Once the Hamiltonian is fully parametrized, standard sampling techniques
(e.g. Monte Carlo, molecular dynamics, or dissipative particle dynamics) can be
used to explore the phase space and to evaluate macroscopic observables. Note that
the coarse-grained hexylthiophenes interact even at a distance of 2σ ¼ 1.2 nm,
which is almost twice the average separation of their centers of mass, estimated
as ρ
À 1=3
0
% 0.63 nm. In order to obtain realistic values for the effective excluded
volume, the interactions for strong overlaps of the density clouds must be weak, i.e.,
κu 0
ð Þ $ k B T. This results in soft potentials and boosts the efficiency of phase-space
sampling, especially if the Monte Carlo algorithm is used.
Simulations using the reptation algorithm [76, 77] showed that one can equilibrate systems on the scale of %50 Â 50 Â 50 nm
3 , containing around 5 Â 10
À5
hexylthiophenes with up to 32 monomers per chain. Depending on the coupling
strength λ and the degree of polymerization, plate-like nematic mesophases (with
normals of the thiophene rings parallel to the director) as well as biaxial phases (see
Fig. 5 for a representative snapshot) have been observed. It has also been shown that
the model predicts reasonable values for the persistence length and Frank elastic
constants [64].
With the large-scale morphology at hand, some insights can be obtained on how
the collective orientation of chains affects the energetic landscape for driftdiffusing charges. This has been done by splitting polymer segments onto conjugated segments (using a simple criterion for the torsion angle [78]; see, however,
Sect. 5.1) and evaluating gas-phase ionization energies of these segments. Even this
crude model predicts that isotropic melts consist of short conjugated segments with
defects uniformly distributed along the chains. In the biaxial nematic case, the
average segment length increases significantly, and the collective orientation of
these segments leads to a spatially correlated energetic landscape, even without
accounting for long-range Coulomb interactions [64].
4 Rate-Based Description of Charge Transport
We will now discuss the semiconducting properties of P3HT. When choosing an
appropriate model for charge transport in this polymer, we have to rely on the
experimentally observed increase in mobility with increasing temperature. This is
interpreted as a sign for temperature-activated hopping transport. In other words,
charges (charged states) are localized and charge transfer reactions that propagate
the localized states are thermally activated. Localization can occur on single
molecules (typically observed in amorphous small-molecule-based organic semiconductors) or molecular assemblies (crystalline materials). In polymers, charge
localizes on molecular, or conjugated, segments, as discussed in Sect. 5.1.
If charge transfer rates are known, the resulting master equation for occupation
probabilities of these localized states describes charge dynamics in the system.
Hence, the solution of the master equation provides information about charge
Morphology and Charge Transport in P3HT: A Theorist’s Perspective
151
(e.g. Monte Carlo, molecular dynamics, or dissipative particle dynamics) can be
used to explore the phase space and to evaluate macroscopic observables. Note that
the coarse-grained hexylthiophenes interact even at a distance of 2σ ¼ 1.2 nm,
which is almost twice the average separation of their centers of mass, estimated
as ρ
À 1=3
0
% 0.63 nm. In order to obtain realistic values for the effective excluded
volume, the interactions for strong overlaps of the density clouds must be weak, i.e.,
κu 0
ð Þ $ k B T. This results in soft potentials and boosts the efficiency of phase-space
sampling, especially if the Monte Carlo algorithm is used.
Simulations using the reptation algorithm [76, 77] showed that one can equilibrate systems on the scale of %50 Â 50 Â 50 nm
3 , containing around 5 Â 10
À5
hexylthiophenes with up to 32 monomers per chain. Depending on the coupling
strength λ and the degree of polymerization, plate-like nematic mesophases (with
normals of the thiophene rings parallel to the director) as well as biaxial phases (see
Fig. 5 for a representative snapshot) have been observed. It has also been shown that
the model predicts reasonable values for the persistence length and Frank elastic
constants [64].
With the large-scale morphology at hand, some insights can be obtained on how
the collective orientation of chains affects the energetic landscape for driftdiffusing charges. This has been done by splitting polymer segments onto conjugated segments (using a simple criterion for the torsion angle [78]; see, however,
Sect. 5.1) and evaluating gas-phase ionization energies of these segments. Even this
crude model predicts that isotropic melts consist of short conjugated segments with
defects uniformly distributed along the chains. In the biaxial nematic case, the
average segment length increases significantly, and the collective orientation of
these segments leads to a spatially correlated energetic landscape, even without
accounting for long-range Coulomb interactions [64].
4 Rate-Based Description of Charge Transport
We will now discuss the semiconducting properties of P3HT. When choosing an
appropriate model for charge transport in this polymer, we have to rely on the
experimentally observed increase in mobility with increasing temperature. This is
interpreted as a sign for temperature-activated hopping transport. In other words,
charges (charged states) are localized and charge transfer reactions that propagate
the localized states are thermally activated. Localization can occur on single
molecules (typically observed in amorphous small-molecule-based organic semiconductors) or molecular assemblies (crystalline materials). In polymers, charge
localizes on molecular, or conjugated, segments, as discussed in Sect. 5.1.
If charge transfer rates are known, the resulting master equation for occupation
probabilities of these localized states describes charge dynamics in the system.
Hence, the solution of the master equation provides information about charge
Morphology and Charge Transport in P3HT: A Theorist’s Perspective
151