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Exciton Dynamics in Disordered Poly(p‑phenylenevinylene). 2. Exciton Diffusion William Barford,*,† Eric R. Bittner,‡ and Alec Ward†,§ †

Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, United Kingdom ‡ Department of Chemistry, University of Houston, Houston, Texas 77204, United States § Christ Church College, University of Oxford, Oxford OX1 1DP, United Kingdom ABSTRACT: We present a first principles theory of exciton diffusion in conformationally disordered conjugated polymers. Central to our theory is that exciton transfer occurs from vibrationally relaxed states (VRSs) to local exciton ground states (LEGSs). LEGSs are determined by the diagonal and off-diagonal disorder induced by static density and torsional fluctuations, and VRSs are further localized by exciton−phonon coupling. The theory is implemented using the Frenkel−Holstein model to calculate the wave functions and energies of the LEGSs and VRSs. The coupling of VRSs and LEGSs via long-range dipole−dipole interactions leads to the familiar line-dipole approximation for the exciton transfer integral. The exciton transfer rates are derived from the Fermi Golden rule. The theory is applied to an ensemble of conformationally disordered poly(pphenylenevinylene) chains using a kinetic Monte Carlo algorithm. The following are shown: (i) Torsional disorder and trans−cis defects reduce the exciton diffusion length. (ii) Radiative recombination occurs from VRSs in the tail of their density of states. (iii) Torsional disorder increases the band gap, the line width of the density of states, and the Stokes shift. As a consequence, it causes a blue shift in the vertical absorption, but a red shift in the emission. (iv) The energy of the radiated photon decreases as −log t, with a gradient that increases with torsional disorder. The predicted exciton diffusion lengths of ∼8−11 nm are in good agreement with experimental values.

I. INTRODUCTION Experimental and theoretical activities to understand exciton migration in conjugated polymer systems are, in part, motivated by the importance of this process in determining the efficiency of polymer electronic devices. In photovoltaic devices, large exciton diffusion lengths are necessary so that excitons can migrate efficiently to regions where charge separation can occur. However, precisely the opposite is required in light emitting devices, because this leads to nonradiative quenching of the exciton. Though coherent, ballistic processes may be important on the ultrafast (1−10 fs) time scale, it is widely recognized that exciton migration is an incoherent or diffusive process. This is because dissipation rates are typically1,2 1012− 1013 s−1, whereas exciton transfer rates are typically 109−1011 s−1. Consequently, most theoretical models of singlet exciton diffusion assume a Coulomb-induced, Förster-like process of exciton transfer between donors and acceptors.3 An early model assumed that the donors and acceptors are point-dipoles whose energy distribution is a Gaussian random variable.4,5 Although this model does reproduce some experimental features, such as the time dependence of spectral diffusion, there is no quantitative link between the model and actual polymer conformations and morphology. More recent approaches have attempted to make the link between random polymer conformations6−9 and the energy © 2012 American Chemical Society

and spatial distributions of the donors and acceptors via the concept of a “chromophore”. A chromophore may be regarded as the irreducible part of a polymer chain that absorbs or emits light.10 Although this definition of a chromophore is probably uncontroversial, the difficulty has been in relating it to polymer conformations.14−16,19 A common definition is that a chromophore is a region of a polymer chain separated by “conjugation breaks”, defined by some minimum threshold in the π-orbital overlap. Those who propose this definition then often proceed to assume that the exciton is freely delocalized on the chromophore, analogously to the lowest particle-in-abox eigenstate. Thus, a distribution of chromophore sizes gives a distribution of exciton energies,8 with the lowest energy excitons being on the longest chromophores. A problem with this definition of a chromophore is that for a singlet exciton the concept of a “conjugation break” is rather meaningless, as exciton transfer between moieties occurs via both a dipole− dipole and superexchange mechanism.15 Before describing the definition of chromophores proposed in this paper and explaining how it differs from the conventional view, we pursue our brief review of the current models. Received: July 16, 2012 Revised: September 26, 2012 Published: October 3, 2012 10319

dx.doi.org/10.1021/jp307041n | J. Phys. Chem. A 2012, 116, 10319−10327

The Journal of Physical Chemistry A

Article

wave functions) and the traditional one (based on “conjugation” breaks) is that in our case lower energy absorbing chromophores (or LEGSs) are shorter than higher energy absorbing chromophores. The derivation of the exciton transfer integrals and transfer rates is contained in section IV. Section V contains details of miscellaneous computational methods, including a discussion of the kinetic Monte Carlo simulation of exciton diffusion. We discuss our results in section VI and conclude in section VII.

