Exciton Dynamics in Conjugated Polymers
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Exciton Dynamics in Conjugated Polymers
William Barford1, 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. In
contrast, 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 correctly
arXiv:2102.06615v1 [physics.chem-ph] 12 Feb 2021
account for ultrafast processes, the exciton and phonon modes must be treated on the same quantum mechanical basis
and the Ehrenfest approximation must be abandoned. This further implies that sophisticated numerical techniques must
be employed to solve these models. This review describes our current theoretical understanding of exciton dynamics
in conjugated polymer systems. We begin by describing the energetic and spatial distribution of excitons in disordered
polymer systems, and define the crucial concept of a ‘chromophore’ in conjugated polymers. We also discuss the
role of exciton-nuclear coupling, emphasizing the distinction between ‘fast’ and ‘slow’ nuclear degrees of freedom in
determining ‘self-trapping’ and ‘self-localization’ of exciton-polarons. Next, we discuss ultrafast intrachain exciton
decoherence caused by exciton-phonon entanglement, which leads to fluorescence depolarization on the timescale of
10-fs. Interactions of the polymer with its environment causes the stochastic relaxation and localization of high-energy
delocalized 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 polymer
planarization), while on post-ps timescales stochastic torsional fluctuations cause exciton-polaron diffusion along the
polymer 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 results
and 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 simulate
exciton dynamics.
I. INTRODUCTION These include fluorescence depolarization1–4 , three-pulse
photon-echo5–8 and coherent electronic two-dimensional
The theoretical study of exciton dynamics in conjugated spectroscopy9 . Some of the timescales extracted from these
polymer systems is both a fascinating and complicated sub- experiments are listed in Table I; the purpose of this review is
ject. One reason for this is that characterizing excitonic to describe their associated physical processes.
states themselves is a challenging task: conjugated polymers As well as being of intrinsic interest, the experimental and
exhibit strong electron-electron interactions and electron- theoretical activities to understand exciton dynamics in conju-
nuclear coupling, and are subject to spatial and temporal dis- gated polymer systems are also motivated by the importance
order. Another reason is that exciton dynamics is charac- of this process in determining the efficiency of polymer elec-
terised by multiple (and often overlapping) time scales; it is tronic devices. In photovoltaic devices, large exciton diffusion
determined by both intrinsic processes (e.g., coupling to nu- lengths are necessary so that excitons can migrate efficiently
clear degrees of freedom and electrostatic interactions) and to regions where charge separation can occur. However, pre-
extrinsic processes (e.g., polymer-solvent interactions); and it cisely the opposite is required in light emitting devices, since
is both an intrachain and interchain process. Consequently, to diffusion leads to non-radiative quenching of the exciton.
make progress in both characterizing exciton states and cor- Perhaps one of the reasons for the failure to fully exploit
rectly describing their dynamics, simplified, but realistic mod- polymer electronic devices has been the difficulty in estab-
els are needed. Moreover, as even these simplified models de- lishing the structure-function relationships which allow the
scribe many quantized degrees of freedom, sophisticated nu- development of rational design strategies. An understand-
merical techniques are required to solve them. Luckily, fun- ing of the principles of exciton dynamics, relating this to
damental theoretical progress in developing numerical tech- multiscale polymer structures, and interpreting the associated
niques means that simplified one-dimensional models of con- spectroscopic signatures are all key ingredients to develop-
jugated polymers are now soluble to a high degree of accuracy. ing structure-function relationships. An earlier review ex-
In addition to the application of various theoretical tech- plored the connection between structure and spectroscopy14 .
niques to understand exciton dynamics, a wide range of time- In this review we describe our current understanding of the
resolved spectroscopic techniques have also been deployed. important 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-
a) Electronic mail: william.barford@chem.ox.ac.uk ble II.
2
Polymer State Timescales Citation
MEH-PPV Solution τ1 = 50 fs, τ2 = 1 − 2 ps Ref2
MEH-PPV Solution τ1 = 5 − 10 fs, τ2 = 100 − 200 fs Ref10
PDOPT Film τ = 0.5 − 4 ps Ref11
PDOPT Solution τ1 < 1 ps, τ2 = 15 − 23 ps Ref11
P3HT Film τ1 = 300 fs, τ2 = 2.5 ps, τ3 = 40 ps Ref11
P3HT Solution τ1 = 700 fs, τ2 = 6 ps, τ3 = 41 ps, τ4 = 530 ps Ref12
P3HT Solution τ1 = 60 − 200 fs, τ2 = 1 − 2 ps, τ3 = 14 − 20 ps Ref13
P3HT Solution τ1 . 100 fs, τ2 ∼ 1 − 10 ps Ref3
TABLE I. Some of the dynamical timescales observed in conjugated polymers whose associated physical processes are summarized in Table
II.
Process Consequences Timescale Section
Exciton-polaron self-trapping via coupling to fast C-C Exciton-site decoherence; ultrafast flu- ∼ 10 fs III A
bond vibrations. orescence depolarization.
Energy relaxation from high-energy quasi-extended ex- Stochastic exciton density localization ∼ 100 − 200 fs III B
citon states (QEESs) to low-energy local exciton ground onto chromophores.
states (LEGSs) via coupling to the environment.
Exciton-polaron self-localization via coupling to slow Exciton density localization on a ∼ 200 − 600 fs III C
bond rotations in the under-damped regime. chromophore; ultrafast fluorescence
depolarization.
Exciton-polaron self-localization via coupling to slow Exciton density localization on a ∼ 1 − 10 ps III C
bond rotations in the over-damped regime. chromophore; post-ps fluorescence
depolarization.
Stochastic torsional fluctuations inducing exciton Intrachain exciton diffusion and energy ∼ 3 − 30 ps IV
‘crawling’ and ‘skipping’ motion. fluctuations.
Interchromophore Förster resonant energy transfer. Interchromophore exciton diffusion; ∼ 10 − 100 ps V
post-ps spectral diffusion and fluores-
cence depolarization.
Radiative decay. ∼ 500 ps
TABLE II. The life and times of an exciton: Some of the key exciton dynamical processes, encompassing over four-orders of magnitude, that
occur in conjugated polymer systems.
