Excitons in Conjugated Polymers: A Tale of Two Particles - The

Review Article pubs.acs.org/JPCA

Excitons in Conjugated Polymers: A Tale of Two Particles William Barford* Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, United Kingdom ABSTRACT: Since the discovery of electroluminescence in the phenyl-based conjugated polymers in 1990, the field of polymer optoelectronics has matured to the extent that presently a wide class of devices have been commercialized. These range from both miniature and wide-area light emitting devices to hybrid photovoltaic devices. Similarly, our understanding of the fundamental processes that determine these optoelectronic properties has also progressed. In particular, owing to insights from both experimental and theoretical investigations, the role of the primary excited states, i.e., excitons, is now considerably clearer. This review discusses these primary excited states and explains how the three key roles of electron−electron interactions, electron−nuclear coupling, and disorder determine their properties. We show that the properties of an exciton are more readily understood by decomposing it into two effective particles. First, a relative particle that describes the size and binding energy of the electron−hole pair. Second, a center-of-mass particle that describes the extent of the delocalization of the electron−hole pair. Disorder and coupling to the normal modes localizes the center-of-mass particle and provides a quantitative definition of chromophores in conjugated polymers, paving the way for a firstprinciples theory of exciton diffusion in these systems.

I. INTRODUCTION Conjugated polymers in the condensed phase are characterized by strong intramolecular and weak intermolecular interactions. Consequently, polymers behave electronically as quasi-onedimensional systems. Because the effects of electron−electron interactions, electron−nuclear coupling, and disorder are all enhanced in one dimension, these effects play a key role in determining the electronic and optical properties of polymers. Indeed, it is the interplay of these effects that makes the study of the excited states of conjugated polymers a challenging, but fascinating, exercise. For example, if the electronic correlations are particularly strong, as is the case in trans-polyacetylene, a reversal of excited state energies renders the polymer nonelectroluminescent. In fact, prior to 1990, this was the expected property of all conjugated polymers, and so the discovery of electroluminescence in poly(p-phenylenevinylene) was largely serendipitous.1 It is now established that electroluminescent conjugated polymers behave as “conventional” band gap semiconductors, albeit with strong electron−electron and electron−nuclear coupling. (In contrast, nonelectroluminescent conjugated polymers should be more properly classified as Mott−Hubbard insulators.) In a conventional semiconductor the ground state is composed of an occupied set of molecular orbitals (that comprise the valence band) and a vacant set of molecular orbitals (that comprise the conduction band). As shown in Figure 1, an electronic excitation corresponds to promoting an electron from a filled orbital to an empty orbital. The vacancy left behind by the electron in an otherwise full set of orbitals is termed a “hole”. A hole has all the attributes of an electron, except for being positively charged, and hence there is a Coulomb attraction between the electron and hole. The linear combination of all pairs of electron−hole excitations is a bound © XXXX American Chemical Society

Figure 1. An electron, promoted from an occupied molecular orbital to a vacant molecular orbital, that is Coulombically bound to the hole it left behind. An exciton is a linear combination of all such electron− hole excitations.

state, whose excitation energy lies below the conduction band threshold. This is an exciton. Excitons are the primary photoexcited states of conjugated polymers.2 To motivate the study of excitons in conjugated polymers, we make a brief digression to discuss excitons in conventional three-dimensional inorganic semiconductors.3,4 Assuming that all the other electrons interact with the exciton as a mean field, then in a typical three-dimensional semiconductor the exciton is analogous to a hydrogen atom (or more precisely, a Received: October 12, 2012 Revised: December 15, 2012

