Theoretical Study of Torsional Disorder in Poly(3-alkylthiophene

Article pubs.acs.org/JPCA

Theoretical Study of Torsional Disorder in Poly(3-alkylthiophene) Single Chains: Intramolecular Charge-Transfer Character and Implications for Photovoltaic Properties Josiah A. Bjorgaard† and Muhammet E. Köse*,‡ †

Department of Chemistry and Biochemistry, North Dakota State University, Fargo, North Dakota, 58108 United States TUBITAK National Metrology Institute, Gebze, Kocaeli, 41470 Turkey

‡

ABSTRACT: The role of polymer chain morphology on the optoelectronic properties of polythiophenes is an ongoing investigation with the promise of improving organic photovoltaic performance. Chain morphology is predominantly affected by torsional disorder, which causes localization of holes and electrons in the conjugated backbone. Using the model compound oligo(3methylthiophene), torsionally disordered oligomers were created to compare with a trans-planar oligomer such as found in crystalline poly(3-hexylthiophene). Low lying electronic excitations are calculated using TD-HF and TD-DFT with various long-range corrected functionals. Probability densities of electron and hole were constructed from natural transition orbitals, giving insight into localization and electron−hole overlap. Overlap is found to be substantially higher in disordered oligomers, indicating a stronger Coulombic interaction between electron and hole. Results suggest that improved photovoltaic performance with increased crystallinity is partially explained by stronger light absorption in crystalline polymers and a higher barrier to charge separation in disordered polymers.

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conjugation.7−10 Several theoretical studies have examined the role of torsional defects and disorder on the optical spectra and excitation localization of prototypical conjugated polymers.9−13 These studies examine some combination of localization in the HOMO, LUMO, or transition density generated from quantum chemical calculations. In the case of a single severe torsional defect (90° from planar) in a large thiophene oligomer, conjugation breaking was not observed in the calculations using the ZINDO Hamiltonian.13 Similarly, localization of the HOMO or LUMO was not observed using the B3LYP/3-21G* method for similar chain morphologies.12 On the contrary, in large disordered chain calculations, the Pariser−Parr−Pople (PPP) Hamiltonian predicts spatial separation of the lowest energy excitations into discrete chromophore units.10 This suggests that multiple torsional defects are required to localize excitations in polythiophenes. In these studies, localization of transition densities and frontier molecular orbitals provides information about the spatial localization of electron and hole for an excited state. In chains with multiple torsional defects, mixing of orbitals in electronic transitions prevents a clear comparison of electron and hole localization. Thus, the amount of overlap

INTRODUCTION Conjugated polymers have enormous promise as a leading class of materials in next-generation photovoltaic devices due to their cheap cost, flexibility, lightweight, and useful optoelectronic properties. However, there is still much to learn about their behavior both in the solid state and in solution. Poly(3hexylthiophene) (P3HT) is a conjugated polymer frequently used in organic photovoltaics (OPVs) as a standard by which to determine the ruling factors in optimizing OPV activity.1−5 Photovoltaic device performance is inherently related to the optoelectronic properties of the conjugated polymers used as light absorbing materials. It is a well accepted fact that the optoelectronic properties of conjugated polymers are governed by polymer conformation and intermolecular interactions. The intramolecular contribution on optoelectronic properties of P3HT are mainly controlled by torsional disorder.6 In crystalline domains torsional disorder is nearly absent, whereas in amorphous domains it plays a significant role. Although the targets of this study are relevant to any poly(3-alkylthiophene) aimed for photovoltaic applications, such as P3HT, computational time is significantly reduced by truncating alkyl groups and performing electronic structure calculations on oligo(3methylthiophene)s. This captures the relevant electronic properties for an isolated system since the extended alkyl chains play a very small part in photophysical properties. Static disorder in conjugated polymers causes localization of the lowest energy excitations through disruption of π© 2013 American Chemical Society