The plan of this review is the following. We begin by II. A BRIEF CRITIQUE OF THEORETICAL TECHNIQUES
briefly describing some theoretical techniques for simulating
exciton dynamics and emphasize the failures of simple meth-
ods. As already mentioned, excitons themselves are fasci- A theoretical description of exciton dynamics in conjugated
nating quasiparticles, so before describing their dynamics, in polymers poses considerable challenges, as it requires a rig-
Section III we start by describing their stationary states. We orous treatment of electronic excited states and their cou-
stress the role of low-dimensionality, disorder and electron- pling to the nuclear degrees of freedom. Furthermore, con-
phonon coupling, and we discuss the fundamental concept of jugated polymers consist of thousands of atoms and tens of
a chromophore. Next, in Section IV, we describe the sub-ps thousands of electrons. Thus, as the Hilbert space grows
processes of intrachain exciton decoherence, relaxation and exponentially with the number of degrees of freedom, ap-
localization, which - starting from an arbitrary photoexcited proximate treatments of excitonic dynamics are therefore in-
state - results in an exciton forming a chromophore. We next evitable. There are two broad approaches to a theoretical treat-
turn to describe the exciton (and energy) transfer processes oc- ment. One approach is to construct ab initio Hamiltonians,
curring on post-ps timescales. First, in Section V, we describe with an exact as possible representation of the degrees of free-
the primarily adiabatic intrachain motion of excitons caused dom, and then to solve these Hamiltonians with various de-
by stochastic torsional fluctuations, and second, in Section VI, grees of accuracy. Another approach (albeit less common in
we describe nonadiabatic interchain exciton transfer. We con- theoretical chemistry) is to construct effective Hamiltonians
clude and address outstanding questions in Section VII. with fewer degrees of freedom, such as the Frenkel-Holstein
model described in Section IV. These effective Hamiltonians
might be parameterized via a direct mapping from ab initio
Hamiltonians (e.g., see Appendix H in ref15 , Appendix A in
ref16 and various papers by Burghardt and coworkers17,18 ) or
3
else semiempirically19 . A significant advantage of effective method, it is not limited by the representation of the PES. It
Hamiltonians over their ab initio counterparts is that they can can, however, only be applied to quantum systems described
be solved for larger systems over longer timescales and to a by one-dimensional lattice Hamiltonians29 . Luckily, as de-
higher level of accuracy. scribed in Section IV, such model Hamiltonians are readily
As the Ehrenfest method is a widely used approximation constructed to describe exciton dynamics in conjugated poly-
to study charge and exciton dynamics in conjugated poly- mers.
mers, we briefly explain this method and describe the impor-
tant ways in which it fails. (For a fuller treatment, see20,21 .)
The Ehrenfest method makes two key approximations. The III. EXCITONS IN CONJUGATED POLYMERS
first approximation is to treat the nuclei classically. This
means that nuclear quantum tunneling and zero-point energies
are neglected, and that exciton-polarons are not correctly de- Before discussing the dynamics of excitons, we begin by
scribed (see Section III C). The second assumption is that the describing exciton stationary states in static conjugated poly-
total wavefunction is a product of the electronic and nuclear mers.
wavefunctions. This means that there is no entanglement be-
tween the electrons and nuclei, and so the nuclei cannot cause
decoherence of the electronic degrees of freedom (see Section A. Two-particle model
IV A). A simple product wavefunction also implies that the
nuclei move in a mean field potential determined by the elec- An exciton is a Coulombically bound electron-hole pair
trons. This means that a splitting of the nuclear wave packet formed by the linear combination of electron-hole excitations
when passing through a conical intersection or an avoided (for further details see15,30,31 ). In a one-dimensional con-
crossing does not occur (see Section IV B), and that there is an jugated polymer an exciton is described by the two-particle
incorrect description of energy transfer between the electronic wavefunction, Φm j (r, R) = ψm (r)Ψ j (R).
and nuclear degrees of freedom (see Section V D). As will be Ψ j (R) is the center-of-mass wavefunction, which will be
discussed in the course of this review, these failures mean that discussed shortly. Before doing that, we first discuss the rela-
in general the Ehrenfest method is not a reliable one to treat tive wavefunction, ψm (r), which describes a particle bound
ultrafast excitonic dynamics in conjugated polymers. to a screened Coulomb potential, where r is the electron-
Various theoretical techniques have been proposed to rec- hole separation and m is the principal quantum number.
tify the failures of the Ehrenfest method; for example, the The electron and hole of an exciton in a one-dimensional
surface-hopping technique22,23 , while still keeping the nuclei semiconducting polymer are more strongly bound than in
classical, partially rectifies the failures at conical interactions. a three-dimensional inorganic semiconductor for two key
More sophisticated approaches, for example the MC-TDHF reasons.15,31 First, because of the low dielectric constant and
and TEBD methods, quantize the nuclear degrees of freedom relatively large electronic effective mass in π-conjugated sys-
and do not assume a product wavefunction. tems the effective Rydberg is typically 50 times larger than
For a given electronic potential energy surface (PES), for inorganic systems. Second, dimensionality plays a role:
the multiconfigurational-time dependent Hartree-Fock (MC- in particular, the one-dimensional Schrödinger equation for
TDHF) method24 is an (in principle) exact treatment of nu- the relative particle32,33 predicts a strongly bound state split-
clear wavepacket propagation, although in practice exponen- off from the Rydberg series. This state is the m = 1 Frenkel
tial scaling of the Hilbert space means that a truncation is re- (‘1Bu ’) exciton, with a binding energy of ∼ 1 eV and an
quired. In addition, this method is only as reliable as the rep- electron-hole wavefunction confined to a single monomer.
resentation of the PES. The first exciton in the ‘Rydberg’ series is the m = 2 charge-
In the time-evolving block decimation (TEBD) method25,26 transfer (‘2Ag ’) exciton.
a quantum state, |Ψi, is represented by a matrix product state
With the exception of donor-acceptor copolymers, conju-
(MPS)27 . Its time evolution is determined via
gated polymers are generally non-polar, which means that
|Ψ(t + δti = exp(−iĤδt/h̄)|Ψ(t)i, (1) each p-orbital has an average occupancy of one electron. This
implies an approximate electron-hole symmetry. Electron-
where Ĥ is the system Hamiltonian and the action of the hole symmetry has a number of consequences for the char-
evolution operator is performed via a Trotter decomposi- acter and properties of excitons. First, it means that the rela-
tion. Since the action of the evolution operator expands the tive wavefunction exhibits electron-hole parity, i.e., ψm (r) =
Hilbert space, |Ψi is subsequently compressed via a singular +ψm (−r) when m is odd and ψm (r) = −ψm (−r) when m is
value decomposition (SVD)28 . Importantly, this approach is even. Second, the transition density, hEX|N̂i |GSi, vanishes
‘numerically exact’ as long as the truncation parameter ex- for odd-parity (i.e., even m) excitons. This means that such
ceeds 2S , where S is the entanglement entropy, defined by excitons are not optically active, and importantly for dynam-
S = − ∑α ωα ln2 ωα and {ω} are the singular values obtained ical processes, their Förster exciton transfer rate (defined in
at the SVD. The TEBD method permits the electronic and nu- Section VI A) vanishes.
clear degrees of freedom to be treated as quantum variables Since Frenkel excitons are the primary photoexcited states
on an equal footing. It thus rectifies all of the failures of the of conjugated polymers, their dynamics is the subject of this
Ehrenfest method described above and, unlike the MC-TDHF review. Their delocalization along the polymer chain of N
4
monomers is described by the Frenkel Hamiltonian,
N N−1
ĤF = ∑ εn N̂n + ∑ Jn T̂n,n+1 , (2)
n=1 n=1
where n = (R/d) labels a monomer and d is the inter-
monomer separation. The energy to excite a Frenkel exciton
on monomer n is εn , where N̂n = |ni hn| is the Frenkel exciton
number operator.
