - 〈∞〉2〉 ) σ2
(7)
Equation 7 implies that at thermal equilibrium, the width of the luminescence band should be equal to σ, i.e., just as broad as the DOS and the associated absorption band. When the relaxation is incomplete (frustrated), theoretical analysis predicts that the width of the fluorescence bands should be considerably smaller than σ and may in certain time domains actually decrease with time. This has recently been discussed by Vissenberg et al.29 who analyzed relaxation of excitations in a Gaussian density of states by Fo¨rster energy transfer theoretically in the T ) 0 K limit. Their results are consistent with earlier predictions.24 Intuitively, the narrowing may be understood in the following terms. The excitations with higher energy have a larger probability for relaxation by energy transfer than the their low energy counterparts. Thus, the relaxation rate is energy dependent and increases for higher energies. This clearly leads to a narrowing of the width of the distribution of energies of the ensemble of quasiparticles. Experimentally, this implies that at low temperature the fluorescence band is more narrow than the corresponding absorption band for films of π-conjugated polymer. This has indeed been observed.15,16,19 Also for phosphorescence from disordered molecular crystals this narrowing has been observed.1 Comprehensive theoretical modeling of the bandwidth as a function of time and excitation energy at finite temperature has not, as far as we know, been performed. The organization of this paper is as follows. After a brief section describing experimental and computational details, we present experimental fluorescence data obtained for a film of polyfluorene 1. After this, results of Monte Carlo simulations of relaxation processes will be discussed. Finally, the limitations of the model used to describe relaxation of excitations in the disordered molecular solid will be evaluated.
2. Experimental and Computational Details A largely amorphous film of 1 was prepared by drop casting from toluene solution (5 mgr:1gr polymer to toluene) on a monocrystalline sapphire substrate. The sample was held in a coldfinger cryostat (15 K). The time-resolved measurements were carried out using a mode-locked picosecond Ti:sapphire laser with a repetition rate of 80 MHz. The excitation wavelength was 370-430 nm, generated by frequency doubling (spectral width ∼0.3 meV). The photoluminescence was detected in the backward scattering direction and was spectrally and time resolved by a 0.32 m imaging spectrometer followed by a synchroscan streakcamera with a two-dimensional readout. A time resolution of 1.5 ps can be achieved with this setup. The excitation beam was focused on an area of 0.2 by 0.2 mm, and the pumpfluence was < 25 nJ/cm2. Given the high repetition rate of our experiment, there may be a build-up of long-lived metastable excited states. Cadby et al.30 found evidence for triplet states in polyfluorene and have estimated the yield of triplet formation via intersystem crossing to be on the order of a few percent. Recently, phosphorescence from films of 1 has been observed. A lifetime up to 1 s at low temperature could be determined,31,32 but a certain fraction of the triplets may decay at a faster rate. On the basis of these numbers one estimates a (steady state) triplet concentration of 10 ps. This implies that the excess photon energy relative to the lowest singlet 0-0 transition hardly influences the relaxation processes taking place. This can be rationalized in terms of the excess energy being released into the vibrational heat bath on a time scale beyond the temporal resolution of the experiment. The local temporal heating of the chain segment has apparently little effect on the spectral diffusion. As the excitation energy is scanned toward the low-energy tail of the absorption band, the relaxation process is delayed, as predicted by theory. Upon pumping at 2.95 eV, the spectral position of the emission is time independent up to 300 ps. After this time some spectral broadening and a small shift are observed (see Figure 3). At the lowest excitation energies used (2.922.90 eV), the emission bands remain almost stationary; for t >1 ns slight changes in the spectra indicate that limited relaxation can occur. Thus, for excitation 2.95 eV. For < 2.95, the maximum of the 0-1 band is at lower energies and shifts with the excitation wavelength. Furthermore, it may be seen in Figure 4 that the width of the fluorescence band is reduced when < 2.95 eV. For > 2.95, the width does not depend significantly on the excitation energy. In Figure 5 the dependence of the width of the 0-0 emission band as a function of time is shown. The solid black squares pertain to low-temperature measurements with 3.35 eV excitation energy. The widths were determined by fitting a Gaussian curve to the high energy side of the 0-0 band of the fluorescence spectrum. (For lower excitation frequencies, measurements of the high energy part of the fluorescence spectrum are hampered by stray light from the laser source.) The bandwidth of the fluorescence decreases as a function of time, going from 45 meV () 0.64 σ) at t ) 1 ns to 23 meV () 0.34 σ) at t ) 2 ns. At room temperature, the width of the 0-0 fluorescence band follows a different time dependence (Figure
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Figure 4. The 0-1 vibronic band in the fluorescence from films of 1. Solid lines correspond to excitation energies of 3.082, 3.046, 2.997, 2.954, 2.914, 2.903 eV (top to bottom). The spectral resolution is 18 meV. The spectra are time integrated from 1.5 to 1.8 ns after excitation. The dashed line shows the spectrum acquired with 6 meV spectral resolution.
