Mobility of Holes in Oligo- and Polyfluorenes of Defined Lengths

Mar 3, 2014 - National Renewable Energy Laboratory, Golden, Colorado 80401, United States. §. Department of Chemistry and Biochemistry, University of...
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Mobility of Holes in Oligo- and Polyfluorenes of Defined Lengths Matthew J. Bird,*,† Obadiah G. Reid,‡ Andrew R. Cook,† Sadayuki Asaoka,∥ Yuki Shibano,⊥ Hiroshi Imahori,⊥,# Garry Rumbles,‡,§ and John R. Miller*,† †

Chemistry Department, Brookhaven National Laboratory, Upton, New York 11973, United States National Renewable Energy Laboratory, Golden, Colorado 80401, United States § Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309-0215, United States ∥ Department of Biomolecular Engineering, Kyoto Institute of Technology, Matsugaskaki, Sakyo-ku, Kyoto 606-8585, Japan ⊥ Department of Molecular Engineering, Graduate School of Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan # Institute for Integrated Cell-Material Sciences (WPI-iCeMS), Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan ‡

S Supporting Information *

ABSTRACT: The high-frequency mobility of positive charges (holes) moving along the backbones of an extensive data set of 7 oligomers (n = 2−16) and 6 polymers (⟨n⟩ = 26−138) of fluorene was measured using pulse-radiolysis time-resolved microwave conductivity (PR-TRMC) in benzene. As expected, at 8.9 GHz, the measured isotropic ac mobility, μiso ac,meas, was observed to be strongly dependent on the lengths of the chains due to the charges encountering chain ends during one microwave cycle. Values of the −4 measured mobility, μiso cm2/(V s) for an n = 2 repeat unit ac,meas, ranged from 5 × 10 2 oligomer to 0.18 cm /(V s) for a polymer with an average length of ⟨n⟩ = 86 repeat units. Global fits to the entire set of lengths extracted the chain-length-independent intramolecular mobility, μintra ac , using the Kubo formula, assuming normal diffusion along the contour of the molecule with reflecting boundary conditions at the ends. The effects of chain conformation, chain defects, polymer length distributions, and a finite polaron length on μiso ac,meas were considered quantitatively. The best fit to the whole data set, 2 taking into account the polymer length distributions, suggests μintra ac = 1.1 cm /(V s). The fit was improved slightly by implementing randomly spaced barriers to transport along the chain with an average spacing of ∼40 repeat units, although at this spacing, the estimate of μintra ac was not affected. These barriers could represent defects in the polymer or dihedral angles between repeat units, giving poor electronic coupling that persists for times greater than the 50 ps period of the microwaves. In this defect model, the best fit for their average spacing depended strongly on the chain conformation used, while the predicted intramolecular mobility did not. This estimate of the frequency of defects, a new aspect of this work, is less accurate because coiling of chains and defects have similar effects on the measured mobilities, and each can mask the effect of the other. The present data set also shows that there are few, if any, traps having depths substantially greater than thermal energy for holes on polyfluorene. The value of 1.1 cm2/(V s) is tightly constrained by the mobility of charges on the oligomers. Should the diffusion model break down for short chains and only be applicable to longer polymers, the intrachain mobility may be able to take both larger and smaller values (∼0.7−3 cm2/(V s)), although the data points to this being reasonably unlikely. It is shown that measurements of the imaginary part of the conductivity could be used in the future to clarify the mobility if this is the case.

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INTRODUCTION Conjugated polymers have been shown to have many useful optoelectronic properties, enabling the realization of devices such as photovoltaics, transistors, and light-emitting diodes. The ability to use low-temperature solution processing, chemically tune their optoelectronic properties, and make use of their flexible mechanical properties provide polymer-based devices with many potential advantages over their inorganic counterparts. Understanding the energetics and transport of excited states and charges on individual chains is crucial to the ultimate realization of efficient molecular scale electronics. Electrical measurements of charge transport are typically performed on polymer films over distances of hundreds of nanometers to tens of micrometers where charges will typically © 2014 American Chemical Society

have to perform a number of intermolecular hops, which are expected to be much slower than transport along the chain itself. While recent surprising results of record mobilities achieved in polymer films exhibiting poor ordering have allowed great strides in understanding the macrostructure that leads to high mobility across such distances,1,2 there are few techniques for measuring mobility along a single chain where contact resistance, interface properties, and intermolecular charge transport do not play a role. Pulse radiolysis can be used to inject known numbers of charges onto polymer Received: January 30, 2014 Revised: February 28, 2014 Published: March 3, 2014 6100

