Ultrafast Dynamical Localization of Photoexcited States in

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Ultrafast Dynamical Localization of Photoexcited States in Conformationally Disordered Poly(p-phenylenevinylene) William Barford,*,†,‡ Igor Boczarow,†,‡ and Thomas Wharram†,§ †

Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford, OX1 3QZ, United Kingdom Balliol College, University of Oxford, Oxford, OX1 3BJ, United Kingdom § University College, University of Oxford, Oxford, OX1 4BH, United Kingdom ‡

ABSTRACT: We consider two types of ultrafast dynamical localization of photoexcited states in conformationally disordered poly(p-phenylenevinylene). First, we discuss nonadiabatic interconversion from higher energy extended exciton states to lower energy more localized local exciton ground states. Second, we calculate the dynamics of local exciton ground states on their BornOppenheimer potential energy surfaces. We show that within the first CC bond oscillation following photoexcitation (∼35 fs) the exciton becomes self-trapped and localized over approximately eight monomers. This process is associated with a CalderiaLeggett type loss of phase coherence owing to the coupling of the polymer to a dissipative environment. Subsequent torsional relaxation (on a time scale of approximately picoseconds) has little effect on the localization. We conclude from this that the initial torsional disorder determines the spatial distribution and localization length of vertical excitations but that electronphonon coupling is largely responsible for the localization length of self-trapped excitons. We next consider the effect of dynamical localization on fluorescence depolarization. We show that exciting higher energy states causes a larger fluorescence depolarization, because these states have a larger initial delocalization. Using the observation that fluorescence depolarization is a function of excitation wavelength and polymer conformation, we show how the models of exciton localization discussed here can be experimentally investigated.

I. INTRODUCTION Photoexcited states in conformationally disordered conjugated polymers exhibit two types of strong, ultrafast dynamical localization. One type of localization is caused by the rapid nonadiabatic interconversion from energetically higher lying quasi-extended exciton states to lower lying more localized “local exciton ground states”. The other type of localization is caused by the rapid vibrational relaxation of the local exciton ground states to self-trapped exciton polarons. In both cases the relaxation is associated with high-frequency CC bond vibrations. As will be explained in this paper, both types of localization can be investigated by observing the fluorescence depolarization as a function of excitation wavelength. A key to understanding these localization processes is first to understand the cause of the localization of the vertical excited states (or excitons) in conformationally disordered polymers. The conventional view of exciton localization is that it is determined by the size of “conjugated segments” or chromophores, which themselves are determined by “breaks” in the π-conjugation between neighboring monomers. In addition, it is usually assumed that between these breaks the exciton may freely delocalize, and thus (by the momentum-position uncertainty principle) the exciton energy is lower on longer conjugated segments. A completely contrary view to this is that the size of a conjugated segment is determined by the value of the exciton localization length, which itself is determined by the underlying disorder.1,2 This disorder is both conformational and environmental in origin. Specifically, the vertical low-energy excited spectrum is composed of r 2011 American Chemical Society

a class of Anderson-localized excitons known as local exciton ground states (LEGS).2 A LEGS is essentially nodeless and is locally the lowest energy excited state.3 LEGS are spatially localized, space filling, and nonoverlapping. Thus, a region of a polymer that supports a LEGS can be regarded as defining a conjugated segment or chromophore. Higher energy states are, however, more extended, being delocalized over a number of LEGS (i.e., chromophores). These are quasi-extended exciton states. We emphasize that this latter description of chromophores, as being defined by the LEGS, is different from the former description of chromophores as being defined by minimum thresholds in the π-conjugation. In particular, it leads to the prediction that as excitons energetically relax they become more, not less, localized. Dynamical localization causes fluorescence depolarization on conformationally disordered polymers, because as the exciton energetically relaxes and its spatial spread changes there is a change in both the magnitude and orientation of its transition dipole moment.4 Generally, the greater the initial delocalization of the exciton compared to its final localization, the greater will be the fluorescence depolarization. Thus, to correctly predict the excitation wavelength dependence of the fluorescence depolarization, a correct theory of exciton localization is required. Indeed, as described in section IV.C, fluorescence depolarization as a function of excitation wavelength and polymer conformation Received: May 24, 2011 Published: July 07, 2011 9111