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dx.doi.org/10.1021/jp310110r | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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positronium atom), albeit with a screened electron−hole interaction. The quantum mechanical solution of the hydrogen atom is “trivial”, because in free space a two-particle problem can be decomposed into two effective one-particle problems. One of the effective particles describes the relative dynamics of the electron and proton, whereas the other effective particle describes the dynamics of the center-of-mass of the electron and proton. Similarly, if we ignore the crystalline structure of an inorganic semiconductor, the quantum mechanical solutions for an exciton are also trivial, because now the electron and proton are simply replaced by an electron and hole. In this case the relative particle is bound to a screened Coulomb potential with a binding energy of

En =

1 ⎛ e2 ⎞ ⎟ ⎜ 2 ⎝ 4πε0εrrn ⎠

(1)

where εr is the relative permittivity. The length scale rn is the mutual radius of the electron−hole orbit, given by

rn = n2a0

(2) Figure 2. Shape and size of excitons in an ordered poly(p-phenylene) chain:6 surface plots of exciton wave functions, Φnj(r,R). r is the electron−hole separation and R is the center-of-mass coordinate, where all distances are in units of the monomer length. The 11B1u (n = 1, j = 1) state is a Frenkel exciton, whereas the 21Ag (n = 2, j = 1) state is a charge-transfer exciton. Notice that odd n excitons have an even parity with respect to r, i.e., Φnj(r,R) = +Φnj(−r,R), whereas even n excitons have an odd parity with respect to r, i.e., Φnj(r,R) = −Φnj(−r,R).

where a0 =

4πε0εr ℏ2 μe 2

(3)

is the exciton Bohr radius and n is the principal quantum number. The reduced mass of the relative particle is μ = m*e m*h /(m*e + m*h ), where m*e and m*h are the band-masses of the electron and hole, respectively. In GaAs, for example, εr = 12.8 and μ = 0.05me, implying small binding energies (4.2 meV) and large electron−hole separations (13 nm) for the n = 1 exciton. Such excitons are termed Mott−Wannier excitons.

Having given a qualitative description of exciton wave functions in conjugated polymers, we now quantify the exciton properties by separately considering first, the relative and second, the center-of-mass particles.

II. EXCITON WAVEFUNCTIONS IN CONJUGATED POLYMERS Excitons are confined to one dimension in conjugated polymers. Thus, the exciton has only two degrees of freedom: the motion of the relative particle with a coordinate r (equal to the electron−hole separation) and the motion of the center-ofmass particle with a coordinate R. The relative and center-ofmass particles are described by the wave functions ψn(r) and Ψj(R), respectively. Associated with these two degrees of freedom are two quantum numbers: the principle quantum number, n, and a center-of-mass quantum number, j. Figure 2 shows the calculated two-dimensional wave functions, Φnj(r,R) = ψn(r) Ψj(R), of the two lowest energy excitons of poly(para-phenylene).5,6 The primary photoexcited 11B1u state is the n = 1, j = 1 exciton. The mean electron−hole separation in this state is less than one phenyl ring in size. This is a Frenkel exciton, where the electron and hole are strongly bound on a single moiety. The center-of-mass of this bound state (i.e., the center-of-mass particle), however, freely delocalizes along a uniform chain, entirely analogously to a particle-in-a-box. The 21Ag state, which is dipole-connected to the 11B1u state, is the n = 2, j = 1 exciton. The n = 2 exciton has a nodal plane at r = 0. This is a charge-transfer exciton, whose mean electron−hole separation is approximately three monomer units. Similarly, the n = 3, j = 1 exciton (or the n1B1u state) has two nodal planes, and the mean electron−hole separation is approximately seven monomer units.

