11104
J. Phys. Chem. C 2007, 111, 11104-11112
Charge Transport along Coiled Conjugated Polymer Chains Paulette Prins,† Ferdinand C. Grozema,*,† Frank Galbrecht,‡ Ullrich Scherf,‡ and Laurens D. A. Siebbeles† Opto-Electronic Materials Section, DelftChemTech, Delft UniVersity of Technology, Julianalaan 136, 2629BL, Delft, The Netherlands, and Department of Chemistry and Institute for Polymer Technology, Bergische UniVersita¨t Wuppertal, Gauβstraβe 20, 42119, Wuppertal, Germany ReceiVed: February 7, 2007; In Final Form: May 18, 2007
The motion of charges on coiled polymer chains was studied using a combination of experimental and theoretical methods. The conductive properties of dilute solutions of polyfluorene and fluorene-binaphthyl copolymers were studied by the pulse-radiolysis time-resolved microwave conductivity technique. This technique enables measurement of the (high frequency, 34 GHz) mobility of charges on isolated polymer chains. The motion of positive charges on the coiled polymer chains was studied theoretically by charge transport simulations with parameters from density functional theory calculations. This combined experimental and theoretical study shows that the mobility of charges decreases with increasing degree of chain coiling. The mobility along (infinitely long) stretched polyfluorene is calculated to be as high as tens of centimeters2/ volt‚second. Our results imply that the performance of conjugated polymers in optoelectronic devices can be significantly improved by optimization of the organization on a molecular scale.
1. Introduction The optoelectronic properties of conjugated polymers are of current interest because of the extensive potential applications for these materials in optoelectronic devices.1,2 Alkyl-substituted conjugated polymers are soluble in organic solvents, which allows convenient processing from solution, which, in turn, enables low-cost production of devices based on conjugated polymers.3 Relatively small changes in the processing conditions, such as choice of solvent or polymer concentration, can lead to large variations in the optoelectronic properties of these devices.4,5 Both the optical and electrical properties of polymer films depend strongly on the organization of the polymer chains on a molecular scale. One well-known example is the red shift of the absorption and emission spectra as a result of a higher degree of intra- or interchain organization.6,7 Another example is the enhanced charge carrier mobility as a result of increased organization upon heat treatment or alignment of polymer chains.8,9 A first step toward understanding the effect of the molecular organization of polymer chains on the optoelectronic properties of devices is to study the optical and electrical properties of isolated polymer chains in the solution from which the polymer layer is processed. In a dilute solution, interchain interactions do not play a role, and the optoelectronic properties are determined by intrachain interactions and the interaction between the polymer and the (processing) solvent. Information about the relation between the optoelectronic properties and the conformation of polymer chains in the processing solution can be used as a starting point to understand the more complicated system of polymer chains after transfer into a polymer film. Conjugated polymer chains can adopt many conformations.10 The specific conformation of a polymer chain depends strongly on the stiffness of the polymer backbone and the properties of * To whom correspondence should
[email protected]. † Delft University of Technology. ‡ Bergische Universita ¨ t Wuppertal.
be
addressed.