dx.doi.org/10.1021/jp204822h | J. Phys. Chem. A 2011, 115, 9111–9119

The Journal of Physical Chemistry A provides a direct verifiable test of the types of exciton localization described in this paper. There are other causes of fluorescence depolarization. An ultrafast mechanism arises from Coulomb-induced excimer formation via the delocalization of an exciton between chromophores.5 Typically the chromophores will have a quasi-parallel orientation giving a quasi-H excimer, whose higher lying state has the higher oscillator strength. Photoexcitation to this state followed by rapid interconversion to the lower state will cause a rotation of the transition dipole moment (TDM) (by 90 if the chromophores have precisely the same TDM magnitudes). The origin of slow fluorescence depolarization (namely on greater than picosecond time scales) is the Coulomb-induced F€orster transfer of vibrationally relaxed excitons between chromophores. The aims of this paper are, first, to describe the two types of ultrafast dynamical localization of photoexcited states and, second, to predict the ultrafast fluorescence depolarization arising from this localization as a function of the conformation of poly(p-phenylenevinylene) (PPV) chains. This second aim is motivated by experimental observations on MEH-PPV4 and also by the hope that it will motivate experimental investigations of the mechanisms for exciton localization described in this paper. The dynamical localization of local exciton ground states is investigated via the PariserParrPoplePeierls (P-P-P-P) model, a realistic model of π-conjugated systems that incorporates bond and torsional degrees of freedom. The P-P-P-P model is solved via an efficient direct configuration interaction-singles algorithm,6 allowing us to propagate the LEGS on their potential energy surfaces for PPV chains of up to 100 monomers and for time scales from 1 fs to 2 ps. We perform ensemble averages over conformationally disordered polymers to determine the experimental observables. Our work is related to and complements a number of earlier investigations. Dynamical localization of photoexcited states via vibrational relaxation was studied in oligo-phenylenevinylenes by Tretiak et al.7 Ground and excited state dynamics have been investigated in PPV8 and MEH-PPV9 chains. Dykstra et al.10 have investigated the ultrafast fluorescence depolarization in MEH-PPV. They assumed that this arises from the interconversion between exciton eigenstates caused by scattering from phonons in a solid state environment. Yaron et al.11 and Beenken12 have speculated that torsional disorder or conformational defects cause the Anderson localization of excitons, and Barford et al. investigated exciton localization in disordered poly(p-phenylenes)1 and polythiophenes.13 Other related work will be mentioned where appropriate. The next section describes the theoretical and computational methods employed in this work. Most of it is rather technical. However, section II.B does justify the important assumption of static conformational disorder. Section III describes exciton localization, starting with the localization of the vertical excited states and then describes the two types of dynamical localization. In section IV we describe the consequences of dynamical localization on fluorescence depolarization, and we conclude in section V.

II. THEORETICAL AND COMPUTATIONAL METHODS A. The PariserParrPoplePeierls Model. Ignoring ring planarization for the moment, the PariserParrPople Peierls (P-P-P-P) model of π-conjugated systems model is

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defined by14 ^ ¼ 2 H

N1

∑

i¼1 N

î þ U ti T

1 1 ^ ^ N iv N iV 2 2 i¼1 N

∑

∑

þ

1 ^ i 1ÞðN ^ j 1Þ Vij ðN 2 i6¼ j

þ

N1 KN1 2 δui 2RΛ δui 2 i¼1 i¼1

∑

∑

ð1Þ

∑σ ð^c†i þ 1σ^ciσ þ ^c†iσ^ciþ1σ Þ

ð2Þ

where î ¼ 1 T 2

is the bond order operator for the ith. bond, ^c†iσ creates a π-elecî = N ^ iv + N ^ iV. δui = (ui+1 ui) is ^ iσ = ^c†iσ ^ciσ, and N tron on site i, N the distortion of the ith bond. {ti} are the H€uckel resonance one-electron integrals, where ti ¼ ðt0 Rδui ÞAi

ð3Þ

and Ai ¼ cos ϕi ,

for phenylenevinylene bridging single bonds

Ai ¼ 1,

otherwise ð4Þ

R is the electronphonon coupling parameter and ϕi is the phenylenevinylene bridging bond torsional angle. The penultimate term on the right-hand side of eq 1 represents the elastic energy of the lattice. The role of the final term in eq 1 will be explained in section II.C. We use the Ohno parametrization for the Coulomb interaction, defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ Vij ¼ U= 1 þ ðUεr rij =14:397Þ2 where rij is the interatomic distance (in Å), U is the on-site Coulomb interaction (in eV), and εr is the relative permittivity. To model the role of torsional relaxation of the phenyl rings for a polymer in solution, we supplement the Hamiltonian by a two-parameter torsional elastic term Vϕ ¼

Kϕ ðϕ ϕi 0 Þ2 2 i ∈ bridgebonds i

∑

ð6Þ

where ϕi0 is the instantaneous, random torsional angle imposed by external fluctuations (in the absence of π-conjugation), and the sum is over the phenylenevinylene bridging single bonds. The P-P-P-P model as defined by eq 1, and eq 6 contains the electronic degrees of freedom and, in addition, the bond {δui} and torsional angle {ϕi} variables which are treated classically. Except for the torsional angles, all angles remain fixed. B. Statistically Generated Disordered Polymer Conformations. Thermally or sterically imposed fluctuations of the torsional angles and transcis defects contribute to the conformational disorder of PPV. We now make the assumption that the conformational disorder is static, that is, the random parameters {ϕi0} are constant. The Condon principle implies that this is a correct assumption for the disorder experienced by an initially photoexcited state. This is also a natural assumption for polymers in the condensed phase, where torsional motion is sterically hindered by other polymers. For polymers in solution, however, the assumption of static conformational disorder requires further 9112

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The Journal of Physical Chemistry A

