A Detailed Analysis of Multiple Photoreactions in a Light-Harvesting

Article pubs.acs.org/JPCC

A Detailed Analysis of Multiple Photoreactions in a Light-Harvesting Molecular Triad with Overlapping Spectra by Utrafast Spectroscopy Thomas Roland,‡ Elodie Heyer,¶ Li Liu,‡ Adrian Ruff,§ Sabine Ludwigs,§ Raymond Ziessel,*,¶ and Stefan Haacke*,‡ ‡

Institut de Physique et Chimie des Matériaux de Strasbourg, Université de Strasbourg - CNRS, 67034 Strasbourg Cedex 2, France Institut de Chimie et Procédés pour l’Énergie, l’Environnement et la Santé, ICPEES-LCOSA, 67087 Strasbourg Cedex 2, France § Institut für Polymerchemie, Universität Stuttgart, Pfaffenwaldring 55, 70569 Stuttgart, Germany ¶

S Supporting Information *

ABSTRACT: The photoreactions that occur in a molecular triad composed of derivatives of boron-dipyrromethene (BOD), diketopyrrolopyrrole (DPP), and of triphenylamine (TPA) specifically designed for the use as a possible donor material for organic solar cells are studied by ultrafast fluorescence and transient absorption spectroscopy with hyper spectral coverage (300−900 nm). While the latter is often sufficient to reveal the reaction scenario in mono- or bicomponent organic molecules, the overlap of ground state absorption and fluorescence spectra of the components present in the TPA-DPP-BOD triad requires a careful and accurate characterization of the excited state differential absorption spectra and kinetics of the individual isolated moieties. In addition, the absorption spectra of the TPA cation and the DPP and BOD anions are determined by in situ spectroelectrochemical experiments. Picosecond fluorescence demonstrates efficient excited state quenching of both DPP and BOD, with downhill energy transfer from TPA occurring in a subpicosecond time scale, leading to its complete excited state quenching. Additionally, an analysis of the multiexponential fluorescence decay indicates the existence of reactive (quenched) and nonreactive (radiative) subpopulations of DPP and BOD, most probably due to structural heterogeneities. In order to treat this complexity of multiple reaction pathways, the transient absorption data are analyzed by a combination of global fitting, providing decayassociated differential spectra (DADS), and a reconstitution of these DADS by linear combinations of the characteristic difference spectra of all molecular species and excited states. This allows us to disentangle the photoreactions that the TPA, DPP and BOD excited state subpopulations undergo on time scales ranging from 200 fs to a few ns, and to clarify the contribution of TPA as an electron donor in the formation of an intramolecular charge transfer (CT) state. DPP has a dual role as an ancillary lightharvesting unit capable of performing ultrafast energy transfer to BOD, and as an electron acceptor for the CT state. The latter has a remarkably long lifetime of 0.5 ns. We conclude that the triad could act as a very efficient electron-donor material in a blend with commonly used acceptors such as phenyl-C61-butyric acid methyl ester (PCBM).

1. INTRODUCTION

in a wheel-like fashion, including bis(phenylethynyl)anthracene (λabs 450 nm), a boron dipyrromethene dye (λabs 513 nm) and a Zn-tetraarylporphyrin (λabs 418 and 598 nm). This latter study reports quantitative electronic energy transfer and 95% quantum efficiency for formation of the P+•−C−• 60 chargeseparated state.20 Other attractive and highly informative systems developed by many different research groups have been reviewed.21−24 Within the recent development of organic materials for photovoltaic applications, blend films composed of poly(3hexylthiophene) (P3HT) and phenyl-C61-butyric acid methyl ester (PCBM) are considered as a reference system.25 The properties of the donor P3HT are well recognized, but limited by an absorption spectrum that, depending of the deposition

In the past decade, tremendous effort has been devoted to the synthesis of artificial scaffoldings capable of mimicking the light harvesting complexes of natural photosynthesis.1,2 Along these lines, wheel-like porphyrin arrays3−7 (diades including a lightharvesting array of metalated porphyrins (P) in which the excitation energy is transferred rapidly to a C60 reservoir) afford a porphyrin-fullerene charge-separated state P+•−C−• 60 with a quantum yield of 70%.8 Self-assembled monolayers of linear ferrocene−porphyrin−fullerene molecular triads and linear boron-dipyrrin dyes have been studied in order to examine both energy and electron transfer in the artificial reaction center (C60).9 There are many reviews dedicated to the design of linear and cyclic porphyrin arrays as artificial models for photosynthetic units.10−17 Single-molecule spectroscopy has been used to probe energy migration in cyclic porphyrin arrays.18,19 A rigid antenna-based system has been designed to include three types of light-absorbing chromophores, organized © XXXX American Chemical Society

Received: July 25, 2014 Revised: September 22, 2014

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dx.doi.org/10.1021/jp507474r | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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a finite number of species with molar concentrations ci and extinction coefficients εi

methods and treatments, does not absorb at wavelengths longer that 550 to 600 nm. As a consequence, a donor dye with a broader spectrum to the IR side able to transfer charge to an acceptor (usually based on fullerenes) is much desired.26,27 The boron-dipyrromethene dye (BODIPY) is a widely studied dye, with a remarkably high extinction coefficient, that proved efficient at charge generation in various systems,28−33 appreciated for its high fluorescence quantum yield, stability, and chemical versatility. For the present work, the BODIPY dye was substituted with phenyl polyoxyethylene styryl fragments (BOD) in order to bathochromically shift the absorption and emission features to the 550−670 nm region. The original fluorine atoms on the boron center were substituted by short polyoxoethylene chains to match the previous observations given by former organic photovoltaic (OPV) studies.32 In order to strengthen the antenna effect of this chromophore, two other residues whose efficiency for charge generation has been shown were covalently linked with unsaturated spacers: a thiophene type diketopyrrolopyrrole (DPP34−38) and a triphenylamine (TPA39−42), resulting in a triad of three moieties with different energetic levels. The molecular structure of this triad is sketched in Figure 1.