III. THE RELATIVE PARTICLE AND EXCITON BINDING ENERGIES As shown by Loudon,7 a particle bound to a Coulomb potential in one dimension has subtly different solutions to the threedimensional solutions. In particular, the even n solutions (i.e., odd parity ψn(r)) have binding energies that satisfy eqs 1−3, but with n replaced by n/2, whereas the odd n solutions (i.e., even parity ψn(r)) for n > 1 are bounded by the even n solutions. Thus, for n > 1, the energy scale for the binding energies is the effective Rydberg, R, given by eq 1 with rn ≡ a0. Because R ∝ εr−2, the much weaker electronic screening in πelectron systems partly explains the stronger binding energy of some excitons in conjugated polymers. However, as also shown by Loudon,7 the energy of the n = 1 exciton is split-off from the Rydberg series, and its binding energy is determined by the energy scale of the Coulomb repulsion between electrons in the same π-orbital. This very large electron−electron repulsion explains the strong electron−hole binding of the Frenkel exciton, as discussed below. The exciton wave functions shown in Figure 2 were obtained from CI-S quantum chemistry calculations.5,6 These calculations also predict a binding energy. However, these predictions are generally unreliable, because the solvation energy of a single charge in a dielectric is both large and difficult to calculate.8 We can, however, make a rough estimate of B



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wave function of the center-of-mass particle on a uniform chain is given by

binding energies from these wave functions, which are consistent with experimental optical observations. A simple estimate of exciton binding energies is obtained via the Virial theorem, which states that a particle bound in a Coulomb potential has an energy equal to one-half of its Coulomb energy. (The Bohr theory of the hydrogen atom automatically predicts this, as shown by eq 1. But, in addition, the Bohr theory also predicts the form of rn (via eq 2).) Thus, using eq 1 with rn=1 ∼ 3 Å for the electron−hole separation in the 11Bu exciton of poly(p-phenylene) and εr ∼ 2.2 gives a binding energy for this exciton of ∼0.9 eV. Similarly, using rn=2 ∼ 13 Å for the 21Ag exciton gives a binding energy of ∼0.2 eV. These energy differences are entirely consistent with the linear and nonlinear optical data, which identifies the 11Bu and 21Ag states as being separated by ∼0.7−0.8 eV,9−11 and the 21Ag and n1Bu states as being separated by ∼0.1−0.2 eV.9,12 It is reasonable to assume that the n1Bu state is near to or at the electron−hole continuum, as its electron−hole separation of ca. seven phenyl rings is comparable to the chromophore size (as discussed in the next section). Thus, a lower bound on the binding energy of the 1Bu exciton is ∼0.8 eV in phenyl-based systems.

Ψj(R ) =

εj = E0 + 2J cos βj

Ψj = 1(R ) =

t(ϕ)2 ΔE

⎛ γ ⎞1/2 ⎜ ⎟ sech(γ(R − L /2)/d) ⎝2⎠

(8)

where L = Nd, γ = A2ℏω/4|J|, A is the dimensionless displacement (related to the Huang−Rhys parameter, S, via S = A2/2), and ℏω is the energy of the vibrational quantum. The width of the center-of-mass wave function is ∼γ−1, and the transition from the free-particle wave function (eq 6) to the self-localized form (eq 8) is dynamical localization. The relaxation energy of the exciton-polaron with respect to the ground state geometry is Er = |J|γ2/3. B. Role of Disorder. In reality, polymers are rarely free from some kind of disorder and thus the form of eq 6 is not valid for the center-of-mass wave function of vertical excitations. Polymers in solution are necessarily conformationally disordered as a consequence of thermal fluctuations. Polymers in the condensed phase usually exhibit glassy, disordered conformations as consequence of being quenched from solution. As well as conformational disorder, polymers are also subject to chemical disorder (arising from the synthesis process) and environmental disorder (arising, for example, from density fluctuations). Conformational disorder renders JSE a random variable and is the source of off-diagonal disorder, whereas environmental disorder is the source of diagonal disorder. Disorder localizes the exciton center-of-mass particle, and determines their energetic and spatial distributions. The origin of localization lies in the wave behavior of particles.15−17 The effect of conformational and environmental disorder on a particle is analogous to a wave being scattered from a series of irregularly spaced barriers of differing heights. The superposition of waves with different phases, caused by the partial reflection and transmission from the barriers, exponentially localizes the particle wave function.