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justification. We make the following a posteriori justification. In section III.C we show that a vertically excited state localizes in the time scale of a single CC bond oscillation, namely, ∼35 fs. The length scale over which the vibrationally relaxed exciton is localized depends only on the strength of the electronphonon coupling and not on the value of the initial disorder. Subsequent torsional relaxation in PPV, on a time scale of approximately a picosecond, has a negligible effect on the localization length. In other words, the time scales for dynamical localization are set by the high frequency CC bond oscillations. The time scales for the fluctuations of the conformational disorder in a solution, on the other hand, are much slower, being on the order of picoseconds. But since disorder has no effect on the localization of the vibrationally relaxed exciton, any dynamical fluctuation of the disorder on time scales J35 fs can also have no effect. Thus, as far as the important excited state dynamics are concerned, the torsional disorder is static. We generate ensembles of statistically generated random conformations of PPV chains (with typically 80100 rings). The phenylvinylene single bond torsional angle, ϕi, is taken to be a Gaussian random variable with a mean value of (15 in the equilibrated ground state7,9 (as discussed in section II.E). The standard deviation, σϕ, is an arbitrary parameter, varying between 0 and 15. (The sign of ϕ is also taken taken to be random.) In addition, in order to generate random coil configurations, we impose transcis defects on the single bond, namely, ϕ f πϕ, with a uniform statistical probability varying between 0 and 16%. A random polymer is typically described by its radius of gyration, Rg, defined as the root-mean-square size of the polymer sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð7Þ Rg ¼ ðR i R j Þ2 Na i > j

∑

where Na is the number of atoms and Ri is the position of the ith atom. Rg is readily determined from neutron scattering experiments. It is, however, a function of molecular weight. A more useful measure of conformation is the number of monomers, m, in a “statistical segment”, defined as m ¼ 6Rg2 =Sa

ð8Þ

where S is the contour length of the polymer and a is the length of a monomer. In section IV we express our calculated dependence of the fluorescence anisotropy on conformation via m. C. Equilibrium Geometries. The equilibrium geometry is determined by the condition that the force per bond fi ¼

∂E ðfδui gÞ ∂δui

ð9Þ

equation for the relaxed bond deformation δui ¼

ϕi ¼ ϕi 0

N1

∑

i¼1

ð14Þ

δui ¼ 0

ð15Þ

which, from the self-consistent equation for the dimerization variable, δui (eq 13), implies that Λ¼

1 N1 ^ ÆT i æAi N 1 i¼1

∑

ð16Þ

D. BornOppenheimer Dynamics. Starting from the initial equilibrium ground state geometry, the local exciton ground states are propagated on their BornOppenheimer potential energy surfaces via Newton’s equations of motion, where the force per bond is given by eq 11. Assuming an effective mass, M, for the CC bond vibration and a damping constant, γ, the bond dynamics are then determined by the coupled first order differential equations,

dυi ðtÞ fi ðtÞ ¼ γυi ðtÞ dt M

ð17Þ

and dδui ¼ υi ðtÞ dt

ð18Þ

These equations are conveniently cast into dimensionless form using the dimensionless electronphonon coupling parameter14,15 λ¼

2R2 πKt0

ð19Þ

and the angular frequency of the CC bond vibrations, ω = (K/M)1/2. They are solved using the fourth order RungeKutta method with an adaptive step size.16 Similarly, the torsional dynamics are solved by the pair of equations dΩi ðtÞ Γi ðtÞ ¼ γϕ Ωi ðtÞ dt Ii

ð10Þ

ð20Þ

and dϕi ðtÞ ¼ Ωi ðtÞ dt

ð11Þ

and ^ i æ sin ϕi Γi ¼ Kϕ ðϕi ϕ0 Þ 2ðt0 Rδui ÞÆT

^ i æ sin ϕi 2ðt0 Rδui ÞÆT Kϕ

The final term in eq 1 enforces a constraint that the overall chain length remains constant. This implies that Λ is determined self-consistently via the condition that

vanish. The HellmannFeynman theorem gives ^ i æAi ΛÞ fi ¼ Kδui 2RðÆT

ð13Þ

Similarly, the condition that Γi = 0 implies the self-consistent equation for the relaxed torsional angle

and the torque per ring ∂E ðfϕi gÞ Γi ¼ ∂ϕi

2R ^ ðÆT i æAi ΛÞ K

ð12Þ

^ i in the relevant state. ^ iæ is the expectation value of T where ÆT Thus, the condition that fi = 0 implies the self-consistent

ð21Þ

where Ωi is the angular frequency and Ii is the moment of inertia of the ith ring, and the torque, Γi, is defined by eq 12. E. Model Parameters. The PariserParrPoplePeierls model is parametrized via the “screened” parameter set derived by Chandross and Mazumdar18 for conjugated molecules to account for solid state solvation effects. These parameters are 9113



