Exciton Dynamics in Disordered Poly(p

Article pubs.acs.org/JPCA

Exciton Dynamics in Disordered Poly(p‑phenylenevinylene). 1. Ultrafast Interconversion and Dynamical Localization Oliver Robert Tozer†,‡ and William Barford*,† †

Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, United Kingdom ‡ University College, University of Oxford, Oxford OX1 4BH, United Kingdom ABSTRACT: The disordered Frenkel−Holstein model is introduced to investigate dynamical relaxation and localization of photoexcited states in conformationally disordered poly(p-phenylenevinylene). It is solved within the Ehrenfest approximation, in which the excited state is treated fully quantum mechanically, but the nuclear displacements are treated classically. The following are shown: (i) Lower energy local exciton ground states (LEGSs) adiabatically relax to vibrationally relaxed states (VRSs) in the time scale of one or two vibrational periods (ca. 40 fs). The relaxation of LEGSs is accompanied by localization and fluorescence depolarization, as the transition dipole moment reduces and rotates. The amount of dynamical localization increases as the torsional disorder decreases, causing an increase in the fluorescence depolarization. (ii) Higher energy quasiextended exciton states (QEESs) interconvert to VRSs via three distinct episodes. A brief initial period of adiabatic relaxation is followed by the time-evolving eigenstate becoming a linear superposition of instantaneous eigenstates of the Frenkel−Holstein Hamiltonian. Typically, after a few hundred femtoseconds, one of the instantaneous eigenstates dominates the linear superposition, and the remaining dynamics is again adiabatic relaxation to a VRS. (iii) Very high energy QEESs, which are delocalized over many chromophores, sometimes exhibit a splitting of the wave function into more than one VRS. This self-localization onto more than one chromophore is assumed to be a failure of the Ehrenfest approximation, as this approximation neglects quantum mechanical coherences between the electronic and nuclear degrees of freedom. (iv) QEESs exhibit larger, but slower, fluorescence depolarization than LEGSs. Thus, ultrafast fluorescence depolarization is a function of excitation energy and conformational disorder.

I. INTRODUCTION There are two broad classes of exciton dynamics in conformationally disordered conjugated polymers. First, there is ultrafast (subpicosecond) intrapolymer dynamical localization of vertically excited states into vibrationally relaxed states (VRSs). Second, following vibrational relaxation, there is exciton transfer between chromophores on the same and different polymers on the time scales of tens of picoseconds to nanoseconds. Both classes of dynamics have differing photophysical signatures, as observed via spectral diffusion and fluorescence depolarization.1−3 Predicating any model of exciton dynamics are the assumptions concerning the energetic and spatial distribution of the excited states. In this and the accompanying paper4 we make the assumption that the energetic and spatial distributions of the vertical excited states are determined by static conformational and environmental disorder. As explained in previous papers,5−7 disorder causes Anderson localization8 of the exciton center-of-mass particle, and in quasi-one-dimensional systems leads to two types of localized exciton wave function. First, there are local exciton ground states (LEGSs), which are the nonoverlapping and space-filling locally lowest energy excited states. Their spatial extent thus defines © 2012 American Chemical Society

absorbing chromophores in the sense that an absorbing chromophore represents the irreducible part of a polymer that supports a vertical excitation.6 Second, there are quasiextended exciton states (QEESs), which, on the other hand, are higher in energy and delocalized over a number of chromophores. These two types of vertically excited states implies two types of ultrafast dynamical relaxation and localization. On coupling to the nuclear degrees of freedom, within a few C−C bond oscillations a LEGS adiabatically relaxes on its Born− Oppenheimer surface to form a VRS (or exciton-polaron). In contrast, because a QEES lies higher in energy than at least one other state with which it spatially overlaps, its relaxation to a VRS implies a nonadiabatic interconversion process. (Because emission occurs from a VRS, in an analogy to an absorbing chromophore, an emissive chromophore is defined by the spatial extent of a VRS wave function.) It is the purpose of this paper to introduce a realistic and computationally tractable model of both relaxation processes Received: July 16, 2012 Revised: October 2, 2012 Published: October 3, 2012 10310

dx.doi.org/10.1021/jp307040d | J. Phys. Chem. A 2012, 116, 10310−10318

The Journal of Physical Chemistry A

Article

model, which describes the coupling of the center-of-mass particle to the nuclei, captures the essential physics of selftrapping.14 The Frenkel exciton model is motivated by the observation that an exciton is a two-particle object, namely an electron bound to a hole. To a good approximation, this two-particle problem can be decomposed into two independent one-particle problems. One particle is the “relative” particle, which is bound to an attractive potential and describes the internal structure of the electron−hole pair. In the phenyl-based light emitting polymers the lowest exciton is strongly bound and the electron−hole pair is typically spread over a single moiety.15 This is a Frenkel exciton. The remaining particle is the centerof-mass particle, which delocalizes along the polymer chain. The Frenkel exciton model is a one-particle description of the delocalization of this tightly bound electron−hole pair along the chain of moieties, where each moiety is treated as a ‘site’ on a one-dimensional lattice. If â†n (ân) creates (destroys) a Frenkel exciton on a phenylene or vinylene moiety, n, then the Frenkel exciton Hamiltonian reads

