Effects of the Environment on Charge Transport in Molecular Wires

Article pubs.acs.org/JPCC

Effects of the Environment on Charge Transport in Molecular Wires Aleksey A. Kocherzhenko,*,† K. Birgitta Whaley,† Giuseppe Sforazzini,‡,⊥ Harry L. Anderson,‡ Michael Wykes,§,# David Beljonne,§ Ferdinand C. Grozema,∥ and Laurens D. A. Siebbeles∥ †

Department of Chemistry, University of California, Berkeley, Berkeley, California 94720, United States Chemistry Research Laboratory, University of Oxford, Oxford OX1 2JD, U.K. § Chemistry of Novel Materials Laboratory, University of Mons, Mons, Belgium ∥ Department of Chemical Engineering, Delft University of Technology, Delft, The Netherlands ‡

S Supporting Information *

ABSTRACT: Supramolecular engineering offers opportunities for creating polymer-based materials with tailored conductive properties. However, this requires an understanding of intermolecular interaction effects on intramolecular charge transport. We present a study of hole transport along molecular wires consisting of fluorene−p-biphenyl or Zn− porphyrin monomer units, in dilute solutions. The intramolecular hole mobility was studied by pulse radiolysis−time-resolved microwave conductivity. Experiments were supplemented by charge transport simulations employing a quantum-mechanical description of the hole and a classical description of the polymer and solvent dynamics. The model was parametrized using ab initio and molecular dynamics calculations. It was found that the solvent-induced energy disorder along a polymer chain in common solvents (benzene, cyclohexane, acetonitrile, water) is ∼1 eV, significantly greater than the values of 0.05−0.2 eV commonly cited in the literature. Environment-induced disorder of this magnitude has profound consequences for intramolecular charge transport. The hole initial state upon injection onto a molecular wire also influences the mobility. Experiments and simulations demonstrate that supramolecular modification of polymers (coordination, rotaxination) can significantly enhance or suppress charge transport. Incorporating a molecular level description of the immediate supramolecular and solvent environment into charge transport models improves their predictive potential, providing a valuable tool for material design.

1. INTRODUCTION Over the past two decades, π-conjugated polymers have gained importance in a variety of applications, including light-emitting diodes,1−3 thin-film field effect transistors,4−6 photovoltaic cells,2,7−9 sensors,6,10,11 and data storage devices.6,12 Polymerbased devices are lightweight and flexible and come in almost any size and shape that may be desired.8,13 The production process is often cheaper, more energy-efficient, and more environment-friendly than for their inorganic counterparts.13,14 Further applications in molecular electronics are envisaged, where conjugated polymers could serve either as a framework for self-assembling electronic circuits, as electrical interconnects between functional electronic devices based on single molecules, or as an integral part of supramolecular electronic devices.12,15 In this context such materials are often referred to as “molecular wires”, even though the conductive properties of single molecules will always differ from those of a macroscopic strand of metal.16,17 Organic materials offer a multitude of opportunities for tuning of their optoelectronic properties. In addition to the techniques of doping and size control that are also available for inorganic crystalline semiconductors,18,19 it is also possible to vary the properties of polymers by chemical functionalization20 or by modifications to their environment. The latter method is surprisingly powerful: it has been shown that even small © 2012 American Chemical Society

changes in the environment of conjugated polymers can lead to variations of an order of magnitude or more in their electrical conductivity.21,22 This places conjugated polymers among the most promising materials for switching and sensing applications. However, such environmental sensitivity also raises the issue of controllability and robustness of their properties. The strong influence of the molecular environment on the conductive properties of organic materials is a well-established fact, particularly in the field of organic photovoltaics.20,23,24 Various methods of polymer sample preparation are available that allow substantial control over the sample morphology whether in thin solid films, solution, or composite materials. In thin film technology, polymer and composite samples with well-defined nanomorphologies (oriented, micropatterned, with percolative pathways of different phases, etc.) have been produced by such methods as Langmuir−Blodgett deposition,25,26 dip coating,27,28 spin coating,29,30 and zone casting,31,32 among others. The morphology of thin films can be further modified by postdeposition treatments, such as stamping33,34 and annealing.30,32 In solution, the conformation and solute−solvent interactions of organic molecules can be Received: July 20, 2012 Revised: November 12, 2012 Published: November 15, 2012 25213

dx.doi.org/10.1021/jp307216q | J. Phys. Chem. C 2012, 116, 25213−25225

The Journal of Physical Chemistry C

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respectively. The values of μ, ΔEb, and μ* are known for several types of polymer subunits.57 Among the characteristics that determine the intramolecular charge carrier mobility in conjugated polymers are the potential energy variations along the polymer chain. In band-like models, these are commonly treated as fitting parameters and their magnitudes usually assumed to be of the order of 0.1 eV, based on estimates of energy variations due to the internal degrees of freedom in a polymer.22,48,49 However, recent studies suggest that environment-induced “outer-sphere” disorder may play a significant role in determining the electronic properties of polymers.58,59 The neglect of such environment-induced disorder (as opposed to the internal orientational disorder) may be the reason why charge carrier mobilities in conjugated polymers calculated with quantum-mechanical models commonly exceed experimental values, sometimes by an order of magnitude or more.22,48,49,60 This paper explores the effects of the molecular environment on charge transport along single conjugated polymer chains in solution for two systems: polyfluorene−p-biphenyl (PFBP)61 and Zn−porphyrin based molecular wires (Chart 1).22,42 Hole

tuned by the choice of solvent.35−37 All these methods achieve some degree of control of the sample structure on mesoscopic length scales (tens to hundreds of nanometers). However, the conductive properties of a polymer may be determined primarily by the direct environment of individual monomer units,22 motivating a more molecular approach to their control. Several methods have been proposed for controlling the environment of polymers on a molecular scale.38 One possible approach involves threading polymers through zeolites or mesoporous materials that provide for a well-defined local environment.39,40 However, an important advantage of organic semiconductors, their ability to be processed in solution, is lost when polymers are threaded into a solid host. This can be avoided by instead encapsulating polymers with an organic material. In dendrimer encapsulation, the polymer acts as a central core, to which regularly branched substituents (dendrons) are covalently attached.38,41 Supramolecular chemistry methods further allow the formation of relatively stable assemblies of two or more molecules held together by noncovalent interactions. In such a supramolecular complex, the local environment of a molecular wire can be substantially modified to induce significant changes to the molecular conductivity, even without encapsulation.22,42 Additional routes to achieve increased control over the local environment are to wrap the molecular wire with a helical polymer, such as amylose43 or schizophyllan,44,45 or to thread it through a series of macrocycles, such as cyclodextrin, to form a polyrotaxane.38,46 This overview shows that modern synthetic and material processing techniques are powerful tools that could be used for tuning the conductive properties of molecular wires by engineering their local environments. To use these tools most effectively, it is essential to go beyond the molecular intuition on which much of the current material design is based and to develop a better theoretical understanding of environmental effects on charge transport. Charge transport in organic materials is generally described using either coherent band-like or incoherent hopping models.22,47−53 Band-like models treat charge carriers quantum-mechanically and follow the temporal evolution of electron (hole) wave functions. In contrast, hopping models separate charge transport into a series of quantum-mechanical charge transfers (hops) and nonunitary decoherence steps. In such models, all memory of coherent wave function evolution is lost between consecutive hops. This allows sequential charge transfers to be treated as fully independent, which leads to a classical diffusive description of the long-range charge carrier motion.54 The choice between these two classes of models is usually based on the relative magnitude of several energy characteristics of the material.22,47,55 The Ioffe−Frohlich−Sewell criterion can serve as a guideline for determining the dominant charge transport mechanism.56,57 Band-like transport is typical of wideband materials,48 where the bandwidth exceeds (ℏ/τs), τs being the characteristic total scattering time due to intramolecular vibrations, as well as static variations in energy and electronic coupling along the polymer chain. The scattering time can be expressed as τs = (m/e)μ, where (m/e) is the charge carrier’s mass-to-charge ratio and μ is the intramolecular drift mobility. Thus, band-like transport along a polymer chain may be expected when μ > μ* = eℏ/(mΔEb), where ΔEb is the width of the conduction or valence band for electron or hole transport,

