Nonadiabatic Simulations of Exciton Dissociation in Poly-p

Jul 2, 2010 - Our results are consistent with a range of apparently conflicting experiments, resolving some controversies regarding the molecular mech...
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J. Phys. Chem. A 2010, 114, 7661–7670

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Nonadiabatic Simulations of Exciton Dissociation in Poly-p-phenylenevinylene Oligomers Michael J. Bedard-Hearn, Fabio Sterpone,† and Peter J. Rossky* Department of Chemistry and Biochemistry and Institute for Computational Engineering and Sciences, The UniVersity of Texas at Austin, Texas 78712 ReceiVed: April 16, 2010

We present simulations of exciton dissociation and charge separation processes in the prototypical conjugated polymer, poly-p-phenylenevinylene. Our mixed quantum/classical simulations focus on the nonadiabatic excited state dynamics of single and π-stacked oligomers of varying length. By applying a constant external electric field, our simulations reveal the details and time scale for exciton dissociation and fluorescence quenching and suggest how those processes relate to charge carrier (polaron) formation in polymer systems. We find that, in such a polarizing environment, sufficiently long chromophores (either single or interacting chains) can form polaron pairs via a delayed exciton dissociation mechanism or nearly instantaneously following photoexcitation. However, we find that these processes are mechanically essentially the same, being highly nonadiabatic in character and requiring transitions through “gateway” states to reach the completely charge separated electronic states. Finally, we observe thermally driven polaron hopping dynamics between chains, similar to the energy transfer dynamics we had described previously (J. Phys. Chem. A 2009, 113 (15), 3427). Our results are consistent with a range of apparently conflicting experiments, resolving some controversies regarding the molecular mechanism for charge carrier photogeneration in conjugated polymers. Introduction Conjugated polymers (CPs) form disordered materials that have been widely tested in various optoelectronic devices, such as field-effect transistors,1 photovoltaics,2 and light-emitting devices (LEDs).3 The materials are desirable because plastics tend to be robust, inexpensive, and easily processable compared to crystalline semiconductors. Despite continued interest in the electronic properties of these materials, the fundamental physics of their excited state dynamics, and charge carrier photogeneration in particular, remains controversial. A detailed understanding of such processes is vital to the development of organic optoelectronics. The primary electronic excited states in CP systems are localized electron/hole pairs (excitons, denoted hereafter by XT),4,5 a result of strong nuclear-electronic coupling. The XT dynamics are exploited in various devices: LEDs rely on the XT’s radiative coupling to the ground state, and free charges may be generated via XT dissociation for photovoltaics. Both processes are expected to be highly dependent on the structure and packing of the chromophores, as well as dynamically nonadiabatic (non-Born-Oppenheimer). However, the connection between charge carrier formation and XT dissociation remains unclear. It is easy to see why a precise mechanism of charge carrier generation has been hard to discern: CP thin films have microscopically heterogeneous morphology, making it difficult to separate the effects of chain conformation and interchain contacts from the intrinsic electronic properties. To avoid such complications, experimentalists have explored dilute solutions of single molecules. However, even relatively short chains can show spectroscopic signals attributed to aggregated chromophores.6 As a result, little consensus exists in the literature, * To whom correspondence should be addressed. E-mail: rossky@ mail.utexas.edu. † Present address: Department of Chemistry, E´cole Normale Supe´riure, Paris, 24 rue Lhomond, F75231 Paris CEDEX 05, France.

and “chemical intuition”sa foundation for physical understandingsfrom which scientists and engineers can optimize polymer optoelectronics is lacking. Molecular dynamics (MD) simulations can offer such a molecular-level picture, but in practice, computing the detailed physics of CP systems presents a computational challenge. Polymer systems are quite large, and methods for simulating the coupling between nuclear and electronic degrees of freedom in disordered systems are an active area of research. Further, one must treat 102-103 correlated quantum mechanical electrons, making dynamics as prescribed by the time-dependent Schro¨dinger equation daunting. Here, we use a mixed quantum/ classical (MQC) electronically nonadiabatic MD scheme to explore how individual and aggregated pairs of CP chromophores respond to photoexcitation. The chromophores are 5-8 repeat unit oligomers of the prototypical CP, poly-pphenylenevinylene (PPV, Figure 1A). Our model includes a fully flexible molecular backbone. To facilitate charge separation, we consider here an external constant electric field that acts as a proxy for the influence of polar solvents, for chemical substitution and defects,7,8 for a donor-acceptor system (as in polymer heterojunctions), or for charge buildup within a device.9 Two different mechanisms to explain the photogeneration of charge carriers in CPs have support in the literature: the “exciton” (indirect)10–12 and “band” (direct)8,13,14 models. For the former, free charges are generated following ultrafast XT relaxation,15 while in the latter scenario, charge-separated (CS) states are optically accessible directly from the ground state and are unrelated to XTs. Resolving this process is an important first step in developing the desired molecular-level picture of charge separation. Here, we refer to a partially CS state as an “exciplex” (EX) to distinguish it from the completely CS state or “polaron pair” (PP), which is comprised of an independent electron and hole.16 Whereas the EX retains some radiative coupling (oscillator strength) to the ground state, the PP does not fluoresce due to