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with respect to the electric field of the microwaves.16 A principal target of our measurement here, μintra ac , might be equal to μintra , the aforementioned mobility that would govern very dc long-distance transport along chains. The two would be unequal at frequencies high enough that the charge will not have time to sample all the possible conformations of the polymer during the measurement. In real polymer chains, the disruption of the one-dimensional diffusional motion by defects, coiling, and encounters with chain ends during a time of a half cycle of field will change the way the net displacement of charge in space relates to the field of the microwaves, thus limiting the observed mobility to μiso ac,meas < 17 18 μintra /3. Theoretical work and experiments with additional ac simulations19 have pointed to a strongly chain-length-dependent microwave mobility in conjugated polymers that is also limited by chain coiling.4 Values of μiso ac,meas equal to 0.25 and 0.23 cm2/(V s) were obtained for the 34 GHz intramolecular mobility using PR-TRMC on long polyfluorenes (ethylhexyl and octyl side groups, respectively) with an average length of over 100 repeat units.4,14 From the electronic coupling between fluorene repeat units, Prins et al.4 developed a charge transport model based on the solution to the time-dependent 2 iso Schrödinger equation, predicting μintra ac,calc = 54 cm /(V s) (μac,calc 2 18 cm /(V s)) for a stretched chain (all dihedral angles ∼45°). For a coiled chain, Monte Carlo calculations predict that μintra ac,calc (at a microwave frequency low enough for the charge to sample all dihedral bond angle conformations) is expected to be only 5% less than that of the stretched chain. However, the effect of coiling was predicted to reduce the expected measured 2 microwave mobility to μiso ac,calc = 3.2 cm /(V s), since the confinement of the charge to the coiled chain limits its possible displacement in space.4 Suggested reasons for the smaller 2 observed mobility μiso ac,meas ∼ 0.24 cm /(V s) included solvent reorganization, solvent effects on the persistence length of the polymer, the large number of shorter polymers in a Flory distribution of polymer lengths, and defects along the backbone.4 A small concentration of defects could significantly reduce the observed mobility in high intrachain mobility polymers.20 An improved model21 for charge transport that includes polaronic effects and a more nuanced understanding of the effect of the initial state of the charge may also address this difference between calculated and measured microwave mobility in polyfluorene. In this contribution, we build on the work of the Delft group, using our newly developed PR-TRMC setup, to measure the 8.9 GHz microwave mobility on a range of oligo- and polyfluorenes spanning from two polymer repeat units (PRU) up to an average of 138 PRU. The oligomers, with lengths of 2−16 PRU, are monodisperse. The polymer fractions were obtained by separation of a batch of polymer with average length of 31 PRU in a preparative gel permeation chromatography (GPC) column to give a series of fractions with more narrowly defined molecular weight distributions. The length distributions and chemical structures are shown in Figure 1, where these narrowed distributions are compared to a Flory distribution of unsorted polymers of average length 31 PRU. The use of this vast range of oligomers and polymers with better defined length distributions enables us to obtain a value for the intramolecular mobility with an increased confidence, enabling the factors effecting mobility to be better understood, and to estimate the role of defects. This is the first use of PRTRMC to examine the role of defects.

chains in solution by charge transfer from ionized solvent molecules. The mobility of these injected charges on polyfluorene has been measured both by optically monitoring transport to end traps3 and by the absorption of microwaves in a technique called pulse-radiolysis time-resolved microwave conductivity (PR-TRMC).4,5 Both techniques inject charges to create radical cations or anions, which delocalize over a few nanometers to form polarons, and while both are subject to ambiguities they can provide direct measures of intrachain mobilities. To date, charge transport to end-caps has been measured in solutions with a high relative dielectric constant (ε = 7.6), where polaronic effects are expected to be more significant compared to films (ε = 2−3) and where the drag from possible ion pairing cannot completely be ruled out. For transport to end traps, the charge must make it all the way to the end of the chain, so a single defect along a chain could stop the charge from ever being able to reach the trap. Indeed, such a measurement on a long polymer may be considered a dc mobility measurement (μintra dc ), as the charge has to sample every conformational possibility along the chain. For singlet exciton transport, conjugated chains have been modeled with relatively short “conjugated segments”,6−10 separated by sites where there is poor electronic coupling or high-energy barriers between the “segments”, so that intramolecular transport over long distances is difficult and much slower than intermolecular transport.8−11 Polarons may have a different sensitivity to breaks in conjugation owing to differences between charge transfer and energy transfer mechanisms. However, the low −3 −4 intramolecular mobility of μintra cm2/(V s) found dc = 10 −10 in this type of polaron transport measurement could be indicative of barriers along the chain. The extent to which defects, polaronic effects, and ion-pairing may affect the observed mobility in this type of measurement are the subject of continuing study. PR-TRMC is a technique that was pioneered12,13 and has been mastered by the Delft group over the decades, more recently being applied to conjugated materials, including polyfluorene,4,14 as summarized recently.15 The effect of the weak electric field, E, from the microwaves of frequency, f, is to only very slightly perturb the preexisting rapid diffusion of charges up and down the chain, resulting in a slight net movement of charge, ⟨xdrift⟩ = μE/2f ∼ 0.0005−0.025 nm for 2 μintra ac = 1 cm /(V s), E = 1−50 V/cm, and f = 10 GHz, due to the field. The upper limit here corresponds to charge displacement from typical field strengths inside the resonant waveguide cavity used in this work and for the case of a polymer being aligned parallel to the field. This field strength was calculated by finite element modeling and reflects the fact that the standing wave produced by the iris used in this work enhances the field strength compared to the field in the rest of the waveguide. This drift motion is superimposed on the larger diffusional motion that occurs during one-half microwave cycle which has a root-mean-squared displacement of approximately 1/2 xrms = (μintrakBT/fe)1/2 ∼ 16 nm for the same diff = (2Dt) parameters, where D is the one-dimensional diffusion constant for a charge along the polymer, kB is Boltzmann’s constant, T is temperature, and e is the charge on an electron. For the idealized infinitely long, straight chain with no defects, PR-TRMC would be able to correctly determine the ac intramolecular mobility, μintra ac . The measured mobility would be intra the isotropic mobility, μiso ac,meas = μac /3, which reflects an average over all the possible orientations of the straight chains 6101