Chart 1. (A) Polyfluorene−p-Biphenyl (PFBP), R′ = Hex; (B) Zn−Porphyrin Based Molecular Wires, R = SiHex3

mobilities along Zn−porphyrin based molecular wires have previously been measured by pulse radiolysis−time-resolved microwave conductivity (PR-TRMC).42 Here, PR-TRMC as well as pulse radiolysis−time-resolved absorption spectroscopy (PR-TRAS) measurements are reported for PFBF and two polyrotaxane molecular wires. The mobility data are analyzed with a quantum-mechanical model of charge transport that predicts charge carrier mobilities in reasonable agreement with the experimental data. The model uses energy variations along the polymer chains estimated from molecular dynamics simulations with a polarizable force field, carried out for polymers in solution. It incorporates polaronic distortion of the quantum state due to polymer and solvent reorganization. The significance of the initial state of the charge carrier, which is largely determined by the charge generation or charge injection process, for subsequent conduction is also demonstrated. Following this quantum-mechanical analysis, the applicability of incoherent hopping models to charge transport along molecular wires is discussed.22,47 Finally, the possibilities for controlling the charge transport properties of polymers by molecular 25214



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generated in solution in the same way as for PR-TRMC measurements. The detection light source was an Osram XBO high-pressure Xe-lamp (450 W), containing a condensor to focus the light on the sample. To avoid photolysis of the sample, a fast shutter was placed between the light source and the sample cell and was only opened during and immediately after the electron pulse. Additionally, short-wave cutoff filters were placed between the light source and the fast shutter. The polymer solution was in a quartz cell with an optical path length of 12.5 mm and a height of 6 mm. After passing the sample, the light was focused by a lens into the entrance slit of a monochromator (Jobin Yvon HL 300). A silicon photodiode was used for detection up to 1000 nm, and a short-wave enhanced InGaAs PIN photodiode G5125-10 (Hamamatsu, Japan) was used for detection at longer wavelengths (up to 1600 nm). After amplification, the time-resolved change in the photodetector signal was monitored with the same LeCroy LT374L digitizer as in PR-TRMC measurements.

environment engineering with supramolecular chemistry methods are considered.

2. EXPERIMENTAL METHODOLOGY The synthetic procedures for molecular wires considered in this paper were previously described in refs 42 and 61. The intramolecular conductivity of all conjugated polymers studied here was measured with pulse radiolysis−time-resolved microwave conductivity (PR-TRMC).62−64 A 1−50 ns pulse of 3 MeV electrons from a van de Graaf accelarator was used to generate free charge carriers in a dilute solution (0.1−5 mM) of the polymers in benzene. The degree of polymerization of the materials studied varied from 10 to 50 (specific values for each material are given in section 4). No indication of polymer aggregation was observed at the concentrations used. Since the solvent in the samples was much more abundant than the solute, the majority of free charges produced upon irradiation were benzene cations and free electrons. Prior to the experiment, the solution was saturated with molecular oxygen that scavanged highly mobile free electrons in the first few nanoseconds after irradiation. Since the ionization potential of benzene is higher than that of the polymers studied, an encounter of these two species could result in the reduction of the benzene cation and a generation of a hole on the polymer chain. About 1 ms after irradiation, all charges in solution would recombine. Irradiation doses were kept sufficiently low that radiolytic products had no observable effect on the properties of the samples. Microwaves with a frequency of 34 GHz, generated by a Gunn diode, were used to probe radiation-induced changes in sample conductivity. The microwaves were guided, via a circulator, into a rectangular waveguide with an internal cross section 7.1 × 3.55 mm2. The microwaves entered the sample cell via a vacuum-tight polyimide window, passed through the polymer solution, were reflected by a metal plate at the other end of the cell, and returned to the circulator, which directed them to a microwave detector (a Schottky barrier diode). The change in the microwave power measured at the detector was amplified and monitored using a LeCroy LT374L digitizer. For small changes in the conductivity, the relative change in the microwave absorption, ΔP/P, due to mobile charges produced upon pulse irradiation of the sample is directly proportional to the change in the sample conductivity, Δσ:

3. THEORETICAL METHODOLOGY 3.1. The Model. The present simulations of charge transport along conjugated polymer chains are based on a hybrid theoretical model in which a quantum-mechanical description of the charge carrier (hole) is combined with a classical description for the dynamic structural disorder in the molecular system.22,47,48,73 The current model is a refinement of earlier band-like models that have been employed for calculations of charge carrier mobilities in several polymers, including derivatives of poly(phenylenevinylene) and polyfluorene.49,67,73 In the present paper, the polaronic contribution to the charge carrier Hamiltonian due to the coupling to the local environment is additionally included. The dependence of charge transport on the initial charge carrier state is also considered. Hole transport along a polymer chain was described using the time-dependent Schrödinger equation: iℏ

d|Ψ(t )⟩ = H(t )|Ψ(t )⟩ dt

(1)

In eq 1, the hole wave vector |Ψ(t)⟩ was expressed as a linear combination of orbitals |ϕi⟩ localized on molecular fragments of the polymer with complex time-dependent coefficients ci(t): |Ψ(t )⟩ =

ΔP = −AΔσ P

∑ ci(t )|ϕi⟩

(2)

i

A time-dependent tight-binding Hamiltonian was used:

where A = 17 Ω·m is a sensitivity factor for the geometric and dielectric properties of the sample and the microwave frequency used.64−66 The value of the charge carrier mobility along the polymer chain contour can be calculated from kinetic fits of the PR-TRMC transients. To account for the random orientation of polymer chains in solution and for possible bending of the chains, the samples were assumed to be isotropic, and the mobility values obtained were multiplied by 3.67 The persistence length of all polymers studied was greater than or of the order of the polymer chain length.68−70 Thus, no significant coiling is expected that would allow charges to hop between any segments of the polymer chain that are not bound covalently. For some samples, the generation and recombination of excess charges were also studied using pulse radiolysis−timeresolved absorption spectroscopy (PR-TRAS) in the visible and near-infrared ranges.71,72 In these experiments, charges were