Received: February 12, 2013 Revised: April 5, 2013 Published: April 12, 2013 3869

dx.doi.org/10.1021/jp401521j | J. Phys. Chem. A 2013, 117, 3869−3876

The Journal of Physical Chemistry A

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between electron and hole, or charge transfer character, has not been addressed so far in the literature. Low lying charge-transfer excitations are well described by long-range corrected functionals (LCFs) in time-dependent DFT (TD-DFT).14−18 Some LCFs predict excitation energies and oscillator strengths with high accuracy.15−19 All LCFs include asymptotically increasing Hartree−Fock (HF) exchange at increased electron−electron separation. The effect of the amount of exact HF exchange incorporated in LCFs on transition density localization has been shown by Tretiak et al. for the case of trans-planar MEH-PPV, whereby localization is increased with increasing long-range HF contribution.20,21 Comparison of different LCFs is necessary in order to perform accurate simulations by evaluating the theoretical results against the experimental data. Various parameters control the amount of HF exchange in LCFs. These include c, the distance at which the long-range HF exchange becomes prevalent, ashort, the amount of HF exchange at short distance, and along, the amount of HF exchange at long distance (Table 1). Several well-known LCFs are chosen to describe the low

B3LYP

CAM-B3LYP

ωB97XD

LC-ωPBE

0 0.19 0.19

0.33 0.19 0.65

0.2 0.22 1

0.4 0.25 1

COMPUTATIONS

Nuclear geometries of P3MT with a total length of 20 α-linked thiophene units, varying in interring torsional angle, were prepared. Then, TD-DFT and TD-HF calculations for the lowest four excitations were performed. The transition density of the lowest excited state is decomposed into electron and hole NTOs23 and summed with associated eigenvalues to create electron and hole wave functions. Finally, the populations are calculated and coarse-grained to individual thiophene units, resulting in electron and hole probability distributions. DFT, TD-DFT, TD-HF, and NTO calculations were performed with Gaussian 09,24 whereas population analysis and other calculations were performed with in-house codes written in Matlab. Nuclear Geometries. To realistically create disordered polymer geometries, the torsional potential (E(Φ) where Φ is the torsional angle between thiophene rings) was calculated for tetra(3-methylthiophene) (T3MT) with the outer thiophene units in the optimized anti conformation (Φ = 30°) as predicted by the B3LYP/6-311G(p,d) level of theory (Figure 1). We note that variations in the interthiophene bond angles or bond lengths increase the total energy of the system by a much larger extent than a change in the torsional angle. The torsional angle, however, drastically affects the optoelectronic properties of the polymer due to its strong effect on the conjugated backbone. Frequently, the torsional potential of 2,2′-bithiophene has been used to model extended polythiophenes, despite the fact that it does not accurately reflect the torsional potential of longer oligomers.25,26 In comparison to 2,2′-bithiophene, the torsional potential of tetrathiophene is predicted to have an increased barrier at 90° due to strengthened conjugation and provides drastic improvement over the study on the 2,2′bithiophene case.27 It has been suggested that both longer oligomers and explicit alkyl side chains are required to derive accurate torsional potentials for P3HT.26 On the contrary, the quaterthiophene torsional potential has been used to derive accurate enthalpies and crystal structures of various poly- and oligo(3-alkylthiophenes) in molecular dynamics simulations.28 This accuracy may be due, in part, to relatively small differences in thermal population of static torsional angles between

Table 1. Parameters for Long Range Corrected Functionals Describing the Distance at Which HF Exchange Contribution Is Switched (c) in Bohrs, the Contribution of HF Exchange at Short Distance (ashort) and at Long Distance (along) c ashort along

Article

lying singlet excitations and compared with those obtained with B3LYP and time dependent HF (TD-HF) calculations. To analyze the results of these calculations, natural transition orbitals (NTOs) are generated by decomposing the resulting transition density matrices into electron and hole components, thus allowing for comparison of electron and hole localization and electron−hole overlap for a large number of chain morphologies.22,23

Figure 1. Torsional potential calculated for tetra(3-methylthiophene) (T3MT) (left) with B3LYP/6-311G* (circles) and associated room temperature Maxwell−Boltzmann distribution (squares). Fourier series were fit to the calculated potential with eq 1 (dashed) and a Boltzman distribution was calculated from the fit (dotted). Twenty thiophene unit long oligomers were then constructed using the geometry of the optimized T3MT internal rings. Some sample oligomer conformations are shown on the righthand side of the figure. 3870