5, open circles). The width remains almost constant with time and is considerably higher than at low temperatures: 52 meV () 0.75 σ). For the room-temperature case we have investigated the timedependent shift of the fluorescence bands. Results for three different excitation wavelengths are show in Figure 6. Also for this temperature, the E ∝ log(t) behavior is observed for excitation with high energy photons. The slope of the E ∝ log(t) curve is somewhat reduced in comparison with the lowtemperature result (see dashed line). For excitation with 2.951 eV photons, the emission band remain stationary up to 300 ps, after which a red shift is observed. Fluorescence measurements with even lower excitation energy are hampered by low signal intensity. Furthermore, the presence of “hot bands” in the absorption spectrum overlapping with the regular absorption bands makes it questionable whether selective excitation to the very lowest excited-state levels in the density of states can be realized experimentally. The photophysical behavior of the polyfluorene at room temperature is more complex than in the low-temperature case. In Figure 7 some fluorescence spectra obtained at room temperature with an acquisition delay of 1.5 ns are shown. As can be seen, an additional broad low energy emission band appears at ∼2.5 eV. The intensity of this band relative to the normal fluorescence at 2.95 eV seems to depend on the excitation energy. At shorter times (0.4 ns), the contribution of the low energy band to the spectrum is negligible (see dotted line d). At low temperatures this additional emission is not detected. In this respect, the decay curve of the integrated emission intensity as a function of time is informative. Shown in Figure 8 is the decay of the spectrally integrated emission at 293 and 15 K, for various excitation energies. At low temperature, the decay depends on the excitation wavelength. It is initially faster for higher excitation energies. At room temperature the lifetime of the emission is reduced in comparison with the low temperature case. Especially with 3.082 eV excitation (dashed line d), the initial decay of the luminescence is quite rapid. The decay features a long-lived tail that contains a contribution of
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Figure 5. Width of the 0-0 fluorescence band from a film of 1 as a function of time after excitation. The standard deviation σ as obtained from a fit of a Gaussian curve to the high energy side of the 0-0 fluorescence band is shown. Open circles: T ) 293 K and 3.111 eV excitation energy. Solid squares: T ) 15 K and 3.35 eV excitation. Dashed lines show results of simulations for e ) -0.5 σ at T ) 15 and T ) 293 K.
Figure 6. Energy corresponding to the first moment of the 0-1 vibronic emission band of a polyfluorene film at room-temperature plotted as a function of time after excitation. The excitation energies are: 9: 3.111, 4: 3.024, 3: 2.951 eV. The dashed line shows the energy of excitations upon 3.081 eV excitation at 15 K (see Figure 2).