H(t ) =

∑ εi(t )ai†ai + ∑ i

i,j>i

Jij (t )(ai†aj + aia†j ) (3)

where εi(t) = ⟨ϕi|H(t)|ϕi⟩ is the energy of a charge localized on the ith molecular fragment, Jij(t) = ⟨ϕi|H(t)|ϕj⟩ is the electronic coupling (charge transfer integral) between localized orbitals i and j, and a†i and ai are the creation and annihilation operators, respectively, for a hole in fragment orbital |ϕi⟩.47 Specifying εi(t) for all fragments and Jij(t) for all pairs of fragments in eq 3, as well as the fragmentation {|ϕi⟩} and the initial state |Ψ(0)⟩ in eq 2, completely defines the charge transport model. The procedure for estimating εi(t) and Jij(t) from a combination of ab initio and molecular dynamics calculations is described in detail in section 3.2. The initial state |Ψ(0)⟩ of a charge carrier on a polymer chain is determined by the charge generation (injection) process. It is 25215



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where ω is the (radial) frequency of the probing electric field, e is the elementary charge, and an implicit convergence factor exp(−ot), lim o → 0 is understood in the integral.80,81 The mobility values calculated according to eq 7 at ω = 34 GHz can be directly compared to the mobilities along the polymer chain contour extracted from PR-TRMC measurements.67 3.2. Hamiltonian Parametrization. Fragmentation of the polymer chain (that is, its subdivision into molecular fragments with orbitals {|ϕi⟩} localized on these fragments, as described in section 3.1) defines a basis set that should span the Hilbert space of all accessible hole states. Generally, hole transport occurs within the valence band (v-band) and electron transport within the conduction band (c-band) of a polymer (a onedimensional crystal).82 For homopolymers, the v-band consists mostly of the HOMOs of individual monomer units and a fragmentation into monomer units produces a reasonable basis set {|ϕi⟩}. In the case of copolymers, the choice of an appropriate fragmentation is more difficult. For instance, in PFBP (Chart 1A), which is fragmented into phenyl and fluorene units, the HOMO of each phenyl unit is coupled to two fluorene orbitals (HOMO and HOMO−1). Thus, to obtain a basis set that spans the v-band, it is necessary to include two orbitals per fluorene unit. This can be achieved, for instance, by taking the HOMO and HOMO−1 of fluorene or by further fragmenting the fluorene unit into two phenyl rings. (The bridging carbon atom and side chains are not part of the π-conjugated system, and their contribution to the two highest occupied orbitals of fluorene is negligible.) The latter fragmentation scheme was used in this paper. The energetic parameters εi(t) and Jij(t) in eq 3 can be calculated using a combination of molecular dynamics and quantum-chemical methods. In the current model, the site energies were assumed to be of the form

commonly assumed that the charge carrier at the initial time t = 0 is localized, i.e., in eq 2.22,47−49 ∃ i:

|ci(0)|2 = 1, cj(0) = 0, ∀ j ≠ i

(4)

However, intermolecular charge transfer (e.g., from a solvent ion to the polymer chain, as in the case of PR-TRMC; see section 3) is usually considered an incoherent process,74,75 with a time scale exceeding the decoherence time of the hole wave vector. Consequently, it may be more reasonable to assume that the wave function of a charge injected into the system has undergone environmentally induced selection of stable preferred basis states (known as “pointer states”).76,77 In the presence of rapid decoherence, pointer states are predominantly composed of eigenstates of the system Hamiltonian, eq 177 H(0)|Ψ(0)⟩ = E|Ψ(0)⟩

(5)

i.e., the instantaneous molecular orbitals of the charge carrier on the polymer chain. Since polymers are significantly more disordered than inorganic crystalline semiconductors, Hamiltonian eigenstates are not delocalized over the entire polymer chain. Nevertheless, it will be shown in section 3.2 that the electronic couplings between fragments of a polymer chain are considerable. Thus, it is important to take into account the delocalization of the eigenstates at both the initial and subsequent times in the evolution.78 Another consequence of the large degree of structural disorder in polymers is a comparatively slow thermalization of “hot” charge carriers. Thus, charge carriers need not necessarily be created and transported only close to the bottom of the c-band or top of the v-band. Formally, the solution to the time-dependent Schrödinger equation, eq 1, is given by

∫0

Ψ(t ) = ; exp(

t

εi(t ) = εi0 + εistat + εidyn(t ) + εipol[ci(t )]

i H(t ′) dt ′)Ψ(0)

The first term in eq 8, is the fragment orbital energy in the optimized fragment geometry (in vacuum). For quantumchemical calculations, the dangling bonds resulting from “cutting” a fragment out of a polymer chain were passivated with hydrogen atoms. The contribution of these atoms to the molecular orbitals that participate in charge transport is negligible. If there is only one type of molecular fragments, then ε0i = ε0, ∀i, and can be set to zero by shifting the origin of energy. The second term in eq 8, εstat i , accounts for structural disorder in the molecular environment. This disorder is assumed to be static, since charge transport over a few molecular fragments is a significantly faster process than environment reorganization (femtoseconds vs picoseconds time scale). To estimate the value of εstat i , molecular dynamics (MD) simulations on chain segments of several polymers (poly-p-phenylene, polyfluorene, PFBP) in a box of solvent were carried out. A polarizable force field, the direct reaction force field (DRF), as implemented in the DRF90 program package, was used for these simulations.83 The systems were equilibrated for 200 ns, followed by a 500 ns MD run. The static disorder, εstat i , was found to be distributed normally, with similar standard deviations σ for all polymers studied here. The standard deviation increased with solvent polarity, from as low as 0.8 eV in the case of cyclohexane to over 2 eV in the case of water; for PFBP in benzene, σ ≈ 1.0 eV. MD simulations for Zn−porphyrin based molecular wires were not carried out due

where ; is the Dyson time-ordering operator. This timeordered integral becomes a time-ordered product if the Hamiltonian parameters εi(t) and Jij(t) are assumed to vary smoothly on the time scale of the simulation so that they may be approximated as piecewise-constant (constant within the integration time step). Solution of the time-dependent Schrödinger equation under these conditions yields explicit numerical solutions for the set of time-dependent coefficients {ci(t)} in eq 2. The mean-squared displacement of the charge carrier along the contour of the polymer chain as a function of time can then be expressed as ⟨Δx 2(t )⟩ =

∑ [|ci(t )|2 − |ci(0)|2 ]i 2a2 i

(6)

where a is the distance between neighboring sites on a polymer chain. The mean-squared displacement as a function of time, eq 6, was averaged over 1000 random realizations of the polymer chain. The total runtime for each realization was 1.2 ns, with an integration time step of 1 atomic time unit (2.4189 × 10−17 s). The mean-squared displacement of the charge carrier was then converted to the charge carrier mobility along the contour of the polymer chain using Kubo’s formula79−81 μ(ω) = −

eω 2 2kBT

∫0

∞

⟨Δx 2(t )⟩ cos(ωt ) dt

(8)