(4)

where μ0 is the transition dipole moment of a single moiety and d is the distance between moieties. Second, for all excitons there is a superexchange (or through bond) mechanism, whose origin lies in a virtual fluctuation from a Frenkel exciton on a single moiety to a charge-transfer exciton spanning two moieties back to a Frenkel exciton on a neighboring moiety. The energy scale for this process, obtained from second order perturbation theory, is JSE ∝ −

(7)

where E0 is the excitation energy of an exciton localized on a single moiety. A. Role of Electron−Nuclear Coupling. Immediately that an exciton is excited, owing to a change in the bond order, the exciton couples to the nuclei and undergoes vibrational relaxation. In one-dimensional systems there is a barrierless process to the formation of self-localized exciton polarons (or vibrationally relaxed states, VRSs).13 Typically, it only takes a couple of C−C bond vibrations (i.e., ~ 40 fs) for the excited state to adiabatically relax on its Born−Oppenheimer potential energy surface. If the center-of-mass particle couples to a normal mode local to each moiety, then the self-trapping transition is described by the Holstein model.14 In the large polaron limit the center-ofmass wave function now takes the form

2μ0 2 4πε0εrd3

(6)

where βj is the center-of-mass pseudo-wavevector, βj = jπ/(N + 1)d, j is the quantum number satisfying 1 ≤ j ≤ N, and N is the number of moieties. (This is illustrated in Figure 2 for j = 1 for a fixed value of r.) These states form an exciton band for each principal quantum number, n, whose kinetic energy on a onedimensional lattice takes the form

IV. THE CENTER-OF-MASS PARTICLE AND CHROMOPHORES The primary, optically active exciton in the phenyl-based lightemitting polymers is the n = 1 exciton, so this will be our focus for the rest of this review. As described in the last section, for this class of polymers the electron and hole are tightly bound onto a single moiety. To a good approximation, therefore, the relative wave function can be considered to be localized onto a single moiety. We now want to consider how this tightly bound electron−hole pair delocalizes along the polymer chain. In principle, excitons delocalize along the chain via two mechanisms. First, for dipole allowed excitons (i.e., singlet excitons with positive electron−hole parity) there is a Coulomb-induced dipolar (or through space) mechanism. This is the familiar mechanism of Förster resonant energy transfer. The exciton transfer integral for this process is JDD = −

⎛ 2 ⎞1/2 ⎜ ⎟ sin(βjR ) ⎝ N + 1⎠

(5)

where t(ϕ) is proportional to the overlap of π-orbitals neighboring a bridging bond, i.e., t(ϕ) ∝ cos ϕ and ϕ is the dihedral angle between neighboring moieties. ΔE is the difference in energy between a charge-transfer and Frenkel exciton. Evidently, JSE vanishes when ϕ = 90°. The total exciton transfer integral is J = JDD + JSE. Because J is negative, it is often useful to regard a conjugated polymer as a Jaggregate, where each moiety in the polymer chain replaces the role of a molecule in an aggregate. In a tight-binding model the C



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extent of a LEGS defines an absorbing chromophores. Similarly, the spatial extent of a VRS defines an emissive chromophore. The effect of conformational disorder on the optical properties of conjugated polymers can be understood by examining its role on the energetic distribution of the vertical and relaxed states. As Figure 4 illustrates, conformational

The role of disorder in localizing excitons is illustrated by Figure 3. This shows the four lowest vertically excited exciton

Figure 3. Exciton center-of-mass wave functions, Ψ(R), for LEGSs (solid curves) and their corresponding VRSs (dotted curves) in conformationally disordered poly(p-phenylenevinylene).24 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.

