Polarons in π-Conjugated Polymers: Anderson or Landau?

Using both analytical expressions and the density matrix renormalization group method, we study the fully quantized disordered Holstein model to investigate the localization of charges and excitons by vibrational or torsional modes-i.e., the formation of polarons-in conformationally disordered π-conjugated polymers. We identify two distinct mechanisms for polaron formation, namely Anderson localization via disorder (causing the formation of Anderson polarons) and self-localization by self-trapping via normal modes (causing the formation of Landau polarons). We identify the regimes where either description is more valid. The key distinction between Anderson and Landau polarons is that for the latter the particle wave function is a strong function of the normal coordinates, and hence the "vertical" and "relaxed" wave functions are different. This has theoretical and experimental consequences for Landau polarons. Theoretically, it means that the Condon approximation is not valid, and so care needs to be taken when evaluating transition rates. Experimentally, it means that the self-localization of the particle as a consequence of its coupling to the normal coordinates may lead to experimental observables, e.g., ultrafast fluorescence depolarization. We apply these ideas to poly(p-phenylenevinylene). We show that the high frequency C-C bond oscillation only causes Landau polarons for a very narrow parameter regime; generally we expect disorder to dominate and Anderson polarons to be a more applicable description. Similarly, for the low frequency torsional fluctuations we show that Anderson polarons are expected for realistic parameters.

Charge mobility induced by Brownian fluctuations in π-conjugated polymers in solution.

We study the motion of a doped charge in a π-conjugated polymer chain in solution subject to Brownian fluctuations. Specifically, we take poly(para-phenylene) to be our model system where the Brownian fluctuations cause rotational motion of the phenylene rings. The instantaneous torsional fluctuations cause Anderson localization of the charge wavefunction, with the lower-energy spectrum being composed of local ground states and the higher-energy spectrum being composed of quasi-extended states. At low temperatures, additional charge localization occurs via torsional relaxation. The dynamical torsional fluctuations lead to two distinct modes of motion of the charge: adiabatic and non-adiabatic. Adiabatic motion is a 'crawling' motion of the charge along the polymer chain while the charge remains in its local ground state. Non-adiabatic motion is a rapid 'hopping' motion as the charge is excited into higher energy quasi-extended states and travels ballistically along the chain before relaxing into a local ground state. The adiabatic motion dominates at low temperatures, and exhibits a linear temperature dependence and thus a constant zero-field charge mobility. Non-adiabatic motion begins to dominate as the temperature is increased, as the charge is thermally excited into higher energy states. At high temperatures the diffusion constant becomes almost temperature independent, indicating a decrease in the charge mobility with increasing temperature, which we attribute to the charge localization length being a decreasing function of temperature at high temperatures.

Intrachain exciton dynamics in conjugated polymer chains in solution.

We investigate exciton dynamics on a polymer chain in solution induced by the Brownian rotational motion of the monomers. Poly(para-phenylene) is chosen as the model system and excitons are modeled via the Frenkel exciton Hamiltonian. The Brownian fluctuations of the torsional modes were modeled via the Langevin equation. The rotation of monomers in polymer chains in solution has a number of important consequences for the excited state properties. First, the dihedral angles assume a thermal equilibrium which causes off-diagonal disorder in the Frenkel Hamiltonian. This disorder Anderson localizes the Frenkel exciton center-of-mass wavefunctions into super-localized local exciton ground states (LEGSs) and higher-energy more delocalized quasi-extended exciton states (QEESs). LEGSs correspond to chromophores on polymer chains. The second consequence of rotations-that are low-frequency-is that their coupling to the exciton wavefunction causes local planarization and the formation of an exciton-polaron. This torsional relaxation causes additional self-localization. Finally, and crucially, the torsional dynamics cause the Frenkel Hamiltonian to be time-dependent, leading to exciton dynamics. We identify two distinct types of dynamics. At low temperatures, the torsional fluctuations act as a perturbation on the polaronic nature of the exciton state. Thus, the exciton dynamics at low temperatures is a small-displacement diffusive adiabatic motion of the exciton-polaron as a whole. The temperature dependence of the diffusion constant has a linear dependence, indicating an activationless process. As the temperature increases, however, the diffusion constant increases at a faster than linear rate, indicating a second non-adiabatic dynamics mechanism begins to dominate. Excitons are thermally activated into higher energy more delocalized exciton states (i.e., LEGSs and QEESs). These states are not self-localized by local torsional planarization. During the exciton's temporary occupation of a LEGS-and particularly a quasi-band QEES-its motion is semi-ballistic with a large group velocity. After a short period of rapid transport, the exciton wavefunction collapses again into an exciton-polaron state. We present a simple model for the activated dynamics which is in agreement with the data.

