Potential Barrier and Excess Energy for Electron–Hole Separation

Article pubs.acs.org/JPCC

Potential Barrier and Excess Energy for Electron−Hole Separation from the Charge-Transfer Exciton at Donor−Acceptor Heterojunctions of Organic Solar Cells Hiroyuki Tamura*,† and Irene Burghardt‡ †

WPI Advanced Institute for Material Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan Institute of Physical and Theoretical Chemistry, Goethe University Frankfurt, Max-von-Laue-Straße 7, 60438 Frankfurt/Main, Germany

‡

ABSTRACT: The mechanism of electron−hole separation overcoming Coulomb attraction is one of the important open questions related to the efficiency of organic solar cells. In this work, we have theoretically predicted the potential curves for electron−hole separation from the bound charge-transfer (CT) state at donor−acceptor heterojunctions. The electron−hole potential was calculated considering an oligothiophene− fullerene donor−acceptor complex using the long-range-corrected density functional theory. The screening effect of the medium was taken into account by scaling with the dielectric constants of fullerene and polythiophene. The potential barrier was found to decrease as the πconjugation length of the donor increased. We propose a simple analytical formula for the Coulomb potential between an electron localized at a fullerene molecule and a hole distributed along a π-conjugated chain. We also discuss the possible role of the excess energy due to the preceding exciton dissociation in facilitating the charge separation.

I. INTRODUCTION Charge separation at donor−acceptor heterojunctions is one of the important processes determining the energy conversion efficiency of organic solar cells.1−18 The photogenerated exciton is thought to decay primarily to a bound electron− hole pair localized at the donor−acceptor interface referred to as a “charge-transfer (CT) state”, “exciplex”, or “charge-transfer exciton”.1−18 The donor species can be a π-conjugated polymer such as poly(3-hexylthiophene) (P3HT) or a small molecule such as a porphyrin derivative. Fullerene derivatives, such as [6,6]-phenyl-C61 butyric acid methyl ester (PCBM), are generally employed as electron acceptors. The anode and cathode should provide a driving force for the charge separation by which the electron in the acceptor and the hole in the donor can descend the chemical potential toward the electrodes (Figure 1). However, the electron−hole Coulomb interaction stabilizes the bound CT state, thereby providing a potential barrier for charge separation.12,13 Furthermore, free carrier formation competes with radiationless decay to the ground state (i.e., charge recombination), which reduces the internal quantum efficiency.1−3 It is not well understood how the free carriers can overcome the Coulomb attraction and be separated from the CT state. Recent pump−probe experiments2,7−9 suggest that the charge separation is enhanced by the excess energy of exciton dissociation, leading to the so-called “hot CT mechanism”. On the other hand, Lee et al.3 suggested that photocurrent can be generated through direct excitation of the CT state by subgap photoabsorption. The charge separation rate would thus © 2013 American Chemical Society

Figure 1. Diagram of the internal electric field, Ex, of organic solar cells induced by the chemical potential difference of the anode and cathode: (a) isolated electrodes (i.e., open circuit) and (b) charge-transferred electrodes resulting from a short circuit. The gray areas indicate the occupied energy levels, where the bold solid lines indicate the Fermi levels of the isolated electrodes. The dashed lines depict the Coulomb potential, ϕ(x), induced by the electrodes.

be determined by various factors including the Coulomb potential barrier, dielectric constant, temperature, vibrational excitations due to the excess energy, molecular relaxation, and energy dissipations. Yuan et al.6 demonstrated experimentally that the charge separation can be accelerated by increasing the Received: June 24, 2013 Published: July 1, 2013 15020

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electric field from the electrodes, indicating the importance of the internal Coulomb potential. The proper estimation of the potential barrier for electron− hole separation from the bound CT state is a starting point for understanding the mechanism of charge separation. The Coulomb potential barrier depends on the internal electric field, the dielectric constants of the donor and acceptor molecules, and the donor−acceptor distance at the moment of exciton dissociation.12,13 The potential curve is not necessarily described by a Coulomb interaction between point charges. Microscopic insight into the potential curve is essential for analyzing the dynamics of charge separation, but such information is not yet available. In this study, we theoretically analyzed the potential profile of charge separation at donor−acceptor interfaces based on density functional theory (DFT) calculations. We considered oligothiophenes and C60 as a model system for P3HT−PCBM donor−acceptor heterojunctions. We investigated the effect of the π-conjugation length of the donor species on the electron− hole potential. Furthermore, to analyze the excess energy, we calculated the relative potentials of the exciton and CT states in a diabatic representation. Finally, we present a perspective on electron−hole separation in stacked molecular aggregates.

