Theory of exciton transfer and diffusion in conjugated polymersWilliam Barford and Oliver Robert Tozer Citation: The Journal of Chemical Physics 141, 164103 (2014); doi: 10.1063/1.4897986 View online: http://dx.doi.org/10.1063/1.4897986 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Theory of optical transitions in conjugated polymers. II. Real systems J. Chem. Phys. 141, 164102 (2014); 10.1063/1.4897985 Theory of optical transitions in conjugated polymers. I. Ideal systems J. Chem. Phys. 141, 164101 (2014); 10.1063/1.4897984 Spin-dependent polaron recombination in conjugated polymers J. Chem. Phys. 136, 244901 (2012); 10.1063/1.4729483 Charge-transfer states in conjugated polymer/fullerene blends: Below-gap weakly bound excitons for polymerphotovoltaics Appl. Phys. Lett. 93, 053307 (2008); 10.1063/1.2969295 Excited-state quenching of a highly luminescent conjugated polymer Appl. Phys. Lett. 78, 1059 (2001); 10.1063/1.1345840

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THE JOURNAL OF CHEMICAL PHYSICS 141, 164103 (2014)

Theory of exciton transfer and diffusion in conjugated polymersWilliam Barford1,a) and Oliver Robert Tozer1,2

1Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford,Oxford OX1 3QZ, United Kingdom2University College, University of Oxford, Oxford OX1 4BH, United Kingdom

(Received 11 August 2014; accepted 1 October 2014; published online 22 October 2014)

We describe a theory of Förster-type exciton transfer between conjugated polymers. The theory isbuilt on three assumptions. First, we assume that the low-lying excited states of conjugated polymersare Frenkel excitons coupled to local normal modes, and described by the Frenkel-Holstein model.Second, we assume that the relevant parameter regime is ¯ω < J, i.e., the adiabatic regime, andthus the Born-Oppenheimer factorization of the electronic and nuclear degrees of freedom is gener-ally applicable. Finally, we assume that the Condon approximation is valid, i.e., the exciton-polaronwavefunction is essentially independent of the normal modes. The resulting expression for the exci-ton transfer rate has a familiar form, being a function of the exciton transfer integral and the effectiveFranck-Condon factors. The effective Franck-Condon factors are functions of the effective Huang-Rhys parameters, which are inversely proportional to the chromophore size. The Born-Oppenheimerexpressions were checked against DMRG calculations, and are found to be within 10% of the exactvalue for a tiny fraction of the computational cost. This theory of exciton transfer is then appliedto model exciton migration in conformationally disordered poly(p-phenylene vinylene). Key to thismodeling is the assumption that the donor and acceptor chromophores are defined by local excitonground states (LEGSs). Since LEGSs are readily determined by the exciton center-of-mass wavefunc-tion, this theory provides a quantitative link between polymer conformation and exciton migration.Our Monte Carlo simulations indicate that the exciton diffusion length depends weakly on the con-formation of the polymer, with the diffusion length increasing slightly as the chromophores becamestraighter and longer. This is largely a geometrical effect: longer and straighter chromophores extendover larger distances. The calculated diffusion lengths of ∼10 nm are in good agreement with exper-iment. The spectral properties of the migrating excitons are also investigated. The emission intensityratio of the 0-0 and 0-1 vibronic peaks is related to the effective Huang-Rhys parameter of the emit-ting state, which in turn is related to the chromophore size. The intensity ratios calculated from theeffective Huang-Rhys parameters are in agreement with experimental spectra, and the time-resolvedtrend for the intensity ratio to decrease with time was also reproduced as the excitation migrates toshorter, lower energy chromophores as a function of time. In addition, the energy of the exciton stateshows a logarithmic decrease with time, in agreement with experimental observations. © 2014 AIPPublishing LLC. [http://dx.doi.org/10.1063/1.4897986]

I. INTRODUCTION

This is the final of a trilogy of papers whose general aimsare to provide a rigorous definition of the term “chromophore”in conjugated polymers and to predict a variety of experi-mental consequences of this definition. As chromophores arethe irreducible parts of a polymer chain that absorb and emitlight, they also represent parts of the polymer chain that actas donors and acceptors in exciton transfer. Thus, a prerequi-site for a first-principles theory of exciton transfer and diffu-sion in conjugated polymers is a rigorous definition of chro-mophores. This definition was given in Paper II1 and will bereprised in Sec. III B.

Having defined (in principle) the energetic and spatialdistributions of the donor and acceptor states, the nextrequirement for a theory of exciton transfer is to derive an

a)Author to whom correspondence should be addressed. Electronic mail:william.barford@chem.ox.ac.uk

expression for the matrix element of the Coulomb opera-tor that induces exciton transfer (from which the rates ofexciton transfer are determined from the Fermi golden ruleexpression). Within the Born-Oppenheimer regime, we canadopt the approach described in Paper I2 to derive simpleexpressions for the rates of exciton transfer. As will bedescribed in Sec. III, the rates are simply the square of theexciton transfer integral (evaluated via the line-dipole ap-proximation) multiplied by effective Franck-Condon factors.The Franck-Condon factors themselves are determined byeffective Huang-Rhys parameters, which are determined bythe chromophore sizes. This theory, therefore, provides aquantitative link between polymer conformations and excitondiffusion. Furthermore, it enables chromophore sizes to beestimated from experimental observables.

Before outlining the scope of this paper, we briefly re-view earlier theories of exciton diffusion in conjugated poly-mers. Most theories of singlet exciton diffusion assume aCoulomb-induced, Förster-like process of exciton transfer

0021-9606/2014/141(16)/164103/14/$30.00 © 2014 AIP Publishing LLC141, 164103-1

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164103-2 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

between donors and acceptors. An early model of excitondiffusion assumed that the donors and acceptors are point-dipoles whose energy distribution is a Gaussian randomvariable.3–5 While this model does reproduce some experi-mental features, such as the time-dependence of spectral dif-fusion, there is no quantitative link between the model andactual polymer conformations and morphology.

More recent approaches have attempted to make thelink between random polymer conformations and the ener-getic and spatial distributions of the donors and acceptors viathe concept of chromophores.6–9 However, the usual practicehas been to arbitrarily define chromophores via a minimumthreshold in the pz-orbital overlaps, and then obtain a distri-bution of energies by assuming that the excitons delocalizefreely on the chromophores thus defined.

As discussed in this series of papers, an unambiguouslink between polymer conformations and chromophores maybe made by defining chromophores via the spatial extent oflocal exciton ground states (LEGSs). (Similar conclusions tothese were recently made by Ma, Qin and Troisi.10) Usingthe Condon approximation, Barford et al.11 used this def-inition of chromophores in their theory of exciton transfer.The Condon approximation implies that exciton transfer oc-curs between a donor state in the relaxed excited state geom-etry (a vibrationally relaxed state, or VRS) and an acceptorstate in the relaxed electronic ground state (a vertical LEGS).They then calculated the vibrationally relaxed and verticalwavefunctions by solving the Born-Oppenheimer limit of theFrenkel-Holstein model. The Born-Oppenheimer limit pre-dicts that the vibrationally relaxed states are self-localizedexciton-polarons, whose spatial extent is much smaller thanthat of the vertical wavefunctions. As recently demonstratedin Ref. 12, however, for the adiabatic parameter regime rel-evant to conjugated polymers self-localized solutions are notexpected, and, as shown explicitly in Sec. IV, vibrationallyrelaxed and vertical wavefunctions are essentially the same.Thus, the distinction made in Ref. 11 between LEGSs andVRSs is incorrect, being based on the false hypothesis thatthe Born-Oppenheimer exciton-polaron solutions are correct.This correction, as will be shown, considerably expedites thecalculation of the exciton transfer integrals.