10.1021/jp103446z  2010 American Chemical Society Published on Web 07/02/2010

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Figure 1. (A) Single repeat unit of the PPV polymer with bond labels in the phenyl ring as used in eq 3. (B and C) Renderings61 of 1 × (8)OPV and 2 × (7)OPV showing the ring indices used throughout this paper and the effect of the field to drive electrons toward the right-hand side of the chains and holes to the left. (C) A particular configuration of the PPV dimer with (temporary) enhanced phenyl-vinyl overlap, a topic discussed in detail previously.40

Franck-Condon effects. By the same argument, the PP should not be accessible from the ground state via one-photon absorption, which seems to fit with the XT model. Further support for the indirect model comes from electric field-induced fluorescence (FL) quenching experiments, showing a delayed effect on LED performance.12 Kersting, Deussen, and co-workers10 studied thin films of a soluble derivative of PPV and found that the electric field effects only the luminescence lifetime, but not the initial FL amplitude. This implies that the initial excited states must transform before dissociating into a nonradiative CS state. On the other hand, there is evidence that CPs follow the band model.8,13,14 Transient excited-state IR absorption experiments show that charge carriers appear ∼100 fs after photoexcitation, with no correlation to the observed XT dynamics in PPV17 and other CP systems.18–20 Moreover, FL quenching in CPs may be attributed to factors other than XT dissociation, such as charge injection,21 internal conversion,22 and chemical defects.23 Although there is little consensus about how PPs are formed, it is generally accepted that XT dissociation and charge separation are extrinsic properties of CPs.24 One way to induce PP formation is via an external electric field. The effect of electric fields on CP solutions and thin films has been the subject of several experimental investigations, and most (but not all14) agree that the macroscopic effects are FL quenching, correlated with increased photoconductivity.11,25 Such field-induced quenching would logically result from polaron formation when the field can overcome the XT binding energy. Understanding XT dynamics also means understanding the effects of chromophore nanomorphology. Conjugated polymers, in effect, consist of arrays of short conjugated segments belonging to the same or different chains; coupling between chromophores plays a fundamental role in their excited state behavior. It has generally been assumed that charge separation only occurs when multiple chromophores are involved, and the electron and hole can reside on separate chains.18,22,24,26,27 Although not universal,23,28–31 interchain interactions generally increase the probability of polaron formation.32 Another aspect of CP nanomorphology to consider is chromophore length. Recent research highlighting the relationship between chromophore length and CP electronic properties includes the use of block copolymers33 to study singlechromophore XT dissociation. In external fields up to 2 MV/ cm, short (three unit) PPV segments showed almost no fieldinduced FL quenching. However, films made with blocks of 20 PPV units showed significant quenching of both the integrated FL quantum yield and lifetime for fields greater than 1.3 MV/cm; smaller fields had no effect.34 The length depen-

dence of field-induced quenching was confirmed independently.22,35 Stark spectroscopy on ultradilute solutions of short PPV-derivative oligomers revealed field-induced single-chain FL quenching that was enhanced in longer chains, more than tripling between seven and nine repeat units. Clearly, the nuclear backbone plays a fundamental role in the excited state properties of CP systems. The strong coupling between the materials’ phonon modes and electronic properties is also well-documented.36 Therefore, any attempt to explain XT relaxation and dissociation necessarily requires a microscopic and dynamical description of the polymer nuclei and, in addition, one that goes beyond the Born-Oppenheimer approximation, as discussed below. The XT model for PP generation is a two-step process involving ultrafast relaxation of the molecular XT followed by dissociation into an electron/hole pair. Transitions from an XT state to a PP may involve a so-called nonadiabatic (NA) transition. Recently, Bittner and co-workers showed the importance of such NA pathways for XT dissociation in polymer heterojunctions.37,38 For the band model to be valid, photoexcitation occurs to an optically active state, followed by an ultrafast nonradiative transition to a PP. Thus, PP formation via either the band or XT models is expected to involve multiple electronic surfaces and be a fundamentally NA process. Our recent MQC MD simulations of oligo-phenylene vinylene (OPV) showed that thermal dynamics of the nuclei largely determines the shape of the optical spectra.39,40 We also explained40 the experimentally observed long radiative lifetimes41 and ultrafast NA relaxation of XTs.10,15 In this paper, we compare the excited-state dynamics of single and π-stacked (interacting) gas phase chromophores with and without an applied electric field. The use of an electric field creates a bias that allows us to study the coupled nuclear/electronic dynamics associated with XT dissociation and PP formation. This paper represents the first simulations of nonadiabatic XT dissociation in CPs with a fully dynamic atomistic description of the interand intramolecular interactions and a quantum description that goes beyond the tight-binding approximation.39,40,42,43 This combination allows us to explore the role of structure (disorder and packing) and dynamics in CP photophysics. Our results are fully consistent with the experiments discussed above, and we find evidence for both delayed and instantaneous charge carrier generation. Finally, we observe and discuss charge transfer dynamics between polymer segments. Computational Details To study XT dissociation in PPV, we performed a series of MQC MD simulations on A × (n)OPVε, where A ∈ [1, 2] is

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TABLE 1: Summary of Results*