ΔA(t , λ) = d · ∑ εi(λ) ·[ci(t ) − ci( −∞)] i

(1)

where d is the sample path length, ci(−∞) is the species concentration without laser excitation, ci(t) is the species concentration at a delay t after excitation, and the sum runs over all photoexcited species, meaning TPA, DPP, and BOD in their ground and excited states, and in relevant anionic and cationic radical forms. Alternatively, we can formulate eq 1 in terms of differential extinction coefficients ε̃i taking into account the ground state bleach for the photoinduced species: ΔA(t , λ) = d · ∑ εĩ (λ) ·ci(t ) i

(2)

Note that eq 2 assumes the time-dependence of TA to be borne only in the formation and decay of the molar concentrations of the species involved. In this case, a combination of singular value decomposition (SVD) and target analysis proved to be successful in providing decay-associated difference spectra (DADS) whose spectral features allow for a clear identification of the photocreated species.49 However, studying the time-resolved spectroscopy of the present three-component molecule is made difficult by the spectral overlap of TPA, BOD, and DPP ground state absorption and fluorescence spectra, and the unknown degree of electronic coupling between them. Can we decompose the tempo-spectral features into a sum of the individual components as eq 1 suggests it ? Recent examples using the broadband excitation capabilities of two-dimensional spectroscopy showed how multicomponent molecular systems with parallel reaction pathways can be studied and their reactions disentangled.50−53 We show in the present contribution, that for a system like the triad drawn in Figure 1, and for regions where electronic coupling can be neglected, the spectral characteristics of all the triad parts can be obtained by a careful selection of model molecules, and using ground state and transient absorption and fluorescence spectroscopy combined with in situ spectroelectrochemistry for the oxidized/reduced species. Time-resolved fluorescence spectroscopy is complementary to TA, as it isolates the excited state dynamics.54 Here, it provides a first hierarchy of the reaction time scales involved. Spectroelectrochemistry has proven earlier to be a suitable complementary method to pump−probe experiments to distinguish between charged and excited species.55 The simultaneous detection of voltammetric (redox information) and spectroscopic (chemical information) data allows the correlation of absorption spectra to distinct oxidation or reduction products. Based on these information, a thorough decomposition of the relevant DADS, based on eq 1 is possible, and provides a clear-cut picture of the multiple photoinduced reactions. The results underscore in particular the importance of structural heterogeneity and the existence of related subpopulations that undergo distinct reaction pathways.

Figure 1. Molecular structure of the triad, composed of derivatives of boron-dipyrromethene (BOD), diketopyrrolopyrrole (DPP), and of triphenylamine (TPA).

The ultimate material combination would feature the conjugation or blend of this triad with PCBM or any other appropriate acceptor material. In this context, the purpose of the present study is to have detailed insight into the molecular photoinduced processes that occur on the triad alone. How fast and efficient is energy transfer among the three components, possibly underscoring the expected antenna effect? Is there any indication of an intratriad charge transfer that could promote formation of a charge-separated state in the blend ? All these questions prompt for a study of the triad by ultrafast spectroscopy. Such studies have been conducted on conjugates such as BOD/PCBM and DPP/PCBM or with other acceptor materials. CT formation in time scales ranging from a few tens to a few hundreds of picoseconds and efficiencies above 4% were observed.25,28,37,43−46 However, it is not straightforward to directly transfer these results to the triad, since it is well-known that the elementary processes depend on the molecular conformation, such as the D−A distance. As the structural complexity of the donor material grows, e.g., as in the present triad, it can not be excluded that the molecule adopts several thermodynamically stable ground state conformations,47 thus leading to heterogeneous reaction kinetics. A simple approach could be to use transient absorption (TA) spectroscopy using a sub-100 fs excitation pulse and probing the TA with a white-light continuum over the entire near-UV to near-IR spectral range.48 Then the total differential absorption can be expressed as a linear combination of the contributions of

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EXPERIMENTAL SECTION Experimental Setups. Static absorption spectra were recorded with a PERKIN ELMER Lambda 950 UV/vis Spectrometer.

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two: the probe that is sent through the sample, and the reference. Relative linear polarization between pump and probe beams is set at the magic angle to avoid any signal from anisotropy decay. By chopping the pump beam, a CCD camera alternatively records the absorption spectrum of the unexcited and excited molecules at 220 Hz. This allows us to record differential absorption (ΔA) spectra as a function of the pump−probe time delay.58 The pump pulse has a width of ∼80 fs and its intensity was set so as to ensure a linear dependence of the ground state bleach signals in all experiments reported below. Data are processed to correct the effects of group velocity dispersion (chirp) in the white light continuum, the contribution of the solvent and the presence of any delayindependent background.58 All transient spectra presented in this work are obtained by merging three data sets covering adjacent spectral windows, from 320 to 990 nm. After chirp correction, the zero time-delay is defined with an error of ±20 fs over the entire observation spectral window. Solutions were prepared in tetrahydrofuran (THF) in order to achieve an optical density of 0.1 in a quartz cell with a d = 1 mm path length (concentrations between 1 and 2 × 10−5 M.L−1), with no aggregation observable in the absorption spectra. For the time-resolved experiments, the cell was mounted on a loud speaker and periodically moved through the excitation beam in order to avoid sample degradation and reexcitation of possible long-lived photoproducts.