Figure 2. Oligothiophene−C60 supermolecule (ball-and-stick model) obtained by DFT calculations, together with the LUMO orbital of C60: (a) bound CT state and (b) separated charges. The surrounding C60 molecules are depicted to illustrate the concept of charge separation. The center-to-center distance, R, and the electron−hole distance, x, are assumed to be identical. a is the effective length of the charge distribution.

generation of free charge carriers depend on the conditions of donor−acceptor heterojunctions.1,14,15 The time scale of the formation of charge-separated (CS) states is still an open issue.1,15,16 III.A. Relative Energies of the Exciton State and the Bound CT State. The relative energies of the exciton state and the bound CT state (i.e., the excess energy) are potentially of key importance for exciton dissociation and subsequent charge separation. Here, we define the excess energy as the sum of the vertical energy difference (without relaxation) and the reorganization energy for the intramolecular vibrational modes. When the energy of the exciton state is higher than the maximum of the electron−hole potential, charge separation is expected to occur unless energy dissipation is so effective that the electron and hole are trapped at the interface as a bound CT state. Figure 3a−c shows the calculated diabatic potentials of the exciton and CT states as functions of the center-to-center distance R. Here, the intramolecular coordinates were fixed at the equilibrium geometry of the ground states for simplicity. The CT state was found to become less stable as the donor− acceptor distance increased (Figure 3a−c). The distance at the CT potential minimum was generally shortened from that at the exciton minimum by 0.2−0.3 Bohr (Figure 3a−c). The distribution range of the donor−acceptor distance is affected by disorder at the heterojunctions.11 The diabatic coupling, V, around the equilibrium donor− acceptor distance, 12−14 Bohr, was found to be large enough to induce an ultrafast exciton dissociation. V vanished at R values greater than ∼18 Bohr (Figure 3d). Here, V was not found to depend significantly on the π-conjugation length of the donor. Our previous quantum dynamics calculations indicated ultrafast exciton dissociations at the oligothiophene−C60 heterojunction,23,24 where the exciton dissociation induces slow oscillations of the intermolecular mode as well as vibrational excitations of the intramolecular modes. We define the vertical potential difference as a function of R as ΔE(R) = Eexciton(R) − ECT(R). The excess energy of exciton dissociation is defined as Eexcess(R) = ΔE(R) + λ, where λ is the reorganization energy along the intramolecular coordinate, r

II. METHODS In this work, we calculated the excited states of oligothiophene−C60 supermolecules using long-range-corrected timedependent DFT (LC-TDDFT)19 with the BLYP functional, where we considered the standard range-separation parameter, μ = 0.33, although there is room for the fine-tuning of μ.18,20 Here, oligothiophenes comprising five, seven, and nine monomer units (denoted as T5, T7, and T9, respectively) were considered. One of the six-membered rings of C60 was assumed to stack with the thiophene ring at the center of Tn. The double-ζ basis set with d functions was employed with the SBKJC pseudopotential.21 The GAMESS code22 was used for all of the DFT calculations. The dominant configuration of the exciton state corresponds to the excitation from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO) within the donor. The configuration of the CT state corresponds to the excitation from the donor’s HOMO to the acceptor’s LUMO. At a long intermolecular distance R, “pure” exciton and charge-transfer states are found in the adiabatic states, where R is the center-to-center distance. At a short distance R, the orbitals of the donor and acceptor are hybridized (Figure 2a), and the adiabatic states are characterized by a superposition of the exciton and charge-transfer states. The adiabatic states were converted to a diabatic representation (i.e., pure exciton and CT states) using the quasidiabatization scheme introduced in ref 23. As the diabatic coupling increases, the mixing of the exciton and CT states increases. III. RESULTS AND DISCUSSION In the following subsections, we address the relative energetics of the exciton and CT states and the electron−hole potential of the oligothiophene−C60 system. Experimental studies of P3HT−PCBM heterojunctions have provided evidence of an ultrafast charge transfer in this type of system, on a time scale of 50−150 fs.15,17 This time scale was confirmed in our recent theoretical investigations.23,24 Quantum efficiencies for the 15021

