EExciton Dynamics in Conjugated Polymers
William Barford a) Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford, OX1 3QZ,United Kingdom
Exciton dynamics in π -conjugated polymers encompass multiple time and length scales. Ultrafast femtosecond pro-cesses are intrachain and involve a quantum mechanical correlation of the exciton and nuclear degrees of freedom. Incontrast, post-picosecond processes involve the incoherent Förster transfer of excitons between polymer chains. Exci-ton dynamics is also strongly determined by the spatial and temporal disorder that is ubiquitous in conjugated polymers.Since excitons are delocalized over hundreds of atoms, a theoretical understanding of these processes is only realis-tically possible by employing suitably parametrized coarse-grained exciton-phonon models. Moreover, to correctlyaccount for ultrafast processes, the exciton and phonon modes must be treated on the same quantum mechanical basisand the Ehrenfest approximation must be abandoned. This further implies that sophisticated numerical techniques mustbe employed to solve these models. This review describes our current theoretical understanding of exciton dynamicsin conjugated polymer systems. We begin by describing the energetic and spatial distribution of excitons in disorderedpolymer systems, and define the crucial concept of a ‘chromophore’ in conjugated polymers. We also discuss therole of exciton-nuclear coupling, emphasizing the distinction between ‘fast’ and ‘slow’ nuclear degrees of freedom indetermining ‘self-trapping’ and ‘self-localization’ of exciton-polarons. Next, we discuss ultrafast intrachain excitondecoherence caused by exciton-phonon entanglement, which leads to fluorescence depolarization on the timescale of10-fs. Interactions of the polymer with its environment causes the stochastic relaxation and localization of high-energydelocalized excitons onto chromophores. The coupling of excitons with torsional modes also leads to various dynam-ical processes. On sub-ps timescales it causes exciton-polaron formation (i.e., exciton localization and local polymerplanarization), while on post-ps timescales stochastic torsional fluctuations cause exciton-polaron diffusion along thepolymer chain. Finally, we describe a first-principles, Förster-type model of intrachain exciton transfer and diffusion,whose starting point is a realistic description of the donor and acceptor chromophores. We survey experimental resultsand explain how they can be understood in terms of our theoretical description of exciton dynamics coupled to informa-tion on polymer multiscale structures. The review also contains a brief critique of computational methods to simulateexciton dynamics. I. INTRODUCTION
The theoretical study of exciton dynamics in conjugatedpolymer systems is both a fascinating and complicated sub-ject. One reason for this is that characterizing excitonicstates themselves is a challenging task: conjugated polymersexhibit strong electron-electron interactions and electron-nuclear coupling, and are subject to spatial and temporal dis-order. Another reason is that exciton dynamics is charac-terised by multiple (and often overlapping) time scales; it isdetermined by both intrinsic processes (e.g., coupling to nu-clear degrees of freedom and electrostatic interactions) andextrinsic processes (e.g., polymer-solvent interactions); and itis both an intrachain and interchain process. Consequently, tomake progress in both characterizing exciton states and cor-rectly describing their dynamics, simplified, but realistic mod-els are needed. Moreover, as even these simplified models de-scribe many quantized degrees of freedom, sophisticated nu-merical techniques are required to solve them. Luckily, fun-damental theoretical progress in developing numerical tech-niques means that simplified one-dimensional models of con-jugated polymers are now soluble to a high degree of accuracy.In addition to the application of various theoretical tech-niques to understand exciton dynamics, a wide range of time-resolved spectroscopic techniques have also been deployed. a) Electronic mail: [email protected]
These include fluorescence depolarization , three-pulsephoton-echo and coherent electronic two-dimensionalspectroscopy . Some of the timescales extracted from theseexperiments are listed in Table I; the purpose of this review isto describe their associated physical processes.As well as being of intrinsic interest, the experimental andtheoretical activities to understand exciton dynamics in conju-gated polymer systems are also motivated by the importanceof this process in determining the efficiency of polymer elec-tronic devices. In photovoltaic devices, large exciton diffusionlengths are necessary so that excitons can migrate efficientlyto regions where charge separation can occur. However, pre-cisely the opposite is required in light emitting devices, sincediffusion leads to non-radiative quenching of the exciton.Perhaps one of the reasons for the failure to fully exploitpolymer electronic devices has been the difficulty in estab-lishing the structure-function relationships which allow thedevelopment of rational design strategies. An understand-ing of the principles of exciton dynamics, relating this tomultiscale polymer structures, and interpreting the associatedspectroscopic signatures are all key ingredients to develop-ing structure-function relationships. An earlier review ex-plored the connection between structure and spectroscopy .In this review we describe our current understanding of theimportant dynamical processes in conjugated polymers, be-ginning with photoexcitation and intrachain relaxation on ul-trafast timescales ( ∼
10 fs) to sub-ns interchain exciton trans-fer and diffusion. These key processes are summarized in Ta-ble II. a r X i v : . [ phy s i c s . c h e m - ph ] F e b Polymer State Timescales CitationMEH-PPV Solution τ =
50 fs, τ = − MEH-PPV Solution τ = −
10 fs, τ = −
200 fs Ref PDOPT Film τ = . − PDOPT Solution τ < τ = −
23 ps Ref P3HT Film τ =
300 fs, τ = . τ =
40 ps Ref P3HT Solution τ =
700 fs, τ = τ =
41 ps, τ =
530 ps Ref P3HT Solution τ = −
200 fs, τ = − τ = −
20 ps Ref P3HT Solution τ (cid:46)
100 fs, τ ∼ −
10 ps Ref TABLE I. Some of the dynamical timescales observed in conjugated polymers whose associated physical processes are summarized in TableII. Process Consequences Timescale SectionExciton-polaron self-trapping via coupling to fast C-Cbond vibrations. Exciton-site decoherence; ultrafast flu-orescence depolarization. ∼
10 fs III AEnergy relaxation from high-energy quasi-extended ex-citon states (QEESs) to low-energy local exciton groundstates (LEGSs) via coupling to the environment. Stochastic exciton density localizationonto chromophores. ∼ −
200 fs III BExciton-polaron self-localization via coupling to slowbond rotations in the under-damped regime. Exciton density localization on achromophore; ultrafast fluorescencedepolarization. ∼ −
600 fs III CExciton-polaron self-localization via coupling to slowbond rotations in the over-damped regime. Exciton density localization on achromophore; post-ps fluorescencedepolarization. ∼ −
10 ps III CStochastic torsional fluctuations inducing exciton‘crawling’ and ‘skipping’ motion. Intrachain exciton diffusion and energyfluctuations. ∼ −
30 ps IVInterchromophore Förster resonant energy transfer. Interchromophore exciton diffusion;post-ps spectral diffusion and fluores-cence depolarization. ∼ −
100 ps VRadiative decay. ∼
500 psTABLE II. The life and times of an exciton: Some of the key exciton dynamical processes, encompassing over four-orders of magnitude, thatoccur in conjugated polymer systems.
The plan of this review is the following. We begin bybriefly describing some theoretical techniques for simulatingexciton dynamics and emphasize the failures of simple meth-ods. As already mentioned, excitons themselves are fasci-nating quasiparticles, so before describing their dynamics, inSection III we start by describing their stationary states. Westress the role of low-dimensionality, disorder and electron-phonon coupling, and we discuss the fundamental concept ofa chromophore. Next, in Section IV, we describe the sub-psprocesses of intrachain exciton decoherence, relaxation andlocalization, which - starting from an arbitrary photoexcitedstate - results in an exciton forming a chromophore. We nextturn to describe the exciton (and energy) transfer processes oc-curring on post-ps timescales. First, in Section V, we describethe primarily adiabatic intrachain motion of excitons causedby stochastic torsional fluctuations, and second, in Section VI,we describe nonadiabatic interchain exciton transfer. We con-clude and address outstanding questions in Section VII.