As in the previous two papers, the model underlying ouranalysis is the Frenkel-Holstein model. This will be intro-duced in Sec. II, along with the definition of the Coulomboperator that induces exciton transfer. Sections III A and III Bcontain the derivations of the Born-Oppenheimer expressionsfor exciton transfer in conjugated polymers, while Sec. III Ctests their validity via density matrix renormalization group(DMRG) calculations. In Sec. IV, the Born-Oppenheimer ex-pressions for exciton transfer are used to model exciton trans-fer in model PPV systems. We discuss our results and con-clude this paper in Sec. VI. Section VII provides an overallsummary of the three related papers.

II. THE FRENKEL-HOLSTEIN MODEL

The Frenkel-Holstein model is a coarse-grained modeldescribing the delocalization of a Frenkel exciton and its cou-pling to local normal modes. For generality, each “site” in the

Frenkel-Holstein model represents a moiety (e.g., a phenylring or vinyl bond). Suppose that the operator a

†n creates a

Frenkel exciton on the nth moiety that couples to the localnormal coordinate, Qn, on the same moiety. If vibrational ex-citations of a normal mode are created by the operator b

†n, then

the Frenkel-Holstein model reads

HFH =∑

n

εna†nan +

∑n

Jn(a†n+1an + a

†nan+1)

− 1√2

∑n

An¯ωn(b†n + bn)a†nan +

∑n

¯ωnb†nbn.

(1)

The excitation energy of the Frenkel exciton onto the nthmoiety is εn. Since the exciton transfer integral, J, scales as(distance)−3 (see Eq. (3)), only the nearest neighbor term isincluded. HFH is thus the unperturbed Hamiltonian, whoseeigenstates form the zeroth-order states.

For realistic polymer systems both εn and Jn are subjectto random fluctuations. As described in Sec. IV, conforma-tional disorder affects Jn, while density fluctuations in eithera solvent or of polymers in the condensed phase change thedispersion interactions and hence affect εn. ωn is the angularfrequency of the normal mode and An is the dimensionlessexciton-phonon coupling constant.

In the Born-Oppenheimer regime, where the nor-mal modes are treated as classical variables, the Born-Oppenheimer Hamiltonian equivalent of Eq. (1) is

H BOFH =

∑n

εna†nan +

∑n

Jn(a†n+1an + a

†nan+1)

−∑

n

An¯ωnQna†nan + 1

2

∑n

¯ωnQ2n, (2)

where Qn is the dimensionless displacement of the nthoscillator.

Transitions between the eigenstates of Eq. (1) on thesame and different chains are induced by the Coulomb inter-actions not included in the nearest neighbor exciton transferintegral, J. Assuming that the Coulomb interaction betweenmoieties can be treated via the point dipole approximation,this interaction is represented by the operator

HDA = 1

4πεrε0

∑n ∈ D

n′ ∈ A

κnn′

R3nn′

μn′ a†n′μnan. (3)

Here, D and A represent the donor and acceptor chro-mophores, respectively, on the same or different chains, whichare determined by LEGSs (as discussed in the Introductionand in Sec. IV). Rnn′ = |Rn − Rn′ |, where Rn is the positionof the nth moiety and μn is the magnitude of the transitiondipole moment of a Frenkel exciton on this moiety. κnn′ is theorientational factor,

κnn′ = rn · rn′ − 3(Rnn′ · rn)(Rnn′ · rn′ ), (4)

where rn is a unit vector parallel to the dipole on moiety n andRnn′ is a unit vector parallel to the vector joining moieties nand n′.

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164103-3 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

The Fermi golden rule expression for the rate of a transi-tion between an initial state, |I〉, to a final state, |F〉, is

kDA =(

2π

¯

)|〈F |HDA|I 〉|2δ(EI − EF ), (5)

where EI and EF are the initial and final energies, respectively.

III. EXCITON TRANSFER RATES

A. Born-Oppenheimer approximation

In this section, we make the Born-Oppenheimer (or adia-batic) approximation, assumed to be valid in the regime whenthe nuclear dynamics are slow compared to the excitonic dy-namics. In conjugated polymers ¯ω/|J| ∼ 0.1, so this approx-imation is generally valid.

The general exciton eigenstate of Eq. (2) is

|�〉 =∑

n

�n|n〉, (6)

where �n is the exciton wavefunction and |n〉 = a†n|0〉 is the

ket representing the Frenkel exciton on monomer n.We define an effective Huang-Rhys parameter for a poly-

mer chain (or more generally, a chromophore) of N moietiesas

S(N ) =N∑

n=1

Sn(1)|�n|4, (7)

where �n is the exciton center-of-mass wavefunction and|�n|2 the exciton density on the nth moiety. Sn(1) ≡ A2

n/2 isthe Huang-Rhys parameter for the nth moiety. For homomoi-ety, polymers Eq. (7) becomes

S(N ) = S(1)/IPR, (8)

where the inverse participation ratio (IPR), defined as

IPR =(

N∑n=1

|�n|4)−1

, (9)

is a measure of the chromophore size.The relaxation energy is

Erelax =N∑

n=1

¯ωnSn(1)|�n|4, (10)

which, from Eq. (7), implies that Erelax = ¯ωS(N) = ¯ωS(1)/IPR for homomoiety polymers.

We begin by deriving exciton transfer rates for ordered,linear polymer chains, for which the lowest energy excitoncarries most of the oscillator strength. Generalizations to dis-ordered systems are described in Sec. III B.

1. 0 − 0 transitions of donor and acceptor

In the Born-Oppenheimer approximation, the many bodywavefunction is a product of the electronic and nuclear wave-functions. Thus, the groundstate of a polymer chain in the ab-sence of electronic and vibrational excitations is the tensor

FIG. 1. The isoenergetic exciton transfer processes considered in this paper:(1) 0 − 0 transitions of donor and acceptor; (2) 0 − 1 transition of donorand 0 − 0 transition of acceptor; (3) 0 − 0 transition of donor and 0 − 1transitions of acceptor; and (4) 0 − 1 transitions of donor and acceptor. Theinitial states of the donor and acceptor are |EX; 0〉 and |GS; 0〉, respectively,because vibrational relaxation occurs on a much faster timescale than excitontransfer.

product

|GS; 0〉 = |0〉 ⊗∏m

|0〉〉m, (11)

where |0〉 represents the vacuum of the Frenkel exciton and|0〉〉m is the vacuum of the undisplaced oscillator on monomerm (here denoted by a double ket). Likewise, the electronicexcited state of a polymer in the absence of vibrational exci-tations is given by the tensor product

|EX; 0〉 = |�0〉 ⊗∏m

|0′〉〉m, (12)

where |�0〉 is given by Eq. (6) with �n ≡ �0n , the exciton

wavefunction associated with the ground vibronic state. |0′〉〉mis the vacuum of the displaced oscillator (denoted by a prime)on monomer m.

For the 0 − 0 transitions of the donor and acceptor, il-lustrated in Fig. 1(1), the initial state, |I〉, corresponds to thedonor in its vibronic groundstate and the acceptor in its elec-tronic groundstate, i.e.,

|I〉 = |EX; 0〉D|GS; 0〉A. (13)

Similarly, the final state, |F〉, corresponds to the donor in itselectronic groundstate and the acceptor in its vibronic ground-state, i.e.,

|F〉 = |GS; 0〉D|EX; 0〉A. (14)

For the 0 − 0 transitions, we therefore require the transitiondensity,

a00n = 〈EX; 0|a†

n|GS; 0〉 ≡ 〈GS; 0|an|EX; 0〉, (15)

for both the donor and acceptor. As shown in Paper I,2

a00n = �0

n exp (−S(N )/2) , (16)

where the effective Huang-Rhys parameter, S(N), is definedby Eq. (7) with �n ≡ �0

n .

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164103-4 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

The required matrix element is thus,

〈F|HDA|I〉 = A〈EX; 0|D〈GS; 0|HDA|GS; 0〉A|EX; 0〉D=

(1

4πεrε0

) ∑n∈D

n′∈A

κnn′

R3nn′

μn′a00n′ μna

00n

= JDA exp(−SD/2) exp(−SA/2), (17)

where

JDA =(

1

4πεrε0

) ∑n∈D

n′∈A

κnn′

R3nn′

μn′�0n′μn�

0n (18)

is the electronic exciton transfer integral. (We note that this isthe line-dipole approximation of the transfer integral.)