Theory of optical transitions in conjugated polymers. II. Real systems.

The theory of optical transitions developed in Barford and Marcus ["Theory of optical transitions in conjugated polymers. I. Ideal systems," J. Chem. Phys. 141, 164101 (2014)] for linear, ordered polymer chains is extended in this paper to model conformationally disordered systems. Our key result is that in the Born-Oppenheimer regime the emission intensities are proportional to S(1)/⟨IPR⟩, where S(1) is the Huang-Rhys parameter for a monomer. ⟨IPR⟩ is the average inverse participation ratio for the emitting species, i.e., local exciton ground states (LEGSs). Since the spatial coherence of LEGSs determines the spatial extent of chromophores, the significance of this result is that it directly relates experimental observables to chromophore sizes (where ⟨IPR⟩ is half the mean chromophore size in monomer units). This result is independent of the chromophore shape, because of the Born-Oppenheimer factorization of the many body wavefunction. We verify this prediction by density matrix renormalization group (DMRG) calculations of the Frenkel-Holstein model in the adiabatic limit for both linear, disordered chains and for coiled, ordered chains. We also model optical spectra for poly(p-phenylene) and poly(p-phenylene-vinylene) oligomers and polymers. For oligomers, we solve the fully quantized Frenkel-Holstein model via the DMRG method. For polymers, we use the much simpler method of solving the one-particle Frenkel model and employ the Born-Oppenheimer expressions relating the effective Franck-Condon factor of a chromophore to its inverse participation ratio. We show that increased disorder decreases chromophore sizes and increases the inhomogeneous broadening, but has a non-monotonic effect on transition energies. We also show that as planarizing the polymer chain increases the exciton band width, it causes the chromophore sizes to increase, the transition energies to decrease, and the broadening to decrease. Finally, we show that the absorption spectra are more broadened than the emission spectra and that the broadening of the absorption spectra increases as the chains become more coiled. This is primarily because absorption occurs to both LEGSs and quasi-extended exciton states (QEESs), and QEES acquire increased intensity as chromophores bend, while emission only occurs from LEGSs.

Theory of exciton transfer and diffusion in conjugated polymers.

We describe a theory of Förster-type exciton transfer between conjugated polymers. The theory is built on three assumptions. First, we assume that the low-lying excited states of conjugated polymers are Frenkel excitons coupled to local normal modes, and 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. The resulting expression for the exciton transfer rate has a familiar form, being a function of the exciton transfer integral and the effective Franck-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-Oppenheimer expressions were checked against DMRG calculations, and are found to be within 10% of the exact value for a tiny fraction of the computational cost. This theory of exciton transfer is then applied to model exciton migration in conformationally disordered poly(p-phenylene vinylene). Key to this modeling is the assumption that the donor and acceptor chromophores are defined by local exciton ground states (LEGSs). Since LEGSs are readily determined by the exciton center-of-mass wavefunction, 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 conformation of the polymer, with the diffusion length increasing slightly as the chromophores became straighter and longer. This is largely a geometrical effect: longer and straighter chromophores extend over larger distances. The calculated diffusion lengths of ~10 nm are in good agreement with experiment. The spectral properties of the migrating excitons are also investigated. The emission intensity ratio of the 0-0 and 0-1 vibronic peaks is related to the effective Huang-Rhys parameter of the emitting state, which in turn is related to the chromophore size. The intensity ratios calculated from the effective Huang-Rhys parameters are in agreement with experimental spectra, and the time-resolved trend for the intensity ratio to decrease with time was also reproduced as the excitation migrates to shorter, lower energy chromophores as a function of time. In addition, the energy of the exciton state shows a logarithmic decrease with time, in agreement with experimental observations.

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.