* Angular brackets 〈...〉 indicate an equilibrium ensemble average; overbars indicate a nonequilibrium average;58 subscripts eq, ver, and ex refer to equilibrium properties, average properties of the simulated photoexcitation events (i.e., the vertical excitation energy), and excited state (nonequilibrium) properties, respectively; 0, k, i, and f refer to the ground electronic excited state, the optically active state (or states), the initially populated excited state, and the equilibrated excited state. a State(s) that contribute strongly to the absorption spectrum. b Average energy gap ((0.1) from S0 to Sk. c Average oscillator strength from S0 to Sk; 1 standard deviation error bars in parentheses. d Number of NA trajectories run. e Average vertical excitation energy, from S0 to Si. f Average oscillator strength for the vertical excitation. g Average excited-state energy gap from Sf to S0. h Average oscillator strength for emission from Sf f S0. i Percentage of NA trajectories showing PP or EX characteristics, otherwise XT. j One trajectory made a NA transition to the ground state about ∼800 fs after excitation; averages do not include this portion of the trajectory. k In the two-chain system, the electron and hole are on the same chain at all times.

the number of oligomer chains, n ∈ [5, 8] is the number of repeat units in a single oligomer (see Figure 1), and ε ∈ [0, 10.4] MV/ cm is the applied field strength. Despite the widely agreed upon importance of nuclear dynamical relaxation in CPs, it is not included in the vast majority of simulations, chiefly because such dynamics are computationally intensive. Those that have included excited state dynamical relaxation tend to rely on onedimensional lattices and tight-binding models, such as those employing the SSH Hamiltonian.27,44,45 Any excited state dynamics simulations are usually performed in the adiabatic46 or Ehrenfest27,45 limits, each with substantial limitations.47,48 Our method combines a fully flexible molecular mechanical model for the polymer backbone and a semiempirical quantum description of the π-electronic system. All simulations are performed using this QCFF/PI framework42,43 that we have discussed in detail elsewhere.39,40 Finally, we use a nonadiabatic MQC MD algorithm49 to compute excited state relaxation dynamics, including transitions between electronic states, following simulated photoexcitation. The classical all-atom backbone of the OPV chains is described by an empirical potential,42 which includes ring puckering and torsional parameters.39 The quantum-mechanical π-electron system is described by the semiempirical Pariser-Parr-Pople (PPP) Hamiltonian,50–52 as parametrized for OPV by Sterpone and Rossky.39 At each time step of the MD simulation, we solve for the ground state of the quantum subsystem. The analytical expressions for each term in the PPP Hamiltonian (which explicitly depend on instantaneous nuclear positions) allow us to calculate

the forces that the π-system exerts on the classical nuclei.39 With the quantum forces added to the interatomic forces from the classical potential, we integrate the nuclear dynamics in a constant-energy velocity Verlet scheme with a 1 fs time step. The ground state adiabatic (Born-Oppenheimer) trajectories are run at a nominal temperature of 300 ( 15 K; all systems include 5-10 ps of velocity rescaling (thermalization) before collecting statistics for 25-50 ps at constant energy. We calculate the electronic excited states and forces at each time step using the configuration interaction (CI)39,42,53 method in a finite active space of 100 single excitations (SCI); further increases in the SCI active space red shift the absorption maximum by a few hundredths of an electron volt, but it does not alter the nature of the underlying states. Our nonequilibrium, NA trajectories emulate one-photon Franck-Condon excitations. The NA ensembles for each system (of varying size and field strength) contain between three and ten NA runs. We wanted the initial conditions to be totally uncorrelated with each other, so they were chosen such that the initial configurations, taken from their respective ground state trajectories, were separated in time by at least 2.5 ps; the nuclei were assigned new (randomized) velocities chosen from a Gaussian distribution about T ) 300 K. In the NA trajectories, each member of the ensemble is excited at t ) 0 from the ground state into one of the SCI excited states, k, such that the energy gap, Ek0 ) Ek - E0, was resonant with the maximum of the ground-state absorption spectrum, (0.1 eV (see Table 1). To

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ensure that each trajectory is experimentally relevant, we also require that the chosen state has significant oscillator strength

Mk0 ) Ek0|〈k|rˆ |0〉|2

(1)

compared to the absorption peak value (see Table 1). In eq 1, |0〉 and |k〉 are the ground and kth SCI excited states, and rˆ is the quantum-mechanical position operator. Following excitation, we calculate the NA dynamics of OPV using the surface hopping algorithm.49 In this method, the system evolves adiabatically (on a single electronic surface) with a stochastic algorithm determining transitions between electronic eigenstates. The probability of a NA transition is proportional to the strength of the NA coupling vector, which relates how the π-electron eigenstates respond to the nuclear dynamics.40,49,54 Following Mu¨ller and Stock,55 “forbidden” upward transitions are ignored instead of being treated as classical turning points. The external electric field considered here changes the energy of the system, which for a constant field is given by

V(F) ) V(0) + f µb F

(2)

where b µ is the molecular dipole moment. Here, the field is applied in the xˆ direction so that b F ) Fx ) εxˆ, with strength ε in units of MV/cm. Given this convention, holes are driven toward the left-hand side and electrons to the right (Figure 1). Since the oligomer ends are free and velocities are thermalized, the orientation of the molecular backbones does not coincide perfectly with the field at all times, though they are aligned initially. The field affects both the classical nuclei and the quantummechanical electrons. We add a one-electron term to the diagonal elements of the PPP Hamiltonian and a classical (nuclear) potential energy term. The forces acting on the nuclei from the electric field are given by the classical gradient of both terms with respect to the nuclear positions. Characterization of Excitons, Exciplexes, and Polaron Pairs. In the Introduction, we qualitatively described the three different electronic excited states that a CP may adopt: a bound exciton (XT), a partially charge-separated exciplex (EX), and a fully charge-separated polaron pair (PP). Before presenting our results, we first describe three metrics that quantitatively characterize these excited states. The first is the extent of electron/hole localization, as measured by nuclear distortions. The second is the degree of charge separation along the chain backbone, which directly addresses the differences between PPs and EXs. Finally, the time dependence of the oscillator strength for a radiative transition to the ground state, Mk0, is an especially sensitive indicator of the nature of CP excited states, as it relates to the FL lifetime. When combined, these three standards provide a framework for interpreting each trajectory from the nonequilibrium ensemble and thus give an intuitive, molecularlevel picture of the CP correlated electronic states. In our previous publications39,40 on PPV oligomers, we showed that the XT undergoes ultrafast NA relaxation, in agreement with time-resolved spectroscopic experiments. The relaxed XT induces benzene to quinone-like distortions of the π-bond network, localized across three to five repeat units. The square of the ring order parameter