ε0i ,

(7) 25216



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solvent reorganization, which is a comparatively slow process. From the induction component of the charge localization energy found in DRF90 MD simulations,83 the external reorganization energy for PFBP molecular wire fragments in benzene solution was estimated to be around 0.5 eV. This quantity decreases with the polarity of the solvent, from over 0.6 eV in cyclohexane to less than 0.3 eV in water (in contrast with the total solvent reorganization energy that increases with solvent polarity).90 The charge transfer integrals, Jij = ⟨ϕi|H|ϕj⟩, in eq 3 were calculated according to the procedure previously described in refs 48 and 91. This takes advantage of the possibility to use molecular fragment orbitals as a basis set in electronic structure calculationsa unique feature of the Amsterdam Density Functional (ADF) program.92 In the first step, the molecular fragment orbitals were calculated with density functional theory (SAOP exchange-correlation functional, DZP atomic basis set).92 In the second step, these fragment orbitals were used to calculate the electronic structure of a system of two molecular fragments at the same level of theory. The standard output of the ADF program provides the overlap matrix, S, the eigenvector matrix, C, and the diagonal eigenvalue matrix, E, defined in terms of the molecular orbitals on the fragment units. The matrix elements of the Kohn−Sham Hamiltonian, ⟨ϕi|hKS|ϕj⟩, are the matrix elements of SCEC−1 (note that zero spatial overlap of fragment orbitals is not assumed here). The matrix elements of S, i.e., the spatial overlaps Sij between orbitals on fragments i and j, may be substantial and must be accounted for by replacing the charge transfer integral Jij in eq 3 with an effective charge transfer integral:21,48,92

to the prohibitively large size of the solvent box required. For this system, transport calculations were carried out with standard deviation σ for the static disorder varying from 0 to 1.0 eV (section 4.2). The third term in eq 8, εdyn i (t), is the dynamic energy disorder from molecular vibrations. In our simulations εdyn i (t) was varied stochastically, with the probability of any εdyn value i given by a thermal distribution P(εidyn) =

⎛ ε dyn ⎞ 1 exp⎜⎜ − i ⎟⎟ Z1 ⎝ kBT ⎠

(9)

and the probability of an energy fluctuation occurring after time τdyn since the previous fluctuation given by i P(τidyn) =

⎛ τ dyn ⎞ 1 exp⎜⎜ − i ⎟⎟ Z2 ⎝ τfluct ⎠

(10)

where Z1 and Z2 in eqs 9 and 10 are partition functions of the respective distributions, and τfluct is the effective bath correlation time. In the present calculations, the value of τfluct is taken to be 100 fs; varying τfluct within the typical range of molecular vibration time scales, 10−200 fs, has little effect on the calculated charge carrier mobility. The values of εstat and εdyn i i (t) were assumed to be spatially uncorrelated in our simulations, since they are largely determined by the structure and dynamics of the polymer chain fragment and the first two solvent coordination spheres. The fourth term in eq 8 accounts for energetic shifts due to polaron formation: εipol(t ) = −|ci(t )|4 (λ int + λext)

⎛ εi + εj ⎞ Jijeff = Jij − Sij⎜ ⎟ ⎝ 2 ⎠

(11)

where λint is the reorganization energy of a molecular fragment (internal) and λext is the reorganization energy of the molecular environment (external). The polaronic term, as defined by eq 11, relies on several implicit approximations. It is local both in the sense that it is only a function of the wave function coefficient ci(t) on the ith site and in the sense that it modifies only the site energy, εi(t), and not the intersite couplings, Jij(t).84 The polaronic term is Markovian, depending only on ci(t) at time t and not on the values at earlier times. Furthermore, εpol i (t) is assumed to depend harmonically on the amount of charge on the ith site, giving rise to the quartic dependence on |ci(t)|.85,86 The internal reorganization energy in eq 11 can be written as22,48,87,88 λ int = [E q(g 0) − E q(g q)] + [E 0(g q) − E 0(g 0)] q

Because of the localized nature of molecular fragment states, it is a good approximation to include only charge transfer integrals between adjacent units, j = i ± 1, in the Hamiltonian, eq 3. Both molecular wires considered in this paper consist of planar fragments. Consequently, the charge transfer integrals between adjacent units can be parametrized as a function of the dihedral angles θi,i+1 between the planes of these units. To a good approximation Jieff (θ ) = J eff (0) cos θi , i + 1 ,i+1 i ,i+1

(13)

For a pair of phenyl rings, Jeff(0) = 0.75 eV; for a pair of Zn− porphyrin units Jeff(0) = 0.27 eV. The dependence of site energies εi on variations in the dihedral angles θi,i±1 is weak:

(12)

(2) |εi(θi(1) , i ± 1) − εi(θi , i ± 1)| < 0.1 eV

0

where E (g ) is the internal energy of a molecular fragment with an excess charge q = ±1 (in elementary charge units) and an equilibrium nuclear geometry g0 of a neutral molecular fragment; the other terms in eq 12 are defined similarly. The values of Eq(g0), etc., were calculated using the Gaussian’09 quantum chemistry package,89 using density functional theory (BLYP density functional, 6-31G(d,p) basis set). For Zn− porphyrin based molecular wire fragments (monomer units), λint was found to be 0.36 eV (the value predicted by Hartree− Fock theory with the same basis set is somewhat higher, 0.50 eV).22 For PFBP molecular wires, λint of a fragment (single benzene ring) is 0.20 eV (the Hartree−Fock value is 0.39 eV). On the time scale of charge transport over a few molecular fragments, the external reorganization energy, λext, in eq 11 is mostly due to polarization of the environment, rather than to

for any two arbitrary values of the dihedral angle θi,i(1)± 1, θ(2) i,i ± 1 ∈ [0, 2π). In our simulations, initial values of the dihedral angles were set according to the thermal distribution P(θi , i + 1) =

⎡ Vrot(θi , i + 1) ⎤ 1 ⎥ exp⎢ − Z kBT ⎦ ⎣

where Vrot(θi,i+1) is the potential barrier for relative rotation of molecular fragments and Z is the corresponding partition function. The potential barriers for rotation of all units in PFBP and Zn−porphyrin based molecular wires were calculated quantum-mechanically (Gaussian’09, BLYP functional, 6-31G(d,p) basis)22,93 and can be found in the Supporting 25217



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the pulse). The initial increase in the sample conductivities, up to about 30 ns after irradiation, is due to free electrons in benzene, ebz− (mobility 0.13 cm2 V−1 s−1). Since the polymer solution is oxygen-saturated, free electrons are quickly scavanged: ebz− + O2 → O2−. This leads to a reduction in the sample conductivity, as the mobility of oxygen anions in benzene is only 10−3 cm2 V−1 s−1.98 The second increase in the sample conductivity, at times above 50 ns after irradiation, is due to hole transfer from benzene cations (bz+) to polymer chains (P):