All current investigations recognize that chromophore sizes are large enough to violate the point-dipole approximation, and thus either a sum over transition densities17 or the line-dipole approximation18,19 are necessary to evaluate the exciton transfer integrals. Finally, it is often assumed that the donor and acceptor chromophores are in their groundstate relaxed geometry, and thus exiton transfer occurs between “vertical” exciton states.20 In this paper we propose a model of exciton transfer that removes the arbitrary definition of the donor and acceptor chromophores. We make two key assumptions. The first assumption is that chromophores are determined directly by the way disordered polymer conformations and exciton− phonon coupling localizes exciton wave functions. The justification for this assumption is that the low-energy vertical excitations of conformationally disordered polymers form a class of superlocalized states known as local exciton ground states (or LEGSs). Because LEGSs are essentially nodeless, nonoverlapping, and space filling exciton center-of-mass wave functions21,23 their spatial extent defines an absorbing chromophore. Within a couple of C−C bond vibrations11,12 (∼40 fs), however, a LEGS adiabatically relaxes on its Born− Oppenheimer potential energy surface to form a self-localized exciton-polaron, or vibrationally relaxed state (VRS). In analogy to an absorbing chromophore, the spatial extent of a VRS defines an emissive chromophore.19 (Torsional relaxation is neglected in this work, as this causes only a small reduction in the size of VRSs in PPV11 on the time scale of ≲ picoseconds.) Now, as already stated (and shown in Figure 7a), the time interval between exciton jumps ranges from tens to hundreds of picoseconds and is much longer than vibrational relaxation times. In contrast, according to the Born−Oppenheimer approximation, the time taken for the exciton transfer is much faster than vibrational relaxation times. This hierarchy of time scales justifies our second assumption, namely that donor chromophores are in their excited state relaxed geometries (and are therefore VRSs), whereas acceptor chromophores are in their groundstate relaxed geometries (and are therefore LEGSs). We implement this theory of exciton transfer using the disordered Frenkel−Holstein model to determine the donor and acceptor states. We perform a Monte Carlo simulation of exciton diffusion in an ensemble of conformationally disordered poly(p-phenylenevinylene) (PPV) polymers. The Frenkel− Holstein model was introduced and motivated in the accompanying paper12 to model the ultrafast interconversion and dynamical localization of photoexcited states to VRSs in PPV. There it was shown that the model reliably reproduces the experimentally determined ultrafast fluorescence depolarization. Here we show that the model reliably predicts exciton diffusion lengths of ∼8−11 nm in PPV (with the value increasing as the conformational disorder decreases). This agreement with experiment over a wide range of time and length scales gives credibility to our model of exciton localization. The next section briefly describes the Frenkel−Holstein model. Section III is devoted to a description of the spatial and energetic distributions of the acceptor states (LEGSs) and donor states (VRSs). We show that the energetic distribution of LEGSs and VRSs resembles a Gaussian density of states, and so this model quantitatively reproduces the assumptions of earlier models.4,5 However, one significant difference between our definiton of chromophores (based on the localization of exciton

II. THE FRENKEL-HOLSTEIN MODEL The motivation for and description of the Frenkel−Holstein model is given more fully in the accompanying paper.12 Here we summarize the key aspects of the model pertinent to this paper. The Frenkel−Holstein model describes both the delocalization of the Frenkel exciton center-of-mass particle along the polymer chain and its coupling to local normal coordinates. The coupling of the exciton to local normal coordinates causes the center-of-mass wave function to self-localize. In PPV the local normal coordinate is associated with the benzenoidquinoid distortion for the phenylene moiety and the C−C bond stretch for the vinylene moiety. When diagonal and off-diagonal disorder is present, the Frenkel−Holstein model also describes the Anderson localization of the exciton center-of-mass wave function.21 If â†n (ân) creates (destroys) a Frenkel exciton on a phenylene or vinylene moiety, n, then the Frenkel−Holstein Hamiltonian reads ĤFH =

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

n

ℏω + 2

∑ Qn

2

n

(1)

where odd and even “sites” n represent phenylene and vinylene moieties, respectively. The on-site excitation energy is εn = E0 + ( −1)n

Δ + αn 2

(2)

where Δ is the difference in excitation energy between vinylene and phenylene moieties and αn represents diagonal disorder, whose physical origin is explained in section VC. {Qn} are the dimensionless classical displacements and ω is the angular frequency of the normal mode. The dimensionless exciton− phonon coupling parameter is A, which is related to the Huang−Rhys parameter, S, for an isolated moiety via S = A2/2. The Frenkel exciton transfer integral is15 Jn = J DD + J SE cos2 ϕn DD

(3)

SE

where J and J are the nearest neighbor dipole−dipole and superexchange contributions, respectively, and ϕn is the torsional (or dihedral) angle between moieties. Random, static fluctuations in the torsional angles from their mean value therefore causes off-diagonal disorder. When disorder is present, the lowest-lying eigenstates of eq 1 with the groundstate geometry (i.e., Qn = 0) are LEGSs.21 As explained in section IV, they are coupled via the long-range dipole−dipole interactions not included in eq 3. Assuming that the Coulomb interaction between moieties can be treated via the point dipole approximation, this interaction is represented by the operator 10320