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U = 8 eV, εr = 2, and t0 = 2.4 eV. For conjugated polymers the value of the electronphonon interaction, λ, is generally taken to be ∼0.1 (see ref 14). We choose a value of λ = 0.11, which gives good agreement with experiment for the vertical transition energy and reorganization energy of the 11Bu state in an ordered PPV chain, namely, 3.0 and ∼0.2 eV, respectively.17 We set Kϕ = 4 eV and a mean value of ϕ0 = (20 in eq 6 to ensure an equilibrium mean ground state torsional angle of (15 and a planarization in the excited state by ∼2 (as predicted in ref 8). The bond dynamics are parametrized by the angular frequency of the CC bond vibrations, ω, and the semiempirical damping constant, γ. We set pω = 0.2 eV17,19 (corresponding to a time period of 21 fs). The value of γ is determined by the coupling of the CC bond phonons to other internal and external degrees of freedom. We take a value consistent with both the time constant observed in the ultrafast fluorescence depolarization of PPV4 and the calculated autocorrelation function of the bond dynamics,8 namely, γ1 = 100 fs. The torsional dynamics are parametrized by the natural angular frequency of ring rotations, ωϕ t (Kϕ/I)1/2, and the torsional damping γϕ. We set pωϕ = 0.01 eV7 (corresponding to a time period of 0.42 ps) and take γϕ1 = 1 ps, consistent with the calculated autocorrelation function of ring dynamics in single polymer chains.9 F. The Exciton Wavefunction. The two-dimensional exciton wave function, Φ(r, R), is defined as a local molecular orbital transition density matrix between the ground state and the excited state20 Φðr, RÞ ¼ ÆEXj^S†r, R jGSæ

ð22Þ

where the projection operator is ^S†r, R ¼ p1ffiffiffi 2

∑σ âL†R þ r=2, σâHR r=2, σ

ð23Þ

L† âR+r/2,σ creates an electron in the LUMO on the moiety at R + r/2, H creates a hole in the HOMO on the moiety at while âRr/2,σ R r/2. For PPV the moieties are phenylene and vinylene units. To a good approximation Φ(r,R) can be factorized as Φ(r,R) = ψ(r)Ψ(R) where ψ(r) is the relative wave function describing the bound electronhole pair and r is the electronhole separation, while Ψ(R) is the center-of-mass wave function and R is the center-of-mass coordinate. The exciton wave function serves to define the exciton probability density function

Pðr, RÞ ¼

Φ2 ðr, RÞ Φ2 ðr, RÞ

∑ r, R

ð24Þ

for the electronhole separation, r, and position, R. The root-meansquare electronhole separation is thus qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð25Þ rrms ¼ Ær 2 æ Æræ2 where Ærnæ = ∑r,RP(r,R)rn. Similarly, the delocalization (or spread) of the bound electronhole pair, i.e., the center-of-mass particle, is described by the root-mean-square size of the center-of-mass wave function. Thus, we define the exciton localization length as L ¼ 2Rrms

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ÆR 2 æ ÆRæ2

ð26Þ

III. EXCITON LOCALIZATION A. Initial Localization of Photoexcited States. As explained in section II.B, as far as the ultrafast excited state dynamics are concerned, it is reasonable to assume static conformational disorder. This disorder causes the initial Anderson localization of excitons. In PPV the root-mean-square electronhole separation of the lowest family of singlet excitons is 1.4 (in units of a moiety), implying that the bound electronhole wave function is essentially localized onto either a phenylene or vinylene unit. These excitons may therefore be regarded as Frenkel-type excitons delocalizing along the polymer chain. As described in section III.A of ref 1 there are (in principle) two types of mechanisms by which Frenkel excitons delocalize. First, there is a superexchange (or through-bond) mechanism, which is mediated via a virtual charge-transfer exciton. The energy scale for this process is ∼t(ϕ)2/ΔE (where ΔE is the energy difference between the charge-transfer and Frenkel exciton and t is given by eq 3). This term is a function of the torsional angle between moieties and vanishes when ϕ = π/2. Second, there is dipole dipole (or through-space) mechanism, whose magnitude is also a function of the relative orientation of neighboring moieties. This term vanishes for triplet excitons. Thus, conformational disorder causes off-diagonal disorder for both through-bond and throughspace delocalization mechanisms. If both terms simultaneously vanish between a pair of moieties, those moieties are not electronically coupled, and there is a true “conjugation” break. In practice, however, this rarely happens. There are two classes of Anderson-localized excitons: local exciton ground states (LEGS) and quasi-extended exciton states (QEES). LEGSs are superlocalized and lie in the Lifshitz tail of the density of states. A LEGS has an essentially nodeless centerof-mass wave function, conveniently defined by3,2 by the “signedvalue” parameter, R

Z R j jΨðRÞjΨðRÞ dRj g 0:95

ð27Þ

As shown in ref 2 LEGS are spatially localized, space filling, and nonoverlapping. Thus, a region of a polymer that supports a LEGS can be regarded as defining a conjugated chromophore. More precisely, the chromophore of size l is the segment of a polymer that supports an exciton of size L, where l = 2.8L. In contrast, a QEES is delocalized over a number of chromophores and has a highly oscillatory center-of-mass wave function. It is defined by R < 0.95. The concepts of LEGS and QEES have been widely used by Malyshev, Knoester, and co-workers21 in their treatment of the optical properties of disordered molecular solids. It is not surprising that these ideas are transferable to conjugated polymers, as polymers are essentially one-dimensional disordered molecular solids, where each moiety represents a “molecule”, with the additional mechanism in polymers of through-bond delocalization of Frenkel excitons.22 In the next two sections we consider the fate of QEES and LEGS. B. Dynamical Localization of Quasi-Extended Exciton States. We do not consider the precise nonadiabatic mechanism for the interconversion (IC) from higher-lying quasi-extended exciton states (QEES) to lower-lying local exciton ground states (LEGS). (Phonon-induced IC between QEES and other QEES 9114