and to describe its predictions. In particular, we show how the different relaxation processes cause different dynamical fluorescence depolarization. To achieve this goal, we introduce an appropriately parametrized disordered Frenkel−Holstein model. This model is solved within the Ehrenfest approximation, whereby the exciton is treated quantum mechanically, whereas the nuclear degrees of freedom are treated classically. Consequently, in the Ehrenfest approximation quantum mechanical coherences between the exciton and phonons are lost. In the accompanying paper4 we use the same model to investigate post picosecond exciton diffusion. As explained in more detail in that paper, we invoke the Condon approximation, implying that exciton transfer occurs from more localized VRSs (donor chromophores) to less localized LEGSs (acceptor chromophores). The same model applied to conformationally disordered poly(p-phenylenevinylene) gives good agreement with experiment for both the ultrafast and slower migratory dynamics, lending support for the theories of exciton localization proposed here. In a previous paper7 the dynamical localization of LEGSs was investigated assuming that this is an adiabatic relaxation on their Born−Oppenheimer potential energy surfaces. The Pariser−Parr−Pople−Peierls model of π-electron systems was solved via the configuration interaction (singles) method. This is a more realistic, albeit practically more limited, model of the photoexcited states of PPV than the Frenkel−Holstein model. Nevertheless, we show here that evolving the excited state wave function via the time-dependent Schrödinger equation with an appropriately parametrized Frenkel−Holstein model accurately reproduces the predictions of ref 7, demonstrating both the adiabatic relaxation of LEGSs and the applicability of the Frenkel−Holstein model in general. This work is related to and complements a number of earlier investigations. Dykstra et al.2 have also investigated the ultrafast fluorescence depolarization in MEH-PPV, although they propose a different mechanism than ours. They assumed that fluorescence depolarization arises from the interconversion between exciton eigenstates caused by scattering from phonons in a solid state environment. Dynamical localization of photoexcited states via vibrational relaxation has been studied in oligo(phenylenevinylene)s,9,10 whereas ground and excited state dynamics have been investigated in PPV11 and MEHPPV12 chains. Exciton delocalization on dimers, induced by nuclear couplings and investigated via the Frenkel−Holstein model with surface hopping algorithms, is discussed in ref 13. The next section contains a motivation for and description of the Frenkel−Holstein model, a description of how the ensemble of PPV conformations are generated, and a discussion of the Ehrenfest dynamics. Section III contains a discussion of the dynamical relaxation and localization of LEGSs and QEESs. Section IV describes the time-dependent ultrafast and timeaveraged fluorescence depolarization arising from exciton localization, where we show that QEESs exhibit larger, but slower fluorescence depolarization than LEGSs. We conclude, with some discussion of further work, in section V.

HF =

∑ εnan̂ †an̂ + ∑ Jmn(am̂ †an̂ + an̂ †am̂ ) n

m>n

(1)

where odd and even “sites” n represent phenylene and vinylene moieties, respectively. The on-site excitation energy is Δ + αn (2) 2 where Δ is the difference in excitation energy between vinylene and phenylene moieties and αn represents diagonal disorder, whose physical origin is explained in ref 4. Jmn is the Frenkel exciton-transfer integral between moieties m and n. This has two contributions.5 First, for all pairs of moieties there is a through-space, dipole−dipole interaction εn = E0 + ( −1)n

DD Jmn =

κmnμ0 2 4πεrε0R mn3

(3)

where μ0 is the magnitude of the transition dipole moment of a Frenkel exciton on a single moiety and Rmn is the distance between the centers of the moieties m and n. κmn is the orientational factor, κmn = rm̂ ·rn̂ − 3(R̂ mn·rm̂ )(R̂ mn·rn̂ )

(4)

where r̂m is a unit vector parallel to the dipole on moiety m and R̂ mn is a unit vector parallel to the vector joining moieties m and n. For nearest neighbor moieties there is also an additional through-bond, or superexchange interaction, JSE, whose origin lies in the virtual mixing of the Frenkel and charge-transfer exciton subspaces. If ΔE is the energy difference between the charge-transfer and Frenkel exiton subspaces and t(ϕ) is the HOMO- and LUMO-transfer integral, then J SE (ϕ) = −

II. THEORETICAL AND COMPUTATIONAL METHODS A. Frenkel−Holstein Model. There are two ingredients to the Frenkel−Holstein model. First, the Frenkel exciton model, which describes the motion of the Frenkel exciton center-ofmass particle. When disorder is introduced, it captures the essential physics of exciton localization.6 Second, the Holstein

2t(ϕ)2 ΔE

(5)