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from the electronic ground-state geometry is larger than that from the minimum of the exciton state (Table 1). That is, the excess energy is reduced by relaxation on the exciton state. Thus, exciton dissociation would be most efficient if it occurred before geometry relaxation right after the photoabsorption or right after the exciton diffusing through the donor layer reached the donor−acceptor interface. As the π-conjugation length of the donor decreased, the exciton potential was found to become higher, and the excess energy increased. The π-conjugation length should play an important role for realistic polymer−fullerene interfaces at which the effective π-conjugation length is affected by the disordered morphology, as discussed by Wu et al.4 and McMahon et al.5 III.B. Potential Curve of Electron−Hole Separation. The electron−hole potential curve was analyzed based on the CT potential in the long-distance region (Figure 4a), where the electron−hole distance, x, was assumed to be identical to the Tn+−C60− distance, R, in the charge-separated complex (Figure 2b). Note that, in the actual material, changes in x involve charge transport between neighboring donor and acceptor species (see the discussion in section III.C). The electron−hole potential calculated by LC-TDDFT in a vacuum was calibrated by taking into account the dielectric constants of fullerene (εr ≈ 425) and P3HT (εr ≈ 326). The present hypothesis is reasonable and practical in that the donor and acceptor in the supermolecule model are separated such that the orbital hybridization is negligible. DFT calculations explicitly considering the surrounding molecules are beyond the scope of this study.27 We propose the following analytical formula for fitting the electron−hole potential in the long-x region (in atomic units)

Figure 3. Calculated diabatic potentials of the exciton and CT states of (a) T5−C60, (b) T7−C60, and (c) T9−C60, as functions of the intermolecular distance, R. (d) Diabatic coupling between the exciton and CT states. (e) Diagram of the potential crossing along the intramolecular coordinate, r, where λ is the reorganization energy, illustrating the vibrational excitation due to exciton dissociation.

ϕ(x) = −

where 2a is the effective length of the charge distribution (Figure 2b), Ex is the electric field from the electrodes (Figure 1), εr is the dielectric constant, and ϕ0 is the potential in the asymptotic region (at = ∞ and Ex = 0). Here, we consider the Coulomb potential between a point charge and a line of uniform charge. The LC-TDDFT calculations predict that the electron−hole potential becomes shallower as the π-conjugation length of the donor increases (Figure 4a). This trend is well reproduced by eq 1, where the potential becomes shallower as the effective charge distribution, a, becomes longer. In typical organic solar cell devices, the electric field due to the chemical potential difference of the anode and cathode (Figure 1) is thought to be no more than ∼10 V/μm.6 The internal electric field can be increased by decreasing the anode−cathode distance, but a thin organic layer is unfavorable for photoabsorption efficiency. Yuan et al. proposed that the electric field can be increased to ∼50 V/μm by introducing ferroelectric layers at the organic−electrode interfaces.6 Figure 4b shows plots of eq 1 fitted to the electron−hole potential for T9, for values of the electric field, Ex, of 0, 10, and 50 V/μm. Here, the plots were calibrated considering the dielectric constants of fullerene (εr ≈ 4) and P3HT (εr ≈ 3). At Ex = 10 and 50 V/μm, the potential barrier was estimated to be 0.4−0.5 and 0.2−0.3 eV, respectively. Thermal activation at room temperature cannot overcome such barriers. According to our calculations (Table 1 and Figure 3), an excess energy of at

