William Barford

Exciton transfer integrals between polymer chains.

The line-dipole approximation for the evaluation of the exciton transfer integral J between conjugated polymer chains is rigorously justified. Using this approximation, as well as the plane-wave approximation for the exciton center-of-mass wave function, it is shown analytically that J approximately L when the chain lengths are smaller than the separation between them, or J approximately L-1 when the chain lengths are larger than their separation, where L is the chain length. Scaling relations are also obtained numerically for the more realistic standing wave approximation for the exciton center-of-mass wave function, where it is found that for chain lengths larger than their separation J approximately L-1.8 or J approximately L-2, for parallel or collinear chains, respectively. These results have important implications for the photophysics of conjugated polymers and self-assembled molecular systems, as the Davydov splitting in aggregates and the Forster transfer rate for exciton migration decrease with chain lengths larger than their separation. This latter result has obvious deleterious consequences for the performance of polymer photovoltaic devices.

Excitons in Conjugated Polymers: A Tale of Two Particles

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 first-principles theory of exciton diffusion in these systems.

Exciton dynamics in disordered poly(p-phenylenevinylene). 1. Ultrafast interconversion and dynamical localization.

The disordered Frenkel-Holstein model is introduced to investigate dynamical relaxation and localization of photoexcited states in conformationally disordered poly(p-phenylenevinylene). It is solved within the Ehrenfest approximation, in which the excited state is treated fully quantum mechanically, but the nuclear displacements are treated classically. The following are shown: (i) Lower energy local exciton ground states (LEGSs) adiabatically relax to vibrationally relaxed states (VRSs) in the time scale of one or two vibrational periods (ca. 40 fs). The relaxation of LEGSs is accompanied by localization and fluorescence depolarization, as the transition dipole moment reduces and rotates. The amount of dynamical localization increases as the torsional disorder decreases, causing an increase in the fluorescence depolarization. (ii) Higher energy quasi-extended exciton states (QEESs) interconvert to VRSs via three distinct episodes. A brief initial period of adiabatic relaxation is followed by the time-evolving eigenstate becoming a linear superposition of instantaneous eigenstates of the Frenkel-Holstein Hamiltonian. Typically, after a few hundred femtoseconds, one of the instantaneous eigenstates dominates the linear superposition, and the remaining dynamics is again adiabatic relaxation to a VRS. (iii) Very high energy QEESs, which are delocalized over many chromophores, sometimes exhibit a splitting of the wave function into more than one VRS. This self-localization onto more than one chromophore is assumed to be a failure of the Ehrenfest approximation, as this approximation neglects quantum mechanical coherences between the electronic and nuclear degrees of freedom. (iv) QEESs exhibit larger, but slower, fluorescence depolarization than LEGSs. Thus, ultrafast fluorescence depolarization is a function of excitation energy and conformational disorder.

Exciton localization in disordered poly(3-hexylthiophene)

Singlet exciton localization in conformationally disordered poly(3-hexylthiophene) (P3HT) is investigated via configuration interaction (singles) calculations of the Pariser-Parr-Pople model. The P3HT structures are generated by molecular dynamics simulations. The lowest-lying excitons are spatially localized, space filling, and nonoverlapping. These define spectroscopic segments or chromophores. The strong conformational disorder in P3HT causes breaks in the pi-conjugation. Depending on the relative values of the disorder-induced localization length and the distances between the pi-conjugation breaks, these breaks sometimes serve to pin the low-lying localized excitons. The exciton confinement also causes a local spectrum of low-lying exciton states. Coulomb-induced intra- or interchain interactions between spectroscopic segments in close spatial proximity can delocalize an exciton across these segments, in principle causing phase coherent transition dipole moments.

Ultrafast dynamical localization of photoexcited states in conformationally disordered poly(p-phenylenevinylene).

We consider two types of ultrafast dynamical localization of photoexcited states in conformationally disordered poly(p-phenylenevinylene). First, we discuss nonadiabatic interconversion from higher energy extended exciton states to lower energy more localized local exciton ground states. Second, we calculate the dynamics of local exciton ground states on their Born-Oppenheimer potential energy surfaces. We show that within the first C-C bond oscillation following photoexcitation (∼35 fs) the exciton becomes self-trapped and localized over approximately eight monomers. This process is associated with a Calderia-Leggett type loss of phase coherence owing to the coupling of the polymer to a dissipative environment. Subsequent torsional relaxation (on a time scale of approximately picoseconds) has little effect on the localization. We conclude from this that the initial torsional disorder determines the spatial distribution and localization length of vertical excitations but that electron-phonon coupling is largely responsible for the localization length of self-trapped excitons. We next consider the effect of dynamical localization on fluorescence depolarization. We show that exciting higher energy states causes a larger fluorescence depolarization, because these states have a larger initial delocalization. Using the observation that fluorescence depolarization is a function of excitation wavelength and polymer conformation, we show how the models of exciton localization discussed here can be experimentally investigated.