Information. The dihedral angles were then propagated in time as θi,i+1(t + Δt) = θi,i+1(t) + Δθi,i+122,48 Δθi , i + 1 = −

dVrot[θi , i + 1(t )] 1 Δt + 2τrotk BT dθi , i + 1

Δt 2τrot

where τrot is the diffusion rotation time, estimated using effective molecular volumes,94,95 with volume increments of individual groups taken from refs 96 and 97. The values of τrot used were 10 ps for a phenyl ring, 400 ps for a fluorene unit, and 2 ns for a Zn−porphyrin unit in benzene.22,67,49

bz + + P → P+ + bz

4. RESULTS AND DISCUSSION Section 4.1 presents PR-TRMC measurements and calculations of the hole mobility in PFBP molecular wires. Section 4.2 presents calculations of the hole mobility in Zn−porphyrin based molecular wires and compares them with mobility values previously measured by PR-TRMC42 and earlier theoretical analysis.22 Section 4.3 discusses changes in the conductive properties of PFBP and Zn−porphyrin molecular wires upon inclusion of these wires in supramolecular complexes. The conductivity of PFBP-based polyrotaxanes is studied here by a combination of PR-TRMC and PR-TRAS measurements as well as theoretically. New theoretical considerations regarding the conductivity of Zn−porphyrin based double-strand ladderlike structures22,42 are also presented. 4.1. Charge Transport in Polyfluorene−p-Biphenyl Molecular Wires. PR-TRMC transients of PFBP molecular wires in benzene solution are shown in Figure 1. These

(14)

This conductivity increase indicates that the hole mobility along polymer chains exceeds the mobility of bz+ ions in solution, 1.2 × 10−3 cm2 V−1 s−1.99 About 1 μs after irradiation, a final decrease of the sample conductivity sets on, due to the recombination of mobile holes on polymer chains with oxygen anions: O2− + P+ → P + O2

(15) +

Fitting the kinetics of charged polymer (P ) formation and recombination according to eqs 14 and 15 yields the experimentally measured mobility of a hole on a polymer chain.100 Fits to the data in Figure 1 show that the hole mobility in PFBP is dependent on the chain length. In particular, a moblity value of 0.043 cm2 V−1 s−1 is obtained for 10 monomer unit long chains (red line in Figure 1) and a value of 0.060 cm2 V−1 s−1 for 15 monomer unit long chains (blue line in Figure 1). Such a chain length dependence of the mobility indicates that charge carriers are sufficiently displaced on polymer chains within a microwave field cycle for wave function reflection at polymer chain ends to influence the mobility. These experimental mobility measurements will now be compared with the results from the hybrid quantum-classical dynamical theory outlined in section 3. The calculations presented here were performed with the initial condition satisfying eq 5. Calculated mean-squared displacements of a hole on PFBP chains of various lengths as a function of time are shown in Figure 2A. Hole mobilities, calculated from the meansquared displacement according to eq 7, are shown in Figure 2B. For values of static disorder in the localization energies at molecular sites estimated from MD simulations (normally distributed with a standard deviation of ∼1 eV), the calculated hole mobilities are less than 0.1 cm2 V−1 s−1 and are thus close to those found from fitting to the experimental data in Figure 1. However, at these large magnitudes of static disorder in the site

Figure 1. PR-TRMC transients of PFBP molecular wires in benzene solution. The degree of polymerization is 10 (red transient) and 15 (blue transient).

transients show the change in sample conductivity Δσ upon irradiation with a pulse of high-energy electrons per absorbed dose D as a function of time (measured from the beginning of

Figure 2. (A) Calculated mean-squared displacement of a hole on a PFBP molecular wire as a function of time for several oligomer chain lengths, standard deviation in the site energy σ = 0.2 eV. (B) Calculated hole mobility along PFBP molecular wires as a function of the amount of static disorder in the environment for several oligomer chain lengths. 25218



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chains. No chain length dependence of the mobility is seen at such disorder values for holes with the initial condition given by a Hamiltonian eigenstate (Figure 2). The assumption that a charge carrier on a polymer chain is initially localized on a single molecular fragment as in eq 4 is common in the literature on charge transport. The present results suggest that this assumption may be the reason why quantum-mechanical band-like models tend to overestimate the charge carrier mobility and to exaggerate its chain length dependence.22,48,49 However, the experimentally observed chain length dependence of the mobility suggests that the initial state of a charge carrier on the polymer chain is not quite a Hamiltonian eigenstate either. Thus, a detailed study of charge injection onto polymer chains is necessary to reproduce all features of molecular conductance correctly. Interestingly, the presence of a polaronic contribution to the site energies, eq 11, seems to have only a minor effect on the calculated charge carrier mobility. Varying the value of the reorganization energy λ between 0 and 0.8 eV produces variations in the calculated charge carrier mobility of 10−30% (depending on the polymer, chain length, and amount of disorder). However, molecular reorganization in real systems is not instantaneous. Thus, polaronic effects on the mobility may be more pronounced when derived from a non-Markovian description of the polymer. 4.2. Charge Transport in Zn−Porphyrin Based Molecular Wires. Hole mobilities along Zn−porphyrin based molecular wires measured by PR-TRMC have been reported in ref 42. The hole mobility is ∼0.1 cm2 V−1 s−1 and is virtually independent of chain length for degrees of polymerization equal to 10 or more. Subsequent theoretical analysis led to the conclusion that charge transport in Zn−porphyrin based molecular wires occurs by incoherent hopping, rather than by a coherent band-like mechanism.22 This conclusion is reexamined here, using refined calculations based on the present model that employs a more realistic description of disorder in the molecular environment and of the initial charge carrier state. The conclusion that charge transport in Zn−porphyrin based molecular wires occurs by incoherent hopping was based on two principal considerations.22 First, to obtain mobilities comparable to experimental values in a quantum-mechanical band-like simulation, it is necessary to introduce static disorder in the site energies with a standard deviation of ∼0.8 eV. Such values of disorder were deemed unreasonable, as they significantly exceeded values previously reported in the literature and commonly used in charge transport simulations (0.05−0.2 eV).21,48,49,106 Second, a quantum-chemical calculation indicated that it is thermodynamically favorable for a hole on a Zn−porphyrin based molecular wire to localize, which was interpreted as an argument in favor of small polaron hopping.22 Calculating the site energy disorder using MD simulations with a polarizable force field, as was done for PFBP, is considerably more difficult in the case of Zn−porphyrin based molecular wires, due to the larger size of the solvent box needed. However, given the comparable linear dimensions of the two monomer units (see Chart 1), it is likely that the energetic disorder along the chains of both polymers may be of the same order of magnitude. Since values of energetic disorder computed in this paper for PFBP wires support the larger values used in ref 22 (standard deviation σ ∼ 1 eV), a band-like transport model cannot be discarded on the grounds of “unreasonably large disorder”.