Table 1. Calculateda Intramolecular Reorganization Energies (eV) of Oligothiophene Cations (Tn), λT, from the Exciton Minimum and from the Neutral Ground-State Minimum, As Well As Calculated Intramolecular Reorganization Energies (eV) of the CT State of the Tn−C60 Supermolecule, λT+C60b (in Parentheses) T5 (T5−C60)

T7 (T5−C60)

T9 (T5−C60)

0.025 (0.094)

0.021 (0.090)

0.021 (0.090)

0.130 (0.199)

0.111 (0.180)

0.093 (0.162)

a 2 + x 2 ) − ln(x)] − Exx + ϕ0 (1)

(Figure 3e). As mentioned earlier, a positive excess energy is a necessary condition for exciton dissociation. When ΔE(R) is negative, that is, the CT state is less stable than the exciton state in terms of the vertical transition, exciton dissociation necessitates a larger reorganization energy than the vertical energy difference (Figure 3e). The calculated intramolecular reorganization energies are summarized in Table 1. The excess energy originates in the vibrational excitations (Figure 3e), which is essential for the hot CT mechanism. The reorganization energy of the CT state

exciton → cation (exciton → CT) neutral → cation (neutral → CT)

1 [ln(a + εra

DFT calculations using the B3LYP functional. bλT+C60 = λT + λC60, where the reorganization energy of the C60 anion, λC60, from the neutral ground-state minimum is 0.069 eV. a

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Figure 5. Schematic illustration of charge-separated (CS) states at a donor−acceptor interface together with the corresponding site potentials. Here, a lamellar stacking of the donor molecules is considered, and the potentials for configurations with the hole localized on the respective donor molecules are shown. The electron is assumed to be localized on a unique fullerene acceptor. The potential at Ex = 50 V/μm, εr = 4, and a = 13 Bohr is considered, where the internal electric field, Ex, is assumed to be perpendicular to the πconjugation plane of the donor.

the chemical interactions are essential for the energetics, the CS states involving more distant donor species are not directly coupled to the exciton state, and the energetics is largely governed by the Coulomb interaction. Figure 5 shows an illustrative example of site potentials for sequential CS states, where a lamellar stacking of oligothiophene is considered and the intermolecular distance is assumed to be 3.8 Å. The zero of energy is set to the minimum of the bound CT state. The energies of the successive CS states involving increasing electron−hole separations approach the potential barrier and then start to decrease beyond the barrier. As the donor−acceptor distance increased, the bound CT state was found to become less stable (Figure 3a−c), that is, the initial CT potential after exciton dissociation became less stable. In particular, the donor−acceptor distance at the moment of exciton dissociation was found to affect the initial CT potential, which, in turn, affects the relative barrier height. A long donor− acceptor distance would be favorable for decreasing the barrier, but the coupling V decreases with increasing distance (Figure 3d). It is curious that photocurrent can be observed even under the conditions where the excess energy is expected to be lower than the Coulomb potential barrier.3 Delocalization of the charge carriers is one factor that might facilitate charge separation.7,28 Delocalized CT states can be analyzed by considering site potentials (Figure 5) in conjunction with intermolecular transfer integrals. As the π-conjugation length of the donor molecule increased, the potential barrier was found to decrease (Figure 4c), but the exciton became more stable and the excess energy decreased (Figure 3a−c). When the exciton is delocalized over π-stacked molecules, the bright exciton states are higher in energy than the single-molecule excitation energy, that is, a delocalized exciton might increase the excess energy. Further theoretical analysis considering the quantum dynamics of exciton dissociation to free carriers is

Figure 4. (a) Electron−hole potentials for charge separation (solid lines) and fitted plots of eq 1 (dashed lines), where εr is unity (in a vacuum) and the effective charge distribution lengths a for T5, T7, and T9 are 9, 11, and 13 Bohr, respectively. The zero of energy is set to the minimum of the bound CT state. (b) Plots of eq 1 for T9 accounting for the internal electric field from the electrodes, Ex, where the minimum of the CT potential is set to zero. Here, the εr values for the upper and lower panels are 3 and 4, respectively. (c) Estimated potential barrier based on eq 1 for Ex values of 10 V/μm (black lines) and 50 V/μm (red lines) as a function of the effective charge distribution length, a, where the limit of a = 0 corresponds to a point charge. Here, the potential at x = 12 Bohr is set to zero (i.e., the initial potential of charge separation).