Now, because t(ϕ) ∝ cos ϕ, where ϕ is the torsional (or dihedral) angle between moieties, JSE(ϕ) = JSE cos2 ϕ, where for convenience we define JSE ≡ JSE(ϕ=0). The dominant contribution to JDD mn is from nearest neighbor interactions. Defining this term as JDD, we now define a nearest neighbor exciton-transfer integral as 10311



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Jn ≡ J(ϕn) = J DD + J SE cos2 ϕn

In this limit the equilibrium local classical displacement is proportional to the local exciton density, i.e.,

(6)

Both JDD and JSE are negative, and thus the low-energy photophysics of a conjugated polymer is analogous to a onedimensional J-aggregate. Static fluctuations in the torsional angles cause off-diagonal disorder in the transfer integrals. This, in addition to diagonal disorder, causes Anderson localization of the Frenkel center-of-mass particle. The analogy of a conformationally disordered polymer to a one-dimensional disordered J-aggregate7,16 means that we can borrow some concepts from that field.17 In particular, the eigenstates of the disordered Frenkel Hamiltonian (with eq 6) can be divided into two types. As discussed in the Introduction, these are LEGSs18,6 and QEESs. The long-range dipole−dipole interactions not included in eq 6 couple chromophores and, in principle, lead to short-lived quantum coherences. For longer time scales the long-range dipole−dipole interactions cause exciton transfer between chromophores, as discussed in the accompanying paper.4 The next step is to couple the Frenkel exciton to the nuclear degrees of freedom. To accomplish this, we adopt the Holstein model,14,19 whereby the Frenkel exciton is linearly coupled to a local normal coordinate, Qn. In PPV the local normal coordinate is associated with the benzenoid-quinoid distortion for the phenylene moiety and the C−C bond stretch for the vinylene moiety. In the present study we make the Ehrenfest approximation, in which the nuclear degrees of freedom are treated classically and are subject to Newton’s equations of motion. The exciton, however, is treated quantum mechanically. Its evolution is determined by the time-dependent Schrödinger equation with the appropriate Hamiltonian defined in the Born−Oppenheimer limit. The Frenkel−Holstein Hamiltonian in the Born− Oppenheimer limit reads HFH =

Q̃ n = Aψn 2

where ψn is the lowest energy eigenstate of eq 8. Thus, ψn satisfies a self-consistent, nonlinear Schrödinger equation, with a “self-trapping” solution ψn =

(11)

describing an exciton-polaron (or a vibrationally relaxed state (VRS)). We expect our nonadiabatic simulations of arbitrary initial vertical exciton eigenstates to evolve into such a form, albeit modified by being on a nonuniform chain. B. Statistically Generated Polymer Conformations. The PPV conformations are generated statistically. A skeleton structure is first constructed similarly to the carbon backbone of trans-polyacetylene. The torsional (or dihedral) angles between the skeleton bonds are taken as Gaussian random variables with a mean of ϕ0 = ±15° and a standard deviation of σϕ. In addition, a fraction, xs, of these angles is assumed to have a trans−cis defect, i.e., ϕ → 180° − ϕ. The centers of the oddnumbered bonds (of length 5.6 Å) represent phenylene moieties, whereas the centers of the even-numbered bonds (of length 1.4 Å) represent vinylene moieties, as illustrated in Figure 1. The structure thus constructed is isomorphic to a coarse-grained PPV conformation, with the orientation of each skeleton bond determining the orientation of the moiety.

n

K + 2

∑ Qn

2

Figure 1. Skeleton trans-polyacetylene structure with the odd (even) bonds decorated with phenylene (vinylene) moieties. These are treated as sites in the Frenkel−Holstein model.

(7)

n

where ϵ is the exciton−phonon coupling strength (for convenience taken to be the same for phenylene and vinlyene moieties), the last term represents the elastic strain, and the nuclear kinetic energy is neglected. To simplify the equations of motion, eq 7 is cast into dimensionless form. This is achieved by setting Q̃ n = (Mω/ ℏ)1/2Qn, A = ϵ/(Mω)1/2 (where S = A2/2 is the Huang−Rhys parameter), and ω = (K/M)1/2. In addition, we scale all energy scales by ℏω (represented by tildes) so that the Hamiltonian becomes

∑ (εñ −

AQ̃ n)an̂ †an̂

n

+

+

∑

Jñ (an̂ †an̂ + 1

+

A random polymer conformation is typically described by its radius of gyration, Rg, defined as the root-mean-square size of the polymer ⎛1 R g = ⎜⎜ ⎝N

n

1 2

∑ Q̃ n2 n

A2 ℏω ≪1 4J

⎞1/2 ∑ (R m − ⟨R⟩) ⎟ ⎠ m=1 N

2⎟

(12)

where Rm is the position of the mth moiety (or “site”), ⟨R⟩ is the center-of-mass position, and N is the number of moieties. A useful (molecular weight-independent) measure of the polymer conformation is the number of moieties in a statistical segment, m, defined as

an̂ †+ 1an̂ )