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Figure 2. Exciton center-of-mass wave function, Ψ, as a function of time. The vertical wave function (t = 0) is a local exciton ground state (LEGS), whose spatial spread defines a chromophore. Regions of negative curvature for the vertical wave function indicate that the Frenkel exciton is tunneling between potential minima determined by the conformational disorder. Within a time period of a single CC bond oscillation the exciton vibrationally relaxes and self-traps into one of the potential minima within a chromophore. N is the monomer label. The corresponding polymer conformation is shown in Figure 3. The standard deviation of the torsional angle is 5.

Figure 1. Exciton center-of-mass wave functions showing (1) interconversion from quasi-extended exciton states (QEES) to local exciton ground states (LEGS), followed by (2) relaxation of LEGS to vibrationally relaxed states (VRS). For this polymer conformation (with 80 monomers and σϕ = 5) there are two LEGS and hence two “conjugated” chromophores. N is the monomer label.

and LEGS is discussed in ref 10.) We simply define the IC from ΨiQEES to ΨjLEGS via the maximum value of the overlap integral Z j jΨiQEES ðRÞjjΨLEGS ðRÞj dR ð28Þ Thus defined, the IC is shown schematically in Figure 1 for two QEES in a polymer conformation that has two LEGS, i.e., two chromophores of ∼40 repeat units in length. According to the single parameter scaling theory of localization in one dimension,24 the localization length of excitons at the band edge (namely, LEGS) satisfies L ∼ (σ/W)2/3, where σ is the energy scale for disorder and W is the bandwidth. At the band center, however, L ∼ (σ/W)3/2. Thus, the localization length increases with energy, as shown in Figure 1 where Ψ6QEES and Ψ9QEES span the entire polymer. Their IC to Ψ2LEGS and Ψ1LEGS, respectively, causes rapid exciton localization onto single chromophores. We return to the effects of this localization on ultrafast fluorescence depolarization in section IV.B. C. Dynamical Localization of Local Exciton Ground States. Figure 2 shows a representative example of the initial (or vertical) center-of-mass (COM) wave function of a local exciton ground state. It exhibits two local maxima and a region of negative

curvature. These features indicate that in general the conformational disorder causes a rugged potential energy surface with multiple minima between which the Frenkel exciton coherently tunnels. The wave functions have negative curvature in the “forbidden” potential energy regions between the potential minima. Electronphonon interactions coupled to a dissipative bath causes the exciton to energetically relax, lose phase coherence, and rapidly become self-trapped into one of the potential minima. This is illustrated in Figure 2 via the time dependence of the COM wave function determined by the BornOppenheimer dynamics on its potential energy surface. This shows that within a time period of ∼20 fs the exciton self-traps into one of the potential minima. This process of a loss of phase coherence is generically modeled for dissipative two-state systems by the CalderiaLeggett model.25 Figure 3 shows the vertical and vibrationally relaxed LEGS wave function superimposed on the associated polymer conformation. The dynamics of the bond deformations at the center of what becomes the exciton-polaron, namely, the 77th monomer, is shown in Figure 4. Evidently, there is a benzenoidquinoid transition whose associated primary normal mode has an initial time period of ∼35 fs, which shortens to ∼20 fs after bond equilibration (indicating a softening of the normal mode during excitonpolaron formation). The time dependence of the exciton localization length (defined by eq 26), averaged over an ensemble of 40 chains, is shown in Figure 5. The localization length decreases from 12 to 2.5 monomers in a time scale of a single bond oscillation. As described in section III.A, the conjugation length—the size of the chromophore over which the exciton is confined—is approximately three times the localization length. So, for this value of torsional disorder, the size of the conjugated segment (or chromophore) has reduced from ∼35 to just under eight monomers in less than 100 fs. (This size of the exciton-polaron is consistent with a value of six monomers in ref 7 and 10 monomers in ref 19.) As 9115



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Figure 5. The ensemble averaged localization length, ÆLæ (in units of the repeat distance), as a function of time for local exciton ground states (LEGS): with torsional relaxation, black squares; with fixed torsional angles, red circles. Oscillations in ÆLæ arising from the vibrational normal modes and the torsional oscillations are visible. The inset shows the average vertical (t = 0) (squares) and relaxed (t f ∞) (circles) LEGS localization length versus the standard deviation of the torsional disorder, σϕ. (The chromophore size, l = 3ÆLæ.) The ensemble contains 40 polymers, each of 60 monomer units. The standard deviation of the torsional angle is 5, and the probability of a transcis defect is 8%.

discussed in the next section, this rapid “dynamic localization” of an exciton, shown schematically in Figure 3, causes ultrafast

)

)

Figure 4. The normalized bond dynamics at the center of the excitonpolaron (shown in Figure 3), showing that the benzenoid quinoid distortion that causes the dynamical localization of the exciton occurs within the period of a single oscillation of the associated normal mode. The aromatic bond length, r0 = 1.40 Å.