most ∼0.3 eV is not large enough to overcome the barrier for Ex = 10 V/μm (Figure 4c), that is, under typical conditions of organic solar cells. At Ex = 50 V/μm, the potential barrier can be lower than the excess energy. This is consistent with the experimentally observed improvement in energy conversion efficiency with increasing electric field.6 III.C. Electron−Hole Separation in Stacked Aggregates. The electron−hole potential as discussed so far provides a basis for analyzing the energetics of charge-separated (CS) states in stacked molecular aggregates (see Figure 5). In a sitebased representation, the energy of each CS state (A− D1+, A− D2+, ..., A− DN+), where A and D denote acceptor and donor species, respectively, is given by the electron−hole potential ϕ(x = xn), where xn denotes the electron−hole distance in the relevant configuration. Whereas the bound CT state at the donor−acceptor interface is coupled to the exciton state and 15023

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(7) Bakulin, A. A.; Rao, A.; Pavelyev, V. G.; van Loosdrecht, P. H. M.; Pshenichnikov, M. S.; Niedzialek, D.; Cornil, J.; Beljonne, D.; Friend, R. H. The Role of Driving Energy and Delocalized States for Charge Separation in Organic Semiconductors. Science 2012, 335, 1340−1344. (8) Grancini1, G.; Maiuri, M.; Fazzi1, D.; Petrozza1, A.; Egelhaaf, H.J.; Brida, D.; Cerullo, G.; Lanzani, G. Hot Exciton Dissociation in Polymer Solar Cells. Nat. Mater. 2013, 12, 29−33. (9) Jailaubekov, A. E.; Willard, A. P.; Tritsch, J. R.; Chan, W.-L.; Sai, N.; Gearba, R.; Kaake, L. G.; Williams, K. J.; Leung, K.; Rossky, P. J.; Zhu, X.-Y. Hot Charge-Transfer Excitons Set the Time Limit for Charge Separation at Donor/Acceptor Interfaces in Organic Photovoltaics. Nat. Mater. 2013, 12, 66−73. (10) Nayak, P. K.; Narasimhan, K. L.; Cahen, D. Separating Charges at Organic Interfaces: Effects of Disorder, Hot States, and Electric Field. J. Phys. Chem. Lett. 2013, 4, 1707−1717. (11) Yost, S. R.; Wang, L.-P.; Voorhis, T. V. Molecular Insight Into the Energy Levels at the Organic Donor/Acceptor Interface: A Quantum Mechanics/Molecular Mechanics Study. J. Phys. Chem. C. 2011, 115, 14431−14436. (12) Blom, P. W. M.; Mihailetchi, V. D.; Koster, L. J. A.; Markov, D. E. Device Physics of Polymer:Fullerene Bulk Heterojunction Solar Cells. Adv. Mater. 2007, 19, 1551−1566. (13) Veldman, D.; Ipek, O.; Meskers, S. C. J.; Sweelssen, J.; Koetse, M. M.; Veenstra, S. C.; Kroon, J. M.; Bavel, S. S.; Loos, J.; Janssen, R. A. J. Compositional and Electric Field Dependence of the Dissociation of Charge Transfer Excitons in Alternating Polyfluorene Copolymer/ Fullerene Blends. J. Am. Chem. Soc. 2008, 130, 7721−7735. (14) Park, S. H.; Roy, A.; Beaupré, S.; Cho, S.; Coates, N.; Moon, J. S.; Moses, D.; Leclerc, M.; Lee, K.; Heeger, A. J. Bulk Heterojunction Solar Cells with Internal Quantum Efficiency Approaching 100%. Nat. Photonics 2009, 3, 297−303. (15) Guo, J.; Ohkita, H.; Benten, H.; Ito, S. Charge Generation and Recombination Dynamics in Poly(3-hexylthiophene)/Fullerene Blend Films with Different Regioregularities and Morphologies. J. Am. Chem. Soc. 2010, 132, 6154−6164. (16) Behrends, J.; Sperlich, A.; Schnegg, A.; Biskup, T.; Teutloff, C.; Lips, K.; Dyakonov, V.; Bittl, R. Direct Detection of Photoinduced Charge Transfer Complexes in Polymer Fullerene Blends. Phys. Rev. B 2012, 85, 125206-1−125206-6. (17) Brabec, C. J.; Zerza, G.; Cerullo, G.; De Silvestri, S.; Luzzati, S.; Hummelen, J. C.; Sariciftci, S. Tracing Photoinduced Electron Transfer Process in Conjugated Polymer/Fullerene Bulk Heterojunction in Real Time. Chem. Phys. Lett. 2001, 340, 232−236. (18) Sen, K.; Crespo-Otero, R.; Weingart, O.; Thiel, W.; Barbatti, M. Interfacial States in Donor−Acceptor Organic Heterojunctions: Computational Insights into Thiophene-Oligomer/Fullerene Junctions. J. Chem. Theory Comput. 2013, 9, 533−542. (19) Towada, Y.; Tsuneda, T.; Yanagisawa, S.; Yanai, Y.; Hirao, K. A Long-Range-Corrected Time-Dependent Density Functional Theory. J. Chem. Phys. 2004, 120, 8425−8433. (20) Körzdörfer, T.; Sears, J. S.; Sutton, C.; Brédas, J. L. Long-Range Corrected Hybrid Functionals for π-Conjugated Systems: Dependence of the Range-Separation Parameter on Conjugation Length. J. Chem. Phys. 2011, 135, 204107−1−4. (21) Stevens, W. J.; Basch, H.; Krauss, M. Compact Effective Potentials and Efficient Shared-Exponent Basis Sets for the First- and Second-Row Atoms. J. Chem. Phys. 1984, 81, 6026−6033. (22) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. General Atomic and Molecular Electronic Structure System. J. Comput. Chem. 1993, 14, 1347−1363. (23) Tamura, H.; Burghardt, I.; Tsukada, M. Exciton Dissociation at Thiophene/Fullerene Interfaces: The Electronic Structures and Quantum Dynamics. J. Phys. Chem. C 2011, 115, 10205−10210. (24) Tamura, H.; Martinazzo, R.; Ruckenbauer, M.; Burghardt, I. Quantum Dynamics of Ultrafast Charge Transfer at an Oligothiophene-Fullerene Heterojunction. J. Chem. Phys. 2012, 137, 22A540-1− 22A540-8.