(8)

m = 6R g 2/Sa

The Frenkel−Holstein Hamiltonian has been widely studied for uniform chains. The limit applicable to conjugated polymers is the large-polaron limit,14,20 defined by γ=

⎛ γ ⎞1/2 ⎜ ⎟ sech γ(n − n0) ⎝2⎠

∑ (εn − ϵQ n)an̂ †an̂ + ∑ Jn(an̂ †an̂ + 1 + an̂ †+ 1an̂ ) n

H̃FH =

(10)

(13)

where S is the contour length of the chain and a is the length of a moiety. C. Parametrizing the Frenkel−Holstein Model. The Frenkel exciton model for an ordered PPV chain is parametrized via E0, Δ, JDD, and JSE. These parameters are determined by comparing the low-energy spectrum of eq 1 to

(9) 10312



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“site”) means that a complete numerical integration of the timedependent Schrödinger equation is possible for large systems (up to 1000 moieties) and long time scales. The computation is also facilitated by the tridiagonal form of the Hamiltonian. 2. Nuclear Dynamics. The nuclei are assumed to behave classically, so their dynamics are described by

Configuration Interaction-Singles (CI-S) calculations on the Pariser−Parr−Pople model of PPV polymers.7 The two contributions to Jn are found by varying the intermoiety distance, D, and fitting to a form Jn = JSE + a/D3. The value of the moiety transition dipole moment (TDM), μ0, is found by calculating the TDM for a PPV chain via the CI-S method, plotting its square against the number of moieties, and linearly extrapolating to a single moiety. The additional parameter in the Frenkel−Holstein model, the exciton−phonon coupling constant, A, is then determined by fitting its predictions to those of the Pariser−Parr−Pople−Peierls model of PPV polymers, taking the reorganization energy of the 11Bu exciton to be 0.08 eV in the asymptotic limit. The nearest neighbor exciton-transfer integral is determined via eq 6 (with the small orientational dependence of JDD being neglected). The fraction of trans−cis defects is also a free parameter. Finally, the diagonal disorder is a Gaussian random variable, with a standard deviation, σα = 65 meV, as derived in ref 4. The parameters are listed in Table 1.

dvñ = fñ dt ̃ and

dQ̃ n = vñ (18) dt ̃ ̃ where Qn and ṽn are the coordinate and velocity of the vibrational mode at site n, respectively, and fñ is the force on the mode at site n. The tildes indicate that the variables are dimensionless. The force on a classical displacement is given by fñ = −

Table 1. Parameters Used in the Frenkel−Holstein Model, Eq 8, and in the Simulation of the Exciton Dynamicsa parameter

value

E0 Δ JSE JDD ℏω exciton−phonon coupling, A diagonal disorder, σα dissipation rate, γ moiety transition dipole moment, μ0

9.24 eV 3.20 eV −1.96 eV −1.35 eV 0.2 eV 4.00 65 meV 1013 s−1 1.51 × 10−29 C m

(19)

(20)

In addition, a “friction” term is included to damp the nuclear motion and encourage relaxation of the excited state. This term is proportional to the velocity and a dissipation rate, γ, chosen to recreate experimental dissipation rates.1,11 This gives the equation of motion for the classical displacement as dvñ = A |Ψn|2 − Q̃ n − γ ṽ ñ dt ̃

(21)

where γ̃ = γ/ω. Equations 18 and 20 are solved via the fourth-order Runge− Kutta method,24 subject to the initial conditions that Q̃ n(t) = ṽn(t) = 0 at t = 0.

III. DYNAMICAL LOCALIZATION Before describing the dynamical localization of vertical states to VRSs, we first discuss how disorder affects the optical spectrum. Figure 2 shows the optical intensity for vertical transitions (defined by Qn = 0) for weak (σϕ = 2°) and moderate (σϕ = 10°) torsional disorder for both all the states and just LEGSs. A LEGS is an essentially nodeless state, conveniently defined by |∑|ψn|ψn|≳0.95. These states dominate the low-energy optical intensity. The vertical dashed line defines the band edge for a hypothetically ordered system, Eb = E0 − 2|⟨J⟩ϕ|, where ⟨J⟩ϕ is the average value of J(ϕ) in a disordered system. If {ϕ} is a Gaussian random variable, then |⟨J⟩ϕ| < |J| when the mean angle, ϕ0, satisfies 0 < ϕ0 < 45°. This explains the blue shift in the mean vertical excitation energy as the disorder is increased. (We show in ref 4 that torsional disorder red-shifts the emission.) A. Relaxation of Local Exciton Ground States. The relaxation of a typical LEGS is shown in Figure 3. The vertical LEGS is initially spread over approximately 30 moieties. This defines the size of an absorbing chromophore. Within two vibrational oscillations (∼40 fs) this state has relaxed and selflocalized to form a VRS, spanning approximately 10 moieties