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To analyze ψh(e), we performed population analysis by constructing the density matrix (D) from the atomic orbital coefficients (Cμ(ν))

tetrathiophene and longer oligothiophenes with explicit consideration of alkyl chains. In the framework of previous studies, the use of T3MT for prediction of approximate torsional populations stands as a reasonable method for poly(3alkylthiophenes). One can generate an approximate Boltzmann distribution of polymer geometries using the calculated torsional potential, assuming that other geometric parameters (such as intrathiophene bond lengths and angles) change only minimally upon geometric relaxation as Φ is changed and held fixed. The calculated torsional potentials were fit with a Fourier series given by E(Φ) =

∑ Cn(1 − sin(Φ − Φ0)) n

Dμν = CμCν*

and calculated the corresponding probability (P) of finding a hole or electron within basis function μ as

Pμ =

where Sμ,ν is the basis function overlap matrix. To reduce the computational effort, basis function overlap within only monomer units were considered. In order to examine the localization of electron and hole along the polymer chain, we condensed the probabilities to monomer units by summation (P(n) = ∑μ∈nPμ, where n is the position of the monomer unit), which resulted in electron (Pe(n)) or hole (Ph(n)) probability distribution for observing an electron or hole in the nth thiophene unit. From the population of electron and hole wave functions, the electron and hole conjugation length (Le and Lh) is calculated as the number of consecutive monomers containing 95% of the NTO population. The probability of observing an electron and hole on the same monomer unit (Pe‑h) is calculated according to Pe−h = ∑nPe(n)Ph(n)Pe‑h = ∑ Pe(n)Ph(n) for each polymer chain. Gas phase absorption spectra for disordered oligomer chains were simulated by summing transitions broadened by Lorentzian functions with half width at half-maximum of 0.05 eV and then weighted by the associated oscillator strength for the indicated transition. Experimental absorption spectra were collected with a Cary WinUV spectrophotometer from well dissolved dilute solutions of P3HT (Rieke Metals) in chloroform and from a film spin-cast from P3HT in dichlorobenzene.

(1)

(2)

where k is Boltzmann’s constant and T is the temperature set at 298 K. Electronic Structure and Analysis. After generating random oligomer conformations, we performed TD-DFT calculations using the LCFs CAM-B3LYP,16 LC-ωPBE,18,19 and ωB97X-D.15 For comparison, we also performed calculations with B3LYP29 and TD-HF methods. All excited state calculations were performed with the 6-31G(d) basis set. NTOs are the result of singular value decomposition of the transition density matrix (T), defined in the molecular orbital basis (ϕi, ϕa) as Tia = ⟨ϕi|T̂(r )|ϕa⟩

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RESULTS AND DISCUSSION Trans-Planar Conformation. In the planar conformation, transition densities occupy nearly the entire polymer chain, with slightly decreased occupation on terminal thiophene units irrespective of the method used in this study (Figure 2).

(3)

where T̂ (r) is the single particle density matrix operator. NTOs are calculated from the transition density matrix with the unitary transformation T = USV†

(9) 31

exp(E(Φ)/kT )

∫ exp(E(Φ)/kT ) dΦ

∑ Dμ,νSμ,ν ν

The polymeric structure is created from the central two thiophene rings of T3MT with varying Φ (Figure 1). These torsional angles were selected by Monte Carlo sampling from the Boltzmann probability calculated according to P(Φ) =

(8)

(4)

where the elements of V and U are given by TT†ui⃗ = λiui⃗ ;

T†Tvi⃗ = λivi⃗

(5)

and the NTOs are computed by using the resulting transformation matrices according to ϕi NTO = ϕi U ;

ϕaNTO = ϕa V

(6)

Each NTO is expressed in the basis set of the canonical molecular orbitals and constructed either from the occupied or virtual space. Each pair of NTOs (one from occupied space and one from virtual space) is associated with a specific λ which gives its relative contribution to the corresponding transition density.30 In general, transition density matrices for this ensemble of disordered polymer chains are described by 2−3 NTO pairs with λ greater than 0.1. We constructed approximate electron (ψe) and hole (ψh) wave functions from ψh =

; ∑ λiϕi NTO ,u i

ψe =

∑ λaϕaNTO ,v a

Figure 2. Transition densities calculated using TD-HF method and various DFT functionals for trans-planar oligomer.