E-mail:
the direct environment. Consequently, the spatial extent of a particular polymer in solution varies with the type of solvent used. In a poor solvent, the polymer chain self-aggregates, resulting in a collapsed chains with a considerable degree of coiling (although much less than in many nonconjugated polymers), whereas in a good solvent, the chain unfolds, leading to a more open and extended conformation. The optical properties of conjugated polymers in solution have been studied extensively and were found to be affected by such differences in chain conformation.11 In accordance with the increase in effective conjugation length, both the absorption and emission spectra are red-shifted for polymer chains in a good solvent with respect to the spectra for the same polymer chains in a poor solvent.12 The charge transport properties of isolated conjugated polymer chains in solution can be studied by pulseradiolysis in combination with time-resolved microwave conductivity measurements13 (PR-TRMC) on dilute polymer solutions.14 With this technique, charge carriers are generated by irradiation of the solution with high-energy electron pulses. The resulting change in conductivity is monitored as a function of time. In this contribution, we present an experimental study of the high-frequency (34 GHz) mobility of charge carriers on coiled polymer chains for which the spatial extent is systematically varied. We show that the measured charge carrier mobility is strongly affected by the spatial extent or, equivalently, by the amount of coiling of the polymer chains. We substantiate our experimental results with numerical simulations of charge transport using parameters from density functional theory (DFT) calculations and show that the isotropic mobility calculated for a coiled chain is more than an order of magnitude lower than the (one-dimensional) mobility along the polymer backbone, which is calculated to be as high as a few tens of centimeters2/ volt‚second. Polyfluorene (PF, see Figure 1a) is a prototype example of a coiled conjugated polymer. The coiled conformation of PF
10.1021/jp071077d CCC: $37.00 © 2007 American Chemical Society Published on Web 07/03/2007
Charge Transport along Polymer Chains
J. Phys. Chem. C, Vol. 111, No. 29, 2007 11105 TABLE 1: Experimental and Calculated Values for the High Frequency (34 GHz) Isotropic Mobility of a Positive Charge on the Fluorene-Naphtyl Copolymers Used in This Study iso iso µac,calc µac,exp lcalcd Mn,exp (units) pdiexp (cm2/Vs) (cm2/Vs)a (units)
polymer stretched PF PF PF + 0.0422 BN PF + 0.0944 BN PF + 0.121 BN F-BN block copolymer
493 235 409 321 21
2.4 2.0 2.2 1.2 1.7
0.23 0.14 0.08 0.06 0.01
18 3.2 1.0 0.50 0.38 0.14b
∞ 11.7 8.7 6.5 5.8 2.1
a
Mobility calculated for infinitely long polymer chains. b Isotropic mobility for F-BN block copolymer with 21 molecular units: 0.022 cm2/Vs.
Figure 1. Chemical structure of (a) poly(9,9′-dioctylfluorene), (b) (2,2′octyloxy)-1,1′-binaphthalene unit, and (c) F-BN alternating copolymer.
Figure 2. Chemical structure of bifluorene (at θ ) 0). φ and θ denote the angle and dihedral angle between the fluorene units, respectively.
results from the nonzero bend angle (φ) and the nonzero dihedral (torsional) angle (θ) between adjacent fluorene units, as shown in Figure 2. The angle φ is 23.5° and the dihedral angle θ is about 45° from a planar conformation (see below). In accordance with this chain twisting, the persistence length for the PF derivative in Figure 1a was found to be only 10 fluorene units in tetrahydrofuran.15 Incorporation of binaphthyl (BN, see Figure 1b) in the PF backbone is expected to result in an even more coiled chain conformation, since the dihedral angle between alkoxy substituted naphthyl units in the configuration shown in Figure 1b is close to 90° (see below). With this perpendicular orientation, BN is expected to act as an efficient chain bender. Incorporation of BN in the backbone of poly(9,9′-dioctylfluorene) was found to suppress beta-phase formation,16 which confirms that the supramolecular organization of this PF derivative is altered by the presence of BN. 2. Experimental Section The PFO and statistical copolymers studied in the present work were prepared via a nickel-mediated Yamamoto coupling by varying the relative ratios of both comonomers and were characterized as described previously.16,17 The charge transport experiments were performed on dilute solutions of coiled polymers in benzene. Charge carriers were generated by pulse-radiolysis of the dilute polymer solution with a 10 ns pulse of 3 MeV electrons from a Van de Graaff electron accelerator. The high-energy electrons scatter on the solvent molecules and produce a close to uniform distribution of excess electrons and benzene cations with a known concentration. These excess electrons and benzene cations can diffuse toward the polymer chains, where they undergo charge transfer, thus yielding a (negative or positive) charge on the polymer backbone.18,19 The change in conductivity after the generation of charges was monitored by time-resolved microwave conductivity measurements at a microwave frequency of 34 GHz.13
Figure 3. Conductivity due to holes on polyfluorene chains with a fraction of 0.000, 0.0422, 0.0944, and 0.121 statistically distributed binaphthyl units and on a fluorene-binaphthyl alternating copolymer, from top to bottom. The measurements were performed at a repeat unit concentration of 0.315 mM, an irradiation dose (D) of 20 Gy, and a microwave frequency of 34 GHz.