Thus, the exciton transfer rate for this process is

k00(A)00(D) =

(2π

¯

)|JDA|2FD

00FA00δ(EI − EF ), (19)

where F00(N) is the effective 0 − 0 Franck-Condon factor fora chain of N monomers,

F00(N ) = exp(−S(N )). (20)

2. 0 − 1 transition of donor and 0 − 0 transitionof acceptor

The 0 − 1 transition of the donor and 0 − 0 transition ofthe acceptor is illustrated in Fig. 1(2). The 0 − 1 transition ofthe donor corresponds to a transition from its ground vibronicstate (given by Eq. (12)) to its electronic groundstate in a one-phonon vibrational state. We assume that the phonons in theelectronic groundstate are localized, thus a one-phonon vibra-tional state is represented as

|GS; 1m〉 = |0〉 ⊗ |1,m〉〉, (21)

where

|1,m〉〉 = b†m

∏m′

|0〉〉m′ (22)

represents a single vibrational excitation on the mth monomer,and 1 ≤ m ≤ N.

For the donor we need the matrix element

a01n,m = 〈GS; 1m|an|EX; 0〉, (23)

which, as shown in Paper I,2 is

a01n,m = �0

n

(Am√

2

∣∣�0m

∣∣2)

exp (−S/2) . (24)

As the acceptor undergoes a 0 − 0 transition, the relevant ma-trix element of the transition density is given by Eq. (16).

Thus, the exciton transfer matrix element correspondingto the donor final state being in its electronic groundstate witha vibrational excitation on monomer m is

〈F(m)|HDA|I〉 =(

1

4πεrε0

) ∑n∈D

n′∈A

κnn′

R3nn′

μn′a00n′ μna

01n,m

=(

Am√2

∣∣�0m

∣∣2)

exp(−SD/2) exp(−SA/2)JDA.

(25)

Squaring the transfer integral and summing over all theone-phonon final donor states gives the total 0 − 1 excitontransfer rate, i.e.,

k00(A)01(D) =

(2π

¯

) ∑m∈D

(Am√

2

∣∣�0m

∣∣2)2

× exp(−SD) exp(−SA)|JDA|2δ(EI − EF )

=(

2π

¯

)|JDA|2SD exp(−SD) exp(−SA)δ(EI − EF )

=(

2π

¯

)|JDA|2FD

01FA00δ(EI − EF ), (26)

where we now introduce an effective 0 − 1 Franck-Condonfactor for a chain of N monomers,

F01(N ) = S(N ) exp(−S(N )) (27)

and JDA is given by Eq. (18). Evidently, for the 0 − 1 emissionprocess the exciton transfer rate is a factor SD smaller than the0 − 0 transfer process.

3. 0 − 0 transition of donor and 0 − 1 transitionof acceptor

The 0 − 0 transition of the donor and 0 − 1 transition ofthe acceptor is illustrated in Fig. 1(3). The 0 − 1 transitionof the acceptor corresponds to a transition from its electronicand vibrational groundstate (given by Eq. (11)) to a state in itsfirst vibronic manifold. Again, assuming that the vibrationalstates are localized, the excited states are represented as

|EX; 1m〉 = |�1〉 ⊗ |1′,m〉〉, (28)

where

|1′,m〉〉 = b′†m

∏m′

|0′〉〉m′ (29)

represents a single displaced vibrational excitation on the mthmonomer and 1 ≤ m ≤ N. |�1〉 is given by Eq. (6) where �1

n

is the exciton wavefunction associated with the 1 −phononvibronic manifold.

Since

m〈〈0′|1〉〉m = m〈〈1′|0〉〉m, (30)

the expressions for the exciton transfer rates resulting from a0 − 1 transition of the acceptor are the same as those resultingfrom a 0 − 1 transition of the donor, i.e.,

k01(A)00(D) =

(2π

¯

)|JDA|2FD

00FA01δ(EI − EF ), (31)

except that where the acceptor exciton wavefunction appearsin the definition of JDA and SA, �0

n is replaced by �1n .

4. 0 − 1 transitions of donor and acceptor

The 0 − 1 transitions of the donor and acceptor is illus-trated in Fig. 1(4). It is straightforward to generalize the re-sults of Secs. III A 2 and III A 3 to derive the rate for this

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164103-5 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

process as

k01(A)01(D) =

(2π

¯

)|JDA|2FD

01FA01δ(EI − EF ), (32)

where the donor exciton wavefunction appearing in the def-inition of JDA and SD is �0

n , while for the acceptor excitonwavefunction it is �1

n .

5. Remarks

Combining the rates for the four processes describedabove, it is convenient to express the overall rate for excitontransfer from the donor to acceptor in the familiar form as

k =(

2π

¯

)|JDA|2

∫D(E)A(E)dE, (33)

where the spectral functions of the donor and acceptor are

D(E) =∑v=0,1

FD0vδ

(E + ED

0v

)(34)

and

A(E) =∑v=0,1

FA0vδ

(E − EA

0v

), (35)

respectively, and we have used the identity

δ(x + y) =∫ ∞

−∞δ(x − y)δ(y + z)dz. (36)

EA0v is the excitation energy of the acceptor,

EA0v = EA

00 + v¯ω, (37)

and ED0v is the de-excitation energy of the donor

ED0v = −(

ED00 − v¯ω

). (38)

Equation (33) is analogous to the usual Förster equationfor the rate of energy transfer. The differences here are thatfirst, the exciton transfer integral, JDA, is evaluated via theline-dipole approximation (Eq. (18)) and second, the spec-tral functions contain effective Franck-Condon factors whichdescribe the chains (or chromophores) coupling to effectivemodes with reduced Huang-Rhys parameters (Eq. (7)).

B. Generalization of the Born-Oppenheimer ratesto disordered polymers

The generalization of the Born-Oppenheimer rates de-rived in Sec. III A for ordered, linear chains to conforma-tionally disordered polymers is straightforward. On disor-dered polymer chains, the low-energy exciton center-of-masswavefunctions are essentially nodeless, non-overlapping, andspace-filling. These local exciton groundstates (LEGSs) natu-rally segment the polymer chain into chromophores.

LEGSs are defined by the condition that13, 14∣∣∣∣∣∑

n

�jn

∣∣�jn

∣∣∣∣∣∣∣ ≥ 0.95, (39)

where the quantum number j ≥ 1, i.e., in general there is morethan one LEGS (or chromophore) on a polymer chain. The

FIG. 2. k00(A)00(D) against chain length for two uniform, parallel chains, as

calculated using DMRG and from the Born-Oppenheimer approximation(Eq. (19)). The interchain separation is fixed at 10 times the monomer size.

exciton transfer rates between pairs of chromophores are thenevaluated via the previous expressions by replacing �0

n or �1n

by �j,0n or �

j,1n in the definitions of JDA and S(N). (In fact,

there is a simplification described in Sec. IV, whereby �j,0n

and �j,1n are replaced by the vertical wavefunctions.)

C. DMRG results

Although general expressions were obtained inSec. III A for the interchain exciton transfer rates usingthe Born-Oppenheimer approximation, it is prudent toinvestigate the validity of the assumptions made in thesederivations. In this section, we compare the results obtainedfrom the Born-Oppenheimer expressions for parallel, uniformchains with those obtained from a density matrix renormal-ization group (DMRG) method15 (described elsewhere2, 12).In this section, all length scales are in units of the monomersize (set to unity). In addition, we set 4πεrε0 = 1, 2π /¯ = 1,and δ(EI − EF) = 1. We set A = √

2 and the interchaindistance at 10 monomer units, while ¯ω/J and the chainlengths are variables.