IV. ULTRAFAST FLUORESCENCE DEPOLARIZATION The fluorescence anisotropy of an ensemble of molecules is defined as27 I I^ r ¼ ð29Þ I þ 2I^ 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 )

Figure 3. Part of the polymer conformation used to generate the LEGS wave function shown in Figure 2. Also shown schematically is the vertical (t = 0) and vibrationally relaxed (t = 195 fs) LEGS wave function. One end of the chain and the transcis defect at the 73rd monomer are labeled.

depolarization on a coiled polymer, because as the exciton relaxes onto a shorter length of the polymer, its transition dipole moment also rotates.4 The time period for torsional oscillations (∼0.4 ps) is illustrated by the oscillations of the localization length occurring after the vibrational oscillations have been damped. The torsional relaxation causes a small further decrease in the localization length. Generally, the benzoidquinoid distortion of the photoexcited state planarizes the polymer in the vicinity of the selftrapped exciton. It also stiffens the torsional potential and hence reduces the fluctuations in the torsional angle.26 Rather than delocalizing the exciton, however, these effects tend to (slightly) further localize the exciton, as the distortion acts as a potential well for the Frenkel exciton. The effect of torsional relaxation is rather negligible in PPV, because the torque planarizing a ring in the excited state is sin ϕ (see eq 12), where ϕ ∼ 15. Associated with the dynamical localization of the LEGS is a dynamical Stokes red shift. Vibrational relaxation causes the exciton energy to red shift by ∼230 meV within 100 fs. Subsequent torsional relaxation causes an additional Stokes shift by ∼10 meV within a few ps.

ri ¼ P2 ðθi Þ 9116

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Figure 6. The ensemble averaged fluorescence anisotropy as a function of time for local exciton ground states. Oscillations arising from the vibrational normal modes and the torsional oscillations are visible. The ensemble contains 40 polymers, each of 60 monomer units. The standard deviation of the torsional angle is 5 and the probability of a transcis defect is 8%.

where P2(θ) is the second-order Legendre polynomial P2 ðθÞ ¼

1 ð3 cos2 θ 1Þ 2

ð31Þ

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

∑i fi ri ∑i fi

2 Æræ ¼ 5

ð32Þ

where fi 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.27 We note by analogy that the second-order Legendre polynomial also plays the role of the nematic order parameter in liquid crystals, where an average value of unity implies that all the molecules are aligned parallel to the director and an average value of zero implies that the molecules are isotropically distributed. Similarly (allowing for the factor of 0.4), for fluorescence anisotropy a value of Æræ = 0.4 implies that the emitted light is parallel to the initial transition dipole moment, whereas a value of Æræ = 0 implies that the emitted light is isotropically depolarized. A. Band Edge Absorption. Photoexcitation at the absorption band edge excites LEGS. As described in the previous section, owing to their coupling to the nuclear motion, these vertical excitations rapidly self-trap and localize. Figure 6 shows the time dependence of the ensemble-averaged fluorescence anisotropy, Ær(t)æ, for LEGS. The oscillations occurring with an initial time period of ∼35 fs (and being rapidly damped on a time scale of 100 fs) arise from the normal mode associated with the benzenoid quinoid disortion. The slower and weaker oscillations occurring with a time period of ∼0.4 ps (and being damped on a time scale of 1 ps) arise from torsional rotations. As the inset of Figure 5 illustrates, the initial (vertical) localization length is determined by the disorder in the torsional angle, σϕ, whereas the final (relaxed) localization length is almost entirely

Figure 7. Ensemble averaged asymptotic fluorescence anisotropy from vibrationally relaxed excitons (prior to F€orster-type process) as a function of the torsional disorder, σϕ, and the number of monomers in a statistical segment, m: (a) band edge excitation (i.e., to LEGS) and (b) full band excitation (i.e., to both LEGS and QEES). The ensemble contains N = 36 polymers, each of 80 monomer units. The error bars represent the standard error of the mean, σN = σ/N1/2. The calculation of the full band absorption fluorescence anisotropy requires that enough excited states are calculated to cover the absorption spectrum. For the largest torsional disorder of 15, this is 30 states.

determined by the strength of the electronphonon coupling. The difference between the initial and final localization lengths is one of the factors that determine the amount of fluorescence depolarization, as this determines the extent to which the wave function collapses onto a shorter segment. The other factor that determines the degree of fluorescence depolarization is the conformation of the polymer, determined, for example, by fluctuations in the torsional angle, bends, and transcis defects. Intuitively, for the same degree of dynamical localization, a more coiled chain will exhibit a larger fluorescence depolarization as there will be a larger rotation of the transition dipole moment. These expectations are confirmed by Figure 7a, which shows the final fluorescence anisotropy after dynamical localization (but prior to any F€orster-type process) as a function of torsional disorder and the number of monomers in a statistical segment, m. The trends are that increasing the torsional disorder decreases the depolarization as there is a decrease in the amount of 9117