DR2 )

[ 41 (d

1

2 1 + d2 + d3 + d4) - (d5 + d6) 2

]

(3)

(also eq 15 in ref 39) tracks these distortions in the oligomer. In eq 3, di is the ith π bond in the phenyl ring, as indicated in Figure 1A. This measure is akin to the familiar “bond-length alternation” parameter used to study lattice distortions in computational studies of linear polyenes.46,56 Using this metric, we can readily observe time-dependent XT migration along the chain and hopping between chains when a second oligomer is sufficiently close (typically e4 Å). The dynamics of XT dissociation may also be monitored using eq 3; as the electron and hole migrate away from each other, quinoidal distortions, consistent with those observed in benzene ions,57 will appear at the oligomer ends. Although eq 3 is useful for determining electron/hole localization, it cannot distinguish between EXs and PPs since both are spatially separated electron/hole pairs. However, we can calculate the excess charge density, qxs (relative to the neutral XT or ground state), at the oligomer termini to measure the degree of charge separation. Although we are working with closed systems (polarons are neither injected nor removed), a subset of atoms may see more (or less) electrons when in the excited state. In fact, the excess charge density on the 22 carbon atoms that make up three terminal phenyl rings (and the two vinylene junctions between them) ranges from 0 for an XT to 1 for a PP, neither of which significantly fluctuate away from those extreme values (see the Supporting Information for details about calculation of the excess charge density); charge densities somewhat less than 1 and showing large thermal fluctuations (∼15%) indicate EX formation. Here, when we display qxs for the oligomer ends, a value of +1 indicates an extra electron is present. Additionally, EXs are differentiated from PPs by the magnitude and fluctuations of the oscillator strength as a function of time following excitation. The oscillator strength measures the radiative coupling to the ground state, and as we show in the next section, the generation of PPs is identifiable by two features in the temporal evolution of Mk0. The first is a rapid (sub 50 fs) decrease of Mk0 to essentially 0 following a NA transition to S1. Exciplexes, on the other hand, tend to stay on S2 and have a small but finite Mk0. The second signature of PPs is a lack of thermal fluctuations in the radiative coupling, whereas Mk0 for EXs fluctuates by an order of magnitude or more. Using this combination of characteristics, we say an electron and hole are fully separated into a PP when: 1. the lattice distortions are localized on the oligomer ends, 2. |qxs| on the three terminal rings is g0.95 e, 3. Mk0 , 0.01 au, and 4. the above conditions are met for at least 250 fs consecutively. The last point is included because sometimes the first three conditions are observed to be met only briefly. In the present work, we believe this is a finite-size effect: given longer chains, the electron and hole would continue to migrate away from each other, reducing geminate recombination. For our purposes, we only collect statistics (such as those given in Table 1) for PPs over the times when the four-part definition above applies and not otherwise. Similarly, we consider systems to show EX behavior only when the electron and hole are spatially separated; Mk0 is finite (typically ∼0.005-0.5 au); and the system has made a NA transition to S2. If a trajectory labeled PP recombines to form an EX (or XT), we do not include that trajectory’s dynamics in the ensemble of averages for EXs (or XTs).

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Figure 2. Ground state absorption spectra for a single heptamer, 1 × (7)OPV, a single octamer 1 × (8)OPV, and two π-stacked heptamers, 2 × (7)OPV, from ground state trajectories run with the field off (panels A, C, and E) and with ε ) 10.4 MV/cm (panels B, D, and E).

Results Figure 2 shows the calculated ground state electronic absorption spectra (thick solid curves) for different OPV systems: a single heptamer, 1 × (7)OPV (panels A and B); a single octamer, 1 × (8)OPV (C and D); and π-stacked heptamers, 2 × (7)OPV (E and F). The top and bottom row show the fieldoff (ε ) 0) and field-on (ε ) 10 MV/cm) spectra, respectively. All spectra were computed from 25 ps ground-state trajectories in the inhomogeneous limit by binning the instantaneous transition dipole matrix elements (width ) 0.025 eV) between the ground and each of the 100 SCI excited states

A(E) ) 4π2R

∑ Mk0δ(E - Ek0)