Steady-state emission and excitation spectra were recorded on a HORIBA Jobin-Yvon FluoroMax 4P spectrofluorimeter. All fluorescence spectra were corrected. The fluorescence quantum yield ϕ was calculated from eq 3: φexp = φref ·

2 ·Iexp(1 − 10−ODref ) nexp 2 ·Iref (1 − 10−ODexp) nref

(3)

where I denotes the integral of the corrected fluorescence spectrum, OD is the absorption of the sample at excitation wavelength, and n is the refractive index of the medium. Indexes ‘exp’ and ‘ref’ respectively relate to the studied sample and a reference sample, in this case a previously reported BODIPY dye (ϕref = 0.49 in CH2Cl2, excitation at 650 nm).56 In situ UV/vis/NIR absorption spectroelectrochemical measurements were conducted in a gas tight three electrode quartz cell under thin layer conditions (layer thickness ≈ 20 μm) with a PGSTAT101 potentiostat (Metrohm, Germany) and a vis/NIR or a UV/vis spectrometer with CLH600 lamps (both from Zeiss, Germany) equipped with MCS621 vis II and MCS611 NIR2.2 μ or MCS621 vis II spectrometer cassettes, respectively, under argon atmosphere. The UV−vis−NIR spectra were recorded in reflection mode (the absorption during electrochemical cycling was calculated from the difference of the emitted light and the reflected light from the working electrode). The working electrode consisted of a polished Pt disk with a nominal diameter of 4 mm. As counter electrode, a Pt wire was used. The reference electrode consisted of a Ag wire coated with a AgCl layer (Ag/AgCl system). The reference electrode was directly immersed into the electrolyte solution. All potentials are rescaled to the formal potential of the potential standard Fc/Fc+ (Fc = ferrocene, Fc+ = ferrocenium).57 For all experiments, NBu4PF6 was used as the supporting electrolyte (0.1 M). Dichloromethane was dried prior to use with neutral Al2O3 (activated at ∼80 °C in an oven for several days). The solvent was deaerated by argon bubbling. Substrate concentrations were in the 10−4 to 10−3 M range. All voltammograms were recorded with a scan rate of 20 mV s−1 and are not background corrected. The time-resolved fluorescence was obtained with an HAMAMATSU streak camera C10627, the samples being excited at 320 nm with a home-built noncollinear optical parametric amplifier (NOPA), seeded by the third harmonic of an Yb-doped fiber laser (TANGERINE, Amplitude System). This wavelength simultaneously excites all three components in the triad and allows evidencing the subsequent flow of excitation among them. Polarization of the exciting light was set at the magic angle (54.7°) in order to avoid anisotropy effects. The presented spectra are corrected from the reabsorption of the fluorescence by the sample. The dynamics of the fluorescences were recorded on several time windows, from 1 ns (with a time resolution of 10 ps) up to 20 ns, in order to resolve the full dynamics with the best possible resolution and high signal-to-noise ratio. The experimental pump−probe setup was already described.58,59 Briefly, the femtosecond laser source is a Ti:sapphire regenerative amplifier laser system (Pulsar, Amplitude Technology) delivering 40 fs-long pulses, at 5 kHz, centered at 800 nm. A commercial optical parametric amplifier (OPA; TOPAS, Light Conversion) produces a pump beam at a selected wavelength (320 nm) and a thin CaF2 or Sapphire plate generates a broad white-light continuum (300− 1000 nm) from the 800 nm fundamental. The latter is split in

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EXPERIMENTAL RESULTS Synthesis. Detail for the synthetic work will be disclosed in a forthcoming paper. Concisely, the preparation of the triad and key dyes used as references is sketched in Figure 2. Sonogashira cross-coupling reactions were mediated by low valent palladium complexes, between the starting material 160 and either the dibromodiketopyrrolopyrrole 261 or the borondipyrromethene 332 in order to obtain the pivotal intermediate 4 (42%) and the model compounds TPA-DPP-TPA (26%), BOD-TPA (89%). An adequate stoichiometry of 1 vs 2 allowed favoring the formation of 4 against TPA-DPP-TPA; this pivotal intermediate was similarly linked via Sonogashira cross-coupling to the terminal alkyne 5 to afford the target TPA-DPP-BOD triad in good yields (66%). The reference BODIPY derivative BOD was easily prepared from dye 3 under cross-coupling conditions using trimethylsilylacetylene as the substrate. Specific deprotection of the silyl-derivative was achieved using a mineral base in a methanolic solution at room temperature leading to derivative 5 quasi quantitatively. All compounds were purified by state-of-the-art equipment and recrystallized in adequate solvents. The molecular structures and purity were assigned by NMR spectroscopy tools, mass spectroscopy, elemental analysis, and standard steady-state absorption and emission spectroscopies, confirming unambiguously the molecular structures drawn in Figure 2. Selected data are provided in the Supporting Information (SI). Steady-State Properties. Ground state absorption spectra and extinction coefficients of BOD (red line) and TPA-DPPTPA (blue line) are shown in Figure 3. TPA-DPP-TPA data is shown for a DPP disubstituted with two TPA groups (TPADPP-TPA), which mimics best the molecular orbitals of the DPP moiety in the triad. Indeed a mono- or unsubstituted DPP has a blue-shifted absorption spectrum that cannot be used as reference (SI, Figure S7). TPA has only negligible absorption at wavelength longer than 350 nm (Figure 3). One can see here that the triad spectrum (black line) matches well the simulated C



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coupling between DPP and BOD. DPP acts as an auxiliary absorber increasing the extinction coefficient in the 500−650 nm and below 360 nm ranges. The additivity of the individual spectra is not as good in the near-UV, meaning that at 320 nm the extinction coefficient is an undetermined mixture of all three molecular components. In order to identify the spectral signature of possible charged species in the pump−probe signal, in situ UV−vis−NIR spectroelectrochemical measurements in 0.1 M NBu4PF6/ CH2Cl2 were performed on the isolated compounds at a Pt electrode under thin layer conditions. The cyclic voltammograms of the TPA species show a chemically irreversible oxidation with a peak potential of about +0.85 V (at a scan rate of 20 mV s−1). The reductions of TPA-DPP-TPA and BOD are chemically reversible with peak potential values of −1.63 V and −1.71 V (at 20 mV s−1), respectively (potentials given vs Fc/ Fc+). The spectra of the TPA cation, TPA-DPP-TPA anion and the BOD anion are shown in Figure 4. The spectrum of the

Figure 2. Synthetic pathways for the preparation of the triad and pivotal reference dyes.