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benzene with oxygen, the electrons will rapidly (∼0.1 ns) attach to O2 forming oxygen anions (reaction A) which will not reduce the polyfluorene, so only positive charges on the polymer will be observed. In the presence of a polymer, the benzene cation will transfer its charge at diffusion controlled rates to form a polymer cation (commonly referred to as a hole) (reaction B). The rate of attachment depends on the concentration and length of the polymer.26 Finally, on a longer time scale, the polymer cation will recombine with the oxygen anion returning both to their neutral state as seen in reaction C. Various concentrations of polymer and oligomer were used depending on the amounts available, giving concentrations of either 0.2, 0.5, or 1 mM in PRU. Figure 1. Length distributions of the polyfluorenes used in this study from gel permeation chromatography (GPC) and the chemical structures of both oligomers and polymers. The dashed line represents a Flory distribution of unseparated polymers with an average length of 31 PRU.

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EXPERIMENTAL METHODS The PR-TRMC technique has been extensively documented elsewhere.13,22,23 Briefly, the high-frequency conductivity of charged species in solution is measured through the absorption of microwaves, in this case with a frequency of 8.9 GHz. The solution is contained in an iris-coupled section of waveguide forming a resonant cavity (loaded quality factor, Q ∼ 200) to increase the sensitivity of the measurement. Charges are produced by ionization from 50 to 400 ns pulses from a 2 MeV electron beam that are incident on a thinned down section of the waveguide wall to increase the dose delivered to the interior. The fractional change in the reflected microwave power, P, at the resonant frequency of the cavity, f 0, is linearly proportional to changes in the conductivity, σ, inside the cavity and can be closely approximated according to eq 1 Q (1/ R 0 ± 1) ΔP =∓ F Δσ P πf0 ε0εr

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i

(A)

Bz•+ + pF → Bz + pF•+

(B)

O2•− + pF•+ → pF + O2

(C)

Bz•+ + O2•− → Bz + O2

(D)

Bz•+ + I → Bz + I+

(E)

RESULTS Figure 2 shows the time-resolved conductivity per unit dose of radiation from solutions of various oligofluorenes and

(1)

where R0 is the ratio of reflected power to incident power of the filled cavity at the resonance frequency, εr is the relative dielectric constant of the solution (2.28 for benzene), F is a factor describing the overlap of the charge concentration with the microwave field, and all other symbols have their standard meanings. The overlap function, F, can be calculated by considering the distribution of the electric field13 and the region of the waveguide that is irradiated with the electron beam to produce ions. Because of a large diameter electron beam and a well-defined region of the waveguide wall being thinned down, tests on glass slides inside the waveguide showed that half of the cavity used here was exposed uniformly giving an F factor of 0.5. The measured conductivity is related to the mobility, μi, of the i different charged species according to eq 2 σ = e ∑ Niμi

e•− + O2 → O2•−

Figure 2. 8.9 GHz microwave conductivity transients for a range of the oligomers and polymers tested following a 400 ns pulse of electrons with a dose of 11 Gy/pulse. Concentrations are given in PRU. For clarity, not all traces are displayed. See Table 1 for summary of all data.

polyfluorenes in oxygen-saturated benzene following pulse radiolysis. The slight drop in the conductivity at 400 ns corresponds to the end of the electron pulse and is due to both the rapid geminate decay of ions that were formed within their mutual Onsager radius and from recently generated high mobility electrons in the solvent disappearing through reaction A. Geminate recombination of pF•+ with O2•− will typically occur in ∼0.6 ns based on estimated geminate recombination times in neat benzene of ∼4 ps27 and the lower mobility of O2•−. While some geminate ions may survive a decade or two longer, the observed conductivity will arise almost entirely from “free ions” that were generated far enough away from their counterion to escape their mutual Coulombic attraction. The observed growth in Figure 2 is from the diffusional attachment of holes from the benzene cations to the polymers; the conductivity signal from the low mobility benzene cations is barely visible on the scale of this plot. The trend of conductivity

(2)

where Ni is the concentration of the given species and e is the charge of an electron. To understand the charged species that are present, we must consider the radiation chemistry and subsequent charge transfer steps involved in the pulse-radiolysis of liquid benzene. Upon irradiation of liquid benzene, ionization occurs forming benzene cations and highly mobile electrons with mobilities of 10−3 and 0.14 cm2/(V s), respectively.24,25 By saturating the 6102

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obtained from the dose-normalized conductivity and concentration using eq 2 as μ = σ(t)/e[pF•+](t). These determinations used box averages over 10% of the time data around three different points between 1 and 3 μs. The contribution from the mobility of the O2− ion was accounted for based on its reported mobility of 0.85 × 10−3 cm2/(V s) in isooctane,31 which would be equivalent to 0.7 × 10−3 cm2/(V s) in benzene, owing to its higher viscosity (0.65 mPa·s compared to 0.5 mPa·s for isooctane) via the Stokes−Einstein equation. As the concentration of O2− ions should equal the concentration of polymer ions, the mobility can simply be subtracted. The measured mobility is referred to as the isotropic mobility throughout this work, owing to the random orientations of the polymers in space with respect to the uniaxial electric field of the probing microwaves. To compare the isotropic mobility to other publications where the 1D mobility is shown, values given here should be multiplied by three. Results are shown in Table 1.