The Journal of Physical Chemistry A ĤDA =

μ0 2 4πεr ε0

∑ m∈D n∈A

κmn R mn

3

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

(am̂ †an̂ + an̂ †am̂ ) (4)

where Rmn = |Rm − Rn|, Rm is the position of the mth moiety, and μ0 is the magnitude of the transition dipole moment of a Frenkel exciton on a single moiety. κmn is the orientational factor, κmn = rm̂ ·rn̂ − 3(R̂ mn·rm̂ )(R̂ mn·rn̂ )

(5)

where r̂m is a unit vector parallel to the dipole on moiety m and R̂ mn is a unit vector parallel to the vector joining moieties m and n. For a uniform chain (Δ = 0 and Jn = J) and in the largepolaron limit,22 defined by γ=

A2 ℏω ≪1 4J

(6)

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

Q n = Aψn 2

parameter

value

E0 Δ JSE JDD ℏω exciton−phonon coupling, A moiety transition dipole moment, μ0 diagonal disorder, σα relative permittivity, εr refractive index, n = √εr vibronic line width, Γ mean monomer volume no. of moieties per chain no. of polymer chains in the simulation sphere radius of simulation sphere

9.24 eV 3.20 eV −1.96 eV −1.35 eV 0.2 eV 4.00 1.51 × 10−29 C m 65 meV 2.25 1.50 100 meV 1.3 × 102 Å3 201 2000 29.2 nm

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

a

(7)

where ψn is the lowest energy eigenstate of eq 1. Thus, ψn satisfies a self-consistent, nonlinear Schrödinger equation, with a “self-trapping” solution in the asymptotic limit ψnVRS =

⎛ γ ⎞1/2 ⎜ ⎟ sech γ(n − n0) ⎝2⎠

(8)

Equation 8 describes an exciton-polaron (or a vibrationally relaxed state (VRS)). The relaxation energy of the self-trapped state, Er, is Er =

⎛γ ⎞ ⎜ ⎟ℏωS ≪ ℏωS ⎝6⎠

(9)

In contrast, the relaxation energies for a single monomer and a dimer are ℏωS and ℏωS/2, respectively. Thus, the relaxation energy decreases if the original vertical excitation is delocalized over a larger polymer segment. This is a consequence of the nonlinear form of the Frenkel−Holstein model: if the center-ofmass wave function is initially more delocalized the coupling of the Frenkel exciton to each local displacement is reduced. We return to this point in sections III and VI, as it explains the tendency for disorder to red-shift the fluorescence. The parametrization of the Frenkel−Holstein model is described in ref 12; for completeness the parameters are listed in Table 1.

Figure 1. Exciton center-of-mass wave functions for LEGSs (solid curves) and their corresponding VRSs (dotted curves) obtained from eq 1. The torsional disorder, σφ = 10°. The spatial extent of a LEGS represents a chromophore with the ground state geometry. This is an acceptor (or absorbing) chromophore. The spatial extent of a VRS represents a chromophore with the excited state geometry. This is a donor (or emissive) chromophore. See also Figure 4.

shown in Figure 2 of ref 12. The VRSs corresponding to the LEGSs shown in Figure 1 are also illustrated in the same figure. They are determined by iteratively applying eq 7 in eq 1 until convergence is achieved. The adiabatic relaxation of a LEGS to a VRS is accompanied by dynamical localization. We quantify the localization of the center-of-mass wave function by the conjugation length, l. We first define the localization length, L, as the root-mean-square spread of the wave function,

III. DONOR AND ACCEPTOR STATES In this section we discuss the low-energy vertical and relaxed eigenstates of the Frenkel−Holstein model, which are, respectively the acceptor and donor states that participate in the exciton diffusion. A. Spatial Distributions. LEGSs are defined by the condition that21,23 |∑ |ψn|ψn| ≳ 0.95 n

L = 2 ⟨m2⟩ − ⟨m⟩2

(11)

where ⟨m p⟩ =

(10)

where ψn is the Frenkel exciton center-of-mass wave function. Figure 1 shows some LEGS wave functions obtained from the Frenkel−Holstein model with the groundstate geometry (i.e., Qn = 0). LEGSs dominate the low-energy optical intensity, as

∑ |ψm|2 m p

(12)

m

Then the conjugation length is l ≈ 3L. Figure 2 is a scatter plot of the conjugation lengths for LEGSs and VRSs as a function of excitation energy, illustrating 21