The transition densities, a00n,m, a01

n,m, etc., are calculatedexplicitly within the DMRG algorithm, and thus a calculationof the transition rates, kDA, can easily be made. A comparisonof the results of these calculations with those obtained usingthe Born-Oppenheimer results are presented in Figs. 2–4.

In Fig. 2, the DMRG-calculated k00(A)00(D) is plotted against

chain length for various values of ¯ω. Also shown is theBorn-Oppenheimer expression (Eq. (19)). It can be seen thatas the Born-Oppenheimer limit is approached (i.e., as ¯ω/|J|→ 0), the DMRG-calculated results approach the Born-Oppenheimer predictions, as we would expect. For realisticvalues of the parameters for conjugated polymers (i.e., ¯ω/J≈ 0.1), the Born-Oppenheimer approximation overestimatesk

00(A)00(D) by approximately 5%–10%.

Since k00(A)00(D) is proportional to the square of the exci-

ton transfer integral between parallel chains, J, its generalnon-monotonic form arises from the non-monotonic form

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164103-6 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

FIG. 3. k01(A)00(D) against chain length for two uniform, parallel chains, as

calculated using DMRG and from the Born-Oppenheimer approximation(Eq. (31)). The interchain separation is fixed at 10 times the monomer size.

of J.16, 17 If R is the separation between parallel chains andL is the chain length, then as shown in Ref. 17, J ∼ L/R3 forL � R and J ∼ 1/LR for L � R, with a maximum value atL/R = ((1 + √

5)/2)1/2. As can be seen from Fig. 2, this be-havior is replicated by k

00(A)00(D).

Similarly, the DMRG-calculated and Born-Oppenheimerexpression (Eq. (31)) for k

01(A)00(D) is plotted against chain length

in Fig. 3 for various values of ¯ω. As the chain length in-creases, it is seen that there is a breakdown in the similar-ity of the Born-Oppenheimer results to those calculated fromDMRG. This issue arises because the energy separation be-tween the exciton pseudo-momentum states becomes compa-rable to, or smaller than, the phonon energy as the size of thechain increases. If the j = 1, v = 1 (i.e., the vibrationally ex-cited, exciton ground) states were separable from the j > 1,

v = 0 (i.e., excited exciton, vibrational ground) states thenno issue would arise. However, there is mixing of thesestates as their energies become comparable, meaning thatthis separation becomes impossible. This causes the Born-Oppenheimer approximation to break down for large chains

FIG. 4. k00(A)01(D) against chain length for two uniform, parallel chains, as

calculated using DMRG and from the Born-Oppenheimer approximation(Eq. (26)). The interchain separation is fixed at 10 times the monomer size.

and high ¯ω/J, as is shown in Fig. 3. For small chains andsmaller phonon frequencies, however, the Born-Oppenheimerapproximation does hold. (An equivalent breakdown of theBorn-Oppenheimer approximation for optical absorption isdiscussed in Paper I.2)

In general, the exciton states that are considered will beon disordered polymer chains, and as such they will be An-derson localized such that the chromophore sizes are about 20monomer units or less. For these chromophore sizes the Born-Oppenheimer predictions are similar to the exact DMRG cal-culations for realistic values of ¯ω/|J|.

Figure 4 shows a plot of the DMRG-calculated and Born-Oppenheimer expression (Eq. (26)) for k

00(A)01(D) against chain

length for various values of ¯ω. In this case, the vibrationallyexcited states involved are the exciton vacuum, meaning thatthere is no mixing of vibronic states (as for k

01(A)00(D)). The

DMRG calculated values of k00(A)01(D), therefore, do not diverge

from the Born-Oppenheimer prediction as the chain lengthincreases. In fact, the Born-Oppenheimer approximation be-comes a better approximation to the correct values as thechain length increases, and is very good for small values of¯ω/|J|.

Although the Born-Oppenheimer calculations show somedeviation from the more correct DMRG calculations, the re-sults in the realistic parameter regime (i.e., ¯ω/|J| ≈ 0.1) showthe same qualitative behavior, and are in fact quantitativelyrather similar, with an error of only about 5%–10% in the ex-citon transfer rates. As a result, the Born-Oppenheimer ex-pressions for the exciton transfer rates are used in the model-ing of exciton transfer in Sec. IV.

IV. MODELING EXCITON DIFFUSION

Having established the validity of the expressions for theexciton transfer rates derived using the Born-Oppenheimerapproximation, we now use them to model exciton diffusionin the condensed phase. The procedure for doing this is asfollows.

First, an ensemble of polymer conformations was statis-tically generated within a simulation sphere with a radius of29 nm. The first monomer of each chain was positioned ran-domly in the sphere, while the chain was grown by choosingthe dihedral angles randomly from a Gaussian distribution.Any conformation which collided with itself, another chain,or the boundary of the sphere was rejected.

Second, the energies and wavefunctions of the donor andacceptor states are determined. This is accomplished for eachchain by solving the disordered Frenkel model, defined as

HF =∑

n

εna†nan +

∑n

Jn(a†n+1an + a

†nan+1). (40)

This model describes a Frenkel exciton delocalized on a poly-mer chain in the absence of exciton-phonon coupling. Asexplained in Appendix A of Paper I,2 the nearest-neighborFrenkel exciton transfer integral, Jn, contains a through-spacecomponent, JDD, and a through-bond component, JSE,

Jn = JDD + JSE cos2 φn, (41)

where φn is the dihedral angle between neigboring moieties.

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164103-7 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

FIG. 5. The two local exciton groundstate wavefunctions on a disorderedPPV chain of 63 moieties. �1, 0 and �2, 0 are the two lowest eigenstates ofthe fully quantized Frenkel-Holstein model (Eq. (1)) obtained via a DMRGcalculation. They are thus the ground vibronic states (or vibrationally relaxedstates) of their corresponding chromophore. �1, vert and �2, vert are the verti-cal states (obtained from the Frenkel model (Eq. (40))) corresponding to thesame chromophores. (Site alternation in � occurs because of the differentFrenkel excitation energies on the phenyl-ring and vinyl-bond.)

The eigenvalues of the Frenkel model, {Evert}, corre-spond to vertical excitation energies. The associated 0 − 0transition energies are therefore

E00 = Evert − Erelax + ¯ω/2, (42)

where Erelax is defined by Eq. (10). The eigenfunctions ofthe Frenkel model, {�vert}, correspond to vertical excitoncenter-of-mass wavefunctions. However, since self-localizedexciton-polarons are not formed in conjugated polymers,12

the vertical wavefunctions closely resemble the vibrationallyrelaxed wavefunctions. This observation is demonstratedin Fig. 5, which shows the vertical wavefunctions fromthe Frenkel model (Eq. (40)) and the wavefunctions fromDMRG solutions of the fully quantized Frenkel-Holsteinmodel (Eq. (1)) corresponding to the ground vibrational state.Figure 9 of Ref. 1 also shows that the differences between thecenter-of-mass wavefunctions associated with the ground andfirst vibronic manifolds are negligible. Thus, with negligibleloss of accuracy, we can assume that the eigenfunctions of theFrenkel model correspond to the vibrationally relaxed eigen-functions of the Frenkel-Holstein model, i.e.,

�0n ≈ �1

n ≈ �vertn . (43)

Exciton transfer occurs between polymer chromophoresand, as has been discussed in earlier papers, these are deter-mined by the local exciton ground states (LEGSs), defined byEq. (39). The LEGSs form a space-filling, non-overlappingset of states, and are also the main states from which absorp-tion and emission are likely to occur, meaning they fulfil allthe criteria required in the definition of a chromophore. Oncethe LEGSs have been determined, the effective Huang-Rhysparameters of the states can be determined (via Eq. (7)), andthis can be combined with knowledge of the wavefunction

TABLE I. The parameters used in the Frenkel-Holstein model in the mod-eling of exciton diffusion in PPV (derived in Paper II1).