The high-frequency conductivity as a function of time is related to the high-frequency mobility of all charged species present in the solution (µac,i) according to
∆σac(t) ) e
∑i µac,ini(t)
(1)
where ni(t) is the number density of charged species i at time t and e is the elementary charge. The initial number density of excess electrons and benzene cations is determined by the irradiation dose (D) and the yield of free charges.18 The polymers used in this study are the PF derivatives depicted in Figure 1a, fluorene-binaphthyl copolymers with different fractions of statistically distributed BN (Figure 1b) incorporated in the PF backbone and the fluorene-binaphthyl (F-BN) alternating copolymer depicted in Figure 1c. The weight averaged molecular weight (Mw) and polydispersity index (pdi) of the polymers are listed in Table 1. 3. Experimental Results In Figure 3, we present the change in conductivity as a function of time for dilute solutions of the polymers listed in Table 1 upon the generation of charge carriers. In order to selectively study the mobility of holes along the polymer chains, the solutions were saturated with oxygen. Due to the relatively high concentration (12 mM) and electron affinity of oxygen, the excess electrons generated during the electron pulse (ebz-) rapidly react with the oxygen molecules (O2) forming the oxygen anion (O2-).
ebz- + O2 f O2-
(A)
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Consequently, the transfer of negative charges to the polymer chains is prevented. Since the mobility of the oxygen anion in benzene is low (10-3 cm2/Vs)20 compared to the mobility of the positive charges along the polymer chain (see below), the contribution of the oxygen anion to the observed change in conductivity is negligible. The benzene cations generated during the 10 ns electron pulse (bz+) react with the polyfluorene chains (PF) by a diffusion-controlled reaction, yielding positively charged polymer chains (PF+).
bz+ + PF f PF+ + bz
(B)
As this reaction proceeds, an increase in the transient conductivity is observed on a time scale of microseconds. This increase directly indicates that the positive charge on the polymer chain is more mobile than the benzene cation (1.2 × 10-3 cm2/Vs).18 On a time scale of hundreds of microseconds, a decrease in the conductivity signal is observed. The time scale of this decay decreases with increasing initial concentration of charge carriers (not shown) and is therefore attributed to charge recombination between the oxygen anion and the positive charge on the polymer chain.
O2- + PF+ f PF + O2
(C)
A more extensive description of the reactions that occur upon irradiation with high-energy electrons and the resulting transient conductivity of dilute polymer solutions has been presented previously.18,19 The experimental isotropic high-frequency mobility for holes (µiso ac,exp) can be obtained from the measured conductivity shown in Figure 3 using eq 1, when the time-dependent number densities of charged species are known. These densities were obtained from a kinetic analysis of the conductivity transients on the basis of reactions A, B, and C. The calculated results for the conductivity as a function of time are shown in Figure 3 as smooth curves. The mobility values obtained from the kinetic analysis are presented in Table 1. The magnitude of the isotropic high-frequency mobility found here for holes on PF chains is comparable to values found previously with TRMC for other conjugated polymers in dilute solution.14,18,19,21-23 The mobility is found to decrease with an increasing fraction of BN incorporated in the PF backbone, from 0.23 cm2/Vs for PF to 0.06 cm2/Vs for PF with a BN fraction of 0.121, indicating that charge transport is less efficient when BN is present. Interestingly, the F-BN alternating copolymer shows a nonzero conductivity signal, indicating that charges can still move along these copolymers. This shows that BN does not break the conjugated pathway along the polymer backbone completely. It should be noted that the charge carrier mobility listed in Table 1 for the alternating F-BN copolymer maybe be considerably limited by the chain ends, since this polymer is much shorter than the random copolymers in Table 1. This is actually shown directly by the calculation presented below and referred to in Table 1. On basis of these experimental results, it is not possible to conclude whether the presence of BN affects the mobility of charges on PF chains due to a modification of the chain conformation or because BN is merely a barrier to charge transport as a result of the close to perpendicular orientation of the naphthyl units. To gain fundamental insight into the charge transport on coiled polymer chains, we have modeled the chain conformation of fluorene-binaphthyl copolymers and we have simulated the charge transport along these polymer chains.