center-of-mass wave functions on conformationally disordered poly(p-phenylenevinylene). Rather than there being a set of particle-in-the-box wave functions, each of which spans the entire chain with an increasing number of nodes, as given by eq 6, there are four essentially nodeless, nonoverlapping wave functions that together spatially span the entire chain. Because of their nodeless qualities, these states have been termed local exciton ground states (or LEGSs).18,19 LEGSs are localized by the static conformational disorder present in the ground state geometry and their size is determined by the amount of disorder. As described above, however, immediately that an exciton is created, it undergoes vibrational relaxation.20−23 The VRSs associated with their vertical LEGSs are also shown in Figure 3, and take the from of eq 8. The concepts of LEGSs and VRSs introduced here resolves a long-standing controversy concerning the definition of chromophores in conjugated polymers. A chromophore may be regarded as the irreducible part of a polymer chain that absorbs or emits light. Although this definition of a chromophore is probably uncontroversial, a difficulty has been in relating it to polymer conformations.4,25−28 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. A problem with this definition of a chromophore is that for a singlet exciton the concept of a “conjugation break” is not useful, because, as described above, exciton transfer between moieties occurs via both a dipole−dipole and superexchange mechanism. Thus, even if JSE vanishes because of negligible π-orbital overlap, JDD will not, and the exciton can retain phase coherence over the “conjugation break”.29 However, as LEGSs are virtually nodeless, nonoverlapping, and space filling, they precisely satisfy the properties of the lowest energy excited states of the chromophores of a polymer chain. In particular, the nodeless qualities implies that the square of the transition moment of a LEGS scales with its localization length.19 Thus, the spatial

Figure 4. Density of states for LEGS (squares) and VRS (circles) in conformationally disordered poly(p-phenylenevinylene).24 Also shown is the density of states of emissive (or trap) states (triangles), defined as those VRSs from which one or more excitons recombines during a simulation of exciton diffusion (see section VB). The standard deviation of the conformational disorder is (a) σϕ = 2° and (b) σϕ = 10°.

disorder has three important effects. First, an increase in disorder raises the average energy of the vertical transitions; second, it increases the Stokes shift; and third, it increases the inhomogeneous line width. The first effect implies that the absorption energy is blue-shifted by disorder. As will be explained below, the second and third effects imply that disorder causes a red shift in the emission.

V. FLUORESCENCE DEPOLARIZATION AND EXCITON DIFFUSION We have already shown how the identification of the n = 1 and n = 2 excitons by linear and nonlinear optical spectroscopies is consistent with the calculated size of the electron−hole pair (or, equivalently, the spatial extent of the relative particle), as illustrated in Figure 2. Now we turn to a discussion of the experimental signatures that are consistent with the concept of a chromophores as being defined by the static and dynamical localization of the center-of-mass particle. A. Fluorescence Depolarization. First, we consider ultrafast fluorescence depolarization, which is an experimental D



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Figure 5. Left panel: Part of a conformationally disordered PPV chain. Also shown schematically is the vertical (LEGS at t = 0) and relaxed (VRS at t = 195 fs) wave function.22 Right panel: calculated time-dependent fluorescence anisotropy for an ensemble 50 PPV chains containing 500 phenylene moieties each for various values of the torsional disorder, σϕ.23 The inset shows the experimentally determined fluorescence anisotropy in PPV31 (reproduced with permission).

a Coulomb-induced, Förster-like process of exciton transfer between donors and acceptors.32 An early model of exciton diffusion assumed that the donors and acceptors are point-dipoles whose energy distribution is a Gaussian random variable.33−35 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 conformations and the energetic and spatial distributions of the donors and acceptors via the concept of a chromophore.36−40 The usual practice has been to define chromophores via a minimum threshold in the π-orbital overlaps and then obtain a distribution of energies by assuming that the excitons delocalize freely on the chromophores thus defined. As discussed above, however, this procedure is at best arbitrary. An unambiguous link between polymer conformations and chromophores may be made by defining absorbing and emissive chromophores via the spatial extent of LEGSs and VRSs, respectively. This leads to a quantitative determination of the energetic and spatial distributions of chromophores. To make the link between these definitions of chromophores and a theory of exciton diffusion, we need to apply the Condon approximation. This is motivated as follows. 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 the assumption that donor chromophores are in their excited state relaxed geometries (and are therefore VRSs), whereas acceptor chromophores are in their ground state relaxed geometries (and are therefore LEGSs). This process of exciton transfer between donor and acceptor chromophores is illustrated in Figure 6. As an exciton diffuses through a polymer system, it does so via hopping between chromophores that become steadily further apart and lower in energy. Because there are progressively fewer chromophores that satisfy the correct