II. A BRIEF CRITIQUE OF THEORETICAL TECHNIQUES
A theoretical description of exciton dynamics in conjugatedpolymers poses considerable challenges, as it requires a rig-orous treatment of electronic excited states and their cou-pling to the nuclear degrees of freedom. Furthermore, con-jugated polymers consist of thousands of atoms and tens ofthousands of electrons. Thus, as the Hilbert space growsexponentially with the number of degrees of freedom, ap-proximate treatments of excitonic dynamics are therefore in-evitable. There are two broad approaches to a theoretical treat-ment. One approach is to construct ab initio
Hamiltonians,with an exact as possible representation of the degrees of free-dom, and then to solve these Hamiltonians with various de-grees of accuracy. Another approach (albeit less common intheoretical chemistry) is to construct effective Hamiltonianswith fewer degrees of freedom, such as the Frenkel-Holsteinmodel described in Section IV. These effective Hamiltoniansmight be parameterized via a direct mapping from ab initio
Hamiltonians (e.g., see Appendix H in ref , Appendix A inref and various papers by Burghardt and coworkers ) orelse semiempirically . A significant advantage of effectiveHamiltonians over their ab initio counterparts is that they canbe solved for larger systems over longer timescales and to ahigher level of accuracy.As the Ehrenfest method is a widely used approximationto study charge and exciton dynamics in conjugated poly-mers, we briefly explain this method and describe the impor-tant ways in which it fails. (For a fuller treatment, see .)The Ehrenfest method makes two key approximations. Thefirst approximation is to treat the nuclei classically. Thismeans that nuclear quantum tunneling and zero-point energiesare neglected, and that exciton-polarons are not correctly de-scribed (see Section III C). The second assumption is that thetotal wavefunction is a product of the electronic and nuclearwavefunctions. This means that there is no entanglement be-tween the electrons and nuclei, and so the nuclei cannot causedecoherence of the electronic degrees of freedom (see SectionIV A). A simple product wavefunction also implies that thenuclei move in a mean field potential determined by the elec-trons. This means that a splitting of the nuclear wave packetwhen passing through a conical intersection or an avoidedcrossing does not occur (see Section IV B), and that there is anincorrect description of energy transfer between the electronicand nuclear degrees of freedom (see Section V D). As will bediscussed in the course of this review, these failures mean thatin general the Ehrenfest method is not a reliable one to treatultrafast excitonic dynamics in conjugated polymers.Various theoretical techniques have been proposed to rec-tify the failures of the Ehrenfest method; for example, thesurface-hopping technique , while still keeping the nucleiclassical, partially rectifies the failures at conical interactions.More sophisticated approaches, for example the MC-TDHFand TEBD methods, quantize the nuclear degrees of freedomand do not assume a product wavefunction.For a given electronic potential energy surface (PES),the multiconfigurational-time dependent Hartree-Fock (MC-TDHF) method is an (in principle) exact treatment of nu-clear wavepacket propagation, although in practice exponen-tial scaling of the Hilbert space means that a truncation is re-quired. In addition, this method is only as reliable as the rep-resentation of the PES.In the time-evolving block decimation (TEBD) method a quantum state, | Ψ (cid:105) , is represented by a matrix product state(MPS) . Its time evolution is determined via | Ψ ( t + δ t (cid:105) = exp ( − i ˆ H δ t / ¯ h ) | Ψ ( t ) (cid:105) , (1)where ˆ H is the system Hamiltonian and the action of theevolution operator is performed via a Trotter decomposi-tion. Since the action of the evolution operator expands theHilbert space, | Ψ (cid:105) is subsequently compressed via a singularvalue decomposition (SVD) . Importantly, this approach is‘numerically exact’ as long as the truncation parameter ex-ceeds 2 S , where S is the entanglement entropy, defined by S = − ∑ α ω α ln ω α and { ω } are the singular values obtainedat the SVD. The TEBD method permits the electronic and nu-clear degrees of freedom to be treated as quantum variableson an equal footing. It thus rectifies all of the failures of theEhrenfest method described above and, unlike the MC-TDHF method, it is not limited by the representation of the PES. Itcan, however, only be applied to quantum systems describedby one-dimensional lattice Hamiltonians . Luckily, as de-scribed in Section IV, such model Hamiltonians are readilyconstructed to describe exciton dynamics in conjugated poly-mers. III. EXCITONS IN CONJUGATED POLYMERS
Before discussing the dynamics of excitons, we begin bydescribing exciton stationary states in static conjugated poly-mers.
A. Two-particle model
An exciton is a Coulombically bound electron-hole pairformed by the linear combination of electron-hole excitations(for further details see ). In a one-dimensional con-jugated polymer an exciton is described by the two-particlewavefunction, Φ m j ( r , R ) = ψ m ( r ) Ψ j ( R ) . Ψ j ( R ) is the center-of-mass wavefunction, which will bediscussed shortly. Before doing that, we first discuss the rela-tive wavefunction, ψ m ( r ) , which describes a particle boundto a screened Coulomb potential, where r is the electron-hole separation and m is the principal quantum number.The electron and hole of an exciton in a one-dimensionalsemiconducting polymer are more strongly bound than ina three-dimensional inorganic semiconductor for two keyreasons. First, because of the low dielectric constant andrelatively large electronic effective mass in π -conjugated sys-tems the effective Rydberg is typically 50 times larger thanfor inorganic systems. Second, dimensionality plays a role:in particular, the one-dimensional Schrödinger equation forthe relative particle predicts a strongly bound state split-off from the Rydberg series. This state is the m = B u ’) exciton, with a binding energy of ∼ m = A g ’) exciton.With the exception of donor-acceptor copolymers, conju-gated polymers are generally non-polar, which means thateach p -orbital has an average occupancy of one electron. Thisimplies an approximate electron-hole symmetry. Electron-hole symmetry has a number of consequences for the char-acter and properties of excitons. First, it means that the rela-tive wavefunction exhibits electron-hole parity, i.e., ψ m ( r ) =+ ψ m ( − r ) when m is odd and ψ m ( r ) = − ψ m ( − r ) when m iseven. Second, the transition density, (cid:104) EX | ˆ N i | GS (cid:105) , vanishesfor odd-parity (i.e., even m ) excitons. This means that suchexcitons are not optically active, and importantly for dynam-ical processes, their Förster exciton transfer rate (defined inSection VI A) vanishes.Since Frenkel excitons are the primary photoexcited statesof conjugated polymers, their dynamics is the subject of thisreview. Their delocalization along the polymer chain of N monomers is described by the Frenkel Hamiltonian,ˆ H F = N ∑ n = ε n ˆ N n + N − ∑ n = J n ˆ T n , n + , (2)where n = ( R / d ) labels a monomer and d is the inter-monomer separation. The energy to excite a Frenkel excitonon monomer n is ε n , where ˆ N n = | n (cid:105) (cid:104) n | is the Frenkel excitonnumber operator.In principle, excitons delocalize along the chain via twomechanisms . First, for even-parity (odd m ) singlet ex-citons there is a Coulomb-induced (or through space) mecha-nism. This is the familiar mechanism of Förster energy trans-fer. The exciton transfer integral for this process is J DA = ∑ i ∈ Dj ∈ A V i j (cid:2) D (cid:104) GS | ˆ N i | EX (cid:105) D (cid:3) (cid:2) A (cid:104) EX | ˆ N j | GS (cid:105) A (cid:3) . (3)The sum is over sites i in the donor monomer and j in the ac-ceptor monomer, and V i j is the Coulomb interaction betweenthese sites. In the point-dipole approximation Eq. (3) becomes J DA = κ mn µ πε r ε R mn , (4)where µ is the transition dipole moment of a single monomerand R mn is the distance between the monomers m and n . κ mn is the orientational factor, κ mn = ˆ r m · ˆ r n − ( ˆ R mn · ˆ r m )( ˆ R mn · ˆ r n ) , (5)where ˆ r m is a unit vector parallel to the dipole on monomer m and ˆ R mn is a unit vector parallel to the vector joiningmonomers m and n . For colinear monomers, the nearestneighbor through space transfer integral is J DA = − µ πε r ε d . (6)Second, for all excitons there is a super-exchange (orthrough bond) mechanism, whose origin lies in a virtual fluc-tuation from a Frenkel exciton on a single monomer to acharge-transfer exciton spanning two monomers back to aFrenkel exciton on a neighboring monomer. The energyscale for this process, obtained from second order perturba-tion theory , is J SE ( θ ) ∝ − t ( θ ) ∆ E , (7)where t ( θ ) (defined in Eq. (12)) is proportional to the overlapof p -orbitals neighboring a bridging bond, i.e., t ( φ ) ∝ cos θ and θ is the torsional (or dihedral) angle between neighboringmonomers. ∆ E is the difference in energy between a charge-transfer and Frenkel exciton.The total exciton transfer integral is thus J n = J DA + J cos θ n . (8)The bond-order operator,ˆ T n , n + = ( | n (cid:105) (cid:104) n + | + | n + (cid:105) (cid:104) n | ) , (9) FIG. 1. The mapping of a polymer chain conformation to a coarse-grained linear site model. Each site corresponds to a moiety alongthe polymer chain, with the connection between sites characterisedby the torsional (or dihedral) angle, θ . represents the hopping of the Frenkel exciton betweenmonomers n and n +