In principle, excitons delocalize along the chain via two
mechanisms31,34 . 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
FIG. 1. The mapping of a polymer chain conformation to a coarse-
JDA = ∑ Vi j D hGS|N̂i |EXiD A hEX|N̂ j |GSiA . (3) grained linear site model. Each site corresponds to a moiety along
i∈D the polymer chain, with the connection between sites characterised
j∈A
by the torsional (or dihedral) angle, θ .
The sum is over sites i in the donor monomer and j in the ac-
ceptor monomer, and Vi j is the Coulomb interaction between
these sites. In the point-dipole approximation Eq. (3) becomes represents the hopping of the Frenkel exciton between
monomers n and n + 1. Evidently, JSE vanishes when θ = 0,
κmn µ02 but JDA will not. Therefore, even if JSE vanishes because of
JDA = , (4) negligible p-orbital overlap between neighboring monomers,
4πεr ε0 R3mn
singlet even-parity excitons can still retain phase coherence
where µ0 is the transition dipole moment of a single monomer over the ‘conjugation break’35 . This observation has impor-
and Rmn is the distance between the monomers m and n. κmn tant implications for the definition of chromophores, as dis-
is the orientational factor, cussed in Section III B.
Eq. (2) represents a ‘coarse-graining’ of the exciton degrees
κmn = r̂m · r̂n − 3(R̂mn · r̂m )(R̂mn · r̂n ), (5) of freedom. The key assumption is that we can replace the
where r̂m is a unit vector parallel to the dipole on monomer atomist detail of each monomer (or moiety) and replace it by a
m and R̂mn is a unit vector parallel to the vector joining ‘coarse-grained’ site, as illustrated in Fig. 1. All that remains
monomers m and n. For colinear monomers, the nearest is to describe how the Frenkel exciton delocalises along the
neighbor through space transfer integral is chain, which is controlled by the two sets of parameters, {ε}
and {J}. Since J is negative, a conjugated polymer is equiva-
2µ02 lent to a molecular J-aggregate.
JDA = − . (6) The eigenfunctions of ĤF are the center-of-mass wavefunc-
4πεr ε0 d 3
tions, Ψ j (n), where j is the associated quantum number. For
Second, for all excitons there is a super-exchange (or a linear, uniform polymer (i.e., εn ≡ ε0 and Jn ≡ J0 )
through bond) mechanism, whose origin lies in a virtual fluc- 1/2
N
tuation from a Frenkel exciton on a single monomer to a 2 π jn
Ψ j (n) = ∑ sin , (10)
charge-transfer exciton spanning two monomers back to a N +1 N +1
n=1
Frenkel exciton on a neighboring monomer. The energy
scale for this process, obtained from second order perturba- forming a band of states with energy
tion theory15 , is
πj
E j = ε0 + 2J0 cos . (11)
t(θ )2 N +1
JSE (θ ) ∝ − , (7)
∆E
The family of excitons with different j values corresponds to
where t(θ ) (defined in Eq. (12)) is proportional to the overlap the Frenkel exciton band with different center-of-mass mo-
of p-orbitals neighboring a bridging bond, i.e., t(φ ) ∝ cos θ menta. In emissive polymers the j = 1 Frenkel exciton is gen-
and θ is the torsional (or dihedral) angle between neighboring erally labeled the 11 Bu state.
monomers. ∆E is the difference in energy between a charge-
transfer and Frenkel exciton.
The total exciton transfer integral is thus B. Role of static disorder: local exciton ground states and
quasiextended exciton states
0
Jn = JDA + JSE cos2 θn . (8)
The bond-order operator, Polymers are rarely free from some kind of disorder and
thus the form of Eq. (10) is not valid for the center-of-mass
T̂n,n+1 = (|ni hn + 1| + |n + 1i hn|) , (9) wavefunction in realistic systems. Polymers in solution are
5
FIG. 3. (a) The energy density of states and (b) the optical absorp-
tion (neglecting the vibronic progression) of the manifold of Frenkel
excitons (where |σJ /J0 | = 0.1). The width of the LEGSs density of
states ∼ |J0 ||σJ /J0 |4/3 . Similarly, the width of the optical absorption
from both the LEGSs and all states ∼ |J0 ||σJ /J0 |4/3 . The band edge
for an ordered chain is at 2|J0 | (indicated by the dashed lines), so
LEGSs generally lie in the Lifshitz (or Urbach) tail of the density of
states, i.e., E < 2|J0 |.
tum particle and the constructive and destructive interference
it experiences as it scatters off a random potential. Malyshev
and Malyshev37,38 further observed that in one-dimensional
systems there are a class of states in the low energy tail of the
density of states that are superlocalized, named local exciton
ground states (LEGSs37–39 ). LEGSs are essentially nodeless,
non-overlapping wavefunctions that together spatially span
the entire chain. They are local ground states, because for the
individual parts of the chain that they span there are no lower
energy states. A consequence of the essentially nodeless qual-
ity of LEGSs is that the square of their transition dipole mo-
FIG. 2. (a) The density of three local exciton ground states (LEGSs, ment scales as their size39 . Thus, LEGSs define chromophores
dotted curves) and the three vibrationally relaxed states (VRSs, solid (or spectroscopic segments), namely the irreducible parts of a
curves) for one particular static conformation of a PPV polymer polymer chain that absorb and emit light. Fig. 2(a) illustrates
chain made up of 50 monomers. The exciton center-of-mass quan- the three LEGSs for a particular conformation of PPV with 50
tum number, j, for each state is also shown. (b) The exciton den- monomers.
sity of a quasiextended exciton state (QEES), with quantum number Some researchers claim that ‘conjugation-breaks’ (or more
j = 7. Reproduced from J. Chem. Phys. 148, 034901 (2018) with correctly, minimum thresholds in the pz -orbital overlap) de-
the permission of AIP publishing.
fine the boundaries of chromophores40 . In contrast, we sug-
gest that it is the disorder that determines the average chro-
mophore size, but ‘conjugation-breaks’ can ‘pin’ the chro-
necessarily conformationally disordered as a consequence of mophore boundaries. Thus, if the average distance between
thermal fluctuations (as described in Section V). Polymers in conjugation breaks is smaller than the chromophore size,
the condensed phase usually exhibit glassy, disordered con- chromophores will span conjugation breaks but they may also
formations as consequence of being quenched from solution. be separated by them. Conversely, if average distance between
Conformational disorder implies that the dihedral angles, {θ } conjugation breaks is larger than the chromophore size the
are disordered, which by virtue of Eq. (8) implies that the ex- chromophore boundaries are largely unaffected by the breaks.
citon transfer integrals are also disordered. The former scenario occurs in polymers with shallow tor-
As well as conformational disorder, polymers are also sub- sional potentials, e.g., polythiophene35 .
ject to chemical and environmental disorder (arising, for ex- Higher energy lying states are also localized, but are node-
ample, from density fluctuations). This type of disorder af- ful and generally spatially overlap a number of low-lying
fects the energy to excite a Frenkel exciton on a monomer (or LEGSs. These states are named quasiextended exciton states
coarse-grained site). (QEESs) and an example is illustrated in Fig. 2(b).