(14)

â†n|0⟩

where |n⟩ = is a Frenkel exciton basis state and Ψn(t) is the Frenkel exciton center-of-mass wave function. Splitting the state vector into real and imaginary parts, |ΨR⟩ and |ΨI⟩, respectively, yields a pair of coupled differential equations d|ΨR ⟩ = H̃FH|Ψ⟩ I dt ̃

n

fñ = A |Ψn|2 − Q̃ n

D. Dynamics. 1. Wave function Dynamics. The exciton dynamics are simulated by numerically solving the timedependent Schrö dinger equation for the exciton state vector,21,22 n

∂H̃ ∂Q̃

where ⟨···⟩ represents the expectation value with respect to |Ψ⟩. Using eq 8 it is easily shown that

The torsional disorder, σϕ, and the fraction of trans−cis defects are free parameters.

∑ Ψn(t )|n⟩

∂E ̃ =− ∂Q̃ n

a

|Ψ⟩(t ) =

(17)

(15)

and

d|Ψ⟩ I = −H̃FH|ΨR ⟩ (16) dt ̃ where H̃ FH is given in eq 8 and t ̃ = ωt (where ω is the vibrational angular frequency). For instantaneous values of {Q̃ (t)} eqs 15 and 16 are solved via the eighth order Runge− Kutta method.23 The coarse-grained nature of our model (the Frenkel exciton is treated as a single particle delocalized over “sites” and the nuclear dynamics are treated as a single normal mode per 10313



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entirely in its initial evolving eigenstate, adiabatically evolving on its Born−Oppenheimer potential energy surface. This is to be expected, as LEGSs are locally the lowest energy state, and confirms the assumption of adiabatic relaxation of ref 7. B. Relaxation of Quasi-Extended Exciton States. The relaxation of QEESs is a more complex process than the relaxation of LEGSs, as a QEES cannot simply evolve on its potential energy surface to form a VRS; there must be an interconversion via intermediate states. In fact, there are three distinct episodes in the dynamical relaxation of a QEES, as the following examples illustrate. As a first example, consider Figure 4. This depicts a typical vertical QEES and the VRS (or exciton-polaron) into which it

Figure 2. Ensemble averaged (1000 chains of 201 moieties) optical intensity, I(E), for vertical transitions. I(E) = ∑α fαδ(E − Eα), where fα and Eα are the oscillator strength and excitation energy of state ψα, respectively, and normalized so that ∫ I(E) dE = 1. For all states (squares) and LEGSs (circles). (a) σϕ = 2°, where the fraction of LEGSs is 1.1% and their fractional contribution to the optical intensity is 26%. (b) σϕ = 10°, where the fraction of LEGSs is 2.5% and their fractional contribution is 36%. The vertical lines indicate the band edge for an ordered system with an exciton-transfer integral, J = ⟨J⟩ϕ.

Figure 4. (a) LEGS center-of-mass wave functions on a region of polymer chain, with the absorbing chromophore boundaries defined by these LEGSs marked by the vertical lines. (b) Exciton center-ofmass wave functions for (green) a QEES on this same region of polymer chain and (blue) the exciton-polaron state that forms from the QEES as it dynamically localizes.

relaxes. The LEGSs on this part of the polymer are also shown, and it can be seen that the exciton density has migrated almost entirely into the chromophore of the LEGS with which it has the largest initial overlap. The final exciton-polaron state has a shape and width similar to those predicted by eq 11. The three episodes in the dynamical relaxation of a QEES are shown in Figure 5. First, there is a brief initial period (≈20 fs) during which there is little migration of the exciton as it adiabatically relaxes on its potential energy surface. Following this there is a period during which there is larger migration of the exciton density. Projection of the evolving wave function onto the instantaneous eigenstates of the Hamiltonian indicates

Figure 3. Magnitude of the exciton center-of-mass wave function per moiety as a function of time for the relaxation of a LEGS into a VRS (or exciton-polaron). The torsional disorder σϕ = 5°.

within the original chromophore. This defines the size of an emissive chromophore. Projections of the evolving wave function onto the instantaneous eigenstates of the Frenkel− Holstein Hamiltonian show that the exciton remains almost 10314



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Figure 7. Magnitude of the exciton center-of-mass wave function per moiety as a function of time for the relaxation of the high energy QEES shown in Figure 6. The horizontal (green) lines indicate the absorbing chromophore boundaries, defined by the vertical LEGSs.