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Figure 2. Scatter plot of the conjugation lengths for LEGSs and VRSs as a function of excitation energy. σφ = 10°. The inset shows the ensemble averaged conjugation lengths as a function of torsional disorder, σφ. The length scales are in units of the monomer unit (i.e., a phenylene and vinylene moiety). The diagonal disorder, σα = 65 meV.

that although there is large scatter in the conjugation length of the LEGSs, the general trend is that the conjugation length decreases as the excitation energy decreases. In contrast, the size of a VRS is independent of energy. As the inset of Figure 2 illustrates, the average conjugation length of a LEGS is a decreasing function of torsional disorder, although for a VRS it is virtually independent of disorder (in this case being principally determined by the value of the exciton−phonon coupling). B. Energetic Distributions. Static torsional disorder affects the energetic distribution of vertical and relaxed states in a number of ways, as illustrated in Figure 3. First, increased torsional disorder causes an increase in the width of the density of states (and concomitantly, an increase in the inhomogeneous line width12,21). Second, it causes an increase in the average energy of the LEGSs, implying a blue shift in the vertical absorption energy. The source of this effect can be traced to the behavior of the exciton transfer integral as a function of disorder. For a disordered polymer the transfer integral is

J = ⟨J ⟩ϕ + σJ

Figure 3. Density of states for LEGSs (squares) and VRSs (circles). Also shown is the density of states of emissive (or trap) states (triangles), defined as those VRSs from which one or more excitons recombine during the entire simulation. (a) σϕ = 2° and (b) σϕ = 10°. The inset shows the ensemble averaged relaxation energy (defined as the difference between the average LEGSs and VRSs energies) as a function of torsional disorder, σϕ.

IV. EXCITON TRANSFER RATES In this section we derive an approximate expression for the exciton transfer rate with the aid of two assumptions. First, we invoke the Condon approximation, which is based on the assumption that the electronic transition occurs much faster than nuclear time scales. Thus, the electronic matrix element is assumed to be parametrized by the instantaneous nuclear coordinates and is taken out of the integral over the vibrational coordinates. Second, we simplify the remaining integral over the vibrational wave functions by assuming that donor and acceptor chromophores are described by single, generalized normal coordinates. The resulting expression, which captures the key physical aspects of exciton transfer in polymers, reduces to the usual Förster expression in the limit of point dipoles. Transitions between the eigenstates of the Frenkel−Holstein Hamiltonian are induced by the Coulomb interaction, Ĥ DA, given in eq 6. The Fermi golden rule expression for the rate of a transition between an initial state, |I⟩, to a final state, |F⟩, is

(13)

where ⟨J⟩ϕ is the transfer integral averaged over the torsional angles and σJ is its standard deviation. From eq 3 it is easily shown that ⟨J ⟩ϕ ≈ J(ϕ0) − J SE σϕ 2 cos 2ϕ0

(14)

and σJ ≈ J SE σϕ sin 2ϕ0

(15)

Thus, provided that 0 ≤ φ0 ≤ 45°, the magnitude of the transfer integral decreases with increasing disorder, implying a smaller exciton bandwidth and an increase in the vertical excitation energy. The final effect of torsional disorder is to increase the relaxation energy (or Stokes shift). The source of this effect is the decrease in the average LEGS conjugation length as a function of disorder, which, as discussed in section II, increases the relaxation energy. We show in section VI that the increase in the width of the density of states and the relaxation energy leads to a red shift in the fluorescence as the torsional disorder increases (in spite of the blue shift in vertical excitation).

kDA =

⎛ 2π ⎞ 2 ⎜ ⎟|⟨I |Ĥ DA |F ⟩| δ(E I − E F) ⎝ℏ⎠

(16)

where EI and EF are the initial and final energies, respectively. Suppose initially that the donor chromophore, D, is in an excited state and the acceptor chromophore, A, is in its ground state. According to the Born−Oppenheimer approximation the ket state for each chromophore is a direct product of the electronic and the nuclear kets. The initial excited state of the donor chromophore is therefore 10322


The Journal of Physical Chemistry A |I ⟩D = |EX;Q D⟩D |φ0EX ⟩⟩D

Article

JDA = D⟨ψ VRS|ĤDA|ψ LEGS⟩A

(17)

where |EX;QD⟩D is the excited electronic ket parametrized by the nuclear displacments, QD, and |φEX 0 ⟩⟩D is ground level nuclear ket associated with this excited electronic state. Similarly, for the acceptor chromophore |I ⟩A =

|GS;Q A ⟩A |φ0GS⟩⟩A

=

(18)

kDA =

|F⟩ =

X=

(19)

(20)

=

(22)

(23)

The Condon approximation is to assume that JDA(QD,QA) is determined by particular values of QD and QA, denoted as Q0D and Q0A. In this case, this is the relaxed excited state geometry of the donor chromophore and the relaxed ground state geometry of the acceptor chromophore. Thus, the donor excited state is a VRS, (24)

n

F0Dvδ(ΔEvD + E) F0Av δ(ΔEvA′ − E) dE ′

(30)

∫ φ0EX(Q D) φvGS(Q D) dQ D|2

exp(−SD)(SD)v v!