1. Evidently, J SE vanishes when θ = J DA will not. Therefore, even if J SE vanishes because ofnegligible p -orbital overlap between neighboring monomers,singlet even-parity excitons can still retain phase coherenceover the ‘conjugation break’ . This observation has impor-tant implications for the definition of chromophores, as dis-cussed in Section III B.Eq. (2) represents a ‘coarse-graining’ of the exciton degreesof freedom. The key assumption is that we can replace theatomist detail of each monomer (or moiety) and replace it by a‘coarse-grained’ site, as illustrated in Fig. 1. All that remainsis to describe how the Frenkel exciton delocalises along thechain, which is controlled by the two sets of parameters, { ε } and { J } . Since J is negative, a conjugated polymer is equiva-lent to a molecular J-aggregate.The eigenfunctions of ˆ H F are the center-of-mass wavefunc-tions, Ψ j ( n ) , where j is the associated quantum number. Fora linear, uniform polymer (i.e., ε n ≡ ε and J n ≡ J ) Ψ j ( n ) = (cid:18) N + (cid:19) / N ∑ n = sin (cid:18) π jnN + (cid:19) , (10)forming a band of states with energy E j = ε + J cos (cid:18) π jN + (cid:19) . (11)The family of excitons with different j values corresponds tothe Frenkel exciton band with different center-of-mass mo-menta. In emissive polymers the j = B u state. B. Role of static disorder: local exciton ground states andquasiextended exciton states
Polymers are rarely free from some kind of disorder andthus the form of Eq. (10) is not valid for the center-of-masswavefunction in realistic systems. Polymers in solution are
FIG. 2. (a) The density of three local exciton ground states (LEGSs,dotted curves) and the three vibrationally relaxed states (VRSs, solidcurves) for one particular static conformation of a PPV polymerchain made up of 50 monomers. The exciton center-of-mass quan-tum number, j , for each state is also shown. (b) The exciton den-sity of a quasiextended exciton state (QEES), with quantum number j =
7. Reproduced from J. Chem. Phys. , 034901 (2018) withthe permission of AIP publishing. necessarily conformationally disordered as a consequence ofthermal fluctuations (as described in Section V). Polymers inthe condensed phase usually exhibit glassy, disordered con-formations as consequence of being quenched from solution.Conformational disorder implies that the dihedral angles, { θ } are disordered, which by virtue of Eq. (8) implies that the ex-citon transfer integrals are also disordered.As well as conformational disorder, polymers are also sub-ject to chemical and environmental disorder (arising, for ex-ample, from density fluctuations). This type of disorder af-fects the energy to excite a Frenkel exciton on a monomer (orcoarse-grained site).As first realized by Anderson , disorder localizes a quan-tum particle (in our case, the exciton center-of-mass particle),and determines their energetic and spatial distributions. Theorigin of this localization is the wave-like nature of a quan- FIG. 3. (a) The energy density of states and (b) the optical absorp-tion (neglecting the vibronic progression) of the manifold of Frenkelexcitons (where | σ J / J | = . ∼ | J || σ J / J | / . Similarly, the width of the optical absorptionfrom both the LEGSs and all states ∼ | J || σ J / J | / . The band edgefor an ordered chain is at 2 | J | (indicated by the dashed lines), soLEGSs generally lie in the Lifshitz (or Urbach) tail of the density ofstates, i.e., E < | J | . tum particle and the constructive and destructive interferenceit experiences as it scatters off a random potential. Malyshevand Malyshev further observed that in one-dimensionalsystems there are a class of states in the low energy tail of thedensity of states that are superlocalized, named local excitonground states (LEGSs ). LEGSs are essentially nodeless,non-overlapping wavefunctions that together spatially spanthe entire chain. They are local ground states, because for theindividual parts of the chain that they span there are no lowerenergy states. A consequence of the essentially nodeless qual-ity of LEGSs is that the square of their transition dipole mo-ment scales as their size . Thus, LEGSs define chromophores(or spectroscopic segments), namely the irreducible parts of apolymer chain that absorb and emit light. Fig. 2(a) illustratesthe three LEGSs for a particular conformation of PPV with 50monomers.Some researchers claim that ‘conjugation-breaks’ (or morecorrectly, minimum thresholds in the p z -orbital overlap) de-fine the boundaries of chromophores . In contrast, we sug-gest that it is the disorder that determines the average chro-mophore size, but ‘conjugation-breaks’ can ‘pin’ the chro-mophore boundaries. Thus, if the average distance betweenconjugation breaks is smaller than the chromophore size,chromophores will span conjugation breaks but they may alsobe separated by them. Conversely, if average distance betweenconjugation breaks is larger than the chromophore size thechromophore boundaries are largely unaffected by the breaks.The former scenario occurs in polymers with shallow tor-sional potentials, e.g., polythiophene .Higher energy lying states are also localized, but are node-ful and generally spatially overlap a number of low-lyingLEGSs. These states are named quasiextended exciton states(QEESs) and an example is illustrated in Fig. 2(b).When the disorder is Gaussian distributed with a standarddeviation σ , single parameter scaling theory provides someexact results about the spatial and energetic distribution of theexciton center-of-mass states:1. The localization length L loc ∼ ( | J | / σ ) / at the bandedge and as L loc ∼ ( | J | / σ ) / at the band center.2. As a consequence of exchange narrowing, the widthof the density of states occupied by LEGSs scales as σ / √ L loc ∼ σ / . Similarly, the optical absorption isinhomogeneously narrowed with a line width ∼ σ / .3. The fraction of LEGSs scales as 1 / L loc ∼ σ / .These points are illustrated in Fig. 3, which shows the Frenkelexciton density of states and optical absorption for a particu-lar value of disorder. Evidently, although LEGSs are a smallfraction of the total number of states, they dominate the opti-cal absorption.This section has described LEGSs (or chromophores) asstatic objects defined by static disorder. However, as discussedin Section V, dynamically torsional fluctuations also renderthe conformational disorder dynamic causing the LEGSs toevolve adiabatically. As a consequence, the chromophores‘crawl’ along the polymer chain. C. Role of electron-nuclear coupling: exciton-polarons
As well as disorder, another important process in determin-ing exciton dynamics and spectroscopy is the coupling of anexciton to nuclear degrees of freedom; in a conjugated poly-mer these are fast C-C bond vibrations and slow monomerrotations. In this section we briefly review the origin of thiscoupling and then discuss exciton-polarons.
1. Origin of electron-nuclear coupling
When a nucleus moves, either by a linear displacement orby a rotation about a fixed point, there is a change in the elec-tronic overlap between neighboring atomic orbitals. Assum-ing that neighboring p -orbitals lie in the same plane normal tothe bond with a relative twist angle of θ , the resonance inte-gral between a pair of orbitals separated by r is t ( θ ) = t ( r ) cos θ = β exp ( − α r ) cos θ , (12)where t ( r ) <
0. The kinetic energy contribution to the Hamil-tonian is ˆ H ke = t ( r ) cos θ × ˆ T , (13)where the bond-order operator, ˆ T , is defined in Eq. (9). Treat-ing r and θ as dynamical variables, suppose that the σ -electrons of a conjugated molecule and steric hinderances pro-vide equilibrium values of r = r and θ = θ , with correspond-ing elastic potentials of V vib = K σ vib ( r − r ) (14)and V rot = K σ rot ( θ − θ ) . (15) FIG. 4. The π -bond order expectation values, (cid:104) ˆ T (cid:105) , for (a) the groundstate and (b) the excited state, showing the benzenoid-quinoid transi-tion. As Eq. (19) and Eq. (20) indicate, the larger bond order of thebridging bond in the excited state implies a smaller dihedral angleand a stiffer torsional potential than the ground state. The coupling of the π -electrons to the nuclei changes theseequilibrium values and the elastic constants.To see this, we use the Hellmann-Feynman to determine theforce on the bond. The linear displacement force is f = − ∂ E ∂ r = − (cid:10) ∂ ˆ H ke ∂ r (cid:11) = α t ( r ) cos θ (cid:104) ˆ T (cid:105) − K σ vib ( r − r ) . (16)Thus, to first order in the change of bond length, δ r = ( r − r ) ,the equilibrium distortion is δ r = α t ( r ) cos θ (cid:104) ˆ T (cid:105) / K σ vib , (17)which is negative because it is favorable to shorten the bondto increase the electronic overlap.Similarly, the torque around the bond is Γ = − ∂ E ∂ θ = − (cid:10) ∂ ˆ H ke ∂ θ (cid:11) = t ( r ) sin θ (cid:104) ˆ T (cid:105) − K σ rot ( θ − θ ) (18)and the equilibrium change of bond angle, δ θ = ( θ − θ ) , is δ θ = t ( r ) sin θ (cid:104) ˆ T (cid:105) / K σ rot , (19)which is also negative, again because it is favorable to increasethe electronic overlap. Thus, the π -electron couplings act toplanarize the chain.The electron-nuclear coupling also changes the elastic con-stants. Assuming a harmonic potential, the new rotationalspring constant is K π rot = ∂ E ∂ θ = − t ( r ) cos θ (cid:104) ˆ T (cid:105) + K σ rot (20)and thus K π rot > K σ rot (because t ( r ) < (cid:104) ˆ T (cid:105) EX > (cid:104) ˆ T (cid:105) GS for the bridging bond in phenyl-based systems, the torsionalangle is smaller and the potential is stiffer in the excited state(as a result of the benzenoid to quinoid distortion) .