4. Theoretical Basis According to the work of Kubo, the (three-dimensional) frequency-dependent mobility of charge carriers is given by24-26
µac(ω) ) -
eω2 6kBT
∫0∞ 〈∆2(t)〉 cos(ωt) dt
(2)
where e is the elementary charge, ω is the (radial) frequency of the probing electric field, kB is Boltzmann’s constant, T is the temperature, and 〈∆2(t)〉 is the mean squared displacement of the charge. An implicit convergence factor, exp(-t) (lim f 0) is understood in the integral.25,26 For normal Gaussian diffusion,the mean squared displacement of charge carriers increases linearly with time,
〈∆x2(t)〉 ) 6Ddct
(3)
where Ddc is the diffusion constant. In this special situation, the mobility is independent of the frequency of the probing electric field, and eq 2 reduces to the Einstein relation
µdc )
e D kBT dc
(4)
In general, the mobility is frequency-dependent in disordered solids, and eqs 3 and 4 are not valid. In such cases, the charge carrier mobility can be obtained from the mean squared displacement using eq 2. In the present work, the mean squared displacement is obtained from a numerical simulation of the motion of charge carriers on coiled polymer chains. The conformation of the polymer chain is defined by the multidimensional energy landscape, that is, the potential energy as a function of all bond lengths, bond angles, and dihedral angles in the polymer chain. The variations in the bond lengths and bond angles are small due to the relatively rigid structure of the fluorene and naphthyl units. As a result, the polymers can be modeled as chains of sites that correspond to the molecular units of the polymers listed in Table 1, that is, a chain of fluorenes for PF, a chain of fluorenes with a fraction of statistically distributed binaphthyls for the F-BN copolymers, and a fluorene-naphthyl-naphthyl sequence for the F-BN alternating copolymer. In this way, the polymers in Table 1 can be modeled with two different molecular units: fluorene and naphthyl. To obtain the three-dimensional isotropic mobility of charges for coiled polymer chains, the actual mean squared displacement in space must be calculated. This can be achieved when the position of the molecular sites in space is known. To obtain the mobility of charges along the polymer chain, the mean squared displacement along the contour of the polymer chain must be considered. This contour mobility is the value that would be measured for a noncoiled, that is, stretched, polymer chain. The position of the molecular sites is determined by the conformation of the polymer chain and, hence, by the angles φ and dihedral angles θ between adjacent molecular sites (see Figure 2). Whereas the variations in the angles are small due to the relatively rigid structure of the molecular units, the dihedral angles between the molecular units can vary considerably. The dihedral angles are distributed according to a Boltzman distribution,
P(θ) )
e-(Etor(θ)/kBT)
∫02π e-(E
tor(θ)/kBT)
(5) dθ
Charge Transport along Polymer Chains
J. Phys. Chem. C, Vol. 111, No. 29, 2007 11107
where Etor(θ) is the torsion potential between adjacent molecular units. The change of the dihedral angles in time is determined by a combination of rotational drift and rotational diffusion. During a small time step (∆t) the change of the angles is given by
∆θ ) -
Drot ∂Etor ∆t + ∆θdiff kBT ∂θ
(6)
The first term describes the rotational drift due to the torsion potential. The second term accounts for the random diffusive rotation, such that 〈∆θdiff2〉 ) 2Drot∆t with Drot ) 1/(2τrot) and τrot the diffusional rotation time of the molecular units. The mean squared displacement can be obtained from the time-dependent wave function27
Ψ(t) )
∑n cn(t,n0)Φn
(7)
with the initial condition cn(t ) 0, n0) ) δn,n0. In eq 7, Φn is an orbital localized at molecular unit n. The time-dependent expansion coefficients of the basis functions on the molecular units in the polymer chain (cn(t)) are obtained by propagation of the wave function according to the time-dependent Schro¨dinger equation:
ip
∂Ψ(t) )H ˆ Ψ(t) ∂t
(8)
In the tight-binding approximation ,the diagonal matrix elements of the Hamiltonian (H ˆ ) correspond to the site energies (i,i), that is, the energy of the charge localized on a single molecular unit in the polymer chain. When only nearest neighbor interactions are taken into account, the off-diagonal matrix elements of the Hamiltonian bi,i(1 are equal to the electronic couplings between adjacent molecular units. The other off-diagonal matrix elements are zero in this approximation. and the Hamiltonian matrix is given by
(
11 b12 0 · · · 0 b21 22 ··· H) 0 ·· ··· · NN 0
)
(9)
The mean squared displacement of the charge as a function of time can be expressed in terms of the time-dependent site coefficients and the distance between the molecular units,
〈∆2(t)〉 )
f(n0)|cn(t, n0)|2|rn(t) - r0|2 ∑ n,n