As first realized by Anderson36 , disorder localizes a quan- When the disorder is Gaussian distributed with a standard
tum particle (in our case, the exciton center-of-mass particle), deviation σ , single parameter scaling theory41 provides some
and determines their energetic and spatial distributions. The exact results about the spatial and energetic distribution of the
origin of this localization is the wave-like nature of a quan- exciton center-of-mass states:
6
1. The localization length Lloc ∼ (|J0 |/σ )2/3 at the band
edge and as Lloc ∼ (|J0 |/σ )4/3 at the band center.
2. As a consequence of exchange narrowing, the width
of the
√ density of states occupied by LEGSs scales as
σ / Lloc ∼ σ 4/3 . Similarly, the optical absorption is
inhomogeneously narrowed with a line width ∼ σ 4/3 .
3. The fraction of LEGSs scales as 1/Lloc ∼ σ 2/3 .
These points are illustrated in Fig. 3, which shows the Frenkel
exciton density of states and optical absorption for a particu-
lar value of disorder. Evidently, although LEGSs are a small
fraction of the total number of states, they dominate the opti-
cal absorption. FIG. 4. The π-bond order expectation values, hT̂ i, for (a) the ground
This section has described LEGSs (or chromophores) as state and (b) the excited state, showing the benzenoid-quinoid transi-
static objects defined by static disorder. However, as discussed tion. As Eq. (19) and Eq. (20) indicate, the larger bond order of the
in Section V, dynamically torsional fluctuations also render bridging bond in the excited state implies a smaller dihedral angle
the conformational disorder dynamic causing the LEGSs to and a stiffer torsional potential than the ground state.
evolve adiabatically. As a consequence, the chromophores
‘crawl’ along the polymer chain.
The coupling of the π-electrons to the nuclei changes these
equilibrium values and the elastic constants.
C. Role of electron-nuclear coupling: exciton-polarons To see this, we use the Hellmann-Feynman to determine the
force on the bond. The linear displacement force is
As well as disorder, another important process in determin- ∂E ∂ Ĥke
ing exciton dynamics and spectroscopy is the coupling of an f =− =−
∂r ∂r
exciton to nuclear degrees of freedom; in a conjugated poly- σ
mer these are fast C-C bond vibrations and slow monomer = αt(r) cos θ hT̂ i − Kvib (r − r0 ). (16)
rotations. In this section we briefly review the origin of this Thus, to first order in the change of bond length, δ r = (r −r0 ),
coupling and then discuss exciton-polarons. the equilibrium distortion is
σ
δ r = αt(r0 ) cos θ hT̂ i/Kvib , (17)
1. Origin of electron-nuclear coupling
which is negative because it is favorable to shorten the bond
to increase the electronic overlap.
When a nucleus moves, either by a linear displacement or Similarly, the torque around the bond is
by a rotation about a fixed point, there is a change in the elec-
tronic overlap between neighboring atomic orbitals. Assum- ∂E ∂ Ĥke
Γ=− =−
ing that neighboring p-orbitals lie in the same plane normal to ∂θ ∂θ
the bond with a relative twist angle of θ , the resonance inte- σ
= t(r) sin θ hT̂ i − Krot (θ − θ0 ) (18)
gral between a pair of orbitals separated by r is42
and the equilibrium change of bond angle, δ θ = (θ − θ0 ), is
t(θ ) = t(r) cos θ = β exp(−αr) cos θ , (12) σ
δ θ = t(r) sin θ0 hT̂ i/Krot , (19)
where t(r) < 0. The kinetic energy contribution to the Hamil- which is also negative, again because it is favorable to increase
tonian is the electronic overlap. Thus, the π-electron couplings act to
Ĥke = t(r) cos θ × T̂ , (13) planarize the chain.
The electron-nuclear coupling also changes the elastic con-
where the bond-order operator, T̂ , is defined in Eq. (9). Treat- stants. Assuming a harmonic potential, the new rotational
ing r and θ as dynamical variables, suppose that the σ - spring constant is
electrons of a conjugated molecule and steric hinderances pro-
∂ 2E
vide equilibrium values of r = r0 and θ = θ0 , with correspond- π
Krot =
ing elastic potentials of ∂θ2
σ
=−t(r0 ) cos θ0 hT̂ i + Krot (20)
1 σ
Vvib = Kvib (r − r0 )2 (14) π > K σ (because t(r ) < 0).
and thus Krot 0
2 rot
Interestingly, as shown in Fig. 4, because hT̂ iEX > hT̂ iGS
and for the bridging bond in phenyl-based systems, the torsional
1 σ angle is smaller and the potential is stiffer in the excited state
Vrot = Krot (θ − θ0 )2 . (15) (as a result of the benzenoid to quinoid distortion)43 .
2
7
2. Exciton-polarons local normal modes (e.g., vinyl-unit bond stretches or phenyl-
ring symmetric breathing modes) to a Frenkel exciton is con-
An exciton that couples to a set of harmonic oscillators, veniently described by the Frenkel-Holstein model19,47 ,
e.g., bond vibrations or torsional oscillations, becomes ‘self- N
h̄ωvib N
Q̃2n + P̃n2 .
trapped’. Self-trapping means that the coupling between the ĤFH = ĤF − Ah̄ωvib ∑ Q̃n N̂n + ∑
exciton and oscillators causes a local displacement of the os- n=1 2 n=1
cillator that is proportional to the local exciton density44–48 (21)
(as illustrated in the next section). Alternatively, it is said that
the exciton is dressed by a cloud of oscillators. Such a quasi- ĤF is the Frenkel Hamiltonian, defined in Eq. (2), while Q̃ =
particle is named an exciton-polaron. As there is no barrier (Kvib /h̄ωvib )1/2 Q and P̃ = (ωvib /h̄Kvib )1/2 P are the dimen-
to self-trapping in one-dimensional systems49 , there is always sionless displacement and momentum of the normal mode.
an associated relaxation energy. The second term on the right-hand-side of Eq. (21) indicates
that the normal mode couples linearly to the local exciton
If the exciton and oscillators are all treated quantum me-
density53 . A is the dimensionless exciton-phonon coupling
chanically, then in a translationally invariant system the
constant, which introduces the important polaronic parameter,
exciton-polaron forms a Bloch state and is not localized.
namely the local Huang-Rhys factor
However, if the oscillators are treated classically, the non-
linear feedback induced by the exciton-oscillator coupling A2
self-localizes the exciton-polaron and ‘spontaneously’ breaks S= . (22)
2
the translational symmetry. This is a self-localized (or auto-
localized) ‘Landau polaron’.50,51 Notice that self-trapping is The final term is the sum of the elastic and kinetic energies
a necessary but not sufficient condition for self-localization. of the harmonic oscillator, where ωvib and Kvib are the angu-
Self-localization always occurs in the limit of vanishing oscil- lar frequency and force constant of the oscillator, respectively.