Figure 5. Magnitude of the exciton center-of-mass wave function per moiety as a function of time for the relaxation of the QEES shown in Figure 4. The horizontal (green) lines indicate the absorbing chromophore boundaries, defined by the vertical LEGSs.

there is more complex wave function dynamics. Again, this dynamical behavior is found to correspond to the wave function becoming a superposition of the instantaneous eigenstates whose probability amplitudes vary in time. After about 300 fs the wave function has predominantly become a mixture of two instantaneous eigenstates localized over two chromophores. Although the exciton has localized significantly the dynamics are not yet complete. The exciton density around site 750 shows Rabi oscillations whose period is ca. 100 fs. The density initially starts entirely at site 740 (in chromophore labeled 2), but with each oscillation more exciton density transfers to the region around site 760 (in chromophore labeled 3) until at long times (shown in Figure 7) the density has migrated entirely to site 760. We can qualitatively understand the Rabi oscillations between chromophores 2 and 3 by the following argument. Suppose that the Frenkel−Holstein Hamiltonian, HFH(Q(t)) (parametrized by Q(t)), varies slowly on time scales, τ. Then at time t + τ the time-dependent wave function can be expressed as a linear superposition of the eigenstates of HFH at time t

that it becomes a linear superposition of these eigenstates. Finally, after about 200 fs the exciton wave function is almost entirely dominated by a single instantaneous eigenstate, which is localized over a single chromophore and evolving on its potential energy surface to form a VRS. The time-dependent coupling between the excited state and the normal coordinates has induced an interconversion between an initial vertical state and a final vibrationally relaxed state. However, this process does not involve discrete transitions between adiabatic states evolving on their potential energy surfaces. Rather, it involves the state vector becoming a linear superposition of adiabatic states whose probability amplitudes also evolve in time, such that (in this instance) one of them finally dominates. As another example of relaxation of a QEES, consider Figure 6. This shows the wave function of a higher energy vertical QEES and the two distinct exciton-polaron states that it relaxes into after 10 ps. The dynamical relaxation of this QEES is depicted in Figure 7. Again, three episodes of relaxation are observed. An initial period of ca. 30 fs, where the exciton remains fairly stationary, is followed by a longer episode where

Ψ(t +τ ;Q (t )) =

∑ aα(t ;Q (t )) ψα(t ;Q (t )) α

exp(− iEα(t ;Q (t ))τ /ℏ)

(22)

where HFH(Q (t )) ψα(t ;Q (t )) = Eα(t ;Q (t )) ψα(t ;Q (t ))

(23)

Rabi oscillations between a pair of quasi-stationary states ψα and ψβ occur on a time scale of h/|Eα − Eβ|. When these time periods are longer than the characteristic vibrational time scales (2π/ω and γ−1), the nuclear displacements quasi-statically follow the oscillations in the exciton probability density, via eq 10. The coupling between the nuclear displacements and the exciton wave function (via HFH(Q(t))) increases the quasistationary probability amplitudes, aα(t;Q(t)), in regions of large nuclear displacements, causing the exciton to self-localize in a particular stationary state. The relaxation to equilibrium in this example is very slow, and indeed after 10 ps the exciton still remains in a nonstationary state localized on chromophores 1 and 3. It is probable that the asymptotic dynamics are incorrectly described via the Ehrenfest approximation, as the classical treatment of the nuclei means that coupled exciton-nuclear quantum

Figure 6. Exciton center-of-mass wave functions for (green) a high energy QEES and (blue) the exciton-polaron state that forms from it after 10 ps. The vertical lines indicate the absorbing chromophore boundaries, defined by the vertical LEGSs. 10315



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mechanical coherences are lost. In principle, the nuclei should also exhibit quantum beatings that, because of the nonlinear form of the exciton Schrödinger equation, become amplified and will force the solution into a unique self-localized excitonpolaron more rapidly than predicted by the Ehrenfest approximation.

IV. FLUORESCENCE ANISOTROPY The process of dynamical localization described in the previous section results in a number of ultrafast experimental observables, which in principle allow for an experimental test of the theories of exciton localization proposed here. In this section we discuss fluorescence depolarization. The fluorescence anisotropy of an ensemble of molecules is defined as25 ⎛ I − I⊥ ⎞ ⎟⎟ r = ⎜⎜ ⎝ I + 2I⊥ ⎠

(24)

where I∥ and I⊥ are the observed fluorescence intensities parallel and perpendicular to the incident polarization, respectively. The fluorescence anisotropy of an individual molecule, i, is defined as

ri = P2(θi)

(25)

where P2(θ) is the second-order Legendre polynomial, P2(θ ) =

1 (3 cos2 θ − 1) 2

(26) Figure 8. (a) Time-dependent fluorescence anisotropy for all LEGSs of 50 PPV chains containing 500 phenylene moieties each for various values of the torsional disorder, σϕ. The cis-defect probability is 8%. (b) The same as (a) but for QEESs.

and θ is the angle by which the transition dipole moment rotates between absorption and emission. Thus, for an ensemble of isotropically oriented molecules the average theoretical fluorescence anisotropy is defined as