increasing with chain length is clear from this data. After peaking, the conductivity can be seen to decay according to reaction C. By comparing the different conductivity from the two pF47 traces in Figure 2, which have polymer concentrations of 0.2 mM and 0.5 mM PRU, it can be seen that the number of charges that attach to the polymer is strongly dependent on its concentration. If the only other process involving benzene cations were the relatively slow recombination with oxygen anions (reaction D), then we would expect the ultimate yield of charges captured by the polymer to be nearly independent of the polymer concentration, as reaction B should proceed more rapidly than reaction D at all concentrations used. Grozema reported that a competing process in the solvent, like reaction E, involving an unknown impurity, I, that removes benzene cations is the likely cause of this concentration-dependent yield of polymer holes.28 This sensitivity to polymer concentration shows that the concentration of charged polymers is not given simply by the known free ion yield of the solvent. Knowledge of the concentrations of these polymer holes can be obtained by kinetic modeling, solving a series of differential equations, and fitting to a series of concentrations for each solute,28 but this becomes particularly difficult with small conductivity signals. An alternative approach adopted here directly determines the concentration of charges in each of the solutions for a given dose by optical absorption measurements, since the extinction coefficient of polyfluorene cations is known (62 000 M−1 cm−1 at 580 nm).29 The advantage of this technique is that it truly measures the concentrations of charges on the polymer at any given time, removing ambiguities that can arise from modeling attachment using literature values for the free ion yield in the solvent. Figure 3 shows a very good agreement between the

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ANALYSIS AND DISCUSSION 1. Kubo Formalism. The framework for understanding and modeling ac mobility used herein is similar to that used by the Delft group and is based around the Kubo formula,32 written in a form useful for hopping models based on mean-squared displacement rather than velocity autocorrelation33 (eq 3). The Kubo formula can relate the mean-squared displacement of a charge, ⟨Δ(t)2⟩, in n-dimensional space to the complex drift mobility, μac(ω), in a weak ac electric field with angular frequency ω, at temperature T, and where Boltzmann’s constant is given by kB: μac ≡ −

eω 2 2nkBT

∫0

∞

⟨Δ(t )2 ⟩e−iωt dt

(3)

For normal Gaussian diffusion, where ⟨Δ(t)2⟩ = 2nDt, the Kubo equation reduces to the frequency-independent Einstein relation for a diffusion constant, D. However, for nonlinear increases in mean-squared displacement with time due to finite chain length, coiling, or defects, the mobility will have a frequency dependence; the real part will be reduced, and a nonzero imaginary component will emerge. A nonzero imaginary part to the mobility corresponds to a change in the real part of the dielectric constant. Rather than absorbing microwaves, an imaginary mobility will therefore shift the resonant frequency of the solution-containing microwave cavity, which can be measured experimentally.22 While we are assuming that motion along the chain can be described by normal diffusion, the exact details of the type of charge transport may affect the validity of this assertion. Both coherent transport, interrupted by scattering, or a hopping model will produce diffusive behavior on long time scales, but deviations may occur at short times.4,34−36 Crucially, the linear ⟨Δ(t)2⟩ random walk should be established in a much shorter time than the microwave period for our assumption to be valid. We believe that an intramolecular mobility of ∼1 cm2/(V s), which would give xrms diff ∼ 19 repeat units during one-half microwave cycle, would mean that the charge is likely to have experienced numerous hops/scattering events on the time scale during which the field is acting on it. Measurements of conductivity over a large range of probing frequencies would be required to improve upon this approximation and establish a more complete experimental data set with which to compare to models of charge transport.

Figure 3. Combined dose-normalized data from microwave conductivity (8.9 GHz) and optical absorption (580 nm) measurements on two solutions of polyfluorene with ⟨n⟩ = 47 PRU. Contributions from pF triplets were removed from the optical absorption by subtracting a scaled trace, taken at 750 nm, using the fact that the triplet absorption ratio at 750 nm compared to 580 nm is 6.4.30

two optical and microwave measurements performed on the two pF47 solutions of different concentrations. An overlapping optical absorption from the polymer triplet affecting the trace in the first hundreds of nanoseconds was subtracted by measuring the triplet decay where it peaks at 750 nm and scaling for its different extinction coefficient at 580 nm, using the ratio of 6.430 to obtain the time-dependent concentration [pF•+](t). The agreement of the two sets of data is a good indication that this dosimetry method works consistently. The mobility is then 6103

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Table 1. Mobilities (Isotropic) for Holes in Oligo- and Polyfluorenes Measured at 8.9 GHz length (PRUs)

concn (mM PRUs)

a 2 μiso ac,meas (cm /(V s))

length (PRUs)

concn (mM PRUs)

a 2 μiso ac,meas (cm /(V s))