(31)

(32)

m

where μ0 is the transition dipole moment for a single moiety and r̂m is the unit vector parallel to the local dipole on the mth moiety. The transition dipole moment is thus

(25)

μTDM = ⟨ψ VRS|μ|̂ GS⟩ = μ0 ∑ rm̂ ψmVRS

(26)

m

(33)

and the radiative rate is then

Now, using the definition of HDA and the general exciton state,

∑ ψnan̂ †|0⟩

−∞

μ ̂ = μ0 ∑ rm̂ am̂ †

The electronic matrix element therefore becomes

|ψ ⟩ =

∞

V. MISCELLANEOUS COMPUTATIONAL DETAILS A. Radiative Rates. The dipole operator connecting the groundstate to an excited state is

whereas the acceptor excited state is a LEGS.

JDA (Q D0 ,Q A0 ) = D⟨ψ VRS|ĤDA|ψ LEGS⟩A

(29)

is the zero-temperature donor chromophore Franck−Condon factor, where SD is the Huang−Rhys parameter for the donor chromophore, defined by SD = EDr /ℏω and EDr is its relaxation energy. Similarly, FA0v′ is the zero-temperature acceptor chromophore Franck−Condon factor and ΔFAv′ is the excitation energy to the v′th vibrational level of the excited acceptor from the zeroth vibrational level of its groundstate. The form of the overlap function ensures that exciton transfer is an energy conserving process, but which only occurs if the relaxed energy of the acceptor chromophore is lower than that of the donor chromophore. Hence, as the new vertical LEGS of the acceptor rapidly relaxes to its VRS, exciton diffusion is inevitably a downhill energy process (at zero temperature). In the evaluation of eq 30 the delta functions are replaced by Lorentzian functions of line width Γ = 100 meV.

(21)

JDA (Q D ,Q A ) = ⟨EX;Q D|⟨GS;Q A|ĤDA|GS;Q D⟩|EX;Q A ⟩

|EX;Q A0 ⟩A ≡ |ψ LEGS⟩A

∑∫

F0Dv = |

where we define the electronic matrix element as

|EX;Q D0⟩D ≡ |ψ VRS⟩D

(28)

is the vibronic-spectral overlap factor. is the de-excitation energy from the zeroth vibrational level of the excited donor chromophore to the vth vibrational level of its ground state.

∫ ∫ JDA (Q D ,Q A) φ0EX(Q D) φ0GS(Q A) φvGS(Q D) φvEX (Q A ) d Q D d Q A ′

ψmVRSψnLEGS

ΔEDv

the full matrix element of the perturbation is ⟨I|ĤDA|F⟩ =

m∈D n∈A

κmn R mn3

⎛ 2π ⎞ 2 ⎜ ⎟J X ⎝ ℏ ⎠ DA

vv ′

′ A. Exciton Transfer Integrals. Changing to a wave function representation for the nuclear states, namely,

φ0EX (Q D) ≡ |φ0EX ⟩⟩D

∑

where

In the final state, |F⟩, the donor chromophore is in its electronic ground state, but (subject to energy conservation) in an arbitrary vibrational level, v, whereas the acceptor is in an excited state, and again (subject to energy conservation) in an arbitrary vibrational level, v′. Thus, |GS;Q D⟩D |φvGS⟩⟩D |EX;Q A ⟩A |φvEX ⟩⟩A

4πεrε0

(Notice that eq 28 is the line-dipole expression for exciton transfer integrals.18,19,24) B. Spectral Overlap. Taking JDA out of the integral over the nuclear coordinates, the transfer rate becomes

where |GS;QA⟩A is the ground electronic ket parametrized by the nuclear displacments, QA, and |φGS 0 ⟩⟩A is the ground level nuclear ket associated with the ground electronic state. In the absence of intermolecular interactions the initial state of the system is |I⟩ = |EX;Q D⟩D |φ0EX ⟩⟩D |GS;Q A ⟩A |φ0GS⟩⟩A

μ0 2

kr =

(27)

nμTDM 2 ΔE3 3πε0ℏ4c 3

(34)

where ΔE is the fluorescence energy and n is the refractive index of the medium.

the electronic coupling between the two chromophores is given by 10323



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• Set up an ensemble of conformationally disordered PPV polymer configurations within the simulation sphere. • Determine all the LEGS wave functions and energies via a full diagonalization of eq 1 (with Qn = 0) with the definition eq 10. • Determine all the corresponding VRS wave functions and energies via iterative diagonalization of eq 1 subject to eq 9. • Calculate the transfer rates between all pairs of VRSs and LEGSs. • Sum through all initial donors (i.e., all the VRSs) and perform the simulation of exciton diffusion subject to the Monte Carlo rules above. A simulation sphere typically contains an ensemble of between 5000 (for σϕ = 2°) and 13 000 (for σϕ = 15°) chromophores. Figure 4 illustrates the process of exciton transfer between chromophores proposed in this paper.