(4)

k

where R is the fine structure constant. In the figure, the dotted curves are the thermally averaged transitions to individual states. For example, the largest dotted peak in panel E shows that the strongest transition for 2 × (7)OPV with the field off is to S2, the second excited state. Table 1 summarizes the results of our calculations for a range of OPV systems and field strengths. The column labeled “Bright St., 〈k〉 ” indicates the state that has, on average, the greatest oscillator strength. For most isolated-chain systems, this corresponds to S1. The exceptions are 1 × (7)OPVε)10 and 1 × (8)OPVε)10, which have two transitions of nearly equal intensity: from the ground state to S1 and S2 in the former and to states S2 and S4 in the latter. For π-stacked systems, the bright state is S2, except for 2 × (7)OPVε)10, which absorbs most strongly to S6. The table also shows the number of NA trajectories and several statistics about those trajectories, such as the average initial excited state, 〈i〉, the average vertical excitation energy for the ensemble, (E0i)ver, and the average oscillator strength for the resonant excitations, (M0i)ver. Following excitation, the trajectories may undergo one or more NA transitions, but each ultimately resides on a single state for the majority of our 1.5 ps runs; we term this state the final state, f. Once on state f, the system re-equilibrates, and we report in the table nonequilibrium ensemble averages58 (indicated by overbars) of the excited-toj f0)ex, as well as the oscillator strength ground-state energy gap, (E j f0)ex. Finally, for a radiative transition to the ground state, (M

we report the fraction of trajectories that, upon relaxation, exhibit characteristics of either EXs or PPs. The spectra for 1 × (5)OPV and 2 × (5)OPV are not shown since no changes are evident upon application of the field. Both 1 × (8)OPV and 2 × (7)OPV, however, undergo abrupt changes in their electronic properties as a function of field strength. For weaker fields, the nature of the low-lying excited states are indistinguishable from their field-off counterparts (i.e., they are neutral XTs). However, both 1 × (8)OPVε)10 and 2 × (7)OPVε)10 see an increase in the low-lying density of states for ε ) 10 MV/cm, with several relatively weak transitions appearing below the dominant peak. Importantly, we will show that for these two systems the low-lying excited states are CS states and that the relaxed S1 is a PP. Single-Chain Exciton Dynamics. Previously, we discussed the dynamics of photoexcited short single-chain OPV,39 focusing particularly on exciton-phonon coupling. We observed large nuclear distortions that characterize the influence of the neutral XT as a perturbation to the phenyl rings and vinylene junctions. We also described sub-100 fs relaxation of the electronic system and nuclear geometry, in agreement with ultrafast spectroscopic experiments.10 As Table 1 shows, the dynamical time scales of short oligomers are unchanged by the presence of a polarizing field. The equilibrium ground (eq) and equilibrated excited (ex) state properties of 1 × (5)OPVε)0 are nearly identical to 1 × (5)OPVε)10. The primary difference is a field-induced red shift of the excited-to-ground-state energy gap, (Ef0)ex, by ∼0.2 eV. This is a feature common to all systems we studied and results from a slight polarization of the XT charge density. For field-on and field-off trajectories, we find the same sort of XT-induced benzoid-to-quinoid distortions across the three central rings following NA relaxation; as before,40 the 1 × (5)OPV system remains quasi-equilibrated on S1 because the NA (nonradiative) coupling to S0 is extremely small. The excited state dynamics of 1 × (7)OPV in weak and moderate electric fields is identical in character to the field-off trajectories and to those of the pentamer. Changes only appear for the largest applied field, ε ) MV/cm, when we observe a mixture of both EX and XT formation. Our results then suggest that the outcome of photoexcitation depends on the length of the oligomer backbone. Figure 3 shows a typical example of XT localization for 1 × (7)OPVε)10; the dynamics observed and the influence of the XT on the nuclear coordinates are both comparable to those in the field-off trajectories. Panel A shows the square of the ring order parameter, eq 3, as a contour plot for each ring as a function of time following excitation (cf. Figure 1 B). Panel B shows the time average. The XT induces quinoidal distortions in rings 3-5, precisely what we had seen in field-off simulations,39 while the terminal rings (numbers 1, 2, 6, and 7) are not substantially perturbed from their ground state geometries. In the seven trajectories classified as XTs, the average magnitude of the excess charge density on the 22 π-carbon atoms that make up the three terminal monomers is |qjxs| ∼ 0.7 e. Again, this is the result of a slight polarization of the XT by the electric field, but the large oscillator strength (2.5 au) and central-ring distortions indicate a bound XT. Finally, we find that, on average, the length of the chain contracts, relative to the ground state, from 33.6 ( 3 Å to 31.4 ( 0.6 Å in the presence of the XT, where the range reflects the fluctuations (and the difference in the ranges reflects the short NA simulations, which do not sample large fluctuations in oligomer length).

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Figure 3. Exciton localization on 1 × (7)OPV. (A) contour plot of the ring order parameter, eq 3 DR2 (in atomic units, a20), as a function of time following excitation into S1. The XT initially perturbs rings 5-6 from ideal benzoid hexagons (DR2 ) 0) but it rapidly localizes in the center of the oligomer, causing quinoidal geometric distortions mainly across rings 3-5. The temporal resolution of the contour plot is 20 fs, where each point is a boxcar average of two configurations separated by 10 fs. (B) The time-averaged ring order parameter, peaked at ring #4, the center of the heptamer (see Figure 1B).