The Journal of Physical Chemistry A dynamical localization, whereas decreasing m increases the depolarization as the chain becomes more coiled. For all values of the disorder there is a trend to a nonzero asymptotic value as the chain becomes more coiled, indicating that for LEGSs the transition dipole moments never become fully depolarized relative to their initial values. B. Full Band Absorption. As shown in ref 2, the set of LEGS and set of quasi-extended exciton states (QEES) contribute equally to the overall optical absorption. Thus, photoexcitation over the entire absorption band excites both lower energy LEGS and higher energy QEES. As described in section III.B, the QEES will undergo rapid nonadiabatic interconversion to low-lying LEGS, thus undergoing rapid dynamical localization, as illustrated schematically in Figure 1. On a coiled polymer this localization will cause an initial rotation of the transition dipole moment. The LEGS thus populated (or populated by direct photoexcitation) will then undergo the dynamical localization described in section III.C, causing another rotation of the transition dipole moment before emission. Not surprisingly, as shown in Figure 7b, the additional initial depolarization, caused by interconversion from QEES to LEGS, causes a larger total fluorescence depolarization than that observed by just band edge absorption. For small values of torsional disorder in highly coiled chains, Æræ vanishes, implying fully depolarized emission. C. Comparison with Experiment. For a given statistical polymer conformation (defined by the number of monomers in a statistical segment, m) and a given off-diagonal disorder (defined by standard deviation of the torsional disorder, σϕ) the amount of fluorescence depolarization is determined by the excitation wavelength. Thus, measurements of the fluorescence anisotropy for both band edge and full band absorption allow the variables m and σϕ to be determined. Since m may also be determined directly via the radius of gyration (see eq 8), there is a direct link between our predictions and experimental observables. Our results are qualitatively in agreement with the measurements on MEHPPV by Ruseckas et al.4 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). Since the higher excitation energy will preferentially excite QEES (as it is not a full band absorption), we cannot directly compare their results to our predictions.28

V. CONCLUSIONS This paper has discussed two types of ultrafast dynamical localization: first, nonadiabatic interconversion (IC) from higher energy quasi-extended exciton states (QEES) to lower energy more localized local exciton ground states (LEGS) and, second, vibrational relaxation of LEGS to self-trapped polarons within the period of the first CC bond oscillation following photoexcitation. In the latter case there is CalderiaLeggett type loss of phase coherence owing to the coupling of the polymer to a dissipative environment. Dynamical localization causes fluorescence depolarization, because as the exciton relaxes its transition dipole moment rotates. Exciting higher energy states causes a larger fluorescence depolarization, because these states have a larger initial delocalization. Thus, in general, fluorescence depolarization is a function of excitation wavelength. Fluorescence depolarization is also a function of the conformational disorder that controls off-diagonal disorder, namely, torsional disorder (as this determines the initial

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exciton localization) and the conformational disorder that controls the radius of gyration, namely, transcis defects. Measuring the fluorescence depolarization as a function of excitation wavelength therefore enables the two variables of torsional disorder and the fraction of transcis defects to be determined. So, by using another independent measure of polymer conformation, e.g., neutron diffraction measurements of the radius of gyration, the descriptions of exciton localization made in this paper may be tested. We do not expect an exact replication of our predictions by observations, because there are a number of processes that have been neglected in our treatment of fluorescence depolarization. First, we assumed that IC from QEES occurs directly to LEGS. However, as discussed in ref 10, for a polymer strongly coupled to a solid state environment there is a probability of phononinduced IC between QEES. Second, as discussed in the Introduction, for polymers with strong interchromophore interactions ultrafast fluorescence depolarization may occur via excimer formation.5 Finally, we have neglected the role of density fluctuations in the environment. These cause diagonal disorder and hence Anderson localization of Frenkel excitons, independently of the polymer conformation.1 Another conclusion from this work is that the initial torsional disorder determines the spatial distribution and localization length of vertical excitations but that electronphonon coupling is largely responsible for the localization length of self-trapped excitons (as shown by the inset of Figure 5). (For example, a torsional disorder of 10 implies a chromophore (defined by the size of a vertical LEGS) of ∼25 monomers, while the vibrationally relaxed state is spread over ∼8 monomers.) In PPV, at least, torsional relaxation caused by the coupling of the electronic, vibrational, and torsional degrees of freedom seems to play little role in the subsequent dynamical localization. This is a convenient conclusion, because as explained in section II.B, we have assumed that the torsional disorder is static. Since long-range F€orster resonance energy transfer (FRET) takes place after vibrational relaxation has occurred,10 these conclusions about exciton localization have implications for the modeling of such processes because of the dependence of FRET on conjugation length. In particular, the rate, k, for FRET is determined by the ratio of the conjugation length, l, to R, the distance of nearest separation between the donor and acceptor chromophores.2933 Specifically, for parallel collinear chromophores of the same length30,31,33 when l , R, k ∼ l2/R6, while when l . R, k ∼ 1/(lR)2. As just noted, however, FRET does not occur between chromophores of the same length, because the initial state is a vibrationally relaxed excitonpolaron, while the final state is a vertical LEGS. Indeed, FRET between the initial and final states initially causes exciton delocalization, before the onset of vibrational relaxation localizes the final state. A correct theory of FRET should include such processes. Finally, we have emphasized that conjugation breaks (or minimum thresholds of the hybridization integral) are not required to cause exciton localization and they do not necessarily define chromophores. In fact, we propose that such a picture is probably wrong, as singlet Frenkel excitons can delocalize via a dipole dipole (or through-space) mechanism. It also leads to the incorrect conclusion that higher energy states are more localized than lower energy states. Such a conclusion would (naively) predict that fluorescence depolarization is a decreasing function of excitation energy, contrary to experimental observations and to the conclusions presented here. 9118