energies, the calculated mobility is found to be independent of chain length, contrary to what is seen experimentally (Figure 1). As discussed below, this difference appears to indicate a significant role of the detailed nature of the initial condition for transport after charge injection. A chain length dependence of the hole mobility becomes evident only at smaller standard deviations in the site energies, below about 0.5 eV. Further reducing site energy disorder makes this chain length dependence even more pronounced: in the absence of any site energy disorder, the hole mobility on a 10 monomer unit long chain of PFBP is 0.15 cm2 V−1 s−1, while the infinite-chain mobility (reached for chain lengths of ∼50 monomer units) is an order of magnitude higher, namely 1.5 cm2 V−1 s−1. The hole mobility in PFBP was found to depend on the initial state of the charge (energy within the v-band). In crystalline inorganic semiconductors charge transport is usually assumed to occur close to the bottom of the c-band or top of the v-band: “hot” charge carriers can quickly thermalize, due to the high density of states within the band. In contrast to this, the high degree of disorder in organic semiconductors creates a large number of localized states close to the band edge. These states are eigenstates of the hole Hamiltonian and are, generally, not localized in the sense of eq 4 (i.e., they are not localized on a single molecular fragment). Instead, they are Anderson localized states,47,101 extending over several molecular fragments. Because of the narrow bandwidth of organic materials, such states may account for a significant fraction of all states in the material. A charge carrier initially confined to a pointer state is unlikely to thermalize on the time scale of microwave field oscillations used in TRMC measurements (ω = 34 GHz). Thus, TRMC measurements probe charge transport near the state in which the charge carrier is initially created.102 To calculate the precise value and correct chain length dependence of the charge carrier mobility, it is important to know the probabilities of initially creating a charge in each eigenstate of the Hamiltonian, eq 3. These probabilities likely depend on the specific process by which charges are generated in or injected into the material.103−105 Exploration of this issue is beyond the scope of the current paper. Starting a simulation with the initial wave function localized on a single molecular fragment, with coefficients ci(0) in eq 2 given by eq 4, increases the calculated hole mobility values relative to the situation in Figure 2, where the initial wave function is an eigenstate of the Hamiltonian at t = 0. For a 10 monomer unit long PFBP chain, the hole mobility calculated using eq 4 as the initial condition is ∼0.4 cm2 V−1 s−1. This value is roughly independent of site energy disorder and is ∼4 times higher than the mobility calculated with the initial condition given by eq 5 (see Figure 2B). This is consistent with expectations that initial states that are further from a Hamiltonian eigenstate would show an increased tendency for delocalization, resulting in higher charge carrier mobility. The mobility for an initially localized charge, eq 4, also shows a stronger chain length dependence than for a charge that is initially in a Hamiltonian eigenstate, eq 5. Thus, on increasing the PFBP chain length from 10 to 50 monomer units or more, the mobility of an initially localized hole in the absence of site energy disorder increases from 0.4 to over 20 cm2 V−1 s−1 (vs 0.1−1.5 cm2 V−1 s−1 for a hole initially in a Hamiltonian eigenstate, Figure 2A). At site energy standard deviations σ ∼ 1 eV, the mobility of initially localized holes on 50 monomer unit long chains is up to 15% higher than on 10 monomer unit long 25219



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Figure 3. (A) Calculated mean-squared displacement of a hole on a Zn−porphyrin based molecular wire as a function of time for several oligomer chain lengths, standard deviation in the site energy σ = 0.4 eV. (B) Calculated hole mobility along Zn−porphyrin based molecular wires as a function of the amount of static disorder in the environment for several oligomer chain lengths.

The question as to whether band-like models should be preferred over hopping models for simulating charge transport in Zn−porphyrin molecular wires and related materials will now be discussed. Small polaron hopping models were intended for simulating charge transport in materials where charge states are sufficiently localized.109 Generally, their applicability to polymers is questionable. For instance, commonly used models that rely on Fermi’s Golden Rule21,110 are strictly valid only for very weak electronic couplings between hopping sites.54 Electronic coupling values commonly encountered in polymers (up to 0.27 eV in Zn−porphyrin based molecular wires, up to 0.75 eV in PFBP; see section 3.2) are comparable to both the site energy disorder σ and the reorganization energy λ for these materials. Consequently, it is unlikely that charge carriers ever become completely localized on single molecular fragments. Indeed, even for large values of site energy disorder, σ ∼ 1.0 eV, most eigenstates of the hole Hamiltonian for Zn-porphyrin based molecular wires are delocalized over 3−5 molecular fragments. (This can be immediately seen by diagonalizing the hole Hamiltonian, eq 3.) In such cases, the approximation of a “small” polaron109 becomes overly simplistic. It should be noted that environment-induced disorder along a polymer chain in solution, described by the term εistat in eq 8 is, in reality, not static. Typical correlation times for this disorder, estimated from MD simulations, are of the order of picoseconds. Consequently, on longer time scales, states that are delocalized over a few molecular fragments can become coupled to other states, further along the polymer chain. Charge transport in this case could, in principle, be viewed as variable-range large polaron hopping with picosecond lifetimes at localization sites. However, while it is possible to formally define hopping models for extended charge states, these are usually both impractical and unphysical.109 Variations in the degree of charge delocalization makes extended state hopping models difficult to parametrize (a large number of parameters is required). Furthermore, the quantum-mechanical nature of a delocalized charge carrier makes the separation of its transport into an instantaneous coherent step (“hop”) and a decoherence step, during which the charge carrier resides at a hopping site, unreasonable: the time scales of charge transfer between molecular fragments and decoherence are, in this case, comparable.77 In this situation, a quantum-mechanical description of charge transport becomes imperative. 4.3. Charge Transport in Polyrotaxanes and Zn− Porphyrin Based Ladders. The molecular wires studied in this paper and related materials have already been used in supramolecular complexes.38,42,61 Supramolecular chemistry offers exciting opportunities for modifying the immediate

As for the second argument in favor of small polaron hopping transport presented in ref 22, it should be noted that the energy difference of a localized and a delocalized charge states is sensitive to the quantum-chemical method employed for its calculation. For molecules as large as a Zn−porphyrin dimer, no method is currently capable of producing a reliable molecular charge distribution.107 (Notoriously, charge distributions calculated with density functional theory are strongly dependent on the density functional used, with most density functionals overly delocalizing and a few overly localizing the charge.108) Furthermore, since charge transport is a nonequilibrium process, kinetic effects may induce charge delocalization even when localized charge states are thermodynamically favorable. Thus, thermodynamic predictions for charge localization should be treated with caution, particularly when dealing with nonequilibrium processes in complex molecular environments. Figure 3A shows calculated mean-squared displacements of a hole on Zn−porphyrin based molecular wires of various lengths as a function of time for site energy disorder σ = 0.4 eV. The calculations were performed according to the hybrid quantumclassical dynamics described in section 3, with incorporation of polaronic effects and the initial condition corresponding to an instantaneous charge carrier eigenstate, eq 5. Similar calculations were made for a range of site energy disorder values σ, and the corresponding hole mobilities were then determined from these calculations according to eq 7. The resulting mobilities are shown in Figure 3B as a function of the static disorder standard deviation σ for a range of polymer chain lengths. From Figure 3B, it is evident that the calculated mobilities at standard deviation in the site energies of 0.8−1.0 eV are close to the experimental values (0.1 cm2 V−1 s−1).42 In agreement with experiments, the calculated mobility for static disorder of this magnitude is independent of the polymer chain length. Thus, for Zn−porphyrin based molecular wires it is possible to reproduce the experimental mobility values within a band-like description of the charge carrier dynamics, without resorting to incoherent hopping models. As for PFBP (section 4.1), the calculated hole mobility depends on the initial hole state |Ψ(0)⟩. Selecting different eigenstates of the Hamiltonian (energies within the band) as the initial state can change the calculated mobility by up to a factor of 3−4. Selecting a localized initial state, eq 4, and reducing the energetic disorder to σ = 0, results in a calculated hole mobility of 12.4 cm2 V−1 s−1.22 The hole mobility is much less sensitive to disorder when the initial hole state is an eigenstate of the Hamiltonian, eq 3, always varying between 0.1 and 0.5 cm2 V−1 s−1 (see Figure 3B). 25220