B. Polymer Conformations. An explanation of how the PPV polymer conformations are generated is given in ref 12. Here we give a brief summary of the key details. A coarsegrained approach is adopted, whereby the phenylene and vinylene moieties are represented as beads, where each bead maps onto a “site” in the Frenkel−Holstein model. The variables in generating the statistical conformations are the torsional disorder and the fraction of trans−cis defects. We define as a measure of the polymer conformation the number of moieties in a statistical segment, m. The radius of gyration of a polymer, Rg, is defined as its root-mean-square size ⎛1 R g = ⎜⎜ ⎝N

⎞1/2 2⎟ ( R R ) − ⟨ ⟩ ∑ m ⎟ ⎠ m=1 N

(35)

where Rm is the position of the mth moiety (or “site”), ⟨R⟩ is the center-of-mass position, and N is the number of moieties. Then m is defined as m = 6R g 2/Sa

(36)

where S is the contour length of the chain and a is the length of a moiety. A simulation sphere is set up whose radius is determined by the number of polymer chains (typically, 2000), the number of moieties per chain (typically, 201), and the mean volume occupied by a monomer unit (typically, 1.3 × 102 Å3),25 implying a typical sphere radius of 29.2 nm. The polymers are randomly generated in this sphere, subject to the constraint of no collisions with themselves and the surface of the sphere. Because the largest calculated mean exciton diffusion length is ∼11 nm, the size of the simulation sphere is large enough for finite-size effects to be negligible. C. Origin of Diagonal Disorder. The origin of diagonal disorder is the interpolymer dispersion interaction that red shifts the excitation energies.26 Because these pairwise interactions scale as R−6, where R is the intermoiety distance, the dispersion interaction experienced by an individual moiety is proportional to the local density. Thus, static density fluctuations cause fluctuations in the excitation energy, or diagonal disorder in the Frenkel−Holstein model. The typical red shift of the 1Bu exciton in conjugated polymers in the solid state is ∼0.1 eV.26,27 Using the random polymer conformations in the simulation sphere, we estimate the relative density fluctuations over a length scale of ∼20 Å to be ∼0.65, and thus the fluctuations in the excitation energy are σα = 65 meV. D. Kinetic Monte Carlo Algorithm and Simulation. If k is the probability of a transition occurring per unit time, then from Poisson statistics the probability of the transition not happening in a time interval, Δt, is8 P(t ) = exp( −k Δt )

Figure 4. Cartoon of exciton transfer. (1) Transfer occurs from a donor chromophore (i.e., a VRS) to an acceptor chromophore (i.e., a LEGS), which (2) adiabatic relaxes to a VRS in ∼40 fs. The exciton subsequently (3) transfers to another acceptor chromophore (LEGS) or (4) radiatively recombines. The time interval between hops for a “young” (recently photoexcited) exciton is tens of picoseconds, whereas it is over hundreds of picoseconds for an “old” exciton (close to radiative recombination); see Figure 7a.

VI. RESULTS Our key results are shown in Figure 5, which shows the exciton diffusion length as a function of torsional disorder and the number of moieties in a statistical segment, m. For these ranges of conformational disorder the exciton diffusion length is ∼8− 11 nm, in excellent agreement with experimental values for PPV and its derivatives.28−31 There are two general trends: increasing the torsional disorder or decreasing m decreases the diffusion length. The reason for the decrease in the exciton diffusion length with torsional disorder can be qualitatively understood via Figures 6 and 7. Figure 6 shows that the fraction of excitons performing n hops prior to radiative recombination is independent of the torsional disorder for a fixed fraction of trans−cis defects. The reason for this can be understood via Figure 7a, which shows that the average time interval between hops is also independent of the torsional disorder. Thus, the transfer rates (although

(37)

Thus, if x is a uniform deviate 0 ≤ x ≤ 1, we can define a (random) time interval, Δt, for a transition to occur via exp( −k Δt ) = x

(38)

For all possible transitions (nonradiative and radiative) out of a VRS, the rates are calculated and a time interval for the transitions are determined via eq 38. The transition with the smallest time interval is chosen and the overall time is updated by Δt. The simulation of exciton diffusion proceeds as follows: 10324



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Figure 5. Ensemble averaged exciton diffusion length (in nm) as a function of the size of a statistical segment for various torsional disorders. The four points for each value of σϕ correspond to a fractional trans−cis defects of 0.16, 0.08, 0.04, and 0.02 (in order of increasing m); i.e., a larger m implies a less coiled chain.