the low energy anomalous emission band (see Figure 8). For low excitation energies (dashed line c), the “normal” fluorescence is longer lived and so a possible contribution of the anomalous bands may be buried under the normal fluorescence and may not be clearly discernible in the spectra (Figure 7, curve c). Simulations. In Figure 9 we show the simulated average energy E of an ensemble of excitations in a random medium as a function of time t. The energies of the localized states available to the excitation are given by a Gaussian distribution and the probabilities for hopping to another site by Fo¨rster rates (eqs 2 and 4). Results for various initial energies of the excitations are displayed. The upper part of Figure 9 depicts results for low temperature (T ) 15 K). For t > 10 ps and initial energies > -1.5 σ, the experimentally observed logarithmic dependence of the average energy E on time is reproduced: E ∝ log-
(t). For start energies below -1.5 σ, the initial relaxation does not follow E ∝ log(t) behavior. Instead, a certain waiting time has to elapse before the first downhill transfer steps take place. This waiting time increases for decreasing . For ) -3.5 σ, practically no energy transfer takes place in the time interval studied. The energy reached after t ) 1.5 ns is predicted to be independent of the start energy when > -2.7σ. For < -2.7σ, the energy at t ) 1.5 ns depends (sublinearly) on the start energy. An energy of < -2.7σ corresponds to an excitation energy of 2.93 eV (0-0 transition). This is in good agreement with experimental observations (see Figure 4) where it was observed that the position of the 0-1 fluorescence band at t ) 1.5 ns was independent of excitation energy for g 2.95 eV. Comparing theoretical predictions for E(,t) with experimental data (Figure 2) we notice the following. The slope of the E vs log(t) plot (for t > 10 ps) is steeper in the simulations than in the experimental data. The absolute value of the calculated slope is about 30% too high. The predicted mean energy at t ) 2 ns for excitations with > -2.5 σ is off by about 10 meV. Changing the value of R0 in the simulations only shifts the curves along the log(t) axis and does not influence the slope. Furthermore, the calculated curves are insensitive to changes in temperature for T < 50 K. Thus, there are not many options to arrange for a better match between the predicted and observed values within the current model. The only option left would be to assume that the actual value for σ is lower than the one determined from the absorption spectrum. Agreement between theoretical prediction and the experiment can most likely be improved by taking into account a dependence of the transfer rates on the magnitude of the energy difference between donating and accepting chromophore. The lower part of Figure 9 shows the simulated relaxation of the excitations at high temperature (T ) 293 K). When > -1.5 σ, the behavior predicted is very similar to that for T ) 15 K. The energy reached at t ) 2 ns is slightly higher than in the low-temperature case. The “thermal noise” is also evident from the simulations. The behavior of excitations with initial energies < -2.5 σ differs more strongly from the dynamics at low temperature. At high temperature, the excitations can be
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Figure 7. Decay of the spectrally integrated fluorescence from polyfluorene film. Solid lines: T ) 15 K, with 2.912 eV (a) and 3.111 eV (b) excitation. Dashed lines: T ) 293 K, with 2.951 eV (c), 3.082 eV (d) excitation.
Figure 8. Fluorescence spectra of a polyfluorene film at T ) 293 K acquired with a time window 1.4-1.8 ns after excitation. Excitation energies: 3.111 (a), 3.024 (b), 2.951 eV (c). The dashed line (d) shows the fluorescence obtained with 0.41-0.45 ns acquisition window and 3.111 eV excitation.
thermally excited and their energy gradually increases with time. At t ) 2 ns the energies of the quasiparticles with initial energies in the interval -0.5 < < -3.5 have practically converged. Interestingly, the energy reached is not yet equal to the value expected for thermal equilibrium (∞ , eq 6) but slightly higher. To illustrate the temperature dependence of the simulated relaxation process more clearly, we have plotted various physical quantities as a function of temperature (Figure 10). The upper panel shows the energy reached after t ) 2 ns for two different start energies. For the two highest temperatures considered, the value for ∞ falls within the energy range displayed. For temperatures T < 250 K, ∞ < -3.6 σ when σ is around 70 meV or higher. For T ) 350 K the difference between energies reached at t ) 2 ns and ∞ is quite small, indicating that the
Figure 9. Monte Carlo simulation of the average energy as a function of time after excitation for an ensemble of photoexcitations in a disordered medium with a Gaussian distribution N(µ ) 0, σ). Energy values on the right axis have been calculated using µ ) 3.12-0.029, σ ) 70 meV. Results for various start energies are shown. Upper part: T ) 15 K, lower: T ) 293 K.
relaxation time for the excitations is short (∼ ns). At lower temperatures, this relaxation time is considerably longer. In the middle panel the variance of the energies of the ensemble of excitations at t ) 2 ns is plotted. It should be remembered that in these simulations the excitations have a sharply defined initial energy at t ) 0 and hence the variance is zero at this time. At high temperatures the variance for both two initial energies studied is close to 1 (when expressed in units of σ2), i.e., the value expected for thermal equilibrium. At lower temperatures the variance reduces sharply and becomes dependent on the initial energy . For lower , smaller variances
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Figure 11. Monte Carlo simulation of the time dependence of the width of the energy distribution of 1000 photoexcitationsin a regular cubic lattice at T ) 15 K with energetic disorder σ ) 70 meV (for further details see legend for figures 9, 10). Results for different start energies are shown.