lator frequency (i.e., the adiabatic or classical limit) and van- The Frenkel-Holstein model is another example of a coarse-
ishing disorder.52 grained Hamiltonian which, in addition to coarse-graining the
Whether or not an exciton-polaron is self-localized in prac- exciton motion, assumes that the atomistic motion of the car-
tice, however, depends on the strength of the disorder and the bon nuclei can be replaced by appropriate local normal modes.
vibrational frequency of the oscillators. Qualitatively, an ex- Exciton-nuclear dynamics is often modeled via the Ehren-
citon coupling to fast oscillators (e.g., C-C bond vibrations) fest approximation, which treats the nuclear coordinates as
forms an exciton-polaron with an effective mass only slightly classical variables moving in a mean field determined by the
larger than a bare exciton52 . For realistic values of disorder, exciton. However, as described in Section II, the Ehrenfest
such an exciton-polaron is not self-localized. This is illus- approximation fails to correctly describe ultrafast dynami-
trated in Fig. 2(a), which shows the three lowest solutions cal processes. A correct description of the coupled exciton-
of the Frenkel-Holstein model (described in Section IV A), nuclear dynamics therefore requires a full quantum mechani-
known as vibrationally relaxed states (VRSs). As we see, the cal treatment of the system. This is achieved by introducing
density of the VRSs mirrors that of the Anderson-localized the harmonic oscillator raising and lowering operators, b̂†n and
√
LEGSs. Conversely, an exciton coupling to slow oscilla- b̂n , for the normal modes i.e., Q̃n → Q̃ˆ n = (b̂†n + b̂n )/ 2 and
√
tors (e.g., bridging-bond rotations) forms an exciton-polaron P̃n → P̃ˆn = i(b̂†n − b̂n ) 2. The time evolution of the quan-
with a large effective mass. Such an exction-polaron is self- tum system can then conveniently be simulated via the TEBD
localized (as described in Section IV C and shown in Fig. 6). method, as briefly described in Section II.
Since the photoexcited system has a different electronic
bond order than the ground state, an instantaneous force is
IV. INTRACHAIN DECOHERENCE, RELAXATION AND established on the nuclei. As described in Section III C, this
LOCALIZATION force creates an exciton-polaron, whose spatial size is quanti-
fied by the exciton-phonon correlation function54
Having qualitatively described the stationary states of exci-
tons in conjugated polymers, we now turn to a discussion of Cnex-ph (t) ∝ ∑hN̂m Q̃ˆ m+n i. (23)
m
exciton dynamics.
ex-ph
Cn correlates the local phonon displacement, Q, with the
ex-ph
instantaneous exciton density, N, n monomers away. Cn (t),
A. Role of fast C-C bond vibrations illustrated in Fig. 5, shows that the exciton-polaron is es-
tablished within 10 fs (i.e., within half the period of a C-
After photoexcitation or charge combination after injection, C bond vibration) of photoexcitation. The temporal oscilla-
the fastest process is the coupling of the exciton to C-C bond tions, determined by the C-C bond vibrations, are damped as
stretches. We now describe the resulting exciton-polaron for- energy is dissipated into the vibrational degrees of freedom,
mation and the loss of exciton-site coherence. which acts as a heat bath for the exciton. The exciton-phonon
As we saw in Section III C, bond distortions couple to elec- spatial correlations decay exponentially, extending over ca.
trons. Using Eq. (13), it can be shown19 that the coupling of 10 monomers. This short range correlation occurs because
8
FIG. 5. The time-dependence of the exciton-phonon correlation
function, Eq. (23), after photoexcitation at time t = 0. It fits the form
ex-ph
Cn = C0 exp(−n/ξ ) as t → ∞, where ξ ∼ 10. n is a monomer
index. The vibrational period is 20 fs. FIG. 6. The time dependence of the exciton coherence correlation
function, Cncoh , Eq. (24). The time dependence of the associated co-
herence number, N coh (Eq. (25)), is shown in the inset. N coh decays
the C-C bond can respond relatively quickly to the exciton’s within 10 fs, i.e., within half a vibrational period. Reproduced from
motion.55 J. Chem. Phys. 148, 034901 (2018) with the permission of AIP pub-
The ultrafast establishment of quantum mechanically corre- lishing.
lated exciton-phonon motion causes an ultrafast decay of off-
diagonal-long-range-order (ODLRO) in the exciton site-basis
density matrix. This is quantified via56,57 3(a), however, for a kinetically hot exciton (i.e., a QEES) this
relaxation is through a dense manifold of states and is neces-
Cncoh (t) = ∑ |ρm,m+n | , (24) sarily a nonadiabatic interconversion between different poten-
m tial energy surfaces. As already stated in Section II, the Ehren-
fest approximation fails to correctly describe this process.62
where ρm,m0 is the exciton reduced density matrix obtained
Dissipation of energy from an open quantum system arising
by tracing over the vibrational degrees of freedom. Cncoh (t)
from system-environment coupling is commonly described by
is displayed in Fig. 6, showing that ODLRO is lost within 10
a Lindblad master equation63
fs. The loss of ODLRO is further quantified by the coherence
number, defined by ∂ ρ̂ i γ
= − Ĥ, ρ̂ − ∑ L̂n† L̂n ρ̂ + ρ̂ L̂n† L̂n − 2L̂n ρ̂ L̂n† , (26)
N coh
=∑ Cncoh , (25) ∂t h̄ 2 n
n
where L̂n† and L̂n are the Linblad operators, and ρ̂ is the system
and shown in the inset of Fig. 6. Again, N coh decays to ca. 10 density operator. In practice, a direct solution of the Lindblad
monomers in ca. 10 fs, reflecting the localization of exciton master equation is usually prohibitively expensive, as the size
coherence resulting from the short range exciton-phonon cor- of Liouville space scales as the square of the size of the as-
relations. As discussed in Section IV E, the loss of ODLRO sociated Hilbert space. Instead, Hilbert space scaling can be
leads to ultrafast fluorescence depolarization29 . maintained by performing ensemble averages over quantum
We emphasise that the prediction of an electron-polaron trajectories (evaluated via the TEBD method), where the ac-
with short range correlations is a consequence of treating the tion of the Linblad dissipator is modeled by quantum jumps.64
phonons quantum mechanically, while the decay of exciton- In this section we assume that the C-C bond vibrations cou-
site coherences is a consequence of the exciton and phonons ple directly with the environment29,65 , in which case the Lin-
being quantum mechanically entangled. Neither of these pre- blad operators are the associated raising and lowering opera-
dictions are possible within the Ehrenfest approximation. tors (i.e., L̂n ≡ b̂n , introduced in the last section). In addition,
γ h̄ ˆ ˆ ˆ Q̃ˆ .