2. Exciton-polarons
An exciton that couples to a set of harmonic oscillators,e.g., bond vibrations or torsional oscillations, becomes ‘self-trapped’. Self-trapping means that the coupling between theexciton and oscillators causes a local displacement of the os-cillator that is proportional to the local exciton density (as illustrated in the next section). Alternatively, it is said thatthe exciton is dressed by a cloud of oscillators. Such a quasi-particle is named an exciton-polaron. As there is no barrierto self-trapping in one-dimensional systems , there is alwaysan associated relaxation energy.If the exciton and oscillators are all treated quantum me-chanically, then in a translationally invariant system theexciton-polaron forms a Bloch state and is not localized.However, if the oscillators are treated classically, the non-linear feedback induced by the exciton-oscillator couplingself-localizes the exciton-polaron and ‘spontaneously’ breaksthe translational symmetry. This is a self-localized (or auto-localized) ‘Landau polaron’. Notice that self-trapping isa necessary but not sufficient condition for self-localization.Self-localization always occurs in the limit of vanishing oscil-lator frequency (i.e., the adiabatic or classical limit) and van-ishing disorder. Whether or not an exciton-polaron is self-localized in prac-tice, however, depends on the strength of the disorder and thevibrational frequency of the oscillators. Qualitatively, an ex-citon coupling to fast oscillators (e.g., C-C bond vibrations)forms an exciton-polaron with an effective mass only slightlylarger than a bare exciton . For realistic values of disorder,such an exciton-polaron is not self-localized. This is illus-trated in Fig. 2(a), which shows the three lowest solutionsof the Frenkel-Holstein model (described in Section IV A),known as vibrationally relaxed states (VRSs). As we see, thedensity of the VRSs mirrors that of the Anderson-localizedLEGSs. Conversely, an exciton coupling to slow oscilla-tors (e.g., bridging-bond rotations) forms an exciton-polaronwith a large effective mass. Such an exction-polaron is self-localized (as described in Section IV C and shown in Fig. 6). IV. INTRACHAIN DECOHERENCE, RELAXATION ANDLOCALIZATION
Having qualitatively described the stationary states of exci-tons in conjugated polymers, we now turn to a discussion ofexciton dynamics.
A. Role of fast C-C bond vibrations
After photoexcitation or charge combination after injection,the fastest process is the coupling of the exciton to C-C bondstretches. We now describe the resulting exciton-polaron for-mation and the loss of exciton-site coherence.As we saw in Section III C, bond distortions couple to elec-trons. Using Eq. (13), it can be shown that the coupling of local normal modes (e.g., vinyl-unit bond stretches or phenyl-ring symmetric breathing modes) to a Frenkel exciton is con-veniently described by the Frenkel-Holstein model ,ˆ H FH = ˆ H F − A ¯ h ω vib N ∑ n = ˜ Q n ˆ N n + ¯ h ω vib N ∑ n = (cid:0) ˜ Q n + ˜ P n (cid:1) . (21)ˆ H F is the Frenkel Hamiltonian, defined in Eq. (2), while ˜ Q =( K vib / ¯ h ω vib ) / Q and ˜ P = ( ω vib / ¯ hK vib ) / P are the dimen-sionless displacement and momentum of the normal mode.The second term on the right-hand-side of Eq. (21) indicatesthat the normal mode couples linearly to the local excitondensity . A is the dimensionless exciton-phonon couplingconstant, which introduces the important polaronic parameter,namely the local Huang-Rhys factor S = A . (22)The final term is the sum of the elastic and kinetic energiesof the harmonic oscillator, where ω vib and K vib are the angu-lar frequency and force constant of the oscillator, respectively.The Frenkel-Holstein model is another example of a coarse-grained Hamiltonian which, in addition to coarse-graining theexciton motion, assumes that the atomistic motion of the car-bon nuclei can be replaced by appropriate local normal modes.Exciton-nuclear dynamics is often modeled via the Ehren-fest approximation, which treats the nuclear coordinates asclassical variables moving in a mean field determined by theexciton. However, as described in Section II, the Ehrenfestapproximation fails to correctly describe ultrafast dynami-cal processes. A correct description of the coupled exciton-nuclear dynamics therefore requires a full quantum mechani-cal treatment of the system. This is achieved by introducingthe harmonic oscillator raising and lowering operators, ˆ b † n andˆ b n , for the normal modes i.e., ˜ Q n → ˆ˜ Q n = ( ˆ b † n + ˆ b n ) / √ P n → ˆ˜ P n = i ( ˆ b † n − ˆ b n ) √
2. The time evolution of the quan-tum system can then conveniently be simulated via the TEBDmethod, as briefly described in Section II.Since the photoexcited system has a different electronicbond order than the ground state, an instantaneous force isestablished on the nuclei. As described in Section III C, thisforce creates an exciton-polaron, whose spatial size is quanti-fied by the exciton-phonon correlation function C ex-ph n ( t ) ∝ ∑ m (cid:104) ˆ N m ˆ˜ Q m + n (cid:105) . (23) C ex-ph n correlates the local phonon displacement, Q , with theinstantaneous exciton density, N , n monomers away. C ex-ph n ( t ) ,illustrated in Fig. 5, shows that the exciton-polaron is es-tablished within 10 fs (i.e., within half the period of a C-C bond vibration) of photoexcitation. The temporal oscilla-tions, determined by the C-C bond vibrations, are damped asenergy is dissipated into the vibrational degrees of freedom,which acts as a heat bath for the exciton. The exciton-phononspatial correlations decay exponentially, extending over ca.10 monomers. This short range correlation occurs because FIG. 5. The time-dependence of the exciton-phonon correlationfunction, Eq. (23), after photoexcitation at time t =
0. It fits the form C ex-ph n = C exp ( − n / ξ ) as t → ∞ , where ξ ∼ n is a monomerindex. The vibrational period is 20 fs. the C-C bond can respond relatively quickly to the exciton’smotion. The ultrafast establishment of quantum mechanically corre-lated exciton-phonon motion causes an ultrafast decay of off-diagonal-long-range-order (ODLRO) in the exciton site-basisdensity matrix. This is quantified via C coh n ( t ) = ∑ m | ρ m , m + n | , (24)where ρ m , m (cid:48) is the exciton reduced density matrix obtainedby tracing over the vibrational degrees of freedom. C coh n ( t ) is displayed in Fig. 6, showing that ODLRO is lost within 10fs. The loss of ODLRO is further quantified by the coherencenumber, defined by N coh = ∑ n C coh n , (25)and shown in the inset of Fig. 6. Again, N coh decays to ca. 10monomers in ca. 10 fs, reflecting the localization of excitoncoherence resulting from the short range exciton-phonon cor-relations. As discussed in Section IV E, the loss of ODLROleads to ultrafast fluorescence depolarization .We emphasise that the prediction of an electron-polaronwith short range correlations is a consequence of treating thephonons quantum mechanically, while the decay of exciton-site coherences is a consequence of the exciton and phononsbeing quantum mechanically entangled. Neither of these pre-dictions are possible within the Ehrenfest approximation. B. Role of system-environment interactions
For an exciton to dissipate energy it must first couple to fastinternal degrees of freedom (as described in the last section)and then these degrees of freedom must couple to the environ-ment to expell heat. For a low-energy exciton (i.e., a LEGS)this process will cause adiabatic relaxation on a single poten-tial energy surface, forming a VRS . As shown in Fig.