Figure 10. Monte Carlo simulations of the dynamics of photoexcitations in an ordered cubic lattice of chromophores with energetic disorder as a function of temperature. Energies of the chromophores are distributed according to a Gaussian distribution width standard deviation σ. Hopping probabilities are described by a Fo¨rster radius R0 ) 2b, where b is the lattice constant. Results for 1000 excitations have been averaged. (upper panel) Average energy of 1000 photoexcitations at t ) 2 ns after excitation. Results for two different start energies are shown. The asterisks labeled ∞ denote the average energy at thermal equilibrium. (middle) Variance of the energy distribution at t ) 2 ns (in units of σ2) of the 1000 photoexcitations. The dashed line labeled Var∞ indicate the variance at thermal equilibrium. (lower) Average rootmean-square distance of the photoexcitation at t ) 2 ns from its original position at t ) 0 (in units of b).
are obtained. This is of course consistent with the fact that the possibilities for downhill energy transfer are more limited for excitations with low energy. This is also illustrated in the lowest panel which shows the average distance of the photoexcitations at t ) 2 ns from their initial position at t ) 0. At low temperatures the excitations with ) -3σ move on average less than one lattice position. The quasiparticles with ) -1σ enjoy more freedom and move on average over a distance of four times the lattice constant. The distance traveled increases approximately exponentially with temperature. At room temperature the excitations have moved over a distance equal to 10 times the lattice constant, practically independent of their initial energy. The variance of the energies of the quasiparticles in the ensemble deserves some more attention. In Figure 11 the standard deviation calculated as the square root of this variance is shown as a function of time for T ) 15 (upper part) and 293 K (lower part). At low temperature a curious behavior is predicted by the simulation. For particles with > -1.5 σ, the
width of the distribution of energies initially increases and then reduces again to a value of 0.4 at t ) 2 ns. For particles with < 2.0, the width shows a monotonic increase with time. The increase is smaller for lower . For excitations with > -1.5 σ, the relaxation process can apparently be subdivided in two phases. In the initial phase, the particles make their first jump to a site close by. The energies of the neighboring sides are of course distributed statistically so that the variance increases rapidly. In the second phase, the particles that have jumped to a neighboring site with relatively high energy are able to undergo further relaxation steps. The ones that have jumped to a neighboring site with low energy right away remain stationary for a long time. Altogether, the subsequent relaxation steps in the second phase result in a reduction of the width of the distribution of particle energies with respect to the first phase. For excitations with < -1.5 σ, the first phase takes a longer time and the particles apparently do not enter the second phase in the time range studied. At room temperature the behavior is less complex. One observes a steady increase of the width regardless of the initial energy. This increase continues until the thermal equilibrium is reached (standard deviation equal to σ, see eq 7). For particles with high initial , the energies spread very rapidly in the time interval between 0 and 1 ps so that in the interval from 1 to 2000 ps only a very slow increase is observed. Theoretical predictions for the bandwidth of fluorescence are also shown along with the relevant experimental data in Figure 5. For t > 5 ps and T ) 15 K, the simulated behavior agrees quite well with the experiments, apart from an almost constant offset of ∼5 meV. Better agreement between experiment and simulations would be obtained when the actual value for σ is taken to be slightly lower than the value determined from the absorption spectrum. The same holds for the slope for the E ∝ log(t) plot for high (see above). For short times the simulations predict and increase in bandwidth, which is not observed
Dispersive Relaxation Dynamics of Photoexcitations experimentally. From Figure 11 it becomes clear than only for excitation at high energy, a time-dependent spectral narrowing is predicted for t > 10 ps. Instead, for initial energies below -2.7σ a broadening with time is predicted in the time range under study. As can be seen in Figure 3 such a broadening is indeed observed for excitations created by 2.954 eV photons. At room temperature, the simulations predict much broader bands and a small increase for t > 5 ps. The experimental data show a rather constant bandwidth whose average value matches that of the predictions quite well. 4. General Discussion Many features shown by the simulations can be confirmed by experimental data from time-resolved fluorescence measurements on films of polyfluorene 1. We mention the E ∝ log(t) behavior for excitations with high initial energy and the delayed relaxation for excitations with low energy relaxation. Furthermore, the spectral narrowing of emission bands for selective excitation above the localization energy