Electron and hole wave functions, however, occupy smaller regions of the polymer chain (Figure 3), whereas Pe‑h, Le, and Lh depend on the calculation method (Table 2). Pe and Ph predominantly occupy different regions of the oligomer backbone for all calculations except those using B3LYP (Figure 3), which show much more overlap than LCFs and TD-HF calculations. This is an expected result, since the B3LYP functional is known to poorly describe CT excitations because the attractive exchange interaction is not prevalent at long

(7) 3871



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Figure 3. Electron and hole wave functions (top) from eq 7 calculated using the methods outlined in the text. Bottom graph shows electron (dotted) and hole (solid) populations along the conjugated backbone.

Table 2. Effective Sizes for Electron (Le) and Hole (Lh), Probability of Observing Electron and Hole on the Same Thiophene Unit (Pe‑h), and Excited State Energy (ωexc) Determined from Calculations for trans-Planar Oligomer Lh Le Pe‑h ωexc/eV

B3LYP

CAM-B3LYP

ωB97XD

LC-ωPBE

TD-HF

15 13 0.07 2.02

11 11 0.03 2.56

11 11 0.03 2.65

8 11 0.03 2.93

8 9 0.02 2.84

distances.16 Strong similarity is observed between Ph for ωB97X-D and CAM-B3LYP functionals while LC-ωPBE produces Ph nearly identical to TD-HF (Table 2). The functionals ωB97X-D and CAM-B3LYP produce more hole delocalization than LC-ωPBE and TD-HF. On the contrary, Pe is similar between all DFT functionals, whereas more electron localization is observed with TD-HF calculations. Overall, calculations predict that electron and hole are likely to be well separated in planar P3MT, as evident from the low Pe‑h values given in Table 2. Simulated Absorption Spectrum. The simulated absorption spectra (Figure 4) of disordered polymers shows a general trend of decreasing ωexc,max with increasing HF exchange for TD-DFT functionals. However, TD-HF calculations predict a slightly higher excitation energy maximum (ωexc,max) than LCωPBE. The lineshapes for the S0 → S1 transition are approximately symmetric, while lineshapes comprised of the lowest four excitations include a tail at higher excitation energies (Figure 4) due to significant oscillator strength of these higher energy transitions. This is in contrast with the results of the higher energy transitions obtained with the transplanar conformation where those transitions have negligible oscillator strength. Both CAM-B3LYP and ωB97X-D are well suited to describe low lying optical transitions in conjugated molecules,15,16 as also supported in this work.

Figure 4. Simulated gas phase absorbance spectrum for S0 → S1 (left) and lowest four transitions (right) for disordered polymers with experimental absorbance spectrum of P3HT in chloroform and as a thin film.

The simulated absorption spectra of CAM-B3LYP and ωB97X-D have ωexc,max ≈ 0.2 eV higher than the experimental ωexc,max of P3HT in chloroform solution, whereas calculations with the B3LYP hybrid functional underestimate the ωexc,max by ∼0.5 eV (Table 3). It is important to note that these are gasphase calculations. With the inclusion of solvent effects, one Table 3. Average Length of Electron (Le,avg) and Hole (Lh,avg), Mean Probability of Electon and Hole Being Located on the Same Thiophene Unit (Pe‑h,avg), and Maximum Excited State Energy (ωexc,max) Determined from Calculations for Disordered Oligomer Conformations