Parameter Value

Phenyl Frenkel excitation energy, Ep 6.05 eV

Vinyl Frenkel excitation energy, Ev

8.59 eVSuperexchange exciton transfer integral, J SE

PPV −1.68 eVDipole-dipole exciton transfer integral, JDD

PPV −0.60 eVPhonon energy, ¯ω 0.19 eVPhenyl exciton-phonon coupling constant, Ap 2.7

Vinyl exciton-phonon coupling constant, Av

4.0Phenyl Huang-Rhys parameter, S

p(1) = A2

p/2 3.65

Vinyl Huang-Rhys parameter, Sv(1) = A2

v/2 8.0PPVa monomer Huang-Rhys parameter, S(1) 2.53Moiety transition dipole moment, μ0 1.51 × 10−29 C mDiagonal disorder, σ

α65 meV

Relative permittivity, εr 2.25Vibronic line width, � 60 meVMean monomer volume 1.3 × 102 Å3

No. of moieties per chain 201No. of polymer chains in the simulation sphere 2000Radius of simulation sphere 29 nm

aEvaluated via Eq. (7) over a phenyl-vinyl monomer.

to calculate the exciton transfer integrals between the chro-mophores, using the equations derived in Sec. III A.

The probability of exciton transfer from one chro-mophore to another strictly requires the energy of the excitonto be identical on the two chromophores. However, in prac-tice the transfer will actually occur for energies that are notperfectly matched, with a Lorentzian linewidth, given by

δ(ED − EA) = �/2π

(ED − EA)2 + �2/4, (44)

where EA and ED are the acceptor and donor exciton energies,respectively, and � is the linewidth, taken to be 0.06 eV.

Exciton diffusion is then modeled by exciting a chro-mophore and allowing the exciton thus formed to migrate bya Monte Carlo process (as described in Ref. 11). The prob-ability for the exciton to hop to another chromophore or toradiatively recombine is proportional to the square magnitudeof the relevant matrix elements (defined above) multiplied bythe Franck-Condon factors. In particular, the exciton hoppingrate is given by Eq. (33), while the radiative rate is the Ein-stein A coefficient.

The migration of the excitation is continued until radia-tive decay occurs. This process is then performed taking everychromophore of the system as the starting chromophore to ob-tain the ensemble averages. The model parameters are listedin Table I.

V. RESULTS

Throughout this series of papers, two types of disor-der have been introduced to the system, reflecting the twomain physical sources of disorder in realistic systems.18, 19

First, the on-site energy, εn in Eq. (40), has been allowed tovary, reflecting the effect of density fluctuations in the sys-tem on the local polarizability and the resulting change in the

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164103-8 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

energy of the Frenkel exciton.20 This disorder is taken to bea Gaussian random variable with a standard deviation, σα ,of 65 meV, in agreement with previous work on poly(para-phenylenevinylene) (PPV) and its derivatives.18

The second form of disorder included in this work hasbeen introduced by allowing the dihedral angle, φ, betweenadjacent moieties to deviate from its mean value, φ0. This af-fects the exciton hopping parameter, Jn, as the superexchangecomponent of this parameter depends on the pz-orbital over-lap between adjacent sites, as described by Eq. (41).

So far throughout this series of papers, it has been as-sumed that the dihedral angles between moieties form a Gaus-sian distribution about a single value of φ0, with a constantstandard deviation throughout the ensemble of chains. Thiscontinues to be how the geometric disorder is modelled inSec. V A. However, the existence of blue and red phases inMEH-PPV21 suggests that more complex morphologies mayoccur in some preparations of PPV or its derivatives. An at-tempt is made to investigate these more complex distributionsof φ in Sec. V B, where the value of φ0 is taken to be constantfor a given chain, but to have a value that forms a Gaussiandistribution with a standard deviation of 5◦ and a mean of 10◦

across the ensemble of chains.Throughout the following, the systems that have been

modeled are comprised of 2000 PPV chains, each contain-ing 201 moieties, with the mean interchain separation chosento reproduce the expected density of a thin film of PPV.22

A. Homogeneous disorder

In this section, results are presented for ensembles ofPPV chains with a constant mean dihedral angle, φ0, of ei-ther 5◦ or 15◦, representing two different preparations of PPV.The actual dihedral angles between adjacent moieties, φ, arethen taken to be Gaussian random variables with a standarddeviation of σφ about φ0.

The density of states for LEGSs for the two values ofφ0 is shown in Fig. 6. As discussed in Paper II,1 the densityof states is shifted to a higher energy for the larger value ofφ0, and the density of states is broader for the higher value.Figure 6 also shows the density of states for the trap statesfrom which emission occurs after migration of the excitationthrough the system by Förster resonance exciton transfer. Asexpected, the trap states occupy the extreme low-energy tailof the density of states.

As well as the density of states, it is also interesting toidentify how the conjugation length, and thus the inverse par-ticipation ratio and vibronic peak intensity ratio, of a chro-mophore depends on the energy of the LEGS on that chro-mophore. It is often claimed that disorder on a conjugatedpolymer chain leads to conjugation breaks which localizethe exciton, while the exciton is allowed to delocalize freelybetween the breaks, resulting in lower energy states beingshorter through a particle-in-a-box argument. However, asFig. 7 shows, for the type of disorder used in this work theexciton conjugation length is shortest for the lowest energystates, with a monotonic increase in conjugation length as en-ergy increases for the case of homogeneous disorder. Also, as

FIG. 6. Density of states for absorbing LEGSs (solid lines) and emitting trapstates (dashed lines), for φ0 = 5◦ (red) and φ0 = 15◦ (blue), for ensemblesof 2000 PPV chains of 201 moieties, with a standard deviation of dihedralangle, σ

φ= 5◦.

discussed in Sec. V B 2 of Paper II,1 the conjugation length isslightly shorter for less planar chains, because of the reductionin the exciton band width.

The relative planarity and disorder of the polymer chainswill have an effect on the migration of excitations within theensemble. Figures 8 and 9 show how the average time be-tween FRET hops and the mean distance of the hops, re-spectively, depend on hop number for two values of φ0 withthe same value of σφ . It can be seen that the time betweenhops (i.e., the exciton transfer rate) is virtually independentof φ0, and hence virtually independent of the chromophoresize. However, the distance traveled during the hop has asignificant dependence, with longer hops being observed in

FIG. 7. Scatter plot showing the correlation between LEGS conjugationlength (i.e., chromophore size) and exciton energy for different mean valuesof the dihedral angle, φ0. Standard deviation of the dihedral angle, σ

φ= 5◦.

The solid curves are the average conjugation length as a function of energy,showing that for a fixed φ0 shorter chromophores are lower in energy. (Note,however, that on average a more planar chain has longer chromophores oflower energy than a less planar chain.)

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164103-9 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

FIG. 8. Average exciton hop time interval versus hop number for φ0 = 5◦(red, solid line) and φ0 = 15◦ (blue, dashed line), for ensembles of PPVchains with a standard deviation of dihedral angle, σ

φ= 5◦. The time interval

is virtually independent of φ0, and increases with hop number as the numberof lower energy states available in the vicinity of the exciton decreases.

more planar chains. This is largely a geometric effect: longerchromophores facilitate longer hops as they extend further inspace. Additionally, these figures also indicate that the dis-tance (and hence interval time) increases as the hop numberincreases, due to the increasing scarcity of lower energy statesin the vicinity of the exciton.

An important concept for conjugated polymer device ac-tivity is the diffusion length of an exciton within the sys-tem, and this will determine many properties, such as theprobability of the exciton undergoing heteroatom-mediatednon-radiative decay. The dependence of the average diffusionlength on the number of monomers in a statistical segment(i.e., the Kuhn length) of the polymer is presented in Fig. 10for various values of dihedral disorder. The diffusion lengthincreases as the polymer chains become less coiled (i.e., as

FIG. 9. Average exciton hop distance versus hop number for φ0 = 5◦ (red,solid line) and φ0 = 15◦ (blue, dashed line) for ensembles of PPV chains witha standard deviation of dihedral angle, σ

φ= 5◦. The hop distance shows a

strong dependence on the planarity of the chain (i.e., the chromophore size),while it increases with hop number as the number of lower energy statesavailable in the vicinity of the exciton decreases.