Ĥ = ĤFH + Q̃ P̃ + P̃ (27)
4 ∑
B. Role of system-environment interactions n n n n
n
For an exciton to dissipate energy it must first couple to fast (In Section V we discuss coupling of the torsional modes with
internal degrees of freedom (as described in the last section) the environment66 .)
and then these degrees of freedom must couple to the environ- The ultrafast localization of exciton ODLRO (or exciton-
ment to expell heat. For a low-energy exciton (i.e., a LEGS) site decoherence) described in Section IV A occurs via the
this process will cause adiabatic relaxation on a single poten- coupling of the exciton to internal degrees of freedom, namely
tial energy surface, forming a VRS58–61 . As shown in Fig. the C-C bond vibrations. We showed in Section III C (see Fig.
9
2(a)) that this coupling does not cause exciton density local-
ization. However, dissipation of energy to the environment
causes an exciton in a higher energy QEES to relax onto a
lower energy LEGS (i.e., onto a chromophore) and thus the
exciton density becomes localized.
FIG. 8. The time dependence of the exciton density for a single tra-
jectory of the quantum jump trajectory method. The discontinuity in
the density at ca. 20 fs is a ‘quantum jump’ caused by the stochastic
application of a Lindblad jump operator. The dynamics were per-
formed for an initial high energy QEES given in Fig. 2(b), showing
localization onto the LEGSs (i.e., a chromophore) labeled j = 2 in
Fig. 2(a). Reproduced from J. Chem. Phys. 148, 034901 (2018) with
FIG. 7. The time dependence of the exciton localization correlation the permission of AIP publishing.
function, Cnloc (Eq. (28)), for an initial high-energy QEES. The main
figure corresponds to the time evolution with the dissipation time
T = γ −1 = 100 fs. The time dependence of the exciton density lo- in Fig. 2(b)). At a time ca. 20 fs a ‘quantum jump’ caused by
calization number, N loc (Eq. (29)), is given in the lower inset. The the stochastic application of a Lindblad jump operator causes
upper inset corresponds to the time evolution without external dissi- the exciton to localize onto the j = 2 LEGS, shown in Fig.
pation showing that in this case exciton denisty localization does not 2(a), i.e., the high-energy extended state has randomly local-
occur. Reproduced from J. Chem. Phys. 148, 034901 (2018) with ized onto a chromophore because of a ‘measurement’ by the
the permission of AIP publishing.
environment.
The spatial extent of the exciton density, averaged over an
ensemble of quantum trajectories, is quantified by the corre-
C. Role of slow bond rotations
lation function67 , approximated by
Cnloc = ∑ Ψm Ψ∗m+n . (28) By dissipating energy into the environment on sub-ps
m timescales, hot excitons relax into localized LEGSs, i.e., onto
chromophores. The final intrachain relaxation and localiza-
Fig. 7 shows the time dependence of Cnloc with an external dis-
tion process now takes place, namely exciton-polaron forma-
sipation time T = γ −1 = 100 fs. The time scale for localization
tion via coupling to the torsional degrees of freedom. For this
is seen from the time dependence of the exciton localization
relaxation to occur bond rotations must be allowed, which
length68 ,
means that this process is highly dependent on the precise
Nloc = ∑ |n|Cnloc /∑ Cnloc , (29) chemical structure of the polymer and its environment.
n n Assuming that bond rotations are not sterically hindered,
their coupling to the excitons is conveniently modeled (via
which corresponds to the average distance between monomers
Eq. (7) and Eq. (12)) by supplementing the Frenkel-Holstein
for which the exciton wavefunction overlap remains non-zero,
model (i.e., Eq. (21)) by69
and is given in the lower inset of Fig. 7. Evidently, the cou-
pling to the environment - and specifically, the damping rate - N−1 N
1
∑ B(θn0 ) × (φn+1 − φn )T̂n,n+1 + 2 ∑ Krot φn2 + Ln2 /I .
controls the timescale for energy relaxation and exciton den- Ĥrot = −
sity localization onto chromophores. In contrast, the upper n=1 n=1
inset to Fig. 7 shows an absence of localization without exter- (30)
nal dissipation, indicating that exciton density localization is
an extrinsic process. Here, φ is the angular displacement of a monomer from its
Figure 7 is obtained by averaging over an ensemble of tra- groundstate equilibrium value and L is the associated angular
jectories. To understand the physical process of localization momentum of a monomer around its bridging bonds.
onto a chromophore, Fig. 8 illustrates the exciton density of The first term on the right-hand-side of Eq. (30) indicates
a single quantum trajectory for a photoexcited QEES (shown that the change in the dihedral angle, ∆θn = (φn+1 − φn ), cou-
10
ples linearly to the bond-order operator, T̂n,n+1 , where In the underdamped regime70 , defined by γ < 2ωrot ,
B(θn0 ) = JSE sin 2θn0 (31) φ (t) = φeq (1 − cos(ωt) exp(−γt/2)) , (37)
is the exciton-roton coupling constant and θn0 is the ground- 2 −γ 2 /4)1/2 . In this regime, the torsional angle
where ω = (ωrot
state dihedral angle for the nth bridging bond. The final term undergoes damped oscillations with a period T = 2π/ω and a
is the sum of the elastic and kinetic energies of the rotational decay time τ = 2/γ.
harmonic oscillator. Conversely, in the overdamped regime70 , defined by γ >
The natural angular frequency of oscillation is ωrot = 2ωrot ,
(Krot /I)1/2 , where Krot is the elastic constant of the rotational
oscillator and I is the moment of inertia, respectively. As dis- 1
cussed in Section III C 1, Krot is larger for the bridging bond in φ (t) = φeq 1 − (γ1 exp(−γ2t/2) − γ2 exp(−γ1t/2)) ,
4β
the excited state than the groundstate, because of the increase (38)
in bond order. Also notice that both the moment of inertia
(and thus ωrot ) of a rotating monomer and its viscous damp- where γ1 = γ + 2β , γ2 = γ − 2β and β = (γ 2 /4 − ωrot 2 )1/2 .
ing from a solvent are strongly dependent on the side groups Now, the torsional angle undergoes damped biexponential de-
attached to it. As discussed in the next section, this obser- cay with the decay times τ1 = 2/γ1 and τ2 = 2/γ2 . In the
vation has important implications for whether the motion is limit of strong damping, i.e., γ 2ωrot , there is a fast re-
under or over damped and on its characterstic timescales. laxation time τ1 = 1/γ = τ/2 and a slow relaxation time
Unlike C-C bond vibrations, being over 10 times slower tor- τ2 = γ/ωrot2 τ. In this limit, as the slow relaxation domi-
sional oscillations can be treated classically69 . Furthermore, nates at long times, the torsional angle approaches equilibrium
since we are now concerned with adiabatic relaxation on a with an effective mono-exponential decay.
single potential energy surface, we may employ the Ehrenfest For a polymer without alkyl side groups, e.g., PPP and PPV,
approximation. Thus, using Eq. (30), the torque on each ring ωrot ∼ γ ∼ 1013 s−1 and are thus in the underdamped regime
is with sub-ps relaxation. However, polymers with side groups,
∂ hĤrot i e.g., P3HT, MEH-PPV and PFO, have a rotational frequency
Γn = − up to ten times smaller and a larger damping rate, and are thus
∂ φn
in the overdamped regime7 .