Lh,avg Le,avg Pe‑h,avg ωexc,max/eV 3872

B3LYP

CAM-B3LYP

ωB97XD

LC-ωPBE

TD-HF

9.3 9.2 0.14 2.43

7.1 6.9 0.16 2.96

6.7 6.4 0.23 3.02

6.1 6.0 0.18 3.30

5.9 5.9 0.26 3.24



Article

might obtain even better agreement with the experimental results using CAM-B3LYP and/or ωB97X-D functionals. We have not included the effects of vibronic coupling on the simulated absorption spectra which would likely cause asymmetry in the absorption line shape and lead shift of ωexc,max to higher energy. Regardless of these effects, the relationship between disorder and lowest singlet 0−0 transitions can be examined using the calculations on the disordered polymer ensemble. Electron/Hole Populations of Disordered Polymers. Electron/hole populations in disordered polymers have a single maxima for the lowest energy transitions, while higher energy transitions are characterized by distributions with multiple maxima. NTO populations in these higher excited states are predominantly dispersed over multiple segments of the polymer chain and only partially disrupted by torsional defects. In the case of the lowest energy singlet excitation, conjugation length is easily characterized. For LCFs, Le and Lh range from 4 to 12 thiophene units long, with average values of 6−7 units. Le and Lh decrease with increasing HF contribution, whereas Pe‑h increases (Table 3, Figure 5). In the disordered ensemble, Figure 6. Distribution of excited state energy (ωexc), oscillator strength ( f), electron (Le), and hole (Lh) length as a function of average polymer torsional potential energy (black) and moving average of the distributions with a window of 1.8 meV (red).

from ∼4.5−2.5. These trends agree with experimental thermochromic spectra of polythiophene derivatives in both solution and solid state at temperatures where low energy peaks associated with aggregation are absent (i.e., in the melted polymer) or in derivatives which do not display these additional low energy peaks.32−34 In plots of Le and Lh, both variables decrease with increasing average torsional energy such that the moving average of Le and Lh resemble inverse sigmoid functions. The shape of the running average points out that at moderate levels of disorder, the size distribution of electrons and holes is quite stable with average length of 6−7 thiophene units. This stability is observed in a range of average torsional energies between 8 and 12 meV (about 50% of the range of calculated average torsional energies). Similarly, the moving average of Pe‑h increases with average torsional potential (Figure 7), from

Figure 5. Probability densities for disordered ensemble calulations: (a) calculated conjugation length (width of 95% of probability density of electron and hole populations along polymer backbone) and (b) total electron and hole probability overlap (probability that electron and hole are found on the same monomer unit).

electron and hole are both more localized and have a higher probability of being observed on the same thiophene unit than in the trans-planar configuration. Variation Within a Disordered Ensemble. To examine the calculated parameters with the extent of disorder, a larger ensemble of 500 polymer chains was prepared according to the presented methodology. Calculations were performed with the CAM-B3LYP functional because it reproduces the experimental absorbance spectrum most accurately. To quantify the disorder in each oligomer chain, the average torsional potential is calculated using the T3MT profile (Figure 1). Plotting ωexc, oscillator strength ( f), Le, and Lh as functions of this average energy, the effects of increasing disorder can be visualized (Figure 6). Although substantial variance is seen in these plots, trends emerge when moving averages (with a window of 1.8 meV) are calculated. In general, increasing disorder increases ωexc and lowers f. The moving averages of ωexc and f have a near linear dependence on average torsional energy. For ωexc, average values range from ∼2.9−3.1 eV, or ∼30 nm difference in wavelength. At low average torsional energy, the average excited state energy tends toward convergence at ∼2.9 eV. The calculated oscillator strength decreases with increasing disorder

Figure 7. Distribution of total electron−hole probability overlap (Pe‑h) as a function of average polymer torsional potential energy (black) and moving average of the distributions with a window of 1.8 meV (red). 3873



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∼0.13 to ∼0.16, but remains stable at intermediate levels. At 14 meV, Pe‑h rapidly increases to ∼0.19 and then remains fairly stable at higher amounts of disorder within the simulated distribution. The relationship between oligomer length and optoelectronic properties have been investigated experimentally35−37 and extrapolated to predict polymer properties.5,38,39 Taking the average values of ωexc for sets of polymers with the same Le or Lh, a slight decrease in average ωexc is observed as Le and Lh increase from 5 to 7 (Figure 8). Beyond these electron and hole

Figure 9. Comparison of simulated spectra using the CAM-B3LYP functional for the trans-planar conformation and disordered ensemble of oligomers with experimental spectra of P3HT in CHCl3 or in a thin film spin-cast from dichlorobenzene. The intensity of the absorption peak for the trans-planar conformation has been reduced for comparison.