orbitals on the fluorene and naphthyl units as the basis functions Φn in eq 7. The site energies for the HOMO orbitals were approximated by experimental values for the ionization energy for fluorene (7.91 eV)28 and 2-methoxynaphthalene (7.82 eV).29 The electronic coupling between adjacent units in the polymer chains was evaluated for model systems consisting of two molecular units (as depicted in Figure 2 for fluorene-fluorene) by density functional theory calculations in the program ADF30,31 using fragment orbitals.32,33 The calculations were performed using the Becke-Perdew gradient-corrected exchange correlation functional34,35 and a double-ζ plus polarization (DZP) basis set consisting of Slater-type functions. The electronic coupling used in this work is the “effective” coupling that incorporates the effects of the spatial overlap between the fragment orbitals.32,33 The geometry of the molecular units was optimized for each dihedral angle using second-order Møller-Plesset perturbation theory (MP2) with a cc-pVDZ basis set in the program Gaussian.36 In this optimized geometry, the angle between adjacent fluorenes (φ, see Figure 2) was found to be 23.5°. The wave function of the charge and the conformation of the chain were propagated in time with a time step of one atomic unit (2.4189 × 10-17 seconds) using eqs 8 and 6, respectively. The rotation time of dialkyl-substituted fluorene and alkoxysubstituted naphthyl used in eq 6 is taken to be τrot ) 400 ps. Experimental values typically found for freely rotating molecules of comparable size are on the order of 100 ps.37 The rotation time for the molecular units in the polymer chain used here is taken longer to take into account the hindrance of rotational motion due to the nonzero angles φ between adjacent molecular units and merely represents an effective rotation time. The actual value of τrot has only a weak influence on the calculated charge carrier mobilities; the mobility increases by a factor of 2 upon decreasing of the rotation time from 800 to 200 ps. For chains with a finite length, the charge is initially assumed to be equally distributed over the molecular units (f(no) ) 1/n in eq 10). The squared displacement of the charge carrier is averaged over a few hundred realizations of the initial conditions (i.e., angles between molecular units and initial position of the charge) and of the rotational diffusive motion of the molecular units. For infinitely long chains, the charge is initially localized in the middle of a chain with sufficient length such that the chain ends do not affect the mean squared displacement on the time scale of the simulation (200 ps, 2500 molecular units). In this case, the squared displacement of the charge carrier is averaged over a few hundred realizations of the dynamic chain conformation. 5. Calculated Results
(10)
0
where f(n0) describes the initial distribution of the charge and |rn(t) - r0| is the distance between the orbitals at sites n (at time t) and n0 (at t ) 0). cn(t, n0) is the coefficient of the orbital at site n at time t for a state that was initially localized at n0.27 The electronic coupling between adjacent molecular units in the polymer chain strongly depends on the dihedral angle between the molecular units. As a consequence, not only the chain conformation but also the efficiency of charge transport along the contour of polymer chain is determined by the molecular conformation. This structural dependence is taken into account through the angular dependence of the electronic coupling (bi,i(1(θ)). The values for the mobility of positive charges on the polymer chains listed in Table 1 were calculated by taking the HOMO
5.1. Chain Conformation. In Figure 4a, the potential energy of a bifluorene is shown as a function of the dihedral angle between two fluorene units. The minima in the torsion potential are found at dihedral angles θ that deviate about 45° from the planar configuration, where θ is close to 45, 135, 225, and 315°. As a consequence of this rather large dihedral angle and the angle φ of 23.5° between adjacent fluorene units, polyfluorene chains exhibit a coiled conformation. An example of a calculated realization of the coiled chain conformation of polyfluorene is shown in Figure 5a. The coiling is a result of the distribution of dihedral angles, described by eq 5. The persistence length calculated for PF on the basis of the angles and the dihedral angles is 12 fluorene units, which is in good agreement with the persistence length of 10 units found experimentally for PF in solution.15 For comparison, the chain conformation for a PF chain with all dihedral angles equal to 45° is shown in Figure
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Figure 4. (a) Torsion potential and (c) electronic coupling for a hole calculated as a function of dihedral angle between two fluorene units. (b) Torsion potential and (d) electronic coupling between two naphthyl units in binaphthyl (oriented as depicted in Figure 1b).