as well as the broadening of the bands for excitation below the demarcation energy have also been observed experimentally. The measurements further substantiate the interpretation of the site selective fluorescence measurements on the same material by Bauer et al.34 Their estimate of approximately 2.97 eV as localization energy is consistent with our data. In some cases, the agreement between theory and experiment is almost quantitative. Altogether, the experimental data provide strong support for the model in which photoexcitations occur as strongly localized species that can migrate by Fo¨rster energy transfer processes according to the Miller-Abrahams rates for hopping eq 2. The value of R0 used in the simulations is two times the lattice constant b. To relate this to real distances, one needs to equate b with an estimate for the average distance between two neighboring chain segments. From diffraction studies on oriented films of 1 and molecular modeling calculations, Leiser at al.38 determined a value of 17 Å between neighboring, parallel oriented chains. This number represents a lower bound for the average distance between neighboring chromophoric chain segments in the disordered film, and, putting b ) 17 Å, one calculates R0 ) 34 Å. This value seems realistic in view of R0 values known for other molecular systems. For energy transfer between rhodamine 6G molecules in room-temperature glycerol solution, R0 ) 50 Å has been determined from fluorescence depolarization measurements.35 In the following paragraph we briefly comment on the results of the simulations in relation to the previous theoretical work. As mentioned, the simulations for relaxation at low temperature via the Fo¨rster mechanism predict an E ∝ log(t) dependence. The results agree almost quantitatively with exact theory obtained for the limits T f 0 K and t f ∞.27 Application of the effective medium approximation (EMA) in the limit T f 0 K predicts more rapid decay,27 illustrating the breakdown of this approximation. For short times, however, we find the predictions to be in good agreement with our simulations. For the simulations presented, it follows that at low but finite temperatures the relaxation of quasiparticles whose transport obeys the Fo¨rster law (eq 4) differs quite strongly from transport of particles governed by the exchange mechanism (eq 3) with its exponential distance dependence. For the latter case, Monte Carlo simulations pertaining to low temperatures (T < 50 K), predict a “freezing in” of the quasiparticles and a strong deviation of the E ∝ log(t) behavior.26 The average energy decreases more slowly as a function of time than the E ∝ log(t), and one reaches a metastable state in which the particles are trapped with only a vanishingly small probability for relaxation by tunneling
J. Phys. Chem. B, Vol. 105, No. 38, 2001 9147 processes to distant sites. In fact, exact theory for the T ) 0 K limit yields E ∝ log(log(t))1/2.27 At higher temperatures (T > 50 K with σ ) 31 meV), the E ∝ log(t) behavior is restored. In summary, we find that for photoexcitation undergoing Fo¨rster type transfer within a Gaussian DOS, the E ∝ log(t) law holds for all temperatures T < 50 K in the time range studied. A freezing in of the quasiparticles is not predicted, and this absence of frustration in the relaxation results from the rather weak distance dependence (1/R6) of the dipole-dipole energy transfer in comparison to other mechanisms that rely on overlap of the charge density (e-RR). Because of this weaker dependence, tunneling of the particles over larger distances is allowed and the role of thermally activated hopping in the transport reduced. Major discrepancies between simulations and experimental data are observed in the short time domain. Here the experiments give evidence for very fast relaxation processes that lead to a rapid lowering of the average energy and a broadening of the distribution of energies. Apart form the relaxation through intersegment energy transfer, also intrasegment relaxation may contribute to observed rapid dynamics. Exciting the sample with high energy photons, certain chain segments would be excited through a higher vibronic transition. In this case, one expects rapid relaxation on a fs time scale to the lowest vibrational level by intramolecular vibrational relaxation. Other chain segments in the sample may at the same time be excited through their 0-0 transition. Thus, at higher energies where various vibronic transitions overlap, selective excitation of segments with a sharply defined electronic energy level is not possible. Thus, intramolecular vibrational relaxation provides an additional pathway for rapid relaxation of high energy excitations and also contributes to rapid broadening of the energy distribution. Another observation for which the simulations can by no means account is the occurrence of the anomalous red shifted emission that is observed at room temperature in the tail of the decay. Previous studies on photophysical properties of polyfluorene derivatives have shown that the relative intensity of the anomalous emission band depends on the processing conditions of the polymer, the aging of the sample, and the nature of the alkyl side chain of