2 4 4 6 8 8 10 10 12

1 0.5 1 1 0.5 1 0.2 1 0.2

0.0005 0.0016 0.0009 0.0015 0.0033 0.0022 0.0074 0.0074 0.0080

16 26 36 47 47 59 86 138

0.2 0.5 0.5 0.2 0.5 0.5 0.2 0.2

0.016 0.073 0.11 0.13 0.13 0.17 0.18 0.17

These are the isotropic mobilities and are obtained by subtracting μ(O2−) = 0.7 × 10−3 cm2/(V s) from the measured mobility. Isotropic mobility refers to the fact that the polymers have random orientations in space. These numbers should be multiplied by three to compare to “1D mobility” found elsewhere. We estimate random measurement error to be less than 10% for all but the shortest oligomers where the mobility is reaching the limitations of the instrument. a

finite chain length; this ac intramolecular mobility is not affected by the chain ends but is intrinsic to the polymer backbone. For an isotropic distribution of chain orientations, such as in a solution, the observed mobility would be reduced by a factor of 3 and is referred to as the isotropic mobility. An alternative to assuming normal diffusion along the polymer would be to use a predictive transport model4,35 that takes into account factors such as electronic structure, disorder, and polaronic effects to calculate ⟨Δ(t)2⟩ in eq 3 for simulated polymer conformations. While an accurate model that can predict intramolecular mobility for any given polymer structure is an ultimate goal, in this work we are simply interested in making the assumption that the transport can be described with normal diffusion along the polymer and using our extensive data set to obtain an accurate value of μintra for polyfluorene. ac The remaining sections are based on using eq 4 to describe motion along the chain, but taking into account the effects of chain conformation, polymer length distributions, defects, and a finite polaron length. 3. Chain Conformation: Incorporating Coiling. In the case of coiled polymers in three dimensions, the charge displacement in space will not correspond to the displacement along the contour of the polymer. Our method here is to translate the normal one-dimensional diffusion along a finitelength polymer backbone to a mean-squared displacement in three dimensions that can be used in the three-dimensional Kubo equation to give the isotropic ac mobility. To achieve this, the calculation of eq 5 is performed numerically by replacing (x − q)2 with ⟨|r(xcontour) − q|2⟩, the average squared displacement in space corresponding to a distance along a simulated chain of xcontour. The solution to this modified eq 5 now represents the average squared displacement of charge, ⟨Δ(t)2⟩, in three dimensions due to normal one-dimensional diffusion along an average polymer conformation of a given length. The average squared displacement function was found by creating random polymer conformations using the calculated repeat unit geometry (Figure 4a) and dihedral bond angle potential (Figure 4b). Relaxed potential energy surfaces were computed as a function of the central dihedral angle between two neutral oF2 segments with Gaussian 0937 using density functional theory at the B3LYP/6-31g(d) level. Dihedral angles were selected randomly assuming a Boltzmann distribution. From Figure 4c it can be seen that for short chains with coiling, the average squared displacement in space closely matches the displacement squared along a hypothetical stretched chain where all dihedrals are set to 37°. This stretched chain is a

2. Effect of Finite Chain Length. A finite chain of length, L, will disrupt the typical random walk diffusion by modifying the shape of ⟨Δ(t)2⟩, which in turn, through eq 3, will affect the observed ac mobility in response to an external microwave field. Rather than increasing linearly without limit, ⟨Δ(t)2⟩, will flatten out at L2/6 when a charge, initially located at some arbitrary point, will have an even probability of existing at any point on the chain due to reflections from the ends. The observed ac mobility will increase with chain length until the chain is long enough that normal diffusion, during the time of one microwave half-cycle, is not significantly affected by reflections at the endsthe point of this saturation in mobility, for a straight chain, being dependent only on the intramolecular mobility and the frequency of the microwave field. Prins et al. found an analytical solution for charge density, C(x,t), from normal diffusion with a one-dimensional diffusion constant, D, along an object of length L of a point charge starting at x = q with reflecting boundary conditions at the ends as shown in eq 4.17 C(x , t ) =

1 L

∞

∑

cos

k =−∞

πkq −(πk / L)2 Dt πkx cos e L L

(4)

The mean-squared displacement of charge can then be found by integrating (x − q)2, multiplied by the normalized charge distribution, C(x,t), over the length of the chain and over all starting positions, q, as shown in eq 5: ⟨Δ(t )2 ⟩ = ⟨(x − q)2 ⟩ ≡

1 L

L

∫0 ∫0

L

(x − q)2 C(x , t ) dx dq (5)

Equations 4 and 5 can be used to give an analytical solution to eq 3 for ac one-dimensional mobility along a finite straight object, where in the absence of an applied field, the motion of a charge would follow normal diffusion: −1 ⎡ ⎛ k Tμ c 2 ⎞⎤ B k intra 2 + 1⎟⎟⎥ ∑ ⎢⎢ck ⎜⎜ 2 ⎝ ieωL ⎠⎥⎦ k=0 ⎣ ∞

μac1D

= 8μintra

(6)

By assuming normal diffusion along the chain, μintra in eq 6 relates to the one-dimensional diffusion constant, D, in eq 4 by μintra = DkBT/e. The coefficient ck is given by 2π(k + 1/2). This ac mobility was calculated for a one-dimensional object or equivalently for a straight polymer chain that is aligned with the electric field and shows how the measured ac mobility would be smaller than the actual ac intramolecular mobility due to the 6104