needed to clarify the mechanisms of charge separation beyond the barrier.

IV. CONCLUSIONS We have estimated the potential barrier for electron−hole separation from the bound CT state in an oligothiophene−C60 system, based on LC-TDDFT calculations. We propose a simple analytical formula for the electron−hole potential (eq 1) that describes the effect of the charge distribution length along the π-conjugated chain. The potential barrier becomes lower as the π-conjugation length of the donor increases. The potential barrier was estimated to be on the order of 0.4−0.5 eV under typical conditions of organic solar cells. As a general trend, the charge distribution along the π-conjugated chain decreases the potential barrier as compared to the Coulomb potential assuming point charges. The present analysis suggests that efficient electron−hole separation cannot be explained by the excess energy of the exciton dissociation alone if the charge carriers are assumed to be localized on single donor or acceptor species. An important issue to be addressed is therefore the dynamics of charge separation considering the charge carriers to be delocalized over several molecules. The potential curves estimated in the present work can be utilized for parametrizing a suitable model Hamiltonian for quantum dynamical23,24,28,29 and semiclassical30 analyses of charge separation.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +81-80-1278-0559. E-mail: [email protected]. ac.jp. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study was supported by a Grants-in-Aid for Scientific Research (C) from JSPS, Tokyo, Japan. The Advanced Institute for Materials Research, Tohoku University, is supported by World Premier International (WPI) Research Center Initiative, MEXT, Tokyo, Japan. Further, support from the JapaneseGerman NAKAMA funds is gratefully acknowledged.

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