= −Krot φn + λn (32)
where we define
2. A chain of torsional oscillators
0
λn = B(θn−1 )hT̂n−1,n i − B(θn0 )hT̂n,n+1 i. (33)
An exciton delocalized along a polymer chain in a chro-
Setting Γn = 0 gives the equilibrium angular displacements in mophore couples to multiple rotational oscillators resulting
eq
the excited state as φn = λn /Krot . φn is subject to the Ehren- in collective oscillator dynamics. Eq. (31) and Eq. (33) in-
fest equations of motion, dicate that torsional relaxation only occurs if the monomers
are in a staggered arrangement in their groundstate, i.e.,
dφn
I = Ln , (34) θn0 = (−1)n θ 0 . In this case the torque acts to planarize the
dt chain. Furthermore, since the torsional motion is slow, the
and self-trapped exciton-polaron thus formed is ‘heavy’ and in the
under-damped regime becomes self-localized on a timescale
dLn of a single torsional period, i.e., 200 − 600 fs. In this limit
= Γn − γLn , (35)
dt the relaxed staggered bond angle displacement mirrors the
where the final term represents the damping of the rotational exciton density. Thus, the exciton is localized precisely as
motion by the solvent. for a ‘classical’ Landau polaron and is spread over ∼ 10
monomers69 .
The time-evolution of the staggered angular displacement,
1. A single torsional oscillator hφn i × (−1)n , is shown in Fig. 9 illustrating that these dis-
placements reach their equilibrated values after two torsional
Before considering a chain of torsional oscillators, it is in- periods (i.e., t & 400 fs). The inset also displays the time-
structive to review the dynamics of a single, damped oscillator evolution of the exciton density, hNn i, showing exciton den-
subject to both restoring and displacement forces. The equa- sity localization after a single torsional period (∼ 200 fs).
tion of motion for the angular displacement is So far we have described how exciton coupling to tor-
sional modes causes a spatially varying planarization of the
d 2 φ (t) 2 dφ (t) monomers that acts as a one-dimensional potential which self-
2
= −ωrot (φ (t) − φeq ) − γ (36) localizes the exciton. The exciton ‘digs a hole for itself’,
dt dt,
forming an exciton-polaron50 . Some researchers11 , however,
where φeq = λ /Krot is proportional to the displacement force. argue that torsional relaxation causes an exciton to become11
FIG. 9. The time-evolution of the staggered angular displacement, hφn i × (−1)n . The change of dihedral angle is ∆θn = (φn+1 − φn ), showing
local planarization for a PPP chain of 21 monomers. The inset displays the time-evolution of the exciton density, hNn i, showing exciton
density localization after a single torsional period (∼ 200 fs). In the long-time limit (i.e., t & 400 fs) hφn i ∝ hNn i × (−1)n , illustrating classical
(Landau) polaron formation. Reproduced from J. Chem. Phys. 149, 214107 (2018) with the permission of AIP publishing.
more delocalized. A mechanism that can cause exciton delo- E. Time resolved fluorescence anisotropy
calization occurs if the disorder-induced localization length is
shorter than the intrinsic exciton-polaron size. Then, in this For general polymer conformations, the loss of ODLRO (or
case for freely rotating monomers, the stiffer elastic poten- the localization of the exciton coherence function) causes a
tial in the excited state causes a decrease both in the variance reduction and rotation of the transition dipole moment. The
of the dihedral angular distribution, σθ2 = kB T /Krot , and the rotation is quantified by the fluorescence anisotropy, defined
mean dihedral angle, θ0 . This, in turn, means that the exci- by71
ton band width, |4J|, increases and the diagonal disorder19 ,
σJ = JSE σθ sin 2θ0 , decreases. Hence, the disorder-induced Ik − I⊥
localization, Lloc ∼ (|J|/σJ )2/3 , increases (see Section III B). r= , (39)
Ik + 2I⊥
where Ik and I⊥ are the intensities of the fluorescence radiation
polarised parallel and perpendicular to the incident radiation,
respectively.
For an arbitrary state of a quantum system, |Ψi, the inte-
D. Summary grated fluorescence intensity polarised along the x-axis is re-
lated to the x component of the transition dipole operator, µ̂x ,
by
The conclusions that we draw from the previous three sec-
tions are that a band edge excitation (i.e., a LEGS, which is Ix ∝ ∑ |hΨ|µ̂x |GS, vi|2 , (40)
an exciton spanning a single chromophore) undergoes ultra- v
fast exciton site decoherence via its coupling to fast C-C bond
stretches. It subsequently couples to slow torsional modes where |GS, vi corresponds to the system in the ground elec-
causing planarization and exciton density localization on the tronic state, with the nuclear degrees of freedom in the state
chromophore. A hot exciton (i.e., a QEES) also undergoes characterised by the quantum number v.
ultrafast exciton site decoherence. However, exciton density The averaged fluorescence anisotropy is defined by
localization within a chromophore only occurs after localiza-
tion onto the chromophore via a stochastic interaction with the ∑i Ii (t) ri (t)
hr (t)i = 0.4 × , (41)
environment. ∑i Ii (t)12
where Ii (t) is the total fluorescence intensity and ri (t) is the Then, using Eq. (24), Eq. (25), and Eq. (42), we observe that
fluorescence anisotropy, associated with a particular confor- the emission intensity, Ix , is related to the coherence length,
mation i at time t. The factor of 0.4 is included on the as- N coh . Thus, not surprisingly, the dynamics of hr (t)i resem-
sumption that the polymers are oriented uniformly in the bulk bles that of N coh (t) shown in Fig. 6. In particular, we observe
material.71 Fig. 10 shows the simulated hr (t)i for both a high a loss of fluorescence anisotropy within 10 fs, mirroring the
energy QEES and a low energy LEGS for an ensemble of con- reduction of N coh in the same time. Furthermore, since there
formationally disordered polymers. is greater exciton coherence localization for the QEES than
for the LEGS, the former exhibits a greater loss of anisotropy.
This predicted loss of fluorescence anisotropy within 10 fs has
been observed experimentally, as shown in Fig. 11. Slower
sub-ps decay of anisotropy occurs because of exciton density
localization via coupling to torsional modes.72
V. INTRACHAIN EXCITON MOTION
The last section described the relaxation and localization
of higher energy excited states onto chromophores, and the
subsequent torsional relaxation and localization on the chro-
mophore. We now consider the relaxation and dynamics of
these relaxed excitons caused by the stochastic torsional fluc-
tuations experienced by a polymer in a solvent.
Environmentally-induced intrachain exciton relaxation in
FIG. 10. The time dependence of the fluorescence anisotropy, hr (t)i,
poly(phenylene ethynylene) was modeled by Albu and
for two initial Frenkel excitons coupled to C-C bond stretches. The
red curve corresponds to an initial LEGS, while the blue curve cor- Yaron66 using the Frenkel exciton model supplemented by the
responds to a QEES. Reproduced from J. Chem. Phys. 148, 034901 torsional degrees of freedom, i.e., Ĥ = ĤF + Ĥrot (given by Eq.