The other three 1 × (7)OPVε)10 trajectories undergo partial polarization/dissociation into an EX (not shown). Following excitation to either S1 or S2, we observe sub-100 fs dissociation, as indicated by an excess charge density that appears at the oligomer ends (|qjxs| ∼ 0.95 e) and by distortions centered on the penultimate rings. This partially CS state is also characterized by a finite, though small, chance to make a radiative transition to the ground state, approximately 7 times less intense than for emission from the XT state. This is an indication that S1 is not a PP, which has virtually no radiative coupling to the ground state. In contrast to the XT, the EX increases the length of the oligomer chain to 37.0 ( 0.2 Å. Further discussion of the correlation between oligomer length and the excited state properties of 1 × (7)OPV is provided in the Supporting Information. The isolated octamer, 1 × (8)OPV, on the other hand, is about 45 Å long in the ground state, which is evidently enough for the electron and hole to move apart and form a PP, as shown in Figure 4. Whereas at all other fields considered here we find only XTs, each of the 1 × (8)OPVε)10 trajectories shows PP formation via delayed XT dissociation. Figure 4 shows a particular trajectory that was initially excited to S2 (the optically active state). Panel A shows the occupied state, k (solid line), as a function of time, as well as the excess charge density at the oligomer ends, qxs (dotted and dashed curves). The middle panel shows the ring order parameter, and panel C shows the oscillator strength from the occupied state, k, to the ground state, 0. The vertical dotted line through all three panels is at t ∼ 210 fs, when the system makes a NA transition to S1, which coincides with the formation of a PP. Panel B shows that the excited state is initially delocalized across several rings, but after the system makes the transition to S1, the distortions migrate toward the penultimate rings on either end of the oligomer. Such distortions signify the electron on the right-hand side of the oligomer and the hole to the left-hand side (cf. Figure 1). At the same time, Mk0 drops to zero (fluorescence is quenched), in accord with our four-part definition for PPs given above. The time to charge separation for 1 × (8)OPVε)10 is 192 ( 118 fs and is as long as 0.5 ps. Of special note, in each of these simulations, PP formation only occurs following a NA transition

Bedard-Hearn et al.

Figure 4. (A) Occupied electronic excited state (solid line) and excess electron density, qxs, around the penultimate rings for a single octamer, 1 × (8)OPVε)10, with ring indices given by the schematic in Figure 1B. (B) Ring order parameter, DR2 (eq 3) for the same trajectory, shown as a contour plot; units are a20. (C) The oscillator strength for the trajectory. The vertical dotted line through all three panels indicates the time at which the system undergoes a NA transition to the first excited state, S1, and the XT dissociates into a polaron pair.

to S1. Thus, the reason S1 does not contribute to the ground state absorption spectrum in Figure 2 is because the lowest singlet excited state is a Franck-Condon forbidden PP and not the XT as seen for ε e 7.8 MV/cm. π-stacked 2 × (n)OPV Dynamics. We have discussed the structure and dynamics of π-stacked PPV pentamers with no electric field, 2 × (5)OPVε)0, in a previous publication40 but review it briefly here for context. Upon excitation to any of the low-lying optically allowed excited states, 2 × (5)OPV undergoes ultrafast NA relaxation to S1, a bound neutral XT delocalized across 3-4 repeat units. The XT generally resides on a single chain, though “hopping” between chains on a ∼500 fs time scale occurs when another chromophore is nearby. The field-off simulations of 2 × (7)OPV are similar, and as Table 1 shows for all ε e 7.8 MV/cm, we observe only XTs. In contrast, for the simulations of 2 × (7)OPV with the highest fields, we observe both EX and PP formation but not XTs. Figure 5 shows an example of delayed XT dissociation. For dimers, we present DR2 on two panels: The bottom panel has the temporal evolution of the ring distortions for one chain and the top panel for those rings on the other (essentially parallel) chain. In this particular trajectory, an initially delocalized excited state on chain 2 evolves into a state with nuclear distortions primarily on rings 1-3 and 12-14. The dashed line in each panel is drawn at t ∼ 500 fs, which is when the system undergoes a NA transition from S3 to S1. After the NA transition, the excess charge density (panel B) and oscillator strength (panel C) conform to the definition of a PP. As for 1 × (8)OPVε)10 discussed above, we find PP formation in the π-stacked system is also accompanied by a NA transition to S1 (or in one case, direct excitation to S1). Here, PP formation coincides with the observation of the electron and hole on separate chains. After some time, the PP may recombine to form an EX, either on S1 or by making a NA transition to S2; this is accompanied by an increase in Mk0 and is evidence that the electron and hole are residing on the same chain. Figure 6 shows a counter example, with near-instantaneous, but not truly direct, PP formation. The initial excited state is

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Figure 5. Delayed polaron pair formation. (A) Occupied electronic excited state, k, during a NA simulation of 2 × (7)OPVε)10. (B) The excess electron density, qxs, with ring indices given by the schematic in Figure 1B. (C) The oscillator strength for the trajectory. (D) Ring order parameter, DR2 (eq 3) for the same trajectory, shown as two contour plots, one for each of the (essentially parallel) OPV chains; units are a20. The vertical dotted line through the panels is at t ∼ 500 fs and indicates the time at which the system undergoes a NA transition to the first excited state, S1, and the XT dissociates into a polaron pair: the electron and hole each cause quinoidal distortions on the ends of separate chains, and the oscillator strength for a radiative transition to the ground state at t > 0.5 ps is zero.