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by being confined on such a polymer; 98% of the length of an additional repeat unit added to the contour length is added to the displacement in space. For longer coiled chains whose length is comparable to or larger than the persistence length, the function becomes almost linear, similar to random walk and deviates significantly from the stretched chain example. The persistence length that was calculated from simulated polymers, based on the calculated bond angle potentials, was 12 PRU, whereas it has been experimentally found to be 10 PRU.38 By increasing the catenation angle by 10%, the simulated polymers were found to have a persistence length matching the experimental value. While this modification is somewhat artificial, it serves as a useful way to see how sensitive the predicted mobility is to uncertainty in the calculated polymer conformation. The calculated isotropic mobility as a function of the length of the polymer from both these conformations and a stretched chain are shown in Figure 4d for an intramolecular mobility of 1 cm2/(V s). It can be seen that for short chains the predicted isotropic mobility is practically the same for all three conformations. At longer lengths, the stretched chain conformation gives an isotropic mobility approaching μintra ac /3 = 0.33 cm2/(V s), as expected, while coiling limits the maximum observable mobility to 0.2 and 0.18 cm2/(V s) for the coiled chains with persistence length of 12 PRU and 10 PRU, respectively. This limiting effect of coiling becomes more pronounced for higher intramolecular mobilities or when probed at lower frequencies where, in both cases, the charge will sample a longer length of the polymer during one microwave half period and so experience more curvature. 4. Length Distributions. The model developed here gives an isotropic mobility as a function of chain length and microwave frequency for a given intramolecular mobility along polyfluorenes. To apply it to the experimental data, we must consider the distributions of polymer lengths present in our solutions. It has been suggested that the Flory distributions of lengths commonly found with polymers could lower the observed mobility, as it is heavily weighted toward shorter chains.18 Even though the distributions used here have been separated by preparative GPC, they may still result in a different observed mobility than a monodisperse distribution with the same average length, particularly in regions where the predicted mobility varies strongly with chain length. To find the best fit for the intramolecular mobility, the monodisperse mobility function was averaged over each of the polymer distributions from Figure 1, weighted by the probability of attachment; charges are more likely to attach to the longer chains within a distribution due to a length-dependent attachment rate. The attachment rate is proportional to an effective radius, which can be approximated by a prolate spheroid model for a polymer26 (see Supporting Information). The averaged mobility varied only slightly from the monodisperse predictions. The best fit to the whole data set, taking into account the polymer distributions and attachment rates gave an intramolecular mobility of 1.1 cm2/(V s) and is shown in Figure 5. The same value of intramolecular mobility was found for the calculated polymer conformations with persistence lengths of both 10 and 12 PRU, although the fit was slightly better for the former and is the one given in the figure; for straight chains, the fit error is twice as large and does not come close to the measured mobility for the polymers. 5. Defects. This model of one-dimensional normal diffusion of a point-like charge along a coiled polymer with reflecting boundary conditions appears to fit the observed isotropic

Figure 4. (a) Two repeat units of polyfluorene with the dihedral (θ) and catenation (φ) angles marked and showing the length of the vector added to represent each repeat unit in the model. (b) Calculated bond angle potentials and probability distribution used for randomly selecting each dihedral angle to construct the polymer. (c) Average squared displacement in space corresponding to a distance along the contour of the polymer, x, for a stretched chain (triangles) and coiled chains with a persistence length of 12 PRU (circles) and 10 PRU (squares). Inset shows a stretched polymer of 50 repeat units (black) and a typical calculated coiled polymer conformation with 200 repeat units (red) with the displacement along the chain and in space labeled. (d) Calculated real and imaginary components of the isotropic ac mobility at 8.9 GHz for three different chain conformations as a function of polyfluorene length using eqs 3 and 5 as modified according to the text.

useful comparison to see the effect of coiling as the distance traveled by the charge in space becomes practically unaffected 6105

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Figure 5. Best fit with an intramolecular mobility of 1.1 cm2/(V s) for simulated polyfluorenes with a persistence length of 10 PRU. Solid line is calculated for monodisperse polymer lengths and open squares for the actual polymer distributions used.

mobility reasonably well for this entire polyfluorene data set. It is possible, however, that the measured isotropic mobility could be limited by defects along the chain rather than by coiling; defects could mask a much higher intramolecular mobility than is apparent. Measurements of slower than expected charge transport to end-caps indicate that barriers to charge transport may exist along polyfluorene chains.3 Broadened absorption spectra and the exciton dynamics in phenylenevinylenes have also been explained with the concept that a polymer can be divided into weakly interacting conjugated segments due to unfavorable dihedral angles or defects.7,10,39 In such a picture, a charge may be confined to a short segment for a period of time, reducing the observed microwave mobility. Such a clear division into subunits may not be applicable to all polymers, however, as shown for polythiophene.40 To investigate, in a quantitative manner, the role of barriers to transport along the chain and whether they might be relevant in polyfluorene, a statistical model was developed in which a polymer chain is divided into segments by randomly dispersed defects. These defects may be either chemical in nature or dihedral angles that result in particularly poor electronic coupling between adjacent repeat units. In order to calculate the mobility for such a segmented polymer, one must know the probability that a charge is on a segment of a particular length, given a chain of length N with an average segment length between barriers of ⟨ns⟩. The full derivation for this is found in the Supporting Information. Figure 6a shows the probability distributions of the various possible segment lengths for the various polymer lengths used in this work, given an average spacing between defects of 16 PRU. The spikes in probability occur as the polymers become close in length to the average segment length, so there is a reasonable probability of there being zero defects on the polymer, which corresponds to a sharp peak in the distribution. If a chain is broken down into segments separated by impermeable barriers, then the probability of a charge being on any particular segment of a given chain is simply proportional to the length of that segment as shown by the dashed line in Figure 6a. Figure 6b shows the final probability of a charge being on a given segment, taking into account both the probability of the segment length existing and the probability that it is occupied. A calculated curve of isotropic mobility versus chain length, such as the solid lines in Figure 4d or 5 (or eq 6 in the case of straight segments), is taken, and for each length the mobility is recalculated by assuming that that chain is made up of a distribution of segments of the type shown in