Figure 4. Normalized absorption spectra of the charged species in dichloromethane (with NBu4PF6 at 0.1 M acting as the supporting electrolyte): TPA radical cation (black, recorded at a potential of +1.1 V), BOD radical anion (red, −1.8 V), and TPA-DPP-TPA radical anion (blue, −1.82 V). All potentials are given vs Fc/Fc+.

TPA cation (black line) exhibits a broad peak centered at 730 nm (consistent with formerly reported results62). The absorption spectrum of the BOD anion (red line) shows a main peak centered at 595 nm and a weak absorption at 455 nm. The main absorption of the DPP anion is located at 753 nm (cf. Table 2). These spectra are proportional to the extinction coefficients of the relevant charged species needed for an analysis based on eq 1. However, due to variations in the baseline and molar concentrations in the electrochemical cell, a straightforward calibration against the ε’s of the neutral compounds is not accurate enough. We will discuss the determination of the differential ε̃ of the charged species (TPA, TPA-DPP-TPA, and BOD) in the analysis section. A detailed discussion of the spectroelectrochemical results on the triad compound and its isolated constituents will be presented in a separate paper.63 Picosecond Fluorescence Kinetics. TPA-DPP-TPA and BOD-TPA have very similar steady-state fluorescence spectra, with a main band whose maximum is centered at 659 nm for TPA-DPP-TPA and a few nm red-shifted for BOD-TPA

Figure 3. Absorption spectra in THF at room temperature of TPA alone, TPA-DPP-TPA, BOD alone, the full triad (TPA-DPP-BOD), and the sum of the three separated components (minus twice that of TPA). BOD absorption is well-defined by two peaks at 644 and 593 nm for its S1 band and 371 nm for its S2 band. DPP absorption (when disubstituted) is characterized in its S1 band by two nearby peaks at 623 and 585 nm, and a secondary absorption band centered at 354 nm. TPA absorption is defined by a broad band from 280 to 350 nm. As shown by the sum of those three spectra, the triad is basically the sum of those contributions.

spectrum (pink, calculated as the algebraic sum of both TPADPP-TPA and BOD spectra, minus that of TPA) in the longwavelength region (>500 nm), indicating a negligible electronic D



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(Figure 5). A characteristic shoulder centered at 717−720 nm is observed for both molecules (dotted lines in Figure 6). The

Figure 6. Solid lines: Normalized DAS associated with 63 ps, 518 ps, and 2.27 ns obtained by global fitting of the time-resolved fluorescence of the triad. Dotted lines: fluorescence spectra obtained for DPP and BOD. Comparison of these leads to the identification of the three decays: the faster decay is associated with DPP, the intermediate one with BOD, and the longer one with a mix of these species.

Figure 5. Spectrally integrated decay traces of the fluorescences of BOD-TPA (red), TPA-DPP-TPA (black), and the triad (blue), in THF, recorded by the streak camera, and scaled with respect to the absorption of each sample at 320 nm to reveal the quenching observed in the triad. Data from different time scales on the streak camera are merged. Inset: Selected fluorescence spectra of BOD-TPA, highlighting the decay of the fluorescence, and the spectral invariance during decay.

Table 1. Fluorescence Lifetimes (Average Lifetime, if MultiExponential Decay) τfluo and Quantum Yields Φ for the BOD Emission in the Triad and the DPP and BOD Reference Compounds, As a Function of Solvents and Excitation Wavelength λex, Indicated in the Parentheses (Solvent, λex in nm)a

main difference between those two is the relative amplitude of the sideband, about twice as intense for the DPP. The time-resolved fluorescence of the separated components TPA-DPP-TPA and BOD-TPA shows a monoexponential decay for both molecules with no change in the fluorescence spectra (as shown in Figure 5 inset for BOD-TPA), and time constants of 2.2(±0.1) and 4.5(±0.1) ns, respectively. Note that the fluorescence quantum yield of BOD-TPA is only 0.44. For BODIPY compounds bearing a phenyl group, like BOD, this is often attributed to the torsional flexibility of the single bond connecting the phenyl with the dipyrromethene core. The corresponding nonradiative rate is then similar to the radiative one, i.e. in the 108 s−1 range. The fluorescence of the triad is much shorter lived and has more complex dynamics. This is not unexpected since the 320 nm excitation is in resonance with transition for all three molecular components of the triad. Thus, the short excitation pulses create a linear combination of the three possible wave functions, that will break up into three different localized excitations: TPA*, DPP*, and BOD*. The fluorescence measured with 10 ps time resolution in the red spectral region captures only the fluorescence of the latter two. Monitoring the near-UV region, where TPA is known to fluoresce, does not show any signal above the noise floor. This allows us to conclude that TPA is quenched within a very small fraction of the streak camera instrument response function (IRF), and we estimate its excited state lifetime to be 100−200 fs due to resonance energy transfer to BOD and DPP. In the red part of the spectrum, we find that global fitting of the triad’s time-resolved fluorescence yields three time constants (error bars in brackets) of 63 ps (1), 518 ps (4), and 2.27 ns (0.02), associated with three different decay associated spectra (DAS) shown in Figure 6, with respective relative amplitudes of 30%, 45%, and 25%. The averaged lifetime of the triad yields a quenching ratio of 5.5 with respect to the 4.5 ns lifetime of isolated BOD-TPA, in good agreement with a 7-fold reduced fluorescence quantum yield found for the triad in steady-state

molecule

τfluo

Φ

TPA-DPP-TPA (THF, 320) TPA-BOD (THF, 320) triad (THF, 320) triad (THF, 650) triad (THF, 524) triad (toluene, 358)