the polymer.14,36 The relative intensity of the anomalous emission is enhanced in the electrogenerated chemiluminescence spectrum.37 Especially in the case of the polyfluorene with n-octane side chains, the relative contribution of the low energy emission is much lager than for the polymer under study here with its branched side chains.23,38 There seems to be some consensus now that the anomalous emission can be attributed to an interchain species, i.e., an excitation delocalized over more than one chain segment. This delocalization is possible only when the segments have a certain geometrical orientation with respect to each other so that an excimer-like excitation can be formed within the aggregate formed by the oriented chain segments. For the polymer film under study, it seems that the number of sites per unit volume fulfilling the geometrical conditions is rather low. Thus, direct excitation of the aggregates is unlikely. They may be populated indirectly by energy transfer from the normal phase of the polymer to the aggregates. The strong temperature dependence of the mobility may then explain why the aggregates are more efficiently populated at higher temperature. The shortening of the lifetime of the emission observed upon increasing the excitation energy of the temperature may also be related to the higher mobility of the excitations under these conditions if one assumes that the excitation may be quenched by defects present at low concentration in the film. The aggregates mentioned above may also act as such a
9148 J. Phys. Chem. B, Vol. 105, No. 38, 2001
Meskers et al.
Figure 12. Quantum mechanical calculation of distance dependence of the energy transfer rate for two (planar) oligo(para-phenylene) molecules with 11 benzene rings. Results for various orientations are shown. In the diagrams the dashed line indicates the distance varied. The dashed curves represent an R-n functional dependence.
quenching defect. Unfortunately the intensity of the excimer emission band is so weak that its rise cannot be measured directly; it is buried under the intensity of the higher vibronic transitions associated with the normal fluorescence. The observation of a rise of the excimer emission with a time constant similar to that observed for the decay of the normal emission would provide strong experimental evidence for the aggregates acting as quenching defects. Both features discussed above indicate that the formalism used to model the relaxation is in certain respects an oversimplification. The question may arise whether the Fo¨rster or dipole-dipole interaction (eq 4) itself, despite its apparent success in describing the relaxation, is merely an approximation. Klaerner and Miller have given an estimate for the average effective conjugation length in polyfluorene derivatives. They conclude that the chromophoric chain segments comprise on average slightly more than 10 fluorene units39 corresponding to a length of more than 80 Å. Clearly, the minimum distance between two segments (17 Å, see above) is much smaller than the length of the segments, and under these conditions the use of the dipole-dipole or Fo¨rster type interaction (eq 4) to approximate the interaction between two molecular segments is questionable. For distances between the segments equal to or longer than a few times the length of the segments, the dipole-dipole terms still constitute the leading terms in the series expansion of the interaction energy. To get some insight in the magnitude of the errors introduced by assuming dipole-dipole interactions, we have used quantum mechanical methods to calculate the transition charge density associated with the lowest allowed optical transition of a fully planar oligo(para-phenylene) molecule comprising 11 aromatic rings.40 These molecules have a π-electron system very similar to the polyfluorenes. Calculations were made using the INDO/S semiempirical method and include configuration interaction with the lowest 60 singly excited configurations. For each carbon atom, a point-like partial charge representing the transition charge density was calculated and the transfer rate between two molecules was calculated as the square of the interaction energy obtained from calculating the electrostatic energy associated with
interactions between the partial charges on the two molecules. We have neglected exchange of electrons between the two molecules. Some typical results are shown in Figure 12. For the collinear and cofacial dimer (left part), the rates show the expected 1/R6 distance dependence for R > 100 Å. For large distances, the differences in rate for the two orientations amount to a factor of 4, consistent with classical dipole-dipole interaction. At shorter distances, rates for the cofacial dimer are overestimated by the dipole-dipole interaction, whereas for the collinear dimer the rates are underestimated. For the two other orientations studied (crossed and T- shape), the contribution of the dipole-dipole interaction is zero (dipoles at right angle) and the distance dependence of the transfer rate calculated indicates quadrupole-quadrupole interaction. The magnitude of the rates calculated is small in comparison with the rates for the cofacial or collinear dimers (