In trans-planar P3ATs, greater electron−hole separation (lower Pe‑h) is indicative of greater charge-transfer character, leading to a lower energetic barrier for charge separation. Assuming that the total Coulombic interaction between electron and hole is dominated by intramonomer interaction, the Coulomb potential is approximately proportional to Pe‑h. Considering this approximation, calculations predict that the Coulomb potential during charge separation is substantially higher in disordered polymers by approximately a factor of 5 according to the calculations with CAM-B3LYP.

Figure 8. Average values of excited state energy (ωexc) for simulated polymers with differing electron (dotted) and hole (solid) lengths.

lengths, average ωexc remains stable. Since increasing disorder increases excitation energy with near linear dependence above a certain threshold, it might be thought that it affects the electron and hole lengths in a similar fasion. However, it seems that disorder can affect the excited state energy independently. Comparison of trans-Planar and Disordered Polymer Calculations. The average electron and hole lengths (Le,avg and Lh,avg) are between 3 and 6 thiophene units shorter in the disordered ensemble, whereas Pe‑h is much higher in disordered polymer chains. This suggests that torsional disorder causes exciton localization, similar to the results obtained with the PPP Hamiltonian.10 New insight into the role of torsional disorder on the electron−hole overlap is gained through the examination of Pe‑h, whereby torsional disorder causes extensive overlap between electron and hole compared to the near absence of this overlap in the trans-planar conformation. In the trans-planar conformation, ωexc is 0.5 eV higher than the corresponding ωexc,max in the disordered ensemble (Figure 9). This is similar to the difference in maximum absorption energy of solvated polymers and the lowest energy 0−0 absorption peak in thin films. The oscillator strength of transplanar oligomers (8.04 for CAM-B3LYP) is greater than twice that of the average oscillator strength in disordered oligomers, predicting that the absorption cross section of crystalline polymers is higher than that of disordered polymers. Implications for Optoelectronic Device Properties. It has been shown experimentally that increasing P3HT crystallinity in organic photovoltaics results in improved device efficiency.40,41 The computational results herein provide some explanation for this improved device efficiency. The transplanar conformation and disordered ensemble are reflective of crystalline and noncrystalline regions of optoelectronic devices incorporating P3ATs. Higher oscillator strength in trans-planar P3ATs suggests a higher number of excitons generated in crystalline regions.

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CONCLUSION Inspired by experimental evidence for major differences in the optoelectronic properties of crystalline and amorphous P3HT, oligo(3-methylthiophene) has been used as a model compound to reveal polymeric properties and investigated using TD-DFT and TD-HF methods. Of these methods, CAM-B3LYP gives the most accurate absorption profile for both the trans-planar conformation and an ensemble of disordered oligomer chains. These simulations predict greater electron−hole separation in the trans-planar conformation and greater localization in the disordered ensemble. The range of electron and hole sizes in the disordered ensemble is large (4−12 thiophene units) and does not change substantially as disorder increases, whereas the excited state energies increase with increasing disorder except when the disorder is very low. Similarly, the electron and hole length in disordered polymers affect the excited state energy only at electron and hole lengths lower than 7 thiophene units. In comparison with trans-planar oligomers, disordered polymers have decreased oscillator strength, electron and hole size, and electron−hole separation. These results suggest that higher crystallinity in P3HT/PCBM devices results in increased device efficiency due to, in part, a greater number of excitons generated in and more efficient charge separation from crystalline domains.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 3874



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Author Contributions

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The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Special thanks to the Center for Computationally Assisted Science and Technology at North Dakota State University for providing the facilities for this computational study. This work was supported by NSF Grant No. EPS-0814442, DOE under award No. DE-FG52-08NA28921, and TUBITAK BIDEB Fellowship Program.

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Theoretical Study of Torsional Disorder in Poly(3-alkylthiophene

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