⟨r ⟩ =

⎛ 2 ⎞ Σifi ri ⎜ ⎟ ⎝ 5 ⎠ Σif i

QEESs. The former observation can be explained by the fact that the QEESs must undergo electronic transitions in addition to vibrational relaxation, whereas the lower asymptotic value of the anisotropy for the QEESs is a result of the QEESs having larger initial localization lengths. The fluorescence anisotropy of both LEGSs and QEESs has essentially equilibrated within the first 200 fs, in agreement with the experimental data of Figure 3 of ref 1. B. Time-Averaged Fluorescence Anisotropy. The timeaveraged fluorescence anisotropy is an ensemble average of random decay processes occurring within the first picosecond of excitation. The data are presented for various values of torsional disorder and are plotted against the number of moieties in a statistical segment, m, given by eq 13.26 The time-averaged fluorescence anisotropy for an ensemble average of LEGSs of 500 phenylene ring PPV chains is shown in Figure 9a. As for the time-dependent fluorescence anisotropy, the more disordered systems show a smaller decay in fluorescence anisotropy, and again this can be explained by the initial LEGSs being more delocalized in more ordered polymer chains. As the number of moieties in a statistical segment, m, decreases, there is a larger decay in the fluorescence anisotropy, because a smaller m implies a more coiled chain. An exciton on a more coiled chain is likely to have a larger rotation of its transition dipole moment as it localizes. Figure 9b shows the time-averaged fluorescence anisotropy for an ensemble of QEESs. As for time-dependent behavior, there is larger depolarization, and again, the disorder appears to

(27)

where f i is the oscillator strength of the ith molecule and the factor of 0.4 arises from the assumption of an isotropic distribution of initial transition dipole moments.25 A. Time-Dependent Fluorescence Depolarization. Figure 8a shows the time-dependent fluorescence depolarization for the optically excited LEGSs (whose absorption spectra are shown in Figure 2) for various values of torsional disorder, σϕ. The asymptotic limit of the fluorescence anisotropy is different for the varying values of σϕ. As the disorder increases, there is a smaller decay in the anisotropy, which is the trend we expect because the initial localization length for the LEGSs is smaller in more disordered systems. As the more localized LEGSs relax, their change in localization length will be smaller, so it is likely that the rotation in their transition dipole moments will also be smaller, resulting in a smaller reduction of the fluorescence anisotropy The time-dependent fluorescence depolarization for the optically excited QEESs (whose absorption spectra are indicated in Figure 2 as the difference between the total and LEGSs absorption) is given in Figure 8b. The most striking feature is that varying the amount of disorder seems to have a much smaller effect for QEESs than for LEGSs. The anisotropy decays on similar time scales for all values of disorder to a similar asymptotic value. Comparison of the QEES results with those for LEGSs shows that the fluorescence anisotropy decays faster for LEGSs, whereas it decays to a greater extent for 10316



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time-evolving eigenstate becoming a linear superposition of instantaneous eigenstates. Typically, after a few hundred femtoseconds, one of the instantaneous eigenstates dominates the linear superposition, and the remaining dynamics is again adiabatic relaxation to a VRS. 3. Very high energy QEESs, which are delocalized over many chromophores, sometimes exhibit a splitting of the wave function into more than one VRS. This selflocalization onto more than one chromophore is presumably a failure of the Ehrenfest approximation, as this approximation neglects quantum mechanical coherences between the electronic and nuclear degrees of freedom. 4. QEESs exhibit larger, but slower fluorescence depolarization than LEGSs. Thus, ultrafast fluorescence depolarization is a function of excitation energy and conformational disorder. The second episode in the relaxation of a QEES corresponds to its interconversion to a VRS. During this process the evolving wave function becomes a nonstationary state; i.e., it is a linear superposition of the quasi-stationary instantaneous eigenstates of the Frenkel−Holstein Hamiltonian (see eq 22). As a consequence, Rabi oscillations are possible between the quasi-stationary states, as observed in Figure 7. In principle, other quantum coherences are possible. For example, nonstationary states localized over different “chromophores” might be coupled by the long-range dipole−dipole interactions. These have not been observed in our calculations (when long-range dipole−dipole interactions are included), presumably because the polymer configurations are too open. However, more folded polymer conformations are more likely candidates to exhibit these types of coherences, indicating the need for realistic simulations of polymer conformations in the condensed phase. In addition, a quantized treatment of the phonons would correlate exciton−phonon coherences, which are expected to change some of the predictions arising from the Ehrenfest approximation. Our predictions of the time dependence and amount of fluorescence depolarization as a function of excitation energy is in semiquantitative agreement with experiment.1 This agreement provides support for our model of exciton localization, where higher energy vertically photoexcited states are more delocalized than lower energy states. In the accompanying paper4 this model of exciton localization is used to perform a “first-principles” simulation of postps exciton diffusion in conformationally disordered PPV. Our accurate prediction of the exciton diffusion length again lends support to the model.