FIG. 10. Average exciton diffusion length versus mean number of monomersin a statistical segment (i.e., the Kuhn length). The mean dihedral angle be-tween moieties, φ0, is 15◦, for various values of σ

φ.

the Kuhn length (or 〈m〉) increases) while it also increases asthe amount of disorder is reduced. These results can be under-stood from the previous discussion. Smaller disorder implieslarge chromophore sizes and hence longer hops. Straighterchromophores also extend further in space than curved chro-mophores, again facilitating exciton migration. The excitondiffusion lengths of approximately 10 nm that are calculatedare in good agreement with experimentally determined diffu-sion lengths.23–25

How the average energy of the exciton varies with timeafter excitation is presented in Fig. 11, where the energyof emitted photons is plotted for the two values of φ0 con-sidered. The logarithmic dependence on time of the energyof the emitted photons is in agreement with experimen-tal observations.5, 26 It also accords with predictions from

FIG. 11. Average photon emission energy as a function of time for φ0 = 5◦(red, solid line) and φ0 = 15◦ (blue, dashed line), for ensembles of PPVchains with a standard deviation of dihedral angle, σ

φ= 5◦. The logarithmic

decrease in energy as a function of time is in agreement with experimentalresults.5, 26

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164103-10 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

Bässler’s Gaussian Random Disorder model,5 which is notsurprising since the emissive states exhibit a quasi-Gaussiandensity of states (see Fig. 6). The energy relaxation shown inFig. 11 is on the order of a few hundredth of an electron voltfor each order of magnitude of time, in good agreement withexperimental results obtained by Hayes et al.26 and Meskerset al.5 It can also be seen that the amount of relaxation ob-served for more planar chains is smaller, due to the narrowerdensity of states in the more planar ensemble. Again, thisagrees with experiments investigating the differences betweenPPV and MEH-PPV,26 as the less planar MEH-PPV showsgreater changes in average photon energy.

The intensity ratio of the vibronic peaks in the emis-sion spectrum, I00/I01, can be estimated from the ratio of theFranck-Condon factors for the two transitions. This gives anintensity ratio1, 2, 27

I00

I01

∝ 1

S(N )= 〈IPR〉

S(1), (45)

where S(N) is the effective Huang-Rhys parameter of the state,defined in Eq. (8). The value of this ratio as time progressesin the simulation is plotted in Fig. 12, where it is seen thatthis ratio is a decreasing function of time. This observationimplies that the inverse participation ratio of the exciton isalso decreasing with time, as we would expect from the factthat the low energy chromophores have the shortest lengths(as shown in Fig. 7).

The observed intensity ratios of 2.5–4.0 are in agreementwith experimental photoluminescence spectra,18, 28, 29 whilethe observation that the intensity ratio decreases with time isalso suggested by the observed time-resolved photolumines-cence spectra in MEH-PPV (see Fig. 3 of Ref. 26).30

The role of exciton migration on the emission spectra isillustrated in Fig. 13. This shows the calculated optical ab-sorption to all absorbing states and the optical emission fromall LEGSs (which occurs in the absence of exciton migra-tion). As discussed in Paper II,1 the absorption spectrum is

FIG. 12. Emission intensity ratio as a function of time for φ0 = 5◦ (red, solidline) and φ0 = 15◦ (blue, dashed line), for ensembles of PPV chains with astandard deviation of dihedral angle, σ

φ= 5◦. The decrease in intensity ratio

as time progress indicates that excitons migrate through progressively shorterchromophores.

FIG. 13. The calculated optical spectra of PPV. Exciton migration causesa red-shift in energy, a narrowing of the inhomogeneous broadening, and adecrease in I00/I01. φ0 = 15◦ and σ

φ= 5◦.

broader than the emission spectrum (even without migration).Also shown is the time-integrated emission spectrum follow-ing exciton migration. This result encapsulates the key resultsalready discussed:

� The emission after migration is red-shifted, becausethe emissive states are in the low-energy tail of the den-sity of states.

� The inhomogeneous broadening is narrowed, becausethe emissive states have a narrower density of statesthan LEGSs.

� The intensity ratio, I00/I01, decreases, because on aver-age emissive chromophores have shorter conjugationlengths than LEGSs.

B. Heterogeneous disorder

As discussed above, we have also investigated the situa-tion of heterogeneous disorder, where the mean dihedral an-gle, φ0, is allowed to vary between chains in the ensemble,allowing for the possibility of the coexistence of blue and redphases.21 The physical basis of this is that it is likely that PPVin the solid state may contain some regions where the poly-mer is more planar than in other regions, resulting in differentvalues of φ0. To attempt to model this we take the mean valueof the dihedral angle for each chain to be a Gaussian ran-dom variable, with a mean value over the ensemble of chainsof 〈φ0〉 = 10◦ and standard deviation σφ0

= 5◦. As before,the dihedral angles on each chain are also Gaussian randomvariables about their mean value, with a standard deviationσφ = 5◦.

The density of local exciton ground states for 2000 PPVchains of 201 moieties, with heterogeneous values of φ0, isshown in Fig. 14. It can be seen that the density of statesis far broader in this case than in the case of homogeneousmean dihedral angles, which is as we would expect given themean dihedral angle is one of the main factors in determin-ing the energy of the states in the low-energy tail. The highenergy tail of the density of states is also significantly longer

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164103-11 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

FIG. 14. The density of states for absorbing LEGSs (solid line) and emittingtrap states (dashed line) for an ensemble of PPV chains, with a Gaussiandistribution of φ0 and φ, as described in the main text. 〈φ0〉 = 10◦, σ

φ0= 5◦,

and σφ

= 5◦.

than the low energy tail. This is a result of the high energystates arising from less planar chains with larger values of φ0,for which we expect a broader density of states, as shown inFig. 6.

The scatter plot of the size of the chromophores as a func-tion of energy given in Fig. 15 shows that in this ensemblethe properties exhibit a more complex relationship. As wasobserved for the homogeneous value of φ0, the low energytail of states have small conjugation lengths, and the conju-gation length increases with energy. However, as the energyincreases there is an interplay between the larger conjugationlength expected for higher energy states with the same valueof φ0 and the smaller conjugation lengths expected for chainswith a higher value of φ0. This means that rather than see-ing a monotonic increase in the conjugation length as energy

FIG. 15. Scatter plot showing the correlation between LEGS conjugationlength and exciton energy for an ensemble of PPV chains, with a Gaussiandistribution of values of φ0 and σ

φ= 5◦. The solid curve is the average con-

jugation length as a function of energy. Unlike for the case of homogeneousdisorder, the average conjugation length is not monotonically increasing withenergy, due to the decrease in conjugation length for larger values of φ0.

FIG. 16. Emission intensity ratio as a function of time for an ensemble ofPPV chains, with a Gaussian distribution of values of φ0. Despite the non-monotonic dependency of conjugation length on energy, the intensity ratiostill decreases with time, suggesting that there is still a decrease in averagechromophore size as the exciton migrates.

increases, the conjugation length begins to decrease slightlybeyond a certain threshold value of state energy, in this caseapproximately 2.6 eV.

However, as was seen in Fig. 14, the trap (emissive) statesof this system almost exclusively have an energy of less than2.6 eV, meaning that we expect the conjugation length ofemitting states to exhibit similar behavior to that observed inthe case of homogeneous disorder. The ratio of the intensitiesof the vibrational peaks in the emission spectrum, I00/I01, isplotted as a function of time in Fig. 16. Once again we seethat this ratio decreases with time, and the change in inten-sity ratio is comparable to that in the case of homogeneousdisorder, as the dominating trend is that lower energy exci-tons have smaller conjugation lengths, despite the reversal inthis trend for relatively high energies. The average emissionenergy as a function of time is shown in Fig. 17, and it canbe seen that the relaxation occurs at a comparable rate to the

FIG. 17. Emission energy as a function of time for radiative decay of ex-citons after migration within an ensemble of PPV chains, with a Gaussiandistribution of values of φ0.