2.2 ns 4.5 ns 0.25 ns

32% 48% 3% 6% 4% 45%

a

Lifetimes and quantum yields are given with error bars of (respectively) ±0.1 ns and ±2%.

fluorescence (measured fluorescence quantum yields of BOD and TPA-DPP-TPA in THF are respectively of 48% and 32%, while that of the triad is 3−6%, a reported in Table 1). Comparison of the DAS with the fluorescence spectra of TPADPP-TPA and BOD-TPA (dotted lines) allows for an assignment of the different decays involved. With the relative intensity of the 720 nm shoulder as a criterion, the 63 ps DAS follows the shape of the fluorescence of TPA-DPP-TPA, while the 518 ps DAS is BOD-TPA-like. These shorter lifetimes for the DPP and BOD excited states indicate that new nonradiative processes occur. The third DAS with a lifetime of around 2.2 ns displays a temporally invariant spectrum that is best described as a linear combination of DPP and BOD. The 2.2 ns time scale is similar to the fluorescence lifetime of isolated TPA-DPP-TPA, and we are missing the 4.5 ns time scale of the isolated BOD-TPA. Despite a high signalto-noise-ratio, adding a fourth longer time constant does not improve the fit. Clearly, the DPP and BOD populations break up into reactive species with quenched excited lifetimes, and “non-reactive” ones with a lifetime close to the ones recorded for the isolated molecules. Since DPP and BOD contribute with almost equal amplitude to the above 2.2 ns DAS, the amplitudes of the DAS indicate that only a minor portion of both BOD and DPP remain unquenched. We take this as a first E



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Figure 7. Left: Selected differential absorption spectra obtained for TPA-DPP-TPA in THF after excitation at 320 nm. Two characteristic spectra are observed: a first one at short times (black curve), with a minimum at 632 nm, corresponding mostly to the DPP bleach and a weak and sightly blueshifted SE, and a second quasi-stationary spectrum with an increased stimulated emission, shifting the minimum to 654 nm (the two vertical bars are visual helps to highlight this effect), and an intense ESA band at 890 nm. This second spectrum decays in nanoseconds, and is attributed to the relaxed DPP S1 state, whose average spectrum is shown in green. The first (320 fs) and last (average) ΔA spectra are shifted (respectively by +5 and −5 × 10−3) for clarity. Right: Selected differential absorption spectra obtained for BOD in THF after excitation at 320 nm. As for DPP, BOD is excited in its Sn state and relaxes into S1 in less than 1 ps, where it then lives for several ns. Differential spectra from this state are averaged yielding the green curve. The ΔA spectra for 30 ps, 300 ps, and the average spectra are each shifted by +2 with respect to the previous one, in order to highlight the lack of changes past the first few picoseconds.

Selected ΔA spectra for BOD-TPA are shown in Figure 7 right panel. As for DPP, in the first few hundreds of femtoseconds, the excited molecules relax from an Sn state to an S1 state, leading to intense ESA with a characteristic peak at 483 nm, and above 850 nm, SE peaks at 650 and 720 nm. Then on several nanoseconds, corresponding to the 4.4 ns time constant, the whole signal reduces, due to radiative decay. The average spectrum obtained for delays longer than 30 ps (shown in green) is used as our reference for ΔABOD*, again proportional to ε̃BOD*. The spectral characteristics of BOD* and DPP*, vibrationally relaxed in their S1 states, are compiled in Table 2. The same experiment on the triad reveals a quite complex temporal evolution, as shown on Figure 8. During the first picoseconds, the ESA band between 830 and 950 nm rises

indication for conformational heterogeneity of the triads. In order to decipher the nature of the quenching mechanisms at work, we carried out TA experiments reported in the following. Femtosecond Transient Absorption. Pump−probe experiments were first carried out on the BOD-TPA and TPA-DPP-TPA reference compounds under excitation at 320 nm, with the main purpose of obtaining the characteristic extinction coefficients ε̃DPP* and ε̃BOD*. Indeed, after a short excited state thermalization period, time-independent differential spectra of DPP* and BOD* in THF are observed. Figure 7 shows a selection of ΔA spectra for TPA-DPP-TPA (left panel), highlighting two distinct spectral shapes observed. Excitation at 320 nm populates TPA* and a higher excited state Sn of DPP that relaxes on an ≈100 fs time scale via internal conversion into S1. The 320 fs ΔA spectrum shows ground state bleaching (GSB) and stimulated emission (SE) of DPP in the 520−760 nm range (ΔA < 0). Positive features are due to excited state absorption (ESA) from S1. However, due to vibrational relaxation within S1, the shapes of that ESA and of the SE evolve, and a quasi-stationary difference spectrum is observed after 25 ps. This vibrationally relaxed population of S1 lives a few nanoseconds, as the ΔA spectra at longer times indicate. A decay time of 2.2 (±0.1) ns is found in perfect agreement with the picosecond fluorescence data. In the following, we use the average over many ΔA spectra at times >25 ps as our reference for ΔADPP*, as shown in green. This spectrum is proportional to ε̃DPP*, the time-invariant extinction coefficient of the relaxed excited state population of S1, plus the negative ground state bleach. Its amplitude depends on several experimental conditions, and needs to be calibrated against the extinction coefficient of the DPP ground state εDPP. This will be detailed in the analysis section.

Table 2. Spectral Characteristics of the Different Participating Speciesa molecule

state

characteristic peak(s)

TPA-DPP-TPA

GSB ESA SE radical anion GSB ESA SE radical anion radical cation

349, 582, and 622 nm 890 nm 665 and 720 nm 753 nm (narrow) 370, 593, and 645 nm 483 nm and >850 nm 665 and 720 nm 595 nm (narrow) 732 nm (broad)

BOD

TPA a

Central wavelengths of the main peaks are given, except for BOD’s ESA in the near IR, for which the visible limit of the band is given.