Figure 9. (a) Ensemble average of the fluorescence anisotropy versus the number of moieties in a statistical segment for LEGSs. The polymer chains contain 500 phenylene rings, and the data are obtained for various values of σϕ. Each data series represents 10 000 radiative decay events. (b) The same as (a) but for QEESs.

have a much smaller effect on the anisotropy decay of QEESs than the anisotropy decay of LEGSs. These results are in semiquantitative agreement with the measurements on MEH-PPV by Ruseckas et al.1 They found a value for the ultrafast fluorescence anisotropy of ≈0.32 at band edge absorption (at 2.2 eV) and ≈0.2 with higher excitation energy (at 2.9 eV).

V. CONCLUDING REMARKS This paper has introduced the disordered Frenkel−Holstein model as a means of investigating dynamical relaxation and localization of photoexcited states in disordered polymers. It was solved within the Ehrenfest approximation, in which the exciton is treated fully quantum mechanically, but the nuclear displacements are treated classically. The following are shown: 1. Lower energy local exciton ground states (LEGSs) adiabatically relax to vibrationally relaxed states (VRSs) in the time scale of one or two vibrational periods (ca. 40 fs). The relaxation of LEGSs is accompanied by localization and fluorescence depolarization, as the transition dipole moment reduces and rotates. The amount of dynamical localization increases as the torsional disorder decreases, causing an increase in the fluorescence depolarization. 2. Higher energy quasi-extended exciton states (QEESs) interconvert to VRSs via three distinct episodes. A brief initial period of adiabatic relaxation is followed by the

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

We thank the John Fell Fund of the University of Oxford for financial support. 10317



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REFERENCES

(1) Ruseckas, A.; Wood, P.; Samuel, I. D. W.; Webster, G. R.; Mitchell, W. J.; Burn, P. L.; Sundström, V. Phys. Rev. B 2005, 72, 115214. (2) Dykstra, T. E.; Hennebicq, E.; Beljonne, D.; Gierschner, J.; Claudio, G.; Bittner, E. R.; Knoester, J.; Scholes, G. D. J. Phys. Chem. B 2009, 113, 656. (3) For a recent review, see: Huang, I.; Scholes, G. D. Chem. Mater. 2011, 23, 610. (4) Barford, W.; Bittner, E. R.; Ward, A. J. Phys. Chem. 2012, DOI: 10.1021/jp307041m. (5) Barford, W.; Trembath, D. Phys. Rev. B 2009, 80, 165418. (6) Makhov, D. V.; Barford, W. Phys. Rev. B 2010, 81, 165201. (7) Barford, W.; Boczarow, I.; Wharram, T. J. Phys. Chem. A 2011, 115, 111. (8) Anderson, P. W. Phys. Rev. 1958, 109, 1492. (9) Tretiak, S.; Saxena, A.; Martin, R. L.; Bishop, A. R. Phys. Rev. Lett. 2002, 89, 97402. (10) Karabunarliev, S.; Bittner, E. R. J. Chem. Phys. 2003, 118, 4291. (11) Sterpone, F.; Rossky, P. J. J. Phys. Chem. B 2008, 112, 4983. (12) De Leener, C.; Hennebicq, E.; Sancho-Garcia, J.-C.; Beljonne, D. J. Phys. Chem. B 2009, 113, 1311. (13) Beenken, W. J. D.; Dahlbom, M.; Kjellberg, P; Pullerits, T. J. Chem. Phys. 2002, 117, 5810. (14) Holstein, T. Ann. Phys. (N. Y.) 1959, 8, 325. (15) Barford, W.; Paiboonvorachat, N. J. Chem. Phys. 2008, 129, 164716. (16) Yamagata, H.; Spano, F. C. J. Chem. Phys. 2012, 136, 184901. (17) Bednarz, M.; Malyshev, V. A.; Knoester, J. J. Chem. Phys. 2002, 117, 6200. Bednarz, M.; Malyshev, V. A.; Knoester, J. J. Chem. Phys. 2004, 120, 3827. (18) Malyshev, A. V.; Malyshev, V. A. Phys. Rev. B 2001, 63, 195111. (19) Holstein, T. Ann. Phys. (N. Y.) 1959, 8, 343. (20) Using the parameters in Table 1, γ = 0.25. (21) Streitwolf, H. W. Phys. Rev. B 1998, 58, 14356. (22) Stafstrom, S. Chem. Soc. Rev. 2010, 39, 2484. (23) Hairer, E.; Norsett, S.; Wanner, G. Solving Ordinary Differential Equation I: Nonstiff Problems; Springer-Verlag: Berlin, 1986. (24) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran; Cambridge University Press: Cambridge, 1986. (25) Lakowicz, J. R. Principles of Fluorescence Spectroscopy; Plenum Press: New York, 1999. (26) Figure 9 also serves as a correction to Figure 7 of ref 7, as the value of m was incorrectly calculated in that paper (and also the radius of gyration was incorrectly defined in eq 7).

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Exciton Dynamics in Disordered Poly(p

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