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164103-12 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

relaxation in the case of heterogeneous disorder, but in thiscase we observe a slightly accelerated relaxation at shorttimes as the exciton rapidly migrates from less planar, higherenergy chromophores, to more planar, lower energy chro-mophores.

VI. DISCUSSION AND SUMMARY

Building on the theory of optical transitions in conjugatedpolymers developed in Paper I,2 this paper has described atheory of Förster-type exciton transfer between conjugatedpolymers. In Paper I,2 the theory is built on three assumptions.First, we assume that the low-lying excited states of conju-gated polymers are Frenkel excitons coupled to local normalmodes, and described by the Frenkel-Holstein model. Second,we assume that the relevant parameter regime is ¯ω < J, i.e.,the adiabatic regime, and thus the Born-Oppenheimer factor-ization of the electronic and nuclear degrees of freedom isgenerally applicable. Finally, we assume that the Condon ap-proximation is valid, i.e., the exciton-polaron wavefunctionis essentially independent of the normal modes. All three as-sumptions are justified in Paper I.2 As for optical transitions,the resulting expression for the exciton transfer rate (Eq. (33))has a familiar form, being a function of the exciton trans-fer integral and the effective Franck-Condon factors. The ef-fective Franck-Condon factors are functions of the effectiveHuang-Rhys parameters, which are inversely proportional tothe chromophore size. The Born-Oppenheimer expressionswere checked against DMRG calculations, and were foundto be within 10% of the exact value for a tiny fraction of thecomputational cost.

This theory of exciton transfer was then applied to modelexciton migration in conformationally disordered PPV. Keyto this modeling is the assumption that the donor and accep-tor chromophores are defined by local exciton ground states(LEGSs). Since LEGSs are readily determined by the excitoncenter-of-mass wavefunction, this theory provides a quantita-tive link between polymer conformation and exciton migra-tion. (LEGSs are both donor and acceptor chromophores, be-cause exciton-polaron formation does not occur in conjugatedpolymers.12 This observation provides a simplification overprevious work.11)

Our Monte Carlo simulations indicate that the excitondiffusion length (i.e., the mean distance traveled by an excitonbefore recombination) depends weakly on the polymer con-formation, with the diffusion length increasing slightly as thechromophores became straighter and longer. This is largelya geometrical effect: longer and straighter chromophores ex-tend over larger distances. It also indicates the role of phasecoherence, as this determines the chromophore size. The cal-culated diffusion lengths of ∼10 nm are in good agreementwith experiment.

In addition to determining the diffusion length of exci-tons in PPV, the spectral properties of the migrating exci-tons were also investigated. The emission intensity ratio of the0-0 and 0-1 vibronic peaks is related to the effective Huang-Rhys parameter of the emitting state, which in turn is relatedto the chromophore size.1, 2, 27 The intensity ratios calculatedfrom the effective Huang-Rhys parameters are in agreement

with experimental spectra, and the time-resolved trend for theintensity ratio to decrease with time was also reproduced asthe excitation migrates to shorter, lower energy chromophoresas a function of time. In addition, the energy of the exci-ton state shows a logarithmic decrease with time, in agree-ment with experimental observations5, 26 and earlier theoreti-cal predictions.5 The magnitude of this decrease is in broadagreement, too.26 Although very good qualitative agreementwith experiment has been obtained throughout this paper, andnot unreasonable quantitative comparisons have been made,an exact quantitative match is made difficult by the lack ofknowledge of the actual disorder present in the experimentalsamples used.

The obvious next step for work in this field is to investi-gate other possible causes of exciton migration in conjugatedpolymers. For example, in solution it is likely that there willbe significant torsional motion within the polymer chains onthe time scale of our exciton hops.31 If a dilute solution isassumed, then interchain migration will be suppressed, but itis likely that thermal fluctuations within the chain will causeinternal migration of the excitation as the torsional angleschange, altering the eigenstates of the Frenkel Hamiltonian.Two possible types of motion are possible—a smooth, adia-batic transfer of energy along the chain as the eigenstate itselfmoves, or a sudden, non-adiabatic hop as the exciton energy israised by random torsional fluctuations, leading to other statesof the chain becoming lower in energy.

VII. CONCLUDING REMARKS

This section draws together the main conclusions ofthis series of three papers. In 2004, Beenken and Pulleritsposed the rhetorical question, “Spectroscopic Units inConjugated Polymers: A Quantum Chemically FoundedConcept?”32 Their answer to this question was that torsionsand kinks do not cause segmentation. Instead they proposedthat a chromophore is defined by the spatial extent of anexciton-polaron, caused by the coupling of an exciton tothe nuclear coordinates. We believe that this proposal canbe rejected, because recent work by Tozer and Barford12 onthe fully quantized Frenkel-Holstein model has shown thatthe formation of self-localized exciton-polarons does notoccur in conjugated polymers. Self-localized solutions ofBorn-Oppenheimer Hamiltonians only occurs in the limit ofvanishing phonon frequencies; such a condition is not met inconjugated polymers. In addition, there are no experimentalobservations that can uniquely be explained by exciton-polaron formation (an example is ultra-fast fluorescencedepolarization, as explained in the Appendix).

A quantitative proposal that correlates chromophoreswith polymer conformation was made by Rossi, Chance, andSilbey.33 However, this proposal fails to link conformationswith the phase coherence of the exciton center-of-mass wave-function, and again we believe that it should be rejected.

The suggestion that the coherence length of the exci-ton center-of-mass wavefunction is related to chromophoresizes has been discussed in the literature for many years,34 ashas the notion that disorder disrupts the phase coherence andcauses Anderson localization of excitons.28, 35–38 These ideas

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164103-13 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

were put on a quantitative footing with the observation thatin one-dimensional systems the low-energy spectrum is com-posed of super-localized, virtually nodeless, non-overlappingand space filling states (named local ground states13 (LGSs)or local exciton ground states14, 38 (LEGSs)). As explained inSec. I of Ref. 1, these states naturally define chromophores.This then raises the question, what are the experimental pre-dictions of such a definition?

The goal of this and the accompanying two papers hasbeen to address this question by developing theories of opti-cal transitions and exciton transfer in conjugated polymers. Inboth cases, we derived simple extensions of the usual Born-Oppenheimer expressions for optical intensities and excitontransfer rates where an effective Franck-Condon factor of achromophore, which is a function of an effective Huang-Rhysparameter, plays the role of the Franck-Condon factor for asingle normal coordinate. The theory of optical transitions andexciton transfer in conjugated polymers proves to be a ratherstraightforward extension of existing theory, because of thefortuitous region of parameter space occupied by conjugatedpolymers. ¯ω/J is small enough that the Born-Oppenheimerfactorization is valid, but large enough that the self-localizedexciton-polaron solutions are not applicable.12 This meansthat exciton center-of-mass wavefunctions, and hence the spa-tial and energetic distribution of chromophores, are easilyobtained from the (appropriately parametrized) one-particleFrenkel model, while the effective Franck-Condon factors aredetermined by the size of the chromophores.

The link between chromophore sizes and optical transi-tions and exciton transfer is that the effective Huang-Rhys pa-rameter is inversely proportional to the chromophore size. Ashas been observed before, the key experimental observable isthen the emission intensity ratio, I00/I01, as this is proportionalto the chromophore size.1, 2, 27 How the observed intensity ra-tio depends on polymer conformations is then the test for thetheory that LEGSs define chromophores. We believe that thisexperimental test is well met by the theory, as we have at-tempted to show in this paper and Paper II.1

APPENDIX: ULTRA-FAST FLUORESCENCEDEPOLARIZATION

Ultra-fast fluorescence depolarization implies a rotationof the excited state transition dipole moment within tensof femtoseconds of photoexcitation. It has been suggestedthat this process can be explained by the formation of self-localized exciton polarons (a process sometimes called “dy-namical localization”).39, 40 As has been explained, however,there are strong theoretical objections to the concept ofself-localized exciton-polarons in conjugated polymers.12 Wethus need alternative explanations for ultra-fast fluorescencedepolarization.