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combination of the differential extinction coefficients ε̃i(λ) of eq 2. In particular, we strive to extract the spectro-temporal data of excited states or charged species (using ε̃DPP*, ε̃DPP−,ε̃BOD*,ε̃TPA+,...), including ground state bleach recovery, in order to determine the photoreaction scenario. The exact values of the coefficients in the linear combinations are not always relevant for our analysis, but we will comment on them as long as their values have to comply to boundary conditions. Note that the rise of photocreated species comes with a negative contribution, while those disappearing have a positive amplitude (see below). The very first step of spectral evolution (290 fs, SI) corresponds to the intra- and intermolecular vibrational and dielectric relaxation, as observed for BOD and DPP, with an increase of the two ESA bands between 417 and 530 nm, and at wavelengths longer than 820 nm, as well as an increase of the stimulated emission (peaks at 660 and 730 nm). The second DADS is associated with the 6.3 ps time constant (Figure 9B). Its main features are a negative part below 640

Figure 8. Selected differential absorption spectra obtained for the triad in THF after excitation at 320 nm. At the shorter times, we have a strong contribution of both DPP* (with the ESA band around 890 nm) and BOD* (with its ESA at 483 nm). The former ESA decays in picoseconds (red to blue spectra), indicating a decrease of the population of DPP*. A new absorption rises between 700 and 850 nm within 100 ps, as a new species is formed, that disappears in the following nanosecond. The ΔA spectra were assembled in three separated groups for easier comparison, and shifted respectively by +0.01/0/−0.01.

significantly, as seen with DPP and BOD. During the following 10 ps, the latter and the negative ΔA close to 600 nm are reduced (red to blue spectra). At longer times (up to a hundred picoseconds), the negative ΔA below 630 nm reforms, and a new induced absorption (ΔA > 0) forms at wavelengths longer than 680 nm, indicative of a new species. The changes in this long wavelength region include a decrease of the >800 nm ESA and the 720 nm SE. Note that during this stage of evolution, the 483 nm ESA of BOD* hardly decays. Finally at longer times, the total shape of ΔA does not change significantly anymore, and simply reduces in amplitude. Global Analysis and Decomposition of the DADS. In order to reduce the spectro-temporal complexity, we performed a global fitting analysis after SVD filtering, retaining only the first six singular values that have kinetic traces with amplitudes above the noise floor. Global fitting of these traces yields five time constants of 290 (±10) fs, 6.3 (±0.3) ps, 52 (±2) ps, 0.50 (±0.03) ns and 2.2 (±0.2) ns, with their DADS. Five is the minimum number of decay times needed to obtain nonstructured residuals in the fits, randomly distributed around zero (cf. SI). Note that they differ by more than a factor of 4, thus minimizing the interdependence of lifetimes and DADS amplitudes in these multiparameter fits. Global fitting means that instead of using eq 2 where we considered each species and separated their time evolution, this information is cast in the decay-time specific DADS: ⎛ −t ⎞ ΔA(t , λ) = d · ∑ DADSj (λ) ·exp⎜⎜ ⎟⎟ ⎝ τj ⎠ j

Figure 9. DADS associated with the 6.3 and 52 ps time constants from the pump−probe data global analysis. The DADS (B and D, black curves) features are well reproduced by the simulated red curve, corresponding to (B) a FRET from DPP* to BOD* (obtained from the ΔA spectra of DPP* and BOD* as represented in A) or (D) the formation of the CT states through hole transfer from DPP* to TPA (obtained from the ΔA spectra of TPA•+, DPP•−, and DPP*); blue curve is the DADS resulting from a second experiment.

nm, with two peaks at 570 and 615 nm (indicating a bleach recovery of DPP), a narrow positive band at 656 nm, and a smaller one at 730 nm (indicating an increase of the stimulated emission), a broad positive band at wavelengths above 800 nm (similar to the ESA of DPP*) and a negative peak at 483 nm (similar, though negative, to the ESA of BOD*). All those features can be well reproduced as shown with the red curve, obtained by the difference between the differential extinction coefficients of DPP* and BOD*: DADS(6.3 ps) ∝ A1 × εDPP ̃ * + A 2 × εBOD ̃ *

(4)

(5)

with A2 < 0. Panel A of Figure 9 highlights the spectral shapes of both terms used in the analysis. The agreement between the experimental DADS and the fit is satisfactory, as the main spectral features described above are reproduced by the fit, while the small differences may be due to the neglect of excited state intra- and intermolecular thermalization effects that may

In a rate equation scheme, the molar concentration of species ci(t) evolve as a sum of exponentials, and the DADS are linear combinations of the species-specific difference spectra, with coefficients that are complex functions of the decay times.49 This implies that the DADS can be analyzed in terms of the contribution of the relevant molecular species, i.e., as a linear G



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contribute at these short time scales. We can conclude that no other species contributes to this DADS. The decay of DPP* leading to excitation of BOD, i.e., a rise of BOD*, is evidence for energy transfer from DPP* to BOD* within 6.3 ps. Note that this process went unnoticed in the picosecond fluorescence data, due to the limited time resolution (10 ps), and the high spectral overlap of DPP and BOD fluorescence. It then follows from the FRET process that A1 = A2. This allows us to calibrate the amplitude of the ε̃BOD* against the one of ε̃DPP* spectra. The DADS associated with the 52 ps time constant, pictured on Figure 9D, features a broad positive band below 650 nm, a very broad negative band from there to 860 nm with a peak at 745 nm, and again a positive band above 860 nm. The timeresolved fluorescence showed a DPP excited state quenching on this time range (63 ps), thus we include the decay of DPP* in the DADS fit (red curve). It is responsible for the negative shoulder at 650 and the positive signal above 860 nm, respectively corresponding to the DPP’s SE and ESA. The negative peak at 745 nm hints at the formation of the DPP radical anion, sitting on a broader background due to the TPA radical cation (Figure 9C). This indicates the formation of an intramolecular (TPA•+,DPP•−) charge transfer state on a 50− 60 ps time scale:

Figure 10. (B) DADS associated with the 0.50 ns time constant from the pump−probe data global analysis. The DADS (black dots) overlays well with the simulated red curve, corresponding to a decrease of BOD* (as shown by the fluorescence data), and a decrease of the CT state population (reconstructed in A). Blue curve is the DADS resulting from a second experiment. (D) DADS associated with the 2.2 ns (black curve) well fitted by the simulated red curve, corresponding to the sum of DPP* and BOD* ΔA spectra (as represented in C).