Figure 5. Chain conformation for (a) fluorene chain with 500 units, (b) fluorene chain with 500 units with a fraction of 0.0944 binaphthyl units, and (c) stretched fluorene chain with 100 units (for which all dihedral angles are 45°).
5c. This stretched PF chain adopts a helical chain conformation with a length close to (97%) the contour length, or the hypothetical case of the “completely extended PF”, where both the angle φ and dihedral angle θ are zero. The torsion potential calculated for rotation of one naphthyl with respect to the other in BN (see Figure 1b) is displayed in Figure 4b. The steric hindrance of the alkoxy substituents prevents BN from adopting a planar conformation. The potential shows a broad minimum around 90° and rises steeply at 50° from the planar conformation. To calculate the chain conformation of PF with a fraction of BN, the torsion potential of naphthyl-fluorene and naphthyl-naphthyl in the inverse con-
figuration (naphthyls connected via the dotted bonds in Figure 1b) must be known. Both torsion potentials are similar (in shape and magnitude) to the potential shown in Figure 4a. A chain conformation for PF with a BN fraction of 0.0944 is shown in Figure 5b. This polymer adopts a more coiled conformation than PF, indicating that the close-to-perpendicular orientation of the naphthyl units makes BN an effective chain bender. Accordingly, the calculated persistence length (lcalcd) is found to decrease strongly with an increasing fraction of BN present in the polymer backbone (see Table 1). The F-BN alternating copolymer is exceptionally folded with a persistence length of only two units. 5.2. Effect of Chain Conformation on Charge Transport. To calculate the mean squared displacement for a charge on the polymer chains considered here, the electronic coupling between adjacent molecular sites must be known as a function of the dihedral angle. For two fluorene units, the electronic coupling resembles a cosine, similar to the spatial overlap between two p-type orbitals (see Figure 4c). The coupling between two naphthyls is more complex because of the specific shape of the HOMO orbitals on the naphthyl units, as can be seen in Figure 4d. The electronic coupling for naphthylfluorene and naphthyl-naphthyl in the inverse configuration (naphthyls connected via the dotted bonds in Figure 1b) is equal in shape to the coupling shown in Figure 4c. The amplitude is slightly smaller, with values of 0.36 and 0.29 eV at zero dihedral angle, respectively. The mean squared displacement for an infinitely long fluorene chain is shown in Figure 6a. The dotted line shows the mean squared displacement along the contour of the chain, whereas the full line shows the actual mean squared displacement of the charge in space, taking into account the coiled nature of the polymer. The motion of charges along the contour of the chain is determined by the electronic coupling (and hence by the dihedral angles) between adjacent fluorene units. On a short time scale, the mean squared displacement is dominated by charges moving on parts of the polymer chains with relatively high electronic coupling. On a longer time scale, various electronic couplings are encountered according to the distribu-
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Figure 6. (a) Mean squared displacement as a function of time and (b) isotropic mobility as a function of frequency calculated for a hole on infinitely long stretched (dotted line) and coiled (solid line) polyfluorene chains.
tion of dihedral angles in eq 5. As a result, the motion of the charge occurs by normal Gaussian diffusion at longer times; the mean squared displacement along the contour of the chain increases linearly with time and can be described by eq 3. As a consequence of the coiled chain conformation of polyfluorene, the direction of the motion of the charge carrier changes in time, and the actual mean squared displacement increases more slowly with time than the mean squared displacement along the contour of the polymer. Only after ∼100 ps, when the charge has encountered parts of the polymer with various curvatures, does the motion of the charge on the coiled polymer chain become diffusive, and the mean squared displacement can be described by eq 3. The isotropic mobility along the contour of the polymer chain (µiso contour) as a function of frequency is calculated from the mean squared displacement along the contour of the polymer, whereas the actual isotropic mobility (µiso) is calculated from the mean squared displacement in space using eq 2. The results are shown in Figure 6b. The oscillations at high frequency result from fluctuations in the mean squared displacement for different chain conformations (barely visible in Figure 6a). At high frequency (>1012 Hz), the motion of the charges is probed over a small-length scale (