FIG. 6. The time dependence of the exciton coherence correlationfunction, C coh n , Eq. (24). The time dependence of the associated co-herence number, N coh (Eq. (25)), is shown in the inset. N coh decayswithin 10 fs, i.e., within half a vibrational period. Reproduced fromJ. Chem. Phys. , 034901 (2018) with the permission of AIP pub-lishing. Dissipation of energy from an open quantum system arisingfrom system-environment coupling is commonly described bya Lindblad master equation ∂ ˆ ρ∂ t = − i ¯ h (cid:2) ˆ H , ˆ ρ (cid:3) − γ ∑ n (cid:0) ˆ L † n ˆ L n ˆ ρ + ˆ ρ ˆ L † n ˆ L n − L n ˆ ρ ˆ L † n (cid:1) , (26)where ˆ L † n and ˆ L n are the Linblad operators, and ˆ ρ is the systemdensity operator. In practice, a direct solution of the Lindbladmaster equation is usually prohibitively expensive, as the sizeof Liouville space scales as the square of the size of the as-sociated Hilbert space. Instead, Hilbert space scaling can bemaintained by performing ensemble averages over quantumtrajectories (evaluated via the TEBD method), where the ac-tion of the Linblad dissipator is modeled by quantum jumps. In this section we assume that the C-C bond vibrations cou-ple directly with the environment , in which case the Lin-blad operators are the associated raising and lowering opera-tors (i.e., ˆ L n ≡ ˆ b n , introduced in the last section). In addition,ˆ H = ˆ H FH + γ ¯ h ∑ n (cid:16) ˆ˜ Q n ˆ˜ P n + ˆ˜ P n ˆ˜ Q n (cid:17) . (27)(In Section V we discuss coupling of the torsional modes withthe environment .)The ultrafast localization of exciton ODLRO (or exciton-site decoherence) described in Section IV A occurs via thecoupling of the exciton to internal degrees of freedom, namelythe C-C bond vibrations. We showed in Section III C (see Fig.2(a)) that this coupling does not cause exciton density local-ization. However, dissipation of energy to the environmentcauses an exciton in a higher energy QEES to relax onto alower energy LEGS (i.e., onto a chromophore) and thus theexciton density becomes localized. FIG. 7. The time dependence of the exciton localization correlationfunction, C loc n (Eq. (28)), for an initial high-energy QEES. The mainfigure corresponds to the time evolution with the dissipation time T = γ − =
100 fs. The time dependence of the exciton density lo-calization number, N loc (Eq. (29)), is given in the lower inset. Theupper inset corresponds to the time evolution without external dissi-pation showing that in this case exciton denisty localization does notoccur. Reproduced from J. Chem. Phys. , 034901 (2018) withthe permission of AIP publishing. The spatial extent of the exciton density, averaged over anensemble of quantum trajectories, is quantified by the corre-lation function , approximated by C loc n = ∑ m (cid:12)(cid:12) Ψ m Ψ ∗ m + n (cid:12)(cid:12) . (28)Fig. 7 shows the time dependence of C loc n with an external dis-sipation time T = γ − =
100 fs. The time scale for localizationis seen from the time dependence of the exciton localizationlength , N loc = ∑ n | n | C loc n / ∑ n C loc n , (29)which corresponds to the average distance between monomersfor which the exciton wavefunction overlap remains non-zero,and is given in the lower inset of Fig. 7. Evidently, the cou-pling to the environment - and specifically, the damping rate -controls the timescale for energy relaxation and exciton den-sity localization onto chromophores. In contrast, the upperinset to Fig. 7 shows an absence of localization without exter-nal dissipation, indicating that exciton density localization isan extrinsic process.Figure 7 is obtained by averaging over an ensemble of tra-jectories. To understand the physical process of localizationonto a chromophore, Fig. 8 illustrates the exciton density ofa single quantum trajectory for a photoexcited QEES (shown FIG. 8. The time dependence of the exciton density for a single tra-jectory of the quantum jump trajectory method. The discontinuity inthe density at ca. 20 fs is a ‘quantum jump’ caused by the stochasticapplication of a Lindblad jump operator. The dynamics were per-formed for an initial high energy QEES given in Fig. 2(b), showinglocalization onto the LEGSs (i.e., a chromophore) labeled j = , 034901 (2018) withthe permission of AIP publishing. in Fig. 2(b)). At a time ca. 20 fs a ‘quantum jump’ caused bythe stochastic application of a Lindblad jump operator causesthe exciton to localize onto the j = C. Role of slow bond rotations
By dissipating energy into the environment on sub-pstimescales, hot excitons relax into localized LEGSs, i.e., ontochromophores. The final intrachain relaxation and localiza-tion process now takes place, namely exciton-polaron forma-tion via coupling to the torsional degrees of freedom. For thisrelaxation to occur bond rotations must be allowed, whichmeans that this process is highly dependent on the precisechemical structure of the polymer and its environment.Assuming that bond rotations are not sterically hindered,their coupling to the excitons is conveniently modeled (viaEq. (7) and Eq. (12)) by supplementing the Frenkel-Holsteinmodel (i.e., Eq. (21)) by ˆ H rot = − N − ∑ n = B ( θ n ) × ( φ n + − φ n ) ˆ T n , n + + N ∑ n = (cid:0) K rot φ n + L n / I (cid:1) . (30)Here, φ is the angular displacement of a monomer from itsgroundstate equilibrium value and L is the associated angularmomentum of a monomer around its bridging bonds.The first term on the right-hand-side of Eq. (30) indicatesthat the change in the dihedral angle, ∆ θ n = ( φ n + − φ n ) , cou-0ples linearly to the bond-order operator, ˆ T n , n + , where B ( θ n ) = J SE sin 2 θ n (31)is the exciton-roton coupling constant and θ n is the ground-state dihedral angle for the n th bridging bond. The final termis the sum of the elastic and kinetic energies of the rotationalharmonic oscillator.The natural angular frequency of oscillation is ω rot =( K rot / I ) / , where K rot is the elastic constant of the rotationaloscillator and I is the moment of inertia, respectively. As dis-cussed in Section III C 1, K rot is larger for the bridging bond inthe excited state than the groundstate, because of the increasein bond order. Also notice that both the moment of inertia(and thus ω rot ) of a rotating monomer and its viscous damp-ing from a solvent are strongly dependent on the side groupsattached to it. As discussed in the next section, this obser-vation has important implications for whether the motion isunder or over damped and on its characterstic timescales.Unlike C-C bond vibrations, being over 10 times slower tor-sional oscillations can be treated classically . Furthermore,since we are now concerned with adiabatic relaxation on asingle potential energy surface, we may employ the Ehrenfestapproximation. Thus, using Eq. (30), the torque on each ringis Γ n = − ∂ (cid:104) ˆ H rot (cid:105) ∂ φ n = − K rot φ n + λ n (32)where we define λ n = B ( θ n − ) (cid:104) ˆ T n − , n (cid:105) − B ( θ n ) (cid:104) ˆ T n , n + (cid:105) . (33)Setting Γ n = φ eq n = λ n / K rot . φ n is subject to the Ehren-fest equations of motion, I d φ n dt = L n , (34)and dL n dt = Γ n − γ L n , (35)where the final term represents the damping of the rotationalmotion by the solvent.
1. A single torsional oscillator
Before considering a chain of torsional oscillators, it is in-structive to review the dynamics of a single, damped oscillatorsubject to both restoring and displacement forces. The equa-tion of motion for the angular displacement is d φ ( t ) dt = − ω ( φ ( t ) − φ eq ) − γ d φ ( t ) dt , (36)where φ eq = λ / K rot is proportional to the displacement force. In the underdamped regime , defined by γ < ω rot , φ ( t ) = φ eq ( − cos ( ω t ) exp ( − γ t / )) , (37)where ω = ( ω − γ / ) / . In this regime, the torsional angleundergoes damped oscillations with a period T = π / ω and adecay time τ = / γ .Conversely, in the overdamped regime , defined by γ > ω rot , φ ( t ) = φ eq (cid:18) − β ( γ exp ( − γ t / ) − γ exp ( − γ t / )) (cid:19) , (38)where γ = γ + β , γ = γ − β and β = ( γ / − ω ) / .Now, the torsional angle undergoes damped biexponential de-cay with the decay times τ = / γ and τ = / γ . In thelimit of strong damping, i.e., γ (cid:29) ω rot , there is a fast re-laxation time τ = / γ = τ / τ = γ / ω (cid:29) τ . In this limit, as the slow relaxation domi-nates at long times, the torsional angle approaches equilibriumwith an effective mono-exponential decay.For a polymer without alkyl side groups, e.g., PPP and PPV, ω rot ∼ γ ∼ s − and are thus in the underdamped regimewith sub-ps relaxation. However, polymers with side groups,e.g., P3HT, MEH-PPV and PFO, have a rotational frequencyup to ten times smaller and a larger damping rate, and are thusin the overdamped regime .
2. A chain of torsional oscillators
An exciton delocalized along a polymer chain in a chro-mophore couples to multiple rotational oscillators resultingin collective oscillator dynamics. Eq. (31) and Eq. (33) in-dicate that torsional relaxation only occurs if the monomersare in a staggered arrangement in their groundstate, i.e., θ n = ( − ) n θ . In this case the torque acts to planarize thechain. Furthermore, since the torsional motion is slow, theself-trapped exciton-polaron thus formed is ‘heavy’ and in theunder-damped regime becomes self-localized on a timescaleof a single torsional period, i.e., 200 −
600 fs. In this limitthe relaxed staggered bond angle displacement mirrors theexciton density. Thus, the exciton is localized precisely asfor a ‘classical’ Landau polaron and is spread over ∼ .The time-evolution of the staggered angular displacement, (cid:104) φ n (cid:105) × ( − ) n , is shown in Fig. 9 illustrating that these dis-placements reach their equilibrated values after two torsionalperiods (i.e., t (cid:38)
400 fs). The inset also displays the time-evolution of the exciton density, (cid:104) N n (cid:105) , showing exciton den-sity localization after a single torsional period ( ∼
200 fs).So far we have described how exciton coupling to tor-sional modes causes a spatially varying planarization of themonomers that acts as a one-dimensional potential which self-localizes the exciton. The exciton ‘digs a hole for itself’,forming an exciton-polaron . Some researchers , however,argue that torsional relaxation causes an exciton to become1 FIG. 9. The time-evolution of the staggered angular displacement, (cid:104) φ n (cid:105) × ( − ) n . The change of dihedral angle is ∆ θ n = ( φ n + − φ n ) , showinglocal planarization for a PPP chain of 21 monomers. The inset displays the time-evolution of the exciton density, (cid:104) N n (cid:105) , showing excitondensity localization after a single torsional period ( ∼
200 fs). In the long-time limit (i.e., t (cid:38)
400 fs) (cid:104) φ n (cid:105) ∝ (cid:104) N n (cid:105) × ( − ) n , illustrating classical(Landau) polaron formation. Reproduced from J. Chem. Phys. , 214107 (2018) with the permission of AIP publishing. more delocalized . A mechanism that can cause exciton delo-calization occurs if the disorder-induced localization length isshorter than the intrinsic exciton-polaron size. Then, in thiscase for freely rotating monomers, the stiffer elastic poten-tial in the excited state causes a decrease both in the varianceof the dihedral angular distribution, σ θ = k B T / K rot , and themean dihedral angle, θ . This, in turn, means that the exci-ton band width, | J | , increases and the diagonal disorder , σ J = J SE σ θ sin 2 θ , decreases. Hence, the disorder-inducedlocalization, L loc ∼ ( | J | / σ J ) / , increases (see Section III B). D. Summary
The conclusions that we draw from the previous three sec-tions are that a band edge excitation (i.e., a LEGS, which isan exciton spanning a single chromophore) undergoes ultra-fast exciton site decoherence via its coupling to fast C-C bondstretches. It subsequently couples to slow torsional modescausing planarization and exciton density localization on thechromophore. A hot exciton (i.e., a QEES) also undergoesultrafast exciton site decoherence. However, exciton densitylocalization within a chromophore only occurs after localiza-tion onto the chromophore via a stochastic interaction with theenvironment.