signature of dynamical localization. The left panel of Figure 5 shows a schematic representation of an absorbing chromophore (i.e., a LEGS) and the corresponding emissive chromophore (i.e., a VRS) on a strand of conformationally disordered PPV. As the exciton wave function self-localizes onto a smaller strand of the chain due to its coupling to the C−C bond vibrations, the transition dipole moment (TDM) both rotates and changes in magnitude, thus causing fluorescence depolarization. The rotation of the TDM prior to emission manifests itself as ultrafast flouresecence depolarization, which takes place in less than 100 fs. The fluorescence anisotropy of an ensemble of molecules is defined as30

⎛ I − I⊥ ⎞ ⎟⎟ r = ⎜⎜ ⎝ I + 2I⊥ ⎠

(9)

where I∥ and I⊥ are the observed fluorescence intensities parallel and perpendicular to the incident polarization, respectively. The right panel of Figure 5 shows the calculated time-dependent fluorescence anisotropy for an ensemble of disordered PPV chains, which are consistent with the experimentally determined values (shown in the inset).31 Notice that a smaller conformational disorder causes a larger depolarization, because the difference in the spatial sizes of the absorbing and emitting chromophores is larger. B. Exciton Diffusion. 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. Although 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 typically 1012−1013 s−1, whereas exciton transfer rates are typically 109−1011 s−1. Consequently, singlet exciton diffusion is E



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Figure 6. 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).

energy requirements, the time interval between hops rises from ∼20 ps to ca. nanoseconds, prior to radiative recombination. Exciton diffusion via progressively lower energy chromophores results in spectral diffusion: a time-dependent change in the fluorescence wavelength. Spectral diffusion is illustrated in Figure 7, which shows that experimentally the average energy of the emitted photon is approximately proportional to −log t. This behavior is reproduced theoretically by assuming a Gaussian density of states for point-like donors and acceptors.33−35 Perhaps unsurprisingly, it is also reproduced by a more realistic simulation of exciton migration from donor VRSs to acceptor LEGSs, because, as shown in Figure 4, the distribution of energy of these states is quasi-Gaussian. The red-shift in emission with increased disorder shown in Figure 7 can also be explained by Figure 4, which shows the distribution of emissive (or trap) states. These lie in the tail of the density of states of the VRS, and because increased disorder both increases the Stokes shift and broadens the distribution, the energy of the emitted photon decreases as the disorder increases. Kinetic Monte Carlo simulations in disordered PPV, assuming exciton transfer from donor VRSs to acceptor LEGSs, predicts exciton diffusion lengths of ∼8−11 nm, with the higher values for more ordered chains. These predictions are in good agreement with experimental values for PPV and its derivatives41−44 and are typical for conjugated polymers in general.32

Figure 7. Energy of emitted photons as a function of time. Top panel: experimental data from polyfluorene films35 (reproduced with permission). The key denotes the excitation energy. Bottom panel: calculated results from kinetic Monte Carlo simulations in conformationally disordered PPV.24 The radiative recombination occurs from the emissive (or trap) states, whose density of states is shown in Figure 4.