Figure 7. (a) Mean time interval between hops (in ps) and (b) the mean distance traveled in a hop (in nm) as a function of the hop number for different torsional disorders. The fraction of trans−cis defects is 0.08.

Figure 6. Fraction of excitons performing n hops prior to radiative recombination for different torsional disorder. The mean number of hops made by an exciton before radiative recombination is ∼4. The average radiative lifetime is typically 1.0−1.5 ns. The fraction of trans− cis defects is 0.08.

varying with time) are independent of torsional disorder. What is dependent on the torsional disorder, however, is the distance traveled between hops, as illustrated by Figure 7b: smaller torsional disorder implies (on average) larger hop sizes, and hence larger overall diffusion lengths. Figure 8 is a cartoon that qualitatively explains this effect. The acceptor chromophore decreases in size as the torsional disorder increases, whereas the size of the VRS (which becomes the new donor chromophore) is independent of disorder. The position of the VRS within the acceptor chromophore, however, is random. Thus, if the contour length of the acceptor chromophore is l, the root-mean-square distance of the center of the VRS from the center of the acceptor chromophore is ∼l1/2. But, the acceptor chromophore is part of a random coil in space and its spatial extent ∼l1/2. This implies that the spatial root-mean-square distance of the center of the VRS from the center of the acceptor chromophore is ∼l1/4. Thus, the spatial distance traveled in an exciton hop (defined as the distance between the centers of the initial and final VRSs) is ∼l1/4. This argument is consistent with the data. The inset of Figure 2 shows that the conjugation length of an acceptor chromophore

Figure 8. Cartoon demonstrating why the exciton diffusion length reduces as the torsional order increases. A LEGS dynamically localizes into a VRS randomly along its contour length. The left panel shows the three possible positions of a VRS. The right panel, with a shorter LEGS, shows the two possible positions of a VRS. Because torsional disorder reduces the size of the acceptor chromophore (i.e., the LEGS), it also reduces the mean exciton hop distances (shown as the distance between the centers of the VRSs (denoted by dots)).

is 36 and 18 monomer units for values of torsional disorder of 2° and 10°, respectively, whereas (from Figure 5) the diffusion length is ∼21/4 times larger for the smaller value of σφ. In contrast to the previous discussion, for a fixed torsional disorder, the average distance traveled in a hop is a weakly decreasing function of the number of trans−cis defects, whereas the time interval between hops is a weakly increasing function of the number of trans−cis defects. Consequently, the diffusion 10325



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Finally, we return to a discussion of the exciton diffusion length. Although the mean exciton diffusion length is a useful quantity for determining a figure of merit for the efficiency of a polymer PV device, of more significance is the probability that an exciton travels more than a distance D before radiative recombination. This is illustrated in Figure 10 for various values

length is an increasing function of the size of a statistical segment. The average time taken for the first exciton hop to occur after photoexcitation is ∼20 ps, whereas the time intervals between hops just prior to radiative recombination is over 20 times longer, and indeed becoming so long that a radiative transition is competitive. Similarly, the average hop distances are increasing with the hop number. These trends are a consequence of the fact that as an exciton diffuses through the polymer system it continuously looses energy. Thus, the energetic condition for exciton transfer to occur, namely EVRS ≤ EVRS A D , becomes harder to satisfy. This means that to find an acceptable acceptor the exciton has to jump further. In addition, because there are fewer energetically acceptable acceptors to choose from, in general the spectral overlap between the donor and acceptor reduces. These two effects cause the time intervals between jumps to increase rapidly with hop number. Exciton diffusion via progressively lower energy chromophores results in spectral diffusion: a time-dependent change in the fluorescence wavelength. This is illustrated in Figure 9,

Figure 10. Fraction of excitons traveling greater than a distance, D, prior to radiative recombination. The fraction of trans−cis defects is 0.08.

of the torsional disorder. As expected from the previous discussions, the median value of D is a decreasing function of torsional disorder, with more excitons traveling further for smaller disorder.

VII. CONCLUDING REMARKS We have introduced a first-principles theory of exciton diffusion in conformationally disordered conjugated polymers that removes the arbitrary definition of absorbing and emitting chromophores. This theory has been applied to PPV. Except for varying the degree of conformational disorder, there are no adjustable parameters. Our predicted exciton diffusion lengths of ∼8−11 nm are in good agreement with experimental values. The theory is based on a number of key assumptions. First, we assume that the acceptor and donor chromophores are in their ground and excited state geometries, respectively. This is a consequence of the separation of time scales between vibrational relaxation and exciton transfer events, and the Condon approximation, which assumes that the electronic transition is “vertical”. Second, we assume that the spatial and energetic distributions of the acceptor chromophores are determined by diagonal and off-diagonal disorder that localizes the exciton center-of-mass particle. These are local exciton ground states (LEGSs). The diagonal and off-diagonal disorder is caused by static density and torsional fluctuations, respectively. Third, the spatial and energetic distributions of the donor chromophores are determined both by disorder, which determines the LEGSs, and exciton−phonon coupling, which causes self-localization into vibrational relaxed states (VRSs) confined within the acceptor chromophore. Finally, we evaluate the vibronicspectral overlap by assuming a single generalized normal coordinate for the chromophores. We implement this theory using the Frenkel−Holstein model to calculate the wave functions and energies of the LEGSs and VRSs. The coupling of VRSs and LEGSs via longrange dipole−dipole interactions leads to the familiar line-