Figure 6. Ultrafast polaron pair formation. (A) Occupied electronic excited state, k, during a NA simulation of 2 × (7)OPVε)10. (B) The excess electron density, qxs, with ring indices given by the schematic in Figure 1B. (C) The oscillator strength for the trajectory. (D) Same as panel D in Figure 5. The ∼3.2 eV excitation into S4 undergoes a NA transition to S1 at t ) 54 fs (as indicated by the dashed lines and arrows), which results in PP formation. The nearly instantaneous formation of the PP coincides with a rapid decay of the radiative coupling to the ground state, Mk0.

delocalized across chain 1 and parts of chain 2, but after ∼50 fs the nuclear distortions appear on the opposite ends of the two chains. As in Figure 5, the dashed lines and arrows are drawn at the moment of the NA transition, in this case from S4 to S1. After the NA transition, the excess charge density for the terminal rings (panel B) is roughly constant at (1, and the emission probability (panel C) vanishes. This ultrafast process generates free charges from an optically accessible state before a bound XT can form. It would therefore be experimentally indistinguishable from direct optical excitation to a charge transfer state and hence would appear as in the band model. Finally, Figure 7 shows an example of EX formation in 2 × (7)OPVε)10, which happens in five of the NA trajectories. An initially delocalized excitation is polarized by the action of the field and dissociates while on a single chain (chain 2 in this case), but the oscillator strength does not vanish (panel B) as it would for a PP. In addition, we do not observe a NA transition prior to charge separation, and the electron/hole pair occupies the same chain. The large vertical arrows in Figure 7 show that the electron and hole may hop between chains, though always in a concerted fashion. The frequency of these EX hops (once every few hundred femtoseconds) is similar to the interchain XT hopping dynamics we observed in 2 × (5)OPVε)0.40 For the EX, the electron and hole occasionally hop with an interval of 10-20 fs between them, but these brief excursions

do not induce the geometric distortions indicative of PPs. They may, however, represent precursors to PP formation. In both of the delayed PP trajectories, several such transient hops were made with increased frequency just before the NA transition that initiated stable PP formation. (This is seen in Figure 5 B; it is difficult to discern these features in Figure 5D.) These observed interchain hopping dynamics are caused solely by thermal fluctuations of the oligomer backbone that may, albeit temporarily, make one chain more hospitable to charges than the other. In summary, for 2 × (7)OPVε)10, both ultrafast and delayed charge separation occurs. From nine nonequilibrium trajectories, we observe four PPs, two via nearly direct and two via indirect mechanisms, as well as five EX trajectories. Excitation Energy. A question one might ask is if PP formation depends on excitation energy. To answer this, we ran a pair of nonequilibrium ensembles with one excited below the ground state absorption maximum and one above it. The first case, 2 × (7)OPVε)10, is excited 0.3 eV to the red of the absorption maximum (see Table 1). One-third of these trajectories are found to form PPs, with one doing so nearly instantaneously and two via delayed mechanisms. The former is notable because it was excited directly to S1 and met the conditions for a PP within 10 fs. These results are consistent with what we had seen when exciting on resonance, with no

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Figure 7. Exciplex formation. (A) Occupied electronic excited state, k, during a NA simulation of 2 × (7)OPVε)10 initially excited to S7. The system undergoes several NA transitions before relaxing on the final state S2, an EX. (B) The oscillator strength, eq 1, for the radiative transition Sk f S0 for the same trajectory as in Panel A. (C) Contour plot of the ring order parameter, eq 3, for the trajectory shown in panels A and B as a function of time after excitation; units are a20. The ∼3.3 eV excitation initially causes delocalized geometric distortions across both chains, but after ∼200 fs, the distortions are clearly identifiable around the penultimate rings of chain 2 (top contour). The vertical arrows at t ∼ 0.6 and ∼ 1.3 ps indicate two concerted EX hopping events from chain 2 to chain 1 and back again.

evident change in the ability to form PPs or to quench FL, despite evidence suggesting the contrary.26 What about the effects of higher energy excitations? Might the excess photon energy compensate for a weaker field and drive XT dissociation? Previous tight-binding calculations of PPV oligomers in the presence of a field suggested that higher electronic excitation led to more rapid polaron formation.27 To test this hypothesis with our model, we examined the NA dynamics of 1 × (8)OPVε)7.8sa system that showed only XT formation when excited on resonance. When the system was excited with an additional 0.5 eV, we observed exactly the same XT formation dynamics seen previously with on resonance excitation: sub-100 fs NA relaxation to a bound XT on S1. Thus, the excitation energy does not appear to influence the excited state behavior of PPV oligomers, either isolated or aggregated. Rather, it is the nature of the low-lying electronic excited states and associated accessible nuclear distortions that determines the outcome of photoexcitation in these systems, at least for the model used here. Nature of Charge Separated States. It is logical to ask about the nature of the CS states when the field is off. Using a variety of metrics, including transition density matrix analysis,46,59 we are unable to identify any accessible low-lying CS states in the 1 × (8)OPVε)0 system. However, in the field-off π-stacked PPV system, we find that S5 (and to a lesser extent, S4) contains large off-diagonal transition density matrix elements (i.e., a betweenchain transfer of electron density) indicating a CS state and significant excess charge density at the oligomer ends for vertical excitations. On the basis of quantum chemical calculations, Bre´das60 has suggested that the charge transfer states might be S3 and S5. Thus, although the presence of CS excited states appears to be intrinsic to thermally disordered aggregated PPV chromophores, photogeneration of charge carriers requires a bias because the PP states lie above the one-photon-allowed bright XT states. Under bias, 2 × (7)OPVε)10 has CS states that are lower in energy than the XT, as our nonequilibrium results above demonstrate. Moreover, we observe in 2 × (7)OPVε)10 that PPs only form on state S1 and that S2 is always an EX (although S1 can also be an EX owing to thermal fluctuations of the nuclear backbone that can make the PP unfavorable). In 1 × (8)OPVε)10, S1 is exclusively a PP, and in 1 × (7)OPVε)10, those chro-