We propose a mechanism of ultra-fast fluorescence depo-larization that simply relies on interconversion from higher-lying states to low-lying states (i.e., to LEGSs). We identifytwo types of interconversion. First, interconversion from localexciton excited states (LEESs) that are spatially localized overthe same region of polymer as a LEGS. Since LEESs are nec-essarily nodeful, on a bent chromophore their transition dipole

moments will be non-zero and oriented in a different directionthan their corresponding LEGS. (An example of this is shownin Fig. 3 of Ref. 1 which shows the transition dipole momentsof the j = 1 and 2 states on a semicircular chromophore.)A second interconversion occurs from quasiextended excitonstates (QEES), that are spatially extended over many LEGSs,to a particular LEGS.40 This implies dynamical localizationinto the low-energy tail of the density of states where the ex-citon states are more localized, and hence a rotation of thetransition dipole moment.

Another ultra-fast mechanism arises from Coulomb-induced excimer formation via the delocalization of an ex-citon between chromophores.41 Typically the chromophoreswill have a quasi-parallel orientation giving a quasi-H ex-cimer, whose higher lying state has the higher oscillatorstrength. Photoexcitation to this state followed by rapid in-terconversion to the lower state will cause a rotation of thetransition dipole moment (TDM) (by 90◦ if the chromophoreshave precisely the same TDM magnitudes).

1M. Marcus, O. R. Tozer, and W. Barford, J. Chem. Phys. 141, 164102(2014).

2W. Barford and M. Marcus, J. Chem. Phys. 141, 164101 (2014).3B. Movaghar, M. Grünewald, B. Reis, H. Bässler, and D. Wurz, Phys. Rev.B 33, 5545 (1986).

4B. Movaghar, B. Reis, and M. Grünewald, Phys. Rev. B 34, 5574 (1986).5S. C. J. Meskers, J. Hüber, M. Oestreich, and H. Bässler, J. Phys. Chem. B105, 9139 (2001).

6D. Beljonne, G. Pourtois, C. Silva, E. Hennebicq, L. M. Herz, R. H. Friend,G. D. Scholes, K. Müllen, and J. L. Brédas, Proc. Natl. Acad. Sci U.S.A.99, 10982 (2002).

7E. Hennebicq, G. Pourtois, G. D. Scholes, L. M. Herz, D. M. Russell,C. Silva, S. Setayesh, A. C. Grimsdale, K. Müllen, J. L. Brédas, and D.Beljonne, J. Am. Chem. Soc. 127, 4744 (2005).

8S. Athanasopoulos, E. Hennebicq, D. Beljonne, and A. B. Walker, J. Phys.Chem. C 112, 11532 (2008).

9J. Singh, E. R. Bittner, D. Beljonne, and G. D. Scholes, J. Chem. Phys. 131,194905 (2009).

10H. Ma, T. Qin, and A. Troisi, J. Chem. Theory Comput. 10, 1272 (2014).11W. Barford, A. Ward, and E. R. Bittner, J. Phys. Chem. A 116, 10319

(2012).12O. R. Tozer and W. Barford, Phys. Rev. B 89, 155434 (2014).13A. V. Malyshev and V. A. Malyshev, Phys. Rev. B 63, 195111 (2001).14D. V. Makhov and W. Barford, Phys. Rev. B 81, 165201 (2010).15S. R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev. B 48, 10345

(1993).16Z. G. Soos, G. W. Hayden, P. C. M. McWilliams, and S. Etemad, J. Chem.

Phys. 93, 7439 (1990); M. J. McIntire, E. S. Manas, and F. C. Spano, ibid.107, 8152 (1997); E. S. Manas and F. C. Spano, ibid. 109, 8087 (1998); J.Cornil, D. A. dos Santos, X. Crispin, R. Silbey, and J. L. Brédas, J. Am.Chem. Soc. 120, 1289 (1998); D. Beljonne, J. Cornil, R. Silbey, P. Millié,and J. L. Brédas, J. Chem. Phys. 112, 4749 (2000).

17W. Barford, J. Chem. Phys. 126, 134905 (2007).18S. T. Hoffmann, H. Bässler, and A. Köhler, J. Phys. Chem. B. 114, 17037

(2010).19G. D. Scholes, D. S. Larsen, G. R. Fleming, G. Rumbles, and P. L. Burn,

Phys. Rev. B 61, 13670 (2000).20W. Barford, N. Paiboonvorachat, and D. Yaron, J. Chem. Phys. 134, 234101

(2011).21A. Köhler, S. T. Hoffmann, and H. Bässler, J. Am. Chem. Soc. 134, 11594

(2012).22D. Chen, M. J. Winkour, M. A. Masse, and F. E. Karasz, Polymer 33, 3116

(1992).23S. R. Scully and M. D. McGehee, J. Appl. Phys. 100, 034907 (2006).24A. J. Lewis, A. Ruseckas, O. P. M. Gaudin, G. R. Webster, P. L. Burn, and

I. D. W. Samuel, Org. Electron. 7, 452 (2006).25D. E. Markov, E. Amsterdam, P. W. M. Blom, A. B. Sieval, and J. C. Hum-

melen, J. Phys. Chem. A 109, 5266 (2005).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

202.28.191.34 On: Sun, 21 Dec 2014 07:17:41

164103-14 W. Barford and O. R. Tozer J. Chem. Phys. 141, 164103 (2014)

26G. R. Hayes, I. D. W. Samuel, and R. T. Phillips, Phys. Rev. B 52, R11569(1995).

27F. C. Spano and H. Yamagata, J. Phys. Chem. B 115, 5133 (2011); H. Ya-magata and F. C. Spano, Phys. Chem. Lett. 5, 622 (2014).

28R. Chang, J. H. Hsu, W. S. Fann, K. K. Liang, C. H. Chang, M. Hayashi,J. Yu, S. H. Lin, E. C. Chang, K. R. Chuang, and S. A. Chen, Chem. Phys.Lett. 317, 142 (2000).

29G. A. Sherwood, R. Cheng, T. M. Smith, J. H. Werner, A. P. Shreve, L. A.Peteanu, and J. Wildeman, J. Phys. Chem. C 113, 18851 (2009).

30We note, however, that our simulation does not include photo-induced tor-sional relaxation, which is expected to slightly planarize and stiffen thechain, thus increasing the chromophore sizes.

31N. M. Albu and D. J. Yaron, J. Phys. Chem. C 117, 12299 (2013).32W. J. D. Beenken and T. Pullerits, J. Phys. Chem. B 108, 6164 (2004).

33G. Rossi, R. R. Chance, and R. Silbey, J. Chem. Phys. 90, 7594 (1989).34C. De Leener, E. Hennebicq, J.-C. Sancho-Garcia, and D. Beljonne, J.

Phys. Chem. B 113, 1311 (2009).35J. Rissler, Chem. Phys. Lett. 395, 92 (2004).36L. T. Lui, D. Yaron, and M. A. Berg, J. Phys. Chem. C 111, 5770

(2007).37W. J. D. Beenken, Phys. Status Solidi A 206, 2750 (2009).38W. Barford and D. Trembath, Phys. Rev. B 80, 165418 (2009).39A. Ruseckas, P. Wood, I. D. W. Samuel, G. R. Webster, W. J. Mitchell, P.

L. Burn, and V. Sundström, Phys. Rev. B 72, 115214 (2005).40W. Barford, I. Boczarow, and T. Wharram, J. Phys. Chem. A 115, 9111

(2011); O. R. Tozer and W. Barford, ibid. 116, 10310 (2012).41M. H. Chang, M. J. Frampton, H. L. Anderson, and L. M. Herz, Phys. Rev.

Lett. 98, 027402 (2007).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

202.28.191.34 On: Sun, 21 Dec 2014 07:17:41