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The role of the hole concentration will be addressed first. If polyrotaxanes were, indeed, insulated molecular wires, charge transfer from benzene ions to polymer chains, eq 14, would be expected to be hindered by the rotaxane shell. Figure 5 shows pulse radiolysis−time-resolved absorption spectroscopy (PRTRAS) spectra and transients at the absorption maximum for 10 unit long chains of PFBP (Chart 1A) and for the same two polyrotaxanes with a PFBP backbone (Chart 2). PR-TRAS measures the difference in the absorption of a neutral species and a cation. The virtually identical spectra and transients for PFBP and PFBP.Me⊂β-CD.COPr indicate absorption by the same cation species in similar concentrations. Since the solvent cations (bz+) and oxygen anions (O2−) do not absorb in the visible and near-infrared, the absorption can only be due to polymer cations. Thus, it can be concluded that the cyclodextrin ring in PFBP.Me⊂β-CD.COPr has no apparent effect on either the diffusion of bz+ and O2− to the polymer chain or the charge transfer reactions described by eqs 14 and 15. In the case of PFBP.Me⊂β-CD.Si, the shell formed around the PFBP backbone by functionalized cyclodextrin rings is much denser than in PFBP.Me⊂β-CD.COPr.61 The slight shift in the absorption maximum (Figure 5A) may indicate some ionization of the shell around the PFBP backbone. A slower increase in the absorption compared to PFBP and PFBP.Me⊂β-CD.COPr (Figure 5B) indicates that diffusion of bz+ to the backbone is hindered by the shell around it. Consequently, recombination bz+ + O2− → bz + O2 is more likely to occur before bz+ ions reach the PFBP chains. This results in less hole transfer to PFBP, eq 14, and less absorption. However, it can be deduced from the absorption intensity that the number of holes on the backbone of PFBP.Me⊂β-CD.Si is at least 60% of the number for unthreaded PFBP. This analysis shows that polyrotaxanes are “insulated” only in the sense that the shell around the backbone polymer sterically hinders interaction with larger molecules (such as the backbones of other polyrotaxanes in a film or common electron acceptors like methyl viologen2+, dipropyl-4,4′bipyridinium disulfonate, anthraquinone-2,6-disulfonate2−).111,112 Smaller molecules and ions can nevertheless diffuse through the polyrotaxane shell and interact with the polymer backbone. It is now evident that the low conductivity in PFBP.Me⊂βCD.COPr and PFBP.Me⊂β-CD.Si cannot be explained by a decrease in the hole concentration nh relative to nonthreaded PFBP. According to eq 16, this means that the significant decrease in the conductivity of polyrotaxanes relative to nonthreaded PFBP must therefore be ascribed to a change in the hole mobility μh. To calculate the hole mobility in a polyrotaxane, it is necessary to analyze the effect of rotaxination on charge transport parameters that enter the hole Hamiltonian, eq 3. This can be achieved by comparing molecular dynamics simulations data for a polyrotaxane (e.g., PFBP.Me⊂βCD.COPr) and for nonthreaded PFBP. MD simulations for both materials were carried out in a box of benzene with an OPLS-derived force field,113,114 using the GROMACS molecular dynamics package.115 Snapshots from a 50 ns trajectory of each system in its neutral state were postprocessed in an additive electrostatic model to characterize the magnitude of temporal and spatial fluctuations in hole−environment interactions. Full details of the simulations and electrostatic model can be found in ref 116.

environment of molecules. Of particular relevance to this paper is the observation that optoelectronic properties of a molecular wire can be significantly altered in a supramolecular complex. For example, rotaxination (threading of a polymer through a series of cylindrical macrocycles, such as β-cyclodextrin, β-CD, Chart 2) is known to enhance fluorescence in polymer Chart 2. (A) Polyrotaxane with a PFBP Backbone Threaded through Functionalized β-Cyclodextrin, R′ = CO2Me; (B) βCyclodextrin (β-CD), R2, R3, R6 = Functionalization Sitesa

β-CD was modified by either trihexylsilylation (SiHex3) or acylation with butyric anhydride (COPr).61

a

films.38,61,111,112 The macrocycle, particularly when it is functionalized with bulky side chains, hinders interaction of the backbone polymer with other polymer chains or electron acceptors, thus suppressing fluorescence quenching. The fact that threaded polymers appear to be shielded from their environment has led researchers to dub them “insulated molecular wires”.38 Figure 4 shows PR-TRMC transients for 10 unit long chains of PFBP and the two polyrotaxanes with a PFBP backbone

Figure 4. PR-TRMC transients for 10 unit long chains of PFBP (red) and two polyrotaxanes with a PFBP backbone: PFBP.Me⊂β-CD.COPr (blue) and PFBP.Me⊂β-CD.Si (black).

shown in Chart 2. It is evident that the conductivity of both polyrotaxanes is much lower than that of the nonthreaded polymer. In fact, these conductivities are below the sensitivity of the TRMC experiment. The hole conductivity of a polymer can be written as σh = enhμ h (16) where e is the elementary charge, nh is the hole concentration, and μh is the hole mobility. Therefore, the reduced conductivity in polyrotaxanes, as compared to nonthreaded PFBP (Figure 4), could be due to a decrease in either the hole concentration or the hole mobility. 25221



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Figure 5. PR-TRAS spectra (A) and transients (B) for 10 unit long chains of PFBP (red) and two polyrotaxanes with a PFBP backbone: PFBP.Me⊂β-CD.COPr (blue) and PFBP.Me⊂β-CD.Si (black).