particles. First, a relative particle, which is localized by a Coulomb potential. Its localization length is a measure of the size of the electron−hole pair and determines the exciton binding energy. Second, the center-of-mass particle, which describes the extent of the delocalization of the electron−hole pair along the polymer. This particle is localized by both static disorder and electron−nuclear coupling. Localization by the static disorder determines the size of absorbing chromophores, whereas dynamical localization by electron−nuclear coupling determines the size of emissive chromophores. Indeed, by using single particle models whose parameters are derived from higher-level quantum chemistry calculations or are semiempirical inputs, one may make quantitative predictions for large polymer systems. This review has discussed the examples of ultrafast fluorescence depolarization and exciton diffusion. Such quantitative predictions are important if theory is to help provide design principles for more efficient polymer optoelectronic devices.

VI. CONCLUDING REMARKS The combined effects of electron−electron interactions, electron−nuclear coupling, and disorder would seem to present considerable obstacles in any attempt to predict the excited stateand hence, photophysicalproperties of conjugated polymers. However, these effects can be qualitatively understood with the realization that the excited states of polymers, i.e., excitons, can be decomposed into two quasi-independent F



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(34) Movaghar, B.; Reis, B.; Grünewald, M. Phys. Rev. B 1986, 34, 5574. (35) Meskers, S. C. J.; Hüber, J.; Oestreich, M.; Bässler, H. J. Phys. Chem. B 2001, 105, 9139. (36) Beljonne, D.; Pourtois, G.; Silva, C.; Hennebicq, E.; Herz, L. M.; Friend, R. H.; Scholes, G. D.; Müllen, K.; Brédas, J. L. Proc. Natl. Acad. Sci U. S. A. 2002, 99, 10982. (37) Hennebicq, E.; Pourtois, G.; Scholes, G. D.; Herz, L. M.; Russell, D. M.; Silva, C.; Setayesh, S.; Grimsdale, A. C.; Müllen, K.; Brédas, J. L.; Beljonne, D. J. Am. Chem. Soc. 2005, 127, 4744. (38) Athanasopoulos, S.; Hennebicq, E.; Beljonne, D.; Walker, A. B. J. Phys. Chem. C 2008, 112, 11532. (39) Singh, J.; Bittner, E. R.; Beljonne, D.; Scholes, G. D. J. Chem. Phys. 2009, 131, 194905. (40) Kruger, B. P.; Scholes, G. D.; Fleming, G. R. J. Phys. Chem. B 1998, 102, 5378. (41) Halls, J. J. M.; Pilcher, K.; Friend, R. H.; Moratti, S. C.; Holmes, A. B. Appl. Phys. Lett. 1996, 68, 3120. (42) Markhov, D. E.; Amsterdam, E.; Blom, P. W. M.; Sieval, A. B.; Hummelen, J. C. J. Phys. Chem. A 2005, 109, 5266. Markhov, D. E.; Tanase, C.; Blom, P. W. M.; Wildeman, J. Phys. Rev. B 2005, 72, 045217. (43) Scully, S. R.; McGehee, M. D. J. Appl. Phys. 2006, 100, 034907. (44) Lewis, A. J.; Ruseckas, A.; Gaudin, O. P. M.; Webster, G. R.; Burn, P. L.; Samuel, I. D. W. Org. Electron. 2006, 7, 452.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS I thank Nattapong Paiboonvorachat for permission to reproduce Figure 2. REFERENCES