E. Time resolved fluorescence anisotropy
For general polymer conformations, the loss of ODLRO (orthe localization of the exciton coherence function) causes areduction and rotation of the transition dipole moment. Therotation is quantified by the fluorescence anisotropy, definedby r = I (cid:107) − I ⊥ I (cid:107) + I ⊥ , (39)where I (cid:107) and I ⊥ are the intensities of the fluorescence radiationpolarised parallel and perpendicular to the incident radiation,respectively.For an arbitrary state of a quantum system, | Ψ (cid:105) , the inte-grated fluorescence intensity polarised along the x -axis is re-lated to the x component of the transition dipole operator, ˆ µ x ,by I x ∝ ∑ v |(cid:104) Ψ | ˆ µ x | GS , v (cid:105)| , (40)where | GS , v (cid:105) corresponds to the system in the ground elec-tronic state, with the nuclear degrees of freedom in the statecharacterised by the quantum number v .The averaged fluorescence anisotropy is defined by (cid:104) r ( t ) (cid:105) = . × ∑ i I i ( t ) r i ( t ) ∑ i I i ( t ) , (41)2where I i ( t ) is the total fluorescence intensity and r i ( t ) is thefluorescence anisotropy, associated with a particular confor-mation i at time t . The factor of 0.4 is included on the as-sumption that the polymers are oriented uniformly in the bulkmaterial. Fig. 10 shows the simulated (cid:104) r ( t ) (cid:105) for both a highenergy QEES and a low energy LEGS for an ensemble of con-formationally disordered polymers. FIG. 10. The time dependence of the fluorescence anisotropy, (cid:104) r ( t ) (cid:105) ,for two initial Frenkel excitons coupled to C-C bond stretches. Thered curve corresponds to an initial LEGS, while the blue curve cor-responds to a QEES. Reproduced from J. Chem. Phys. , 034901(2018) with the permission of AIP publishing.FIG. 11. The experimental time dependence of the fluorescenceanisotropy, (cid:104) R ( t ) (cid:105) , in polythiophene in solution. (cid:104) R ( t ) (cid:105) has decayedfrom 0 . ∼ .
25 within 10 fs, consistent with the theoretical pre-dictions shown in Fig. 10. Subsequent fluorescence depolarizationis caused by slower torsional relaxation on timescales of 1 −
10 psfollowed by possible conformational changes . Reproduced from J.Phys. Chem. C , 15404 (2007) with the permission of ACS pub-lishing. It is instructive to express Eq. (40) as I x ∝ ∑ m , n s xm s xn ρ mn , (42)where s xm is the x -component of the unit vector for the m thmonomer and ρ mn is the exciton reduced density matrix. Then, using Eq. (24), Eq. (25), and Eq. (42), we observe thatthe emission intensity, I x , is related to the coherence length, N coh . Thus, not surprisingly, the dynamics of (cid:104) r ( t ) (cid:105) resem-bles that of N coh ( t ) shown in Fig. 6. In particular, we observea loss of fluorescence anisotropy within 10 fs, mirroring thereduction of N coh in the same time. Furthermore, since thereis greater exciton coherence localization for the QEES thanfor the LEGS, the former exhibits a greater loss of anisotropy.This predicted loss of fluorescence anisotropy within 10 fs hasbeen observed experimentally, as shown in Fig. 11. Slowersub-ps decay of anisotropy occurs because of exciton densitylocalization via coupling to torsional modes. V. INTRACHAIN EXCITON MOTION
The last section described the relaxation and localizationof higher energy excited states onto chromophores, and thesubsequent torsional relaxation and localization on the chro-mophore. We now consider the relaxation and dynamics ofthese relaxed excitons caused by the stochastic torsional fluc-tuations experienced by a polymer in a solvent.Environmentally-induced intrachain exciton relaxation inpoly(phenylene ethynylene) was modeled by Albu andYaron using the Frenkel exciton model supplemented by thetorsional degrees of freedom, i.e., ˆ H = ˆ H F + ˆ H rot (given by Eq.(2) and Eq. (30), respectively). Fast vibrational modes wereneglected because although they cause self-trapping, they donot cause self-localization, and these modes can be assumed torespond instantaneously to the torsional modes. The polymer-solvent interactions were modeled by the Langevin equation.For chains longer than the exciton localization length theexcited-state relaxation showed biexponential behavior witha shorter relaxation time of a few ps and a longer relaxationtime of tens of ps.After photoexcitation of the n = et al. reported torsional relaxationon sub-100 fs timescales. Since this timescale is faster thanthe natural rotational period of an undamped monomer, theyascribed it to the electronic energy being rapidly converted tokinetic energy via nonadiabatic transitions. They argue thatthis is analogous to inertial solvent reorganization.Tozer and Barford using the same model as Albu andYaron to model intrachain exciton motion in PPP where theexciton dynamics were simulated on the assumption that attime t + δ t the new exciton target state is the eigenstate ofˆ H ( t + δ t ) with the largest overlap with the previous targetstate at time t . A more sophisticated simulation of exciton motion inpoly(p-phenylene vinylene) and oligothiophenes chains wasperformed by Burghardt and coworkers where high-frequency C-C bond stretches were also included, the sol-vent was modeled by a set of harmonic oscillators with anOhmic spectral density, and the system was evolved via themultilayer-MCTDH method. Their results, however, are inquantitative agreement with those of Tozer and Barford in the‘low-temperature’ limit (discussed in Section V C), namelyactivationless, linearly temperature-dependent exciton diffu-3sion with exciton diffusion coefficients larger, but close to ex-perimental values.The Brownian forces excerted by the solvent on the poly-mer monomers have two consequences. First, as alreadynoted in Section III B, the instantaneous spatial dihedral an-gle fluctuations Anderson localize the Frenkel center-of-masswavefunction. Second, the temporal dihedral angle fluctua-tions cause the exciton to migrate via two distinct transportprocesses. At low temperatures there is small-displacement adiabaticmotion of the exciton-polaron as a whole along the polymerchain, which we will characterize as a ‘crawling’ motion.At higher temperatures the torsional modes fluctuate enoughto cause the exciton to be thermally excited out of the self-localized polaron state into a more delocalized LEGS or quasi-band QEES. While in this more delocalized state, the excitonmomentarily exhibits quasi-band ballistic transport, before thewavefunction ‘collapses’ into an exciton-polaron in a differentregion of the polymer chain. We will characterize this large-scale displacement as a non-adiabatic ‘skipping’ motion.Before describing the details of these types of motion, wefirst describe a model of solvent dynamics and consider againexciton-polaron formation in a polymer subject to Brownianfluctuations.