Figure 6. Probability mass function for (a) a segment of length ns existing on a chain of N PRU and (b) a charge being on a segment of length ns on a chain of N PRU if, in both cases, the average segment length on an infinitely long chain, ⟨ns⟩N→∞, is 16 PRU. The dashed line in (a) represents the relative probability that a charge would occupy a particular length segment on a given chain. The probability functions are calculated for monodisperse chains having lengths equal to the averages for the various polymer batches used in this study.

Figure 6. The contribution to the observed mobility from each of these segment lengths, taken from the original isotropic mobility calculation is then averaged together (weighted by its appropriate probability of existing and being occupied) to give a new value of mobility that now takes defects into account. Figure 7 shows the experimentally determined 8.9 GHz isotropic mobility plotted against number of repeat units together with the best fit of the defect model for the three different chain conformations mentioned above, where μintra ac and ⟨ns⟩ are the only fit parameters. Interestingly, all three conformations give μintra to be the same within 10%. This ac

Figure 7. Calculated real (solid) and imaginary (dashed) parts of mobility with the defect model for three different chain conformations. Lines are calculated for monodisperse polymer lengths and open squares for the actual polymer distributions used. Average defect spacing in PRU between defects is given in parentheses. 6106

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7. Validity of Diffusion Model for the Shortest Oligomers. Finally, the applicability of a model that uses normal diffusion of a point-like charge to represent polarons that have a known length shall be assessed. The charges are polarons having lengths of 4−5 fluorene repeat units29 based on a bleach of the neutral absorption or changes of spectra or redox properties with length of oligomers. From the dependence of redox potentials on oligomer length, it is estimated that compressing a polaron by 15% would require 2kT in energy, so it may be that any element of charge density in the polaron is limited to travel a distance along the polymer of Lpolymer − Lpolaron, which corresponds to shifting the mobility function from Figure 5 by ∼4 repeat units to the right. It is clear that, should this be correct, the shape of the calculated mobility would better match the experimental data for the longest oligomers and all the polymers to the significant detriment of the fit for the shortest oligomers. This could mean that the holes occupy less space than reported, but the present data provide only very indirect information about this question. As it is far from clear that a diffusion model would be appropriate for oligomers of a length comparble to the polaron, it may not be reasonable to expect the model to fit well in this region. As a comparison, terahertz spectroscopy of photoexcited CdSe nanoparticles have shown a rapid drop in ac mobility when the nanoparticle becomes sufficiently small that the electron is delocalized over its entire volume.44 Possible explanations for an enhanced observed mobility for static polarons on short oligomers could include the contribution from the diffusion of the molecule itself and the possibility that for oligomers and short polymers the presence of a hole on the chain might alter the structure of the entire chain. For example, a diffusing hole four units long in an eight unit long oligomer might oscillate along the chain faster than relaxation of dihedral angles. The dihedrals would then take on angles halfway between those of the neutral and the positive ion, perhaps substantially altering transport. The Kubo expression assumes diffusion in very small steps. If transport is coherent over short lengths, the observed mobilities for short chains may be modified relative to that for long chains. Other factors are possible alignment of the molecule over time with the direction of the field, small changes in reflected microwave power due to the imaginary part of the conductivity changing the resonant frequency of the cavity, or some other mechanism for microwave dissipation for delocalized charge distributions. Figure 8 shows the best fit, both with and without defects, when the polaron length is taken into account, by fitting only to the polymer data based on the possibility that the model may not be valid in the shorter oligomers. In the absence of defects, such a shift due to the polaron improves the fit to the polymers and matches well to the longest oligomers that were not included when optimizing the fit. The estimate of intramolecular mobility is reduced slightly to 0.8 cm2/(V s) with the addition of this finite polaron effect. Introducing defects without the constraint of the oligomers allows for the higher intramolecular mobility of 2.8 cm2/(V s). It is clear that if this were the case, we would be able to tell from a measurement of the imaginary part of the conductivity, which would be much larger than the defect-free case. However, the fact that the predicted mobility for the longest oligomers is far from the experimental data suggests that such a high mobility is unlikely. In summary, these results show that if it turns out that the diffusion model is not valid for the shortest oligomers, the estimate of intramolecular mobility is not significantly affected.