’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

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(31) Das, M.; Ramesesha, S. J. Chem. Phys. 2010, 132, 124109. (32) Hennebicq, E.; Pourtois, G.; Scholes, G. D.; Herz, L. M.; Russell, D. M.; Silva, C.; Setayesh, S.; Grimsdale, A. C.; M€ullen, K.; Bredas, J. L.; Beljonne, D. J. Am. Chem. Soc. 2005, 127, 4744. (33) Barford, W. J. Phys. Chem. A 2010, 114, 11842.

’ ACKNOWLEDGMENT We thank the John Fell Fund of the University of Oxford for financial support and L. M. Herz for discussions. ’ REFERENCES (1) Barford, W.; Trembath, D. Phys. Rev. B 2009, 80, 165418. (2) Makhov, D. V.; Barford, W. Phys. Rev. B 2010, 81, 165201. (3) Malyshev, A. V.; Malyshev, V. A. Phys. Rev. B 2001, 63, 195111. (4) Ruseckas, A.; Wood, P.; Samuel, I. D. W.; Webster, G. R.; Mitchell, W. J.; Burn, P. L.; Sundstr€om, V. Phys. Rev. B 2005, 72, 115214. (5) Chang, M. H.; Frampton, M. J.; Anderson, H. L.; Herz, L. M. Phys. Rev. Lett. 2007, 98, 27402. (6) Tomlinson, A.; Yaron, D. J. Comput. Chem. 2003, 24, 1782. (7) Tretiak, S.; Saxena, A.; Martin, R. L.; Bishop, A. R. Phys. Rev. Lett. 2002, 89, 97402. (8) Sterpone, F.; Rossky, P. J. J. Phys. Chem. B 2008, 112, 4983. (9) De Leener, C.; Hennebicq, E.; Sancho-Garcia, J.-C.; Beljonne, D. J. Phys. Chem. B 2009, 113, 1311. (10) 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. (11) Liu, L. T.; Yaron, D.; Berg, M. A. J. Phys. Chem. B 2007, 111, 5770. (12) Beenken, W. J. D. Phys. Status Solidi A 2009, 206, 2750. (13) Barford, W.; Lidzey, D. G.; Makhov, D. V.; Meijer, A. J. H. J. Chem. Phys. 2010, 133, 044504. (14) Baeriswyl, D.; Campbell, D. K.; Mazumdar, S. Conjugated Conducting Polymers; Kiess, H., Ed.; Springer-Verlag: Berlin, 1992. (15) Barford, W. Electronic and Optical Properties of Conjugated Polymers; Oxford University Press: Oxford, 2005. (16) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran; Cambridge University Press: Cambridge, 1986. (17) Martin, S. J.; Bradley, D. D. C.; Lane, P. A.; Mellor, H.; Burn, P. L. Phys. Rev. B 1999, 59, 15133. (18) Chandross, M.; Mazumdar, S. Phys. Rev. B 1997, 55, 1497. (19) Karabunarliev, S.; Bittner, E. R. J. Chem. Phys. 2003, 118, 4291. (20) Barford, W.; Bursill, R. J.; Yu Lavrentiev, M. J. Phys.: Condens. Matter 1998, 10, 6429. Barford, W.; Paiboonvorachat, N. J. Chem. Phys. 2008, 129, 164716. (21) 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. (22) The analogy of a linear polymer with a one-dimensional molecular solid (or a J-aggregate) has also recently been made by Yamagata and Spano in their analysis of exciton coherence lengths in a perfectly ordered polydiacetylene chain.23 (23) DYamagata, H.; Spano, F. C.; J. Chem. Phys., in press. (24) Kramer, B.; MacKinnon, A. Rep. Prog. Phys. 1987, 56, 1469. (25) Caldeira, A. O.; Leggett, A. J. Phys. Rev. Lett. 1981, 46, 211. (26) Liu, L. T.; Yaron, D.; Sluch, M. I.; Berg, M. A. J. Phys. Chem. B 2006, 110, 18844. (27) Lakowicz, J. R. Principles of Fluorescence Spectroscopy; Plenum Press: New York, 1999. (28) If we do assume that the high energy excitation at 2.9 eV corresponds to full band absorption, we find the values of m ∼ 50 and σϕ ∼ 25. (29) Scholes, G. D.; Jordanides, X. J.; Fleming, G. R. J. Phys. Chem. B 2001, 105, 1640. Jordanides, X. J.; Scholes, G. D.; Fleming, G. R. J. Phys. Chem. B 2001, 105, 1652. (30) Wong, K. F.; Biman, B.; Rossky, P. J. J. Phys. Chem. B 2004, 108, 5752. 9119


Ultrafast Dynamical Localization of Photoexcited States in

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