Figure 9. Energy of emitted photons as a function of time. The radiative recombination occurs from the emissive (or trap) states, whose density of states is shown in Figure 3. The fraction of trans−cis defects is 0.08. This is an ensemble average over all donors; i.e., experimentally it corresponds to absorption over the entire spectrum of LEGS (as shown in Figure 2 of ref 12).

which shows that average energy of the emitted photon is approximately proportional to −log t, in agreement with experiment5 and also with calculations that assume a Gaussian density of states.5 Fluorescence occurs from the emissive (or trap) chromophores, defined as those VRSs from which one or more excitons recombines during the entire simulation. Their density of states is shown in Figure 3, where we see that they lie in the low energy tail of the density of states of the VRSs. As discussed in section IIIB, increased torsional disorder causes an increase in the line width of the energy distribution and an increase in the Stokes shift. The first observation explains why the gradient of emitted photon energy versus log t increases with torsional disorder, because as time proceeds the recombining excitons fall deeper into the lower part of the energy distribution of the emissive states. The two observations together explain the tendency for a red shift in the fluorescence as a function of torsional disorder. However, this trend is not quite monotonic, because of the competing effect of a blue shift in the vertical absorption with increasing torsional disorder. 10326



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Foundation (CHE-1011894) and Robert A. Welch Foundation (E-1334).

dipole approximation for the exciton transfer integral and the exciton transfer rates are derived from the Fermi Golden rule. The theory is applied to an ensemble of conformationally disordered PPV chains using a kinetic Monte Carlo algorithm. We find that 1. torsional disorder and trans−cis defects reduce the exciton diffusion length, 2. radiative recombination occurs from VRSs in the tail of their density of states, 3. torsional disorder increases the band gap, the line width of the density of states, and the Stokes shift (as a consequence, it causes a blue shift in the vertical absorption, but a red shift in the emission), and 4. the energy of the radiated photon decreases as −log t, with a gradient that increases with torsional disorder. One spectral signature of exciton diffusion that we have not considered here is the fluorescence depolarization as a function of time. The accompanying paper12 discusses ultrafast fluorescence depolarization, but experimentally slower fluorescence depolarization is also observed. In our simulation the ensemble averaged fluorescence anisotropy decays instantly (at the first exciton hop). This is because the polymer conformations are uncorrelated, so a memory of the orientation of the transition dipole moment is lost once the exciton hops onto another randomly oriented chromophore. This discrepancy from experiment indicates that in reality polymer conformations are correlated on short length scales of ∼3 nm. A truly ab initio simulation of exciton diffusion must therefore start from polymer conformations determined from a molecular dynamics (MD) simulation. As well as the need for improved MD simulations, the theory would be improved by lifting the Condon approximation employed in the derivation of the exciton transfer rates. This requires a quantum mechanical treatment of the vibrational modes of the Frenkel−Holstein model. Finally, we note that there is an obvious connection with this theory of exciton diffusion in conformationally disordered polymers and charge transport. Charges are described by the Holstein model, which is eq 1, but with the exciton transfer integral, J, replaced by the charge transfer integral, t. For charges, the transfer integral between neighboring moieties is t ∝ cos ϕ, so torsional fluctuations and “conjugation breaks” play a greater role in localizing charge wave functions. In addition, the “chromophore” for a charged particle is no longer the irreducible absorbing or emitting segment of a chain, but instead, and equivalently, it is the region of the chain over which the particle retains phase coherence. Because the same separation of time scales exist for charges as for excitons, the charge self-localizes and the “donor chromophore” is smaller than the “acceptor chromophore”. Finally, for charge transport, the Förster-type transfer rate (eq 29) is replaced by the Marcus rate expression.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

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Notes

NOTE ADDED AFTER ISSUE PUBLICATION This paper was published on the Web on 10/16/2012 and is in the 10/25/2012 issue. The dissipation rates in the first paragraph were corrected to 1012−1013 and 109−1011 s−1. The corrected version reposted on 10/30/2012.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS W.B. thanks the John Fell Fund of the University of Oxford for financial support. E.R.B. was supported by the National Science 10327


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