A. Solvent dynamics
If the solvent molecules are subject to spatially and tempo-rally uncorrelated Brownian fluctuations, then the monomerrotational dynamics are controlled by the Langevin equation dL n ( t ) dt = Γ n ( t ) + R n ( t ) − γ L n ( t ) , (43)where Γ n ( t ) is the systematic torque given by Eq. (32). R n ( t ) is the stochastic torque on the monomer due to the randomfluctuations in the solvent and γ is the friction coefficient forthe specific solvent. From the fluctuation-dissipation theorem,the distribution of random torques is given by (cid:104) R m ( t ) R n ( ) (cid:105) = I γ k B T δ mn δ ( t ) , (44)which are typically sampled from a Gaussian distribution witha standard deviation of σ R = ( I γ k B T ) . As a consequence ofthese Brownian fluctuations the monomer rotations are char-acterized by the autocorrelation function (cid:104) δ φ ( t ) δ φ ( ) (cid:105) = (cid:104) δ φ (cid:105) (cid:18) cos ( ω rot t ) + (cid:18) γ ω rot (cid:19) sin ( ω rot t ) (cid:19) exp ( − γ t / ) , (45)where (cid:104) δ φ (cid:105) = k B T / K rot , K rot is the stiffness and ω rot = (cid:112) K rot / I is the angular frequency of the torsional mode. B. Polaron formation
As we saw in Section IV C, at zero temperature torsionalmodes couple to the exciton, forming an exciton-polaron. At
FIG. 12. The exciton localization length as a function of temperaturefor the ‘free’ (i.e., ‘untrapped’) exciton (red circles) and exciton-polaron (i.e., ‘self-trapped’) (black squares). The untrapped exci-ton localization length obeys L freeloc ∝ T − / . The lengths coincidewhen k B T ∼ the exciton-polaron binding energy. Reproduced fromJ. Chem. Phys. , 084102 (2015) with the permission of AIP pub-lishing. finite temperatures, however, a combination of factors affectthe localization of the exciton. First, the exciton will stillattempt to form a polaron. However, the thermally inducedfluctuations in the torsional angles will affect the size of thisexciton-polaron, as there is a non-negligible probability thatthe exciton will be excited out of its polaron potential well intoa more delocalized state at high enough temperatures. Second,the exciton states will be Anderson localized by the instanta-neous torsional disorder.Fig. 12 shows how the average localization length varieswith temperature both with and without coupling between theexciton and the torsional modes (i.e., ‘self-trapped’ and ‘free’exciton, respectively). As described in Section III B, the local-ization length for the ‘free’ exciton is determined by Ander-son localization. For small angular displacements from equi-librium a Gaussian distribution of dihedral angles implies aGaussian distribution of exciton transfer integrals. Then, asconfirmed by the simulation results shown in Fig. 12, fromsingle-parameter scaling theory, L freeloc ∝ σ − / θ = (cid:104) δ θ (cid:105) − / ∝ T − / .In contrast, the localization length of the ‘self-trapped’ ex-citon slowly increases with temperature because of the ther-mal excitation of the exciton from the self-localized polaronto a more delocalized LEGS or QEES. The two values co-incide when k B T equals the exciton-polaron binding energy(i.e., T ∼ C. Adiabatic ‘crawling’ motion
At low temperatures ( (cid:46)
100 K) the exciton has only a smallamount of thermal energy, and not enough to regularly breakfree from its polaronic torsional distortions. Thus, the exciton-polaron migrates quasi-adiabatically and diffusively as a sin-gle unit. This is a collective motion of the exciton and the4
FIG. 13. The intrachain mean-square-distance traveled by anexciton-polaron at low temperatures caused by stochastic torsionalfluctuations. The motion is diffusive, as shown by the mean-square-distance increasing linearly with time: (cid:104) L (cid:105) = D A ( T ) t . The gra-dients satisfy the Einstein-Smoluchowski equation, D A ( T ) = µ k B T .Reproduced from J. Chem. Phys. , 084102 (2015) with the per-mission of AIP publishing. torsional degrees of freedom, as the torsional planarizationaccompanies the exciton. The random walk motion is illus-trated in Fig. 13, which shows that the mean-square-distancetraveled by the exciton-polaron is proportional to time, i.e., (cid:104) L (cid:105) = D A ( T ) t , where D A is the diffusion coefficient. Sincethe migration of the exciton-polaron is an activationless pro-cess, as the gradients of Fig. 13 indicate, at low temperatures itobeys the Einstein-Smoluchowski equation, D A ( T ) = µ k B T ,where µ is the mobility of the particle.The time taken for an exciton to diffuse a distance L alongthe chain is determined by the equation for a one-dimensionalrandom walk, i.e., τ D = (cid:104) L (cid:105) / D . As shown in Fig. 12, thetypical exciton-polaron localization length is ∼
12 monomersor ∼ −
30 ps at room tem-perature depending on the solvent friction coefficient, beingshorter at higher temperatures and smaller damping rates. Aswe show in Section V, these timescales are an order of mag-nitude shorter than Förster transfer times in the condensedphase.As the exciton-polaron migrates along the polymer chainit experiences a different potential energy landscape, so itsenergy will also fluctuate on a timescale ∼ τ D . Interest-ingly, these timescales are consistent with the longer timescalefound experimentally in biexponential fits of relaxation pro-cesses of polymers in solution (see Table I) and correspond tothe longer timescale simulated by Albu and Yaron in longerpolymers. D. Nonadiabatic ‘skipping’ motion
At higher temperatures, adiabatic ‘crawling’ migration ofthe exciton-polaron, as described above, still occurs. How- γ (s − ) T (K) D A (cm s − ) τ D (ps)10
300 6 . × −
100 2 . × −
300 2 . × −
100 9 . × −
300 6 . × −
100 2 . × − D A , and times, τ D , in PPP from ref . τ D is the time taken foran exciton to diffuse along a chromophore of linear size 6 nm in a sol-vent at temperature, T , with a damping rate γ . From simulations , τ D ∼ γ / / T . Hegger et al. obtained D A ∼ − cm s − in olig-othiophenes at 300 K and γ = × s − . ever, a second non-adiabatic mechanism for the dynamicsplays an important role. This mechanism involves the exciton-polaron being excited to a high enough energy by the thermalfluctuations to be excited out of the polaron potential well,resulting in a breakdown of the polaron and the exciton toenter an untrapped local exciton ground state (LEGS), or ahigher energy quasi-extended exciton state (QEES). Once inthis more delocalized state the exciton has quasi-band charac-teristics and travels quasi-ballistically.As described in Section IV, however, on a sub-ps timescalethe hot exciton will shed some of its excess kinetic energyand relax back into an exciton-polaron. As a result, the time-averaged exciton localization length calculations of Fig. 12show only a slight increase in localization length with increas-ing temperatures, as the majority of its lifetime is still spent inself-localized exciton-polarons.The requirement that the exciton is excited out of the po-laron potential well means that this process is activated. Thus,from a simple Fermi golden rule analysis it can be shownthat D NA ( T ) ∼ T / exp (cid:32) − ∆ Ek B T (cid:33) , (46)where ∆ E is the exciton-polaron binding energy. At 300 K D NA is approximately twice as large as D A and thus the overalldiffusion coefficient is considerably enhanced by this skippingmotion.The role of exciton transport in disordered one-dimensionalsystems via higher-energy quasi-band states has been dis-cussed in ref , where in that work phonons in the condensedphase environment induced non-adiabatic transitions. VI. INTERCHAIN EXCITON MOTION
The stochastic, torsionally-induced intrachain exciton dif-fusion in polymers in solution described in the last sectionis not expected to be the primary cause of exciton diffusionin polymers in the condensed phase. Instead, owing to re-stricted monomer rotations and the proximity of neighboringchains, exciton transfer is determined by Coulomb-induced,5Förster-like processes. Moreover, since dissipation rates aretypically − s − , whereas exciton transfer ratesare typically 10 − s − , exciton migration is an incoher-ent or diffusive process .Early models of condensed phase diffusion assumed thatthe donors and acceptors are point-dipoles whose energy dis-tribution is a Gaussian random variable . An advantage ofthese models is that they allow for analytical analysis, for ex-ample predicting how the diffusion length varies with disorderand temperature . They also reproduce some experimentalfeatures, such as the time-dependence of spectral diffusion. Adisadvantage, however, is that there is no quantitative link be-tween the model and actual polymer conformations and mor-phology.More recent approaches have attempted to make the linkbetween random polymer conformations and the energetic andspatial distributions of the donors and acceptors via the con-cept of extended chromophores and using transitiondensities to compute transfer integrals. However, the usualpractice has been to arbitrarily define chromophores via a min-imum threshold in the p -orbital overlaps, and then obtain adistribution of energies by assuming that the excitons delocal-ize freely on the chromophores thus defined as a ‘particle-in-a-box’.As discussed in Section III B, an unambiguous link betweenpolymer conformations and chromophores may be made bydefining chromophores via the spatial extent of local exci-ton ground states (LEGSs). Using this insight, a more real-istic first-principles model that accounts for polymer confor-mations can be developed . This is described in SectionVI A, while its predictions and comparisons to experimentalobservations are described in Section VI B. A. Modified Förster theory