(2018) with the permission of AIP publishing. (2) and Eq. (30), respectively). Fast vibrational modes were
neglected because although they cause self-trapping, they do
not cause self-localization, and these modes can be assumed to
respond instantaneously to the torsional modes. The polymer-
solvent interactions were modeled by the Langevin equation.
For chains longer than the exciton localization length the
excited-state relaxation showed biexponential behavior with
a shorter relaxation time of a few ps and a longer relaxation
time of tens of ps.
After photoexcitation of the n = 2 (charge-transfer) exciton
in oligofluorenes, Clark et al.73 reported torsional relaxation
on sub-100 fs timescales. Since this timescale is faster than
the natural rotational period of an undamped monomer, they
ascribed it to the electronic energy being rapidly converted to
kinetic energy via nonadiabatic transitions. They argue that
this is analogous to inertial solvent reorganization.
Tozer and Barford74 using the same model as Albu and
Yaron to model intrachain exciton motion in PPP where the
FIG. 11. The experimental time dependence of the fluorescence exciton dynamics were simulated on the assumption that at
anisotropy, hR (t)i, in polythiophene in solution. hR (t)i has decayed time t + δt the new exciton target state is the eigenstate of
from 0.4 to ∼ 0.25 within 10 fs, consistent with the theoretical pre- Ĥ(t + δt) with the largest overlap with the previous target
dictions shown in Fig. 10. Subsequent fluorescence depolarization
state at time t.75
is caused by slower torsional relaxation on timescales of 1 − 10 ps
followed by possible conformational changes3 . Reproduced from J. A more sophisticated simulation of exciton motion in
Phys. Chem. C 111, 15404 (2007) with the permission of ACS pub- poly(p-phenylene vinylene) and oligothiophenes chains was
lishing. performed by Burghardt and coworkers18,76–78 where high-
frequency C-C bond stretches were also included, the sol-
It is instructive to express Eq. (40) as vent was modeled by a set of harmonic oscillators with an
Ohmic spectral density, and the system was evolved via the
Ix ∝ ∑ sxm sxn ρmn , (42) multilayer-MCTDH method. Their results, however, are in
m,n
quantitative agreement with those of Tozer and Barford in the
where sxm is the x-component of the unit vector for the mth ‘low-temperature’ limit (discussed in Section V C), namely
monomer and ρmn is the exciton reduced density matrix. activationless, linearly temperature-dependent exciton diffu-13
sion 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 already
noted in Section III B, the instantaneous spatial dihedral an-
gle fluctuations Anderson localize the Frenkel center-of-mass
wavefunction. Second, the temporal dihedral angle fluctua-
tions cause the exciton to migrate via two distinct transport
processes.79
At low temperatures there is small-displacement adiabatic
motion of the exciton-polaron as a whole along the polymer
chain, which we will characterize as a ‘crawling’ motion.
At higher temperatures the torsional modes fluctuate enough
to cause the exciton to be thermally excited out of the self-
localized polaron state into a more delocalized LEGS or quasi- FIG. 12. The exciton localization length as a function of temperature
band QEES. While in this more delocalized state, the exciton for the ‘free’ (i.e., ‘untrapped’) exciton (red circles) and exciton-
momentarily exhibits quasi-band ballistic transport, before the polaron (i.e., ‘self-trapped’) (black squares). The untrapped exci-
wavefunction ‘collapses’ into an exciton-polaron in a different free ∝ T −1/3 . The lengths coincide
ton localization length obeys Lloc
region of the polymer chain. We will characterize this large- when kB T ∼ the exciton-polaron binding energy. Reproduced from
scale displacement as a non-adiabatic ‘skipping’ motion. J. Chem. Phys. 143, 084102 (2015) with the permission of AIP pub-
Before describing the details of these types of motion, we lishing.
first describe a model of solvent dynamics and consider again
exciton-polaron formation in a polymer subject to Brownian
fluctuations. finite temperatures, however, a combination of factors affect
the localization of the exciton. First, the exciton will still
attempt to form a polaron. However, the thermally induced
A. Solvent dynamics fluctuations in the torsional angles will affect the size of this
exciton-polaron, as there is a non-negligible probability that
If the solvent molecules are subject to spatially and tempo- the exciton will be excited out of its polaron potential well into
rally uncorrelated Brownian fluctuations, then the monomer a more delocalized state at high enough temperatures. Second,
rotational dynamics are controlled by the Langevin equation the exciton states will be Anderson localized by the instanta-
neous torsional disorder.
dLn (t) Fig. 12 shows how the average localization length varies
= Γn (t) + Rn (t) − γLn (t), (43)
dt with temperature both with and without coupling between the
exciton and the torsional modes (i.e., ‘self-trapped’ and ‘free’
where Γn (t) is the systematic torque given by Eq. (32). Rn (t)
exciton, respectively). As described in Section III B, the local-
is the stochastic torque on the monomer due to the random
ization length for the ‘free’ exciton is determined by Ander-
fluctuations in the solvent and γ is the friction coefficient for
son localization. For small angular displacements from equi-
the specific solvent. From the fluctuation-dissipation theorem,
librium a Gaussian distribution of dihedral angles implies a
the distribution of random torques is given by
Gaussian distribution of exciton transfer integrals. Then, as
hRm (t)Rn (0)i = 2IγkB T δmn δ (t), (44) confirmed by the simulation results shown in Fig. 12, from
single-parameter scaling theory, Llocfree ∝ σ −2/3 = hδ θ 2 i−1/3 ∝
which are typically sampled from a Gaussian distribution with θ
1 T −1/3 .
a standard deviation of σR = (2IγkB T ) 2 . As a consequence of
In contrast, the localization length of the ‘self-trapped’ ex-
these Brownian fluctuations the monomer rotations are char-
citon slowly increases with temperature because of the ther-
acterized by the autocorrelation function80
mal excitation of the exciton from the self-localized polaron
hδ φ (t)δ φ (0)i = to a more delocalized LEGS or QEES. The two values co-
γ
incide when kB T equals the exciton-polaron binding energy
2
hδ φ i cos(ωrot t) + sin(ωrot t) exp(−γt/2),(45) (i.e., T ∼ 1500 K in PPP).
2ωrot
where
p hδ φ 2 i = kB T /Krot , Krot is the stiffness and ωrot =
Krot /I is the angular frequency of the torsional mode. C. Adiabatic ‘crawling’ motion
At low temperatures (. 100 K) the exciton has only a small
B. Polaron formation amount of thermal energy, and not enough to regularly break
free from its polaronic torsional distortions. Thus, the exciton-
As we saw in Section IV C, at zero temperature torsional polaron migrates quasi-adiabatically and diffusively as a sin-
modes couple to the exciton, forming an exciton-polaron. At gle unit. This is a collective motion of the exciton and theYou can also read