mophores that are stretched have low-lying EX states, but we find no low-lying PP states. On the basis of the results of these simulations, we have developed considerable molecular insight into polaron formation and XT dissociation in CPs. Chromophore size and morphology are important considerations for XT dissociation and charge carrier photogeneration. The reason for this is clear: It requires a polarizing environment and sufficient conjugation length to shift the CS states to lower energy. Thus, following photoexcitation to an optically allowed XT state, NA (i.e., nucleardriven, nonradiative) relaxation pathways may deliver the system to an exciplex (correlated) or polaron (uncorrelated) electron/ hole pair. That this can occur in both individual and aggregated systems is an important result for interpreting experiments on single molecules and in dilute solutions. We also find that the electric field does not significantly alter the strength of transitions from the ground state to CS states. For example, transitions to S1 in 1 × (8)OPVε)10 and 2 × (7)OPVε)10 from the ground state are 100 and 20 times less intense than their respective transitions to the “Bright” XT states; thus, one-photon absorption directly to a CS state in both single and aggregated systems remains highly improbable. Instead, the critical primary effect of a polarizing medium is to lower CS state energies, so that they sit below the optically allowed states. In the systems under investigation here, where no molecular environment surrounds the chains, the threshold field is ∼10 MV/cm. One can also now understand why we see both ultrafast and delayed PP photogeneration in 2 × (7)OPV. The “instantaneous formation” trajectories are the result of rapid NA transitions to S1 from a higher-lying excited state, either S3 or S4. In the remaining trajectories, we observe sequential relaxation to S2, where the electronic system remains kinetically trapped for some time. As the system evolves on S2, the high density of electronic excited states mixes, and most trajectories experience at least one upward transition to S3, which is the gateway to the PP, S1. S3 may be occupied only briefly before a NA transition takes the system to either a PP or S2, an EX. Nevertheless, at least for our systems and models, it is clear that transition through an intermediate state (S3 or S4) is a requisite step to reach the fully CS state.38 Such a gateway role for NA transitions deserves further study.

Exciton Dissociation in PPV Oligomers Conclusions In this paper, we have used oligomer systems to investigate some of the controversies surrounding charge carrier photogeneration and exciton dissociation in conjugated polymers. Our mixed quantum/classical molecular dynamics simulations represent the first fully dynamical, NA simulations of conjugated polymers in a polarizing environment that go beyond the tightbinding approximation. Our model reproduces several experimental results, including delayed FL quenching (by XT dissociation) and instantaneous photogeneration of charge carriers unrelated to XT dynamics. By examining both single chromophores and π-stacked chromophore systems, we are able to create a detailed molecular-level picture of the excited state processes in terms of conjugation length, interchain interactions, and NA dynamics involving multiple electronic surfaces for the model systems. The NA relaxation pathways are driven by thermal dynamics of the polymer backbone, which shapes the electronic energy surfaces and induces NA coupling, thus driving transitions between electronic surfaces. Moreover, we saw that interchain interactions play an important role in efficient polaron formation, particularly in the ability of electrons to transfer between chains, though PPs can also form on sufficiently long individual chromophores. We also observed the dynamics of uncorrelated polarons hopping between chromophores and of concerted quasi-particle hopping in the case of an EX. The ability of the aggregated chromophores to share these excitations and rapidly transfer excitation energy and/or charge is clearly an important property of CPs. Finally, we observed relatively rapid (subpicosecond) and frequent geminate recombination of PPs in 2 × (7)OPV. It is clear why two very different mechanismssthe band and exciton modelssof charge carrier generation have experimental support. Our calculations show evidence for both delayed FL quenching and nearly instantaneous generation of PPs, though the results suggest that the same nuclear and electronic dynamics are acting. The two are only distinguished by the time scale. The nuclear dynamics ultimately determines the outcome of electronic relaxation by inducing NA couplings within a relatively dense manifold of electronic excited states. Given a bias and a large enough conjugated backbone region, excitation can lead to either fast or slow charge separation. Our results, which agree with both field-induced fluorescence quenching and time-resolved measurements of photocurrent, help to resolve this controversy. Acknowledgment. This work was supported by the NSF under grant CHE-0910499. P.J.R. also acknowledges the support of the R. A. Welch Foundation (F-0019). Supporting Information Available: 1 × (7)OPV: Correlation of initial geometry and excited state evolution and calculation of excess charge density. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Paloheimo, J.; Kuivalainen, P.; Stubb, H.; Vuorimaa, E.; Yli-Lahti, P. App. Phys. Lett. 1990, 56 (12), 1157. (2) Brabec, C.; Sariciftci, N.; Hummelen, J. AdV. Funct. Mater. 2001, 11, 15. (3) Burroughes, J. H.; Bradley, D. D. C.; Brown, A. R.; Marks, R. N.; Mackay, K.; Friend, R. H.; Burns, P. L.; Holmes, A. B. Nature 1990, 347, 539. (4) Schwartz, B. Annu. ReV. Phys. Chem. 2003, 54, 141. (5) Scholes, G. D.; Rumbles, G. Nat. Mater. 2006, 5, 683. (6) Becker, K.; Da Como, E.; Feldmann, J.; Scheliga, F.; Thorn Csanyi, E.; Tretiak, S.; Lupton, J. M. J. Phys. Chem. B 2008, 112 (16), 4859.

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