The Förster exciton transfer rate from a donor (D) to anacceptor (A) has the general Golden rule form k DA = (cid:18) π ¯ h (cid:19) | J DA | (cid:90) D ( E ) A ( E ) dE , (47)where J DA is the Coulomb-induced donor-acceptor transfer in-tegral defined by Eq. (3). As we remarked in Section III A,the transition density vanishes for odd-parity singlet excitons;it also vanishes for all triplet excitons. D ( E ) and A ( E ) are the donor and acceptor spectral func-tions, respectively, defined by D ( E ) = ∑ v F D v δ ( E + E D v ) (48)and A ( E ) = ∑ v F A v δ ( E − E A v ) , (49)where F v is the effective Franck-Condon factor, defined inEq. (51). E A v = ( E A + v ¯ h ω vib ) is the excitation energy of theacceptor, while E D v = − ( E D − v ¯ h ω vib ) is the de-excitation en-ergy of the donor. The link between actual polymer conformations and a re-alistic model of exciton diffusion is made by realising thatthe donors and acceptors for exciton transfer are LEGSs (i.e.,chromophores). This assumption is based on the observationthat exciton transfer occurs at a much slower rate than stateinterconversion, so the donors are LEGSs, while the spectraloverlap between LEGSs and higher energy QEESs is small,so the acceptors are also LEGSs. Moreover, the energeticand spatial distribution of LEGSs is entirely determined bythe conformational and site disorder, as described in SectionIII B. Finally, polaronic effects are incorporated by an effec-tive Huang-Rhys factor for each chromophore and the Condonapproximation may be assumed as C-C vibrational modes donot cause exciton self-localization.Then, as proved rigorously in ref :1. J DA is evaluated by invoking the Condon approximationand using the line-dipole approximation J DA = (cid:18) πε r ε (cid:19) ∑ n ∈ Dn (cid:48) ∈ A κ nn (cid:48) R nn (cid:48) µ D Ψ D ( n ) µ A Ψ A ( n (cid:48) ) , (50)where Ψ ( n ) is the LEGS center-of-mass wavefunctionon monomer n determined from the disordered Frenkelexciton model (Eq. (2)). Since the spatial extent of Ψ ( n ) defines a chromophore, the sum over n and n (cid:48) is im-plicitly over monomers of a donor and acceptor chro-mophore, respectively. µ X is the transition dipole mo-ment of a single monomer of the donor ( X = D ) or ac-ceptor ( X = A ) chromophores (so µ X Ψ X ( n ) is the tran-sition dipole moment of monomer n as part of the chro-mophore). κ nn (cid:48) is the orientational factor, defined inEq. (5), and R nn (cid:48) is the separation of monomers on thedonor and acceptor chromophores. The line-dipole ap-proximation is valid when the monomer sizes are muchsmaller than their separation on the donor and acceptorchromophores; it becomes the point-dipole approxima-tion when the chromophore sizes are much smaller thantheir separation.2. The spectral functions describe ‘polaronic’ effects, bycontaining effective Franck-Condon factors which de-scribe the chromophores coupling to effective modeswith reduced Huang-Rhys parameters: F v = S v eff exp ( − S eff ) v ! , (51)where S eff = S / PN, S is the local Huang-Rhys parame-ter (defined by Eq. (22)) and PN = (cid:0) ∑ n | Ψ n | (cid:1) − is theparticipation number (or size) of the chromophore .3. Similarly, the 0 − E = ( E vert − E relax ) , where E vert is determined fromthe Frenkel exciton model and E relax = ¯ h ω S eff is the ef-fective reorganisation energy for the effective mode.6 FIG. 14. The density of states for absorbing LEGSs (solid line) andemitting trap states (dashed line) for an ensemble of PPV chains, witha Gaussian distribution of dihedral angles. (cid:104) θ (cid:105) = , σ θ = , and σ θ = . Reproduced from J. Chem. Phys. , 164103 (2014) withthe permission of AIP publishing. B. Condensed-phase exciton diffusion
We might attempt to anticipate the results of the simu-lation of exciton diffusion from the properties of the exci-ton transfer rate, k DA . When the chromophore size, L , ismuch smaller than the donor-acceptor separation, R , the point-dipole approximation is valid. In this limit k DA ∼ L / R and thus the hopping rate increases with increasing chro-mophore size. Conversely, when the chromophore size ismuch larger than the donor-acceptor separation, the line-dipole approximation predicts that for straight, parallel orcollinear chromophores k DA ∼ / ( LR ) and thus the hop-ping rate decreases with increasing chromophore size.In practice, Monte Carlo simulations assuming a statisticalmodel of polymer conformations find that the exciton hop-ping rate is essentially independent of disorder and hence ofchromophore size. This is presumably because neither the as-sumption of straight, parallel chromophores nor point dipolesare valid. These simulations also show that the average timetaken for the first exciton hop to occur after photoexcitation is ∼
10 ps, whereas the time intervals between hops just prior toradiative recombination is over 10 times longer, and indeedbecoming so long that a radiative transition is competitive.This increase in hopping time intervals occurs because as anexciton diffuses through the polymer system it continuouslylooses energy. Thus, the energetic condition for exciton trans-fer to occur, namely E A ≤ E D , becomes harder to satisfy andin general the spectral overlap between the donor and accep-tor decreases. As the excitons diffuse they eventually becometrapped in ‘emissive’ chromophores, from which they radi-ate. As shown in Fig. 14, these emissive chromophores oc-cupy the low energy tail of the LEGSs density of states. Theirquasi-Gaussian distribution explains spectral diffusion : atime-dependent change in the fluorescence energy, satisfying E ∝ − log t .Typically, the average hop distance is between 4 nm (forstrong disorder giving an average chromophore length of 8 FIG. 15. The calculated optical spectra of PPV assuming a statisticalmodel of random polymer conformations. Exciton migration priorto emission causes a red-shift in energy, a narrowing of the inhomo-geneous broadening, and a decrease in I / I . Reproduced from J.Chem. Phys. , 164103 (2014) with the permission of AIP pub-lishing. nm) to 6 nm (for weak disorder giving an average chro-mophore length of 30 nm). On average, an exciton only makesfour hops before radiating, and thus average diffusion lengthsare between ∼ −
12 nm, being longer for more ordered sys-tems. These theoretical predictions are consistent with exper-imental values obtained via various techniques (see Köh-ler and Bässler for further experimental references). Thediffusion length is remarkably insensitive to disorder, andfrom simulation satisfies L D ∼ L / ∼ σ − / ; a result that canbe explained by the spatial distribution of chromophores inrandomly coiled polymers .An interesting prediction of Anderson localization is thatfor the same mean dihedral angle lower energy chromophoresare shorter than higher energy chromophores. Now, as the in-tensity ratio of the vibronic peaks in the emission spectrum, I / I , is proportional to the chromophore size ,i.e., I I ∝ S eff = (cid:104) PN (cid:105) S , (52)spectral diffusion also implies that I / I reduces in time,as observed in time-resolved photoluminescence spectra inMEH-PPV (see Fig. 3 of ref ). According to simulations , I / I ∝ − log t .Some of the key features of exciton relaxation and dynam-ics described in this review are nicely encapsulated by Fig.15. This figure shows the simulated absorption to all ab-sorbing states, the fluorescence via emission from all LEGSs(which occurs in the absence of exciton migration), and thetime-integrated fluorescence following exciton migration andemission from ‘trap’ chromophores. We observe that:• The absorption spectrum and the emission spectrum as-suming no exciton migration are broadly a mirror im-age. However, the absorption is broader and has ahigh energy tail as absorption occurs to both LEGSs7and QEESs (as also shown in Fig. 3(b)), whereas, fromKasha’s law, emission occurs only from LEGSs follow-ing interconversion from QEESs.• The emission following exciton migration is red-shifted, because the emissive states are in the low-energy tail of the density of states (as shown in Fig. 14).• The inhomogeneous broadening of the post-migrationemission is narrowed, because the emissive states havea narrower density of states than LEGSs.• Similarly the intensity ratio, I / I , decreases, becauseon average emissive chromophores have shorter conju-gation lengths than LEGSs. VII. SUMMARY AND CONCLUDING REMARKS
We have reviewed the various exciton dynamical processesin conjugated polymers. In summary, they are:• Following photoexcitation, the initial dynamical pro-cess is the correlation of the exciton and phonons asso-ciated with high-frequency C-C bond vibrations. Thisquantum mechanical entanglement causes exciton-sitedecoherence, which is manifest as sub-10 fs fluores-cence depolarization (see Section IV A).• Next, the energy that is transferred from the excitonto the nuclei is dissipated into the environment on atimescale determined by the strength of the system-bath interactions. For a hot exciton (i.e., a QEES) thesystem-bath interactions cause the entangled exciton-nuclear wavefunction to stochastically ‘collapse’ into aparticular LEGS, causing the exciton density to be lo-calized on a ‘chromophore’ (see Section IV B).• The fate of an exciton on a chromophore is now stronglydependent on the polymer chemical structure and thetype of environment. For underdamped, freely rotat-ing monomers, the coupling of the exciton to the low-frequency torsional modes creates an exciton-polaron,with associated planarization and exciton-density local-ization (see Section IV C).• For a polymer in solution, stochastic torsional fluctu-ations also causes the exciton-polaron to diffuse alongthe polymer chain; a process known as environment-assisted quantum transport . The diffusion coefficientis linearly proportional to temperature (see Section V).• For a polymer in the condensed phase, the dominantpost-ps process is Förster resonant energy transfer andexciton diffusion. An exciton diffusing in the randomenergy landscape soon gets trapped in chromophoresoccupying the low-energy tail of the LEGSs density ofstates, exhibiting log t spectral diffusion. An excitontypically diffuses ∼
10 nm before radiative decay, withthe diffusion length weakly increasing with decreasingdisorder (see Section VI). In this review we have argued that theoretical modelingof exciton dynamics over multiple time and length scales isonly realistically possible by employing suitably parametrizedcoarse-grained exciton-phonon models. Moreover, to cor-rectly account for the ultrafast processes of exciton-site deco-herence and the relaxation of hot excitons onto chromophores,the exciton and vibrational modes must be treated on the samequantum mechanical basis and importantly the Ehrenfest ap-proximation must be abandoned. We have also repeatedlynoted that spatial and temporal disorder play a key role in ex-citon spectroscopy and dynamics; and it is for this reason thatexciton dynamics is conjugated polymers is essentially an in-coherent process.In a previous review we explained how spectroscopicsignatures are highly-dependent on polymer multiscale struc-tures, and how - in principle - good theoretical modeling of ex-citons and spectroscopy can be used as a tool to predict thesepolymer structures. This review builds on that prospectus bydescribing how time-resolved spectroscopy can be understoodvia a theoretical description of exciton dynamics coupled toinformation on polymer multiscale structures. Again, the re-verse proposition follows: time-resolved spectroscopy cou-pled to a theoretical description of exciton dynamics can beused to provide insights into polymer multiscale structures. ACKNOWLEDGMENTS
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