(1) Burroughes, J. H.; Bradley, D. D. C.; Brown, A. R.; Marks, R. N.; Mackay, K.; Friend, R. H.; Burn, P. L.; Holmes, A. D. Nature 1990, 347, 539. (2) Barford, W. Electronic and Optical Properties of Conjugated Polymers, 2nd ed.; Oxford University Press: Oxford, U.K., 2013. (3) Knox, R. S. Solid State Physics; Academic: New York, 1963; Suppl. 5. (4) For a review of excitons in nanoscale systems, see: Scholes, G. D.; Rumbles, G. Nat. Mater. 2006, 5, 683. (5) Barford, W.; Paiboonvorachat, N. J. Chem. Phys. 2008, 129, 164716. (6) Paiboonvorachat, N. Part II thesis, Department of Chemistry, University of Oxford, 2008. (7) Loudon, R. Am. J. Phys. 1959, 27, 649. (8) Moore, E.; Gherman, B.; D. Yaron, D. J. Chem. Phys. 1997, 106, 4216. Moore, E.; Yaron, D. J. Chem. Phys. 1998, 109, 6147. (9) Martin, S. J.; Bradley, D. D. C.; Lane, P. A.; Mellor, H.; Burn, P. L. Phys. Rev. B 1999, 59, 15133. (10) Cadby, A. J.; Lane, P. A.; Mellor, H.; Martin, S. J.; Grell, M.; Giebler, C.; Bradley, D. D. C.; Wohlgenannt, M.; An, C.; Vardeny, Z. V. Phys. Rev. B 2000, 62, 15604. (11) Frolov, S.; Bao, Z.; Wohlgenannt, M.; Z. V. Vardeny, Z. V. Phys. Rev. B 2002, 65, 205209. (12) Mathy, A.; Ueberhofen, K.; Schenk, R.; Gregorius, H.; Garay, R.; Müllen, K.; Bubeck, C. Phys. Rev. B 1996, 53, 4367. (13) Rasba, E. I. Excitons, 1982, edited by Rashba E. I.; Sturge, M. D. North-Holland. (14) Holstein, T. Ann. Phys. (N.Y.) 1959, 8, 325; Ann. Phys. (N.Y.) 1959, 8, 343. (15) Anderson, P. W. Phys. Rev. 1958, 109, 1492. (16) Mott, N. F.; Twose, W. D. Adv. Phys. 1961, 10, 107. (17) Kramer, B.; MacKinnon, A. Rep. Prog. Phys. 1993, 56, 1469. (18) Malyshev, A. V.; Malyshev, V. A. Phys. Rev. B 2001, 63, 195111. (19) Makhov, D. V.; Barford, W. Phys. Rev. B 2010, 81, 165201. (20) Tretiak, S.; Saxena, A.; Martin, R. L.; Bishop, A. R. Phys. Rev. Lett. 2002, 89, 97402. (21) Karabunarliev, S.; Bittner, E. R. J. Chem. Phys. 2003, 118, 4291. (22) Barford, W.; Boczarow, I.; Wharram, T. J. Phys. Chem. A 2011, 115, 111. (23) Tozer, O. R.; Barford, W. J. Phys. Chem. A 2012, 116, 10310. (24) Barford, W.; Bittner, E. R.; Ward, A. J. Phys. Chem. A 2012, 116, 10319. (25) Beenken, W. J. D.; Pullerits, T. J. Phys. Chem. B 2004, 108, 6164. (26) Beenken, W. J. D. Phys. Status Solidi A 2009, 206, 2750. (27) Barford, W.; Trembath, D. Phys. Rev. B 2009, 80, 165418. (28) Lupton, J. M. ChemPhysChem 2012, 13, 901. (29) Barford, W.; Lidzey, D. G.; Makhov, D. V.; Meijer, A. J. H. J. Chem. Phys. 2010, 133, 044504. (30) Lakowicz, J. R. Principles of Fluorescence Spectroscopy; Plenum Press: New York, 1999. (31) Ruseckas, A.; Wood, P.; Samuel, I. D. W.; Webster, G. R.; Mitchell, W. J.; Burn, P. L.; Sundström, V. Phys. Rev. B 2005, 72, 115214. (32) Huang, I.; Scholes, G. D. Chem. Mater. 2011, 23, 610. (33) Movaghar, B.; Grünewald, M.; Reis, B.; Bässler, H.; Wurz, D. Phys. Rev. B 1986, 33, 5545. G


Excitons in Conjugated Polymers: A Tale of Two Particles - The

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