Exciton Dynamics in Disordered Poly(p-phenylenevinylene). 2. Exciton Diffusion

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(p-phenylenevinylene) 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.

Theory of optical transitions in conjugated polymers. I. Ideal systems.

We describe a theory of linear optical transitions in conjugated polymers. The theory is based on three assumptions. The first is that the low-lying excited states of conjugated polymers are Frenkel excitons coupled to local normal modes, described by the Frenkel-Holstein model. Second, we assume that the relevant parameter regime is ℏω ≪ J, i.e., the adiabatic regime, and thus the Born-Oppenheimer factorization of the electronic and nuclear degrees of freedom is generally applicable. Finally, we assume that the Condon approximation is valid, i.e., the exciton-polaron wavefunction is essentially independent of the normal modes. Using these assumptions we derive an expression for an effective Huang-Rhys parameter for a chain (or chromophore) of N monomers, given by S(N) = S(1)/IPR, where S(1) is the Huang-Rhys parameter for an isolated monomer. IPR is the inverse participation ratio, defined by IPR = (∑(n)|Ψ(n)|(4))(-1), where Ψ(n) is the exciton center-of-mass wavefunction. Since the IPR is proportional to the spread of the exciton center-of-mass wavefunction, this is a key result, as it shows that S(N) decreases with chain length. As in molecules, in a polymer S(N) has two interpretations. First, ℏωS(N) is the relaxation energy of an excited state caused by its coupling to the normal modes. Second, S(N) appears in the definition of an effective Franck-Condon factor, F(0v)(N) = S(N)(v)exp ( - S(N))/v! for the vth vibronic manifold. We show that the 0 - 0 and 0 - 1 optical intensities are proportional to F00(N) and F01(N), respectively, and thus the ratio of the 0 - 1 to 0 - 0 absorption and emission intensities are proportional to S(N). These analytical results are checked by extensive DMRG calculations and found to be generally valid, particularly for emission. However, for large chain lengths higher-lying quasimomentum exciton states become degenerate with the lowest vibrational excitation of the lowest exciton state. When this happens there is mixing of the electronic and nuclear states and a partial breakdown of the Born-Oppenheimer approximation, meaning that the ratio of the 0 - 0 to 0 - 1 absorption intensities no longer increases as fast as the IPR. When ℏω/J = 0.1, a value applicable to phenyl-based polymers, the critical value of N is ~20 monomers.

Porphyrins for Probing Electrical Potential Across Lipid Bilayer Membranes by Second Harmonic Generation

Neurons communicate by using electrical signals, mediated by transient changes in the voltage across the plasma membrane. Optical techniques for visualizing these transmembrane potentials could revolutionize the field of neurobiology by allowing the spatial profile of electrical activity to be imaged in real time with high resolution, along individual neurons or groups of neurons within their native networks.1, 2 Second harmonic generation (SHG) is one of the most promising methods for imaging membrane potential, although so far this technique has only been demonstrated with a narrow range of dyes.3 Here we show that SHG from a porphyrin-based membrane probe gives a fast electro-optic response to an electric field which is about 5–10 times greater than that of conventional styryl dyes. Our results indicate that porphyrin dyes are promising probes for imaging membrane potential.

Excitons in conjugated polymers: wavefunctions, symmetries, and quantum numbers.

We introduce a mapping from configuration interaction singles wavefunctions, expressed as linear combinations of particle-hole excitations between Hartree-Fock molecular orbitals, to real-space exciton wavefunctions, expressed as linear combinations of particle-hole excitations between localized Wannier functions. The exciton wavefunction is a two-dimensional amplitude for the exciton center-of-mass coordinate, R, and the electron-hole separation (or relative coordinate), r, having an exact analogy to one-dimensional hydrogenlike wavefunctions. We describe the excitons by their appropriate quantum numbers, namely, the principle quantum number, n, associated with r and the center-of-mass pseudomomentum quantum number, j, associated with R. In addition, for models with particle-hole symmetry, such as the Pariser-Parr-Pople model, we emphasize the connection between particle-hole symmetry and particle-hole parity. The method is applied to the study of excitons in trans-polyacetylene and poly(para-phenylene).

Electron-Lattice Relaxation, and Soliton Structures and Their Interactions in Polyenes

Density matrix renormalisation group calculations of a suitably parametrised model of long polyenes (polyacetylene oligomers), which incorporates both long range Coulomb interactions and adiabatic lattice relaxation, are presented. The triplet and 2Ag states are found to have a 2-soliton and 4-soliton form, respectively, both with large relaxation energies. The 1Bu state forms an exciton-polaron and has a very small relaxation energy. The relaxed energy of the 2Ag state lies below that of the 1Bu state. The soliton/anti-soliton pairs are bound.