similarity is largely due to the fact that neither coiling nor defects have much effect in the oligomer region, which constrains the range of μintra ac values that are possible. 6. Defects vs Coiling. As the chain length increases, the observed mobility reaches a maximum at ∼50 repeat units and does not increase further. Both coiling and defects that act as barriers can limit the measured mobility in this way, saturating it short of the value of the true μintra ac that would be measured for a defect-free, infinitely long chain. Analyses given here find that either coiling (Figure 5) or defects that create barriers (Figure S4) can adequately explain this plateauing of the mobility without any contribution from the other. So, of course, can combinations of the two as illustrated in Figure 7. Coiling is expected and clearly supported experimentally, but plausibly both are present. If so, a question is which sets in first. Of these two effects, the one that first limits observed mobility as chain lengths increase will at least partially mask the effect of the other. Because coiling can be considered inevitable in conjugated chains, certainly in polyfluorenes, it is tempting to neglect defects. On the other hand, measurements of transport to end traps3,41 indicate that some charges can take significantly longer than others to diffuse along the full length of a chain. The fact that for the straight chain model a large fraction of the segments would have to be longer than the known persistence length for polyfluorene suggests that, if they exist on the polymer chain, the defects are sufficiently sparse that they do not precede coiling as the limiting factor determining the measured isotropic mobility in polyfluorene. Indeed, if defects were the limiting factor, then the fitting would have predicted the same defect spacing for all three chain conformations. This means that defects may be present and could have a major effect on transport over long distances, but we are unable to get a clean estimate, owing to the simultaneous effect of coiling, which is sensitive to having simulated the correct chain conformation. Polymers enclosed in macrocycles42,43 may provide a path to gain more control over the chain conformation allowing for a more subtle understanding of ac mobility based upon well-defined straight segment lengths and electronic couplings.35 It is highly likely, however, that defects will be the limiting factor for ac mobility in more rigid polymers. The model developed in this paper would be attractive for that scenario as, to a good approximation, the chain could be considered straight in between defects, and the less computationally intensive analytical expression in eq 6 could be used in conjunction with the defect and occupation statistics. That said, the fact that the fits are improved through the addition of defects allows for the possibility that there are defects along the polyfluorene backbone with an average spacing on the order of 30−40 repeat units. This represents a lower limit as the fit error worsens only marginally as the spacing between defects increases, as shown in Figure S5 of the Supporting Information. Such defects could be static and chemical in nature, leading to breaks in the conjugation or possibly dynamic, such as fleeting rare dihedral angles, which give particularly poor electronic coupling that persists for at least the 50 ps time scale of the microwave half-period. If present, the defects impede transport and are not deep traps that eliminate holes or render them immobile. The near constancy of the observed mobility from 40 to 125 repeat units, with the assumption that the number of deep traps, if present, would increase with chain length, leads to the conclusion that there are probably few, if any, such deep traps. 6107

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98-CH10886 (BNL, including use of the LEAF and Van de Graaff facilities of the BNL Accelerator Center for Energy Research) and Grant DE-AC36-08GO28308 (NREL) for polymer research. Work of M.B. and O.R. to develop the Pulse-Radiolysis TRMC experiment was supported by Laboratory Directed Research Grants 02544 at BNL and 06RF1002 at NREL. M.B. thanks Stephen Albright for useful discussions and assistance during initial testing of the microwave apparatus, Jim Anselmini and Jeff Hoogsteden for fabrication of the microwave cavities, Richard Holroyd for background information on ion mobilities in organic solvents, and James Wishart, Jack Preses, and Sergei Lymar for assistance with the Van de Graaff accelerator.

Figure 8. Calculated real (solid lines) and imaginary (dashed lines) isotropic mobility at 8.9 GHz, with and without defects, when taking account of the finite length of a polaron. Fitting is only done to the polymer data, as this model would be inappropriate for the shorter oligomers.

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CONCLUSION By measuring the 8.9 GHz microwave mobility of a range of oligomers and polymers, we have demonstrated that the intramolecular mobility along a fluorene backbone is 1.1 cm2/ (V s) by fitting to a one-dimensional diffusion model. This model takes account of the curvature of the backbone, the length distributions of the polymers, and the possibility of defects along the chain. Should such defects exist, the data are suggestive that they occur at a rate of at most 1 every 30 or 40 repeat units, with this value being strongly dependent on the conformation of the polymer in solution. Such defects appear to act as barriers or shallow (∼kT) traps; the data offer evidence against the presence of deep traps. The estimate of 1.1 cm2/(V s) for intramolecular transport is larger than the measured isotropic mobility in the longest chains by a factor of ∼6, underscoring the importance of careful analysis to interpret microwave mobility values. The extensive data set available for fitting appears to limit the possibility of the mobility being either significantly smaller or larger than our estimate. Optical absorption was used as a new method for calculating charge concentrations, which overcame complications in calculating dose due to impurities, varying charge capture rates, and reliance on literature radiation dose yields.

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ASSOCIATED CONTENT

S Supporting Information *

Information about the statistical modeling of polymer distributions with defects for calculating average mobility. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected]; Tel 631 344 4354; Fax 631 344 5815 (J.R.M.). *E-mail [email protected] (M.J.B.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge support of the Solar Photochemistry Program, Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences of the U.S. Department of Energy, through Grant DE-AC026108

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