and have a standard deviation of 0.6 eV and a correlation lifetime on the order of picoseconds. The presence of βCD.COPr significantly alters the interaction of the backbone polymer with its environment (right panel). Since β-CD.COPr has a wider and a narrower rim,38 the two phenyl rings in a monomer unit of the backbone now have different environments. This results in a substantial hole localization energy difference (0.6 eV) between the phenyl rings; the interaction of a hole on the fluorene with the environment also changes. The large variations in charge−environment interaction energies along the chain are mainly due to the heterogeneity introduced by β-CD.COPr macrocyles (especially the polar COPr solvation chains) and the benzene-excluded volume they generate. The β-CD.COPr contribution to relative site energy has a correlation time on the order of nanoseconds and a standard deviation of about a quarter of an electronvolt, introducing large shifts in the static time-averaged hole environment potential energy landscape along the chain. To calculate the hole mobility on PFBP.Me⊂β-CD.COPr chains, the site energy variation due to the presence of the βCD.COPr macrocyles estimated from MD simulations (Figure 6) should be included in the hole Hamiltonian, eq 3. This additional site energy variation hinders charge transport, resulting in calculated mobility values μh < 0.01 cm2 V−1 s−1 (in agreement with mobility values found in PR-TRMC measurements on polyrotaxanes). Note that the less polar β-CD.Si macrocycle likely has a smaller impact on hole−environment interactions, which can result in somewhat higher hole mobility in PFBP.Me⊂β-CD.Si than in PFBP.Me⊂β-CD.COPr. However, the charge density for the former polyrotaxane is lower than for the latter (see Figure 5A). The above analysis has shown that charge transport in PFBP is suppressed by supramolecular modification of the polymer’s environment (rotaxination). Yet, for other materials, a conductivity enhancement can occur. For instance, adding 4,4′-bipyridyl to the benzene solution of Zn−porphyrin based molecular wires leads to the formation of ladder-like supramolecular structures (Chart 3). The intramolecular hole mobility is enhanced by an order of magnitude, as compared to single polymer strands in the absence of 4,4′-bipyridyl: from below 0.1 cm2 V−1 s−1 to over 0.9 cm2 V−1 s−1.22,42 The rotation of Zn−porphyrin units in ladder-like structures is restricted, resulting in smaller dihedral angles θi,i+1 between neighboring units and larger charge transfer integrals, eq 13. However, this increases the hole mobility only by a factor of 3−4, rather than by an order of magnitude, as seen in experiments. Further

The interaction energy between a hole and its environment was calculated as the difference between the electrostatic interaction of a charged and a neutral molecular subunit with the environment, where each of these were represented by a distribution of atom-centered point charges. Solvent partial charges were taken from the OPLS force field, while those for the conjugated backbone and the β-CD.COPr macrocycles were assigned by averaging over AM1-BCC117 quantumchemical calculations of 100 snapshots. This provided a single set of representative atomic charges, including intramolecular polarization effects poorly described by the original OPLS force field charges. In particular, these calculations indicated the presence of a dipole moment of 15 D in the CO.Pr macrocycles, aligned principally along the local polymer axis. Our calculations also indicated that the dipole moment of βCD.COPr is not constant but fluctuates with a standard deviation of 3 D. These results are consistent with previous force field and semiempirical studies on cyclodextrin.118 Results of the electrostatic calculations are schematically shown in Figure 6. In PFBP (left panel), both phenyl rings (P) in a monomer unit have, on average, identical solvent environments. A hole localized on either ring interacts with the environment in the same way; the interaction for a hole delocalized over a fluorene (F) is slightly different. Fluctuations of site energies are due almost entirely to benzene dynamics

Figure 6. Schematic representation of the average energy landscape due to hole−environment electrostatic interactions for PFBP (left) and PFBP.Me⊂β-CD.COPr (right). F = fluorene unit; P = phenyl unit; PN and PW = phenyl units near the narrower and wider rim of βCD.COPr, respectively. 25222



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the charge carrier. This could sometimes result in overestimated mobility values and an exaggerated mobility dependence on the polymer chain length.22,47−49 In this paper, the environmentally induced preferred basis states that correspond, under conditions of strong decoherence, to eigenstates of the system Hamiltonian were employed instead. At large values of static disorder, corresponding to the situation in many organic materials, including those considered in this paper, Hamiltonian eigenstates approach localized states, making this distinction less significant. Nevertheless, these results indicate that theoretical investigation of charge generation or injection, the processes determining the initial conditions for charge transport, is an important direction for future research in molecular electronics. Supramolecular engineering can be used to create polymerbased materials with diverse conductive properties for a variety of applications. Intermolecular interactions can either suppress (as in the case of threaded polyfluorene−p-biphenyl) or enhance (as in the case of Zn−porphyrin based ladder-like structures) the polymer conductivity. It is important to realize that characterizing the conductivity of a polymer without regard to its molecular environment is meaningless. The intramolecular conductivity of a polymer chain in a variety of solvents, in polymer films with different nanomorphologies, in composite materials or supramolecular complexes can vary by at least an order of magnitude. Description and characterization of interactions between noncovalently bound entities and investigation of the micro- and nanostructural features should be priorities in electronic materials research. The present study has shown that incorporating a description of the molecular environment into theoretical models for charge transport can significantly improve their predictive potential. It is expected that a detailed description of the molecular environment is also necessary to achieve an accurate understanding of charge generation and charge injection processes.

Chart 3. Ladder-like Supramolecular Arrangement Formed upon Addition of 4,4′-Bipyridyl to the Benzene Solution of Zn−Porphyrin Based Molecular Wires (Ar = metaC6H3[SiHex3]2)

increase in the mobility was previously assigned to changes in the molecular environment.22 An additional factor that may enhance the hole mobility in Zn−porphyrin molecular wires upon ladder-like structure formation is a modification to the charge carrier’s initial state. If the charge carrier is initially placed into an eigenstate of the Hamiltonian, eq 3, as suggested in this paper, the initial state of the charge in the more ordered ladder-like structures is more spatially extended, resulting in enhanced charge transport. This effect is less important in the case of PFBP molecular wires, since MD simulations suggest no significant effect of rotaxination on oligomer conformation and dynamics for these systems.119

5. CONCLUSIONS A hybrid quantum-classical model for charge transport along molecular wires that includes effects of the molecular environment, polaron formation, and the initial quantum state of a charge carrier on charge transport has been presented. Simulations of charge transport based on this model reproduce experimental charge carrier mobilities for several classes of molecular wires reasonably well. The good agreement between theory and experiment achieved here can be attributed to a careful characterization of the molecular environment and its role in this hybrid dynamic model. Molecular dynamics (MD) simulations show that the disorder in charge localization energies along a molecular wire due to the environment (solvent) can significantly exceed the values of 0.05−0.2 eV commonly cited in the literature.21,48,49,106 For several common solvents, including benzene, localization energy disorder of the order of 1 eV is suggested by MD. Disorder of this magnitude was previously deemed “unreasonable” for polymers in solution.22 However, when it is included in the charge carrier Hamiltonian, quantummechanical simulations produce mobility values close to those measured experimentally. Thus, adoption of numerically simpler incoherent small polaron hopping models,21,74,110 the applicability of which to intramolecular charge transport in conjugated polymers is questionable,54 may be neither necessary nor advisable. The initial state of a charge carrier was also found to have a sizable effect on the efficiency of charge transport, particularly at lower values of energy disorder. Previous band-like transport simulations have commonly employed a localized initial state of

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ASSOCIATED CONTENT

S Supporting Information *

Potential barriers for the relative rotation of PFBP and Zn− porphyrin based molecular wire fragment units. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Addresses ⊥

Department of Organic Chemistry, University of Geneva. Cambridge Display Technology, Cambridge, UK.

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Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.A.K. acknowledges the financial support by The Netherlands Organization for Scientific Research (NWO) Rubicon Grant 680-50-1022. K.B.W. was supported in part by DARPA under Award N66001-10-1-4068. The European Union FP6 Marie Curie Research Training Network “THREADMILL” (MRTNCT-2006-036040) is also acknowledged for financial support. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant OCI-1053575 (Project TG-CHE120013). 25223



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