DADS(52 ps) ∝ B1 × εTPA ̃ • + B2 × εDPP ̃ •− + B3 × εDPP ̃ *

Table 3. Amplitudes Used for Fitting the Four Principal DADS, According to Eqs 5−8 in Units of 10−8 M·cm, Indicating the Relative Contributions of Excited State Differential Absorption Spectra and the Extinction Coefficients of Neutral and Charged Speciesa

(6)

This process is at the origin of the drastic changes of ΔA observed in the 10−100 ps time interval (Figure 8). The ion’s differential spectra (ε̃TPA• and ε̃DPP•−) include both their absorption (obtained through the spectro-electro-chemistry) and GSB (cf. SI). However, the exact ratio between these terms is determined here, as well as the amount of GSB in ε̃DPP*, since B1 = B2 = B3 (global neutrality), and due to other constraints related to the spectrovoltametry. Since the 700−800 nm range is central to ascertain the presence of the CT state, and from an experimental point of view fluctuations of the white-light spectrum are frequently observed in that range, a control experiment was performed, with an improved accuracy in this region. The control data yields a similar DADS (Figure 9D), fitted with the same contributions. The DADS associated with the 0.50 ns time constant is displayed in Figure 10B. Consistent with the 518 ps decay found for BOD* in the picosecond fluorescence data, the two negative peaks at 650 and 730 nm are due to BOD’s SE decay, while ESA decay is included for wavelengths above 850 nm and below 550 nm. Beside these peaks, we recognize a negative peak at 590 nm, that with a peak at 640 nm (merging with the 650 nm one) corresponds to BOD bleach, indicating relaxation of BOD* to its ground state. To fit the rest of the DADS (a broad positive band with a peak at 750 nm), we include the contributions of the previously formed anion and cation (cf. Figure 10A), meaning that the CT recombines on this 0.5 ns time scale. This entails the recovery of the GSB of DPP. Therefore, the total DADS is fitted by

DADS lifetime (ps)

ε̃DPP*

ε̃BOD*

6.3 ± 0.3 52 ± 2.3 500 ± 27 (CT recomb.) 500 ± 27 (BOD* dec.) 2200 ns ± 200

A1 = 10 B3 = 4.0

A2 = −10

D1 = 3.0

ε̃TPA+•/ε̃DPP−•

FLUO

B1 = B2 = −4.0 C1 = 4.0

0.30

C2 = 8.0 D2 = 4.5

0.45 0.25

a Typical error bars are ±10%. The column FLUO compiles the amplitudes of the three fluorescence decay times. See text for comments.

relaxation of BOD*, which happen to occur on the same time scale. We recall that the 500 ps decay of BOD* was not observed on the isolated TPA-BOD molecule, meaning that the conjugation with DPP and TPA in the triad induces a new quenching channel. This point will be addressed in the Discussion section. Finally the DADS associated with 2.2 ns as shown on Figure 10D presents a broad positive band below 570 nm, three negative peaks at 593, 651, and 735 nm, and a broad positive peak above 830 nm. All those features are consistent with the spectral characteristics of DPP* and BOD*, as shown in Figure 10C and summarized in Table 2, and the DADS is perfectly reproduced by the sum of the excited species differential spectra (as shown with the red curve): DADS(2.2 ns) ∝ D1 × εDPP ̃ * + D2 × εBOD ̃ *

DADS(0.5 ns) ∝ C1 × (εTPA ̃ • + εDPP ̃ •−) + C2 × εBOD ̃ * (7)

(8)

This is in very good agreement with the time-resolved fluorescence data, from which we assigned the 2.2 ns decay time to nonreactive excited state decay of both red-emitting moieties. The differential spectra used (ε̃TPA, ε̃TPA•+, ε̃DPP, ε̃DPP*, ε̃DPP•−, ε̃BOD, and ε̃BOD*) are reported in the SI, Figures S9 to S11.

As for the 52 ps DADS, we show the DADS resulting from a second control experiment (blue curve). The good agreement with this fit, in particular for the wavelengths below 750 nm, confirms the above scenario, namely a recombination of the previously formed CT state (with C1 = −B1; cf. Table 3) and a H



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Figure 11. Reaction scheme compiled from the quantitative analysis of the DADS. See text for detailed explanations. Percentages indicate the branching ratios for excited state decay paths of BOD* (rad. decay vs internal conversion) and DPP* (rad. decay vs CT).

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• Internal conversion from Sn to S1 and a subsequent vibrational and dielectric relaxation in S1 for both DPP and BOD, in 290 fs. • Equilibration of the excited state populations of BOD* and DPP* within 6.3 ps due to resonant energy transfer. • Excited state decay of DPP*, and formation of TPA+•− DPP−• due to hole transfer from DPP* to TPA within 52 (±2) ps, consistent with the 63 (±1) ps fluorescence quenching of DPP*. Note that this process is favored by the high dipole moment of the used solvent (THF). Indeed, the fluorescence quantum yield is significantly lower in THF than in low polarity toluene (Table 1). • Excited state decay of a fraction of BOD* on a 0.5 ns decay time, consistently observed in both TA and picosecond fluorescence, concomitant with recombination of the TPA+•−DPP−• CT state. The global analysis with five time constants treats both processes to occur with the same decay time, but the CT recombination time may differ by ±30% from the BOD* decay. The limited dynamic range of TA data restrains the resolution in terms of distinct decay times. • Excited state decay of DPP* and BOD* with a 2.2 ns decay time, consistently observed in both TA and picosecond fluorescence. The similarity with the decay times of the isolated DPP and BOD molecules suggest that this process is nonreactive. The multiple photoinduced reactions are sketched in Figure 11, highlighting the remarkable diversity of reactive and nonreactive processes that both BOD and DPP undergo. The 6.3 ps time scale for energy transfer between DPP and BOD warrants a more precise analysis. The Förster radius R0 is calculated via eq 9:

DISCUSSION The transient absorption data of the TPA-DPP-BOD triad in THF indicate the existence of five different lifetimes, the longer three of which match remarkably well with the fluorescence decay times obtained with the streak camera limited to 10 ps time resolution. Given the complexity of the three component triad and the structural heterogeneity hinted on by the fluorescence decays, analyzing the TA data with a target analysis appears to be a hopeless task, since the number of molecular states and species (compartments) is a priori not known, and a too large number of reaction schemes would be needed to be tested. We therefore chose to analyze the five DADS individually according to eq 2, as linear combinations of the known spectral components obtained separately on the relevant isolated BOD and DPP molecules by TA and spectroelectrochemistry experiments. A priori knowledge from the picosecond fluorescence decay of BOD* and DPP* was another invaluable input for the quantitative fits of the DADS. The imperfect agreement of the reconstituted DADS (fits) with the experimentally obtained ones has different origins. First, the DADS shapes result from a multiparameter fit, and even though the lifetimes differ by a factor of >4, the amplitudes carry a systematic error bar. Second, ΔA spectra depend on the overlap of pump and probe spots. Any spatial nonuniformity of the probe beam spectrum introduces variations of ΔA as hinted on by the control experiments. However, within these uncertainties, the overall agreement is good, since the main spectral features are reproduced, indicating that we did not leave out any photochemical species in the analyses of the four DADS. The so-obtained reaction scenario is therefore complete, but there are limitations, such as the exact timing of the CT state recombination (see below), and a determination of the branching ratios due to structural heterogeneities restricted to the slower processes (>10 ps). The photoreaction scheme of the triad is summarized as follows • Ultrafast excited state quenching of TPA*, most likely due to resonance energy transfer to BOD* and DPP*.

R0 =

6

9000· ln(10) ·κ 2·φD 4 ·Nav 128π 5·nsolv

∫ εA ·FD·λ 4 dλ

(9)

where κ2 is the transition dipole orientation factor, ϕD is the donor fluorescence quantum yield, nsolv the refractive index of the solvent, Nav is the Avogadro number, εA is the acceptor I



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extinction coefficient, FD is the donor fluorescence (normalized to unity integral), and λ is the wavelength. Assuming that the transition dipoles are along the symmetry axis of DPP and BOD and lie in the same plane, the structure of the triad indicates that κ2 = 2.5 and R = 1.6 nm are good approximations. We thus obtain R0 = 6.7 ± 0.5 nm for the DPP* → BOD energy transfer. Clearly the small spectral overlap is compensated by the large extinction coefficients of BOD. Notably, BOD* → DPP energy transfer is possible, too! Due to the smaller spectral overlap, the Förster radius is only 6.1 ± 0.5 nm for BOD* → DPP FRET. However, in both cases, R0 is much larger than the DPP-BOD distance R of approximately 1.6 nm. Then the time constant of FRET is given by eq 10, using eq 11 to calculate the efficiency of the energy transfer: τFRET = τD·(1 − φFRET)

2.2 ns radiative decay (cf. Table 3). From the DADS amplitude, one deduces that B3/(B3 + D1) ≃ 60% of DPP* form the CT state, and C2/(C2 + D2) ≃ 65% of molecules that have BOD in the excited state are quenched within 0.5 ns. From the 0.5 ns DADS and its fit with the excited state decay of BOD (Figure 10A, C) we can conclude that we are observing a process of internal conversion and not a triplet formation. The latter would not lead to a decrease of the ground state bleach spectrum. Since this internal conversion is observed in the triad only, and not in the reference compound, it is suggested that BOD adopts in the triad a configuration that modifies the vibrational coupling to the ground state, thus opening a new channel for internal conversion into S0, the exact nature of which remains to be determined. One may speculate that the phenyl group of BOD adopts a larger out-of-plane angle, which could lead to a higher nonradiative decay rate along this torsional mode. Computations of the triad conformations could help resolving this issue. The amplitudes of the 2.2 ns DADS indicate that the partition within the nonreative BOD*/DPP* mixture is D2/D1 = 60/40, while we concluded from the fluorescence DAS an equipartition. However, the faster FRET rates for DPP* → BOD obviously favors the population of BOD* with respect to DPP*. The triad was studied isolated (in solution), as such observations are only an indication that the charge and energy transfer are possible, but they are not necessarily relevant to what would happen in a blend with PCBM. Charge transfer toward PCBM in solar cells made of molecular blends or bulk heterojunctions occur on time scales generally faster than 50 ps, but are of course highly dependent on the molecular arrangement at the nanoscale. It is most likely that excited state quenching of both DPP* and BOD* occurs due to very efficient CT to PCBM well before the observed intramolecular (TPA+•, DPP−•) CT is being formed. However, the 6.3 ps net energy transfer is fast enough to compete with CT to the acceptor, thus validating the antenna effect of this triad. Further studies should be made in films with an acceptor to assess the efficiency of this antenna. Another improvement would be to replace or modify the DPP in order to blue shift its absorption spectra, which would better fill the absorption trough in the blue/near-UV and render the energy transfer unidirectional to BOD.

(10)

where τD is the fluorescence lifetime of the donor (2.2 ns for DPP) 1

φFRET = 1+

6

( ) R R0

(11)

The above values for R0 would entail FRET to occur on subpicosecond time scales, which is much shorter than the observed 6.3 ps. The slower net FRET rate is a result of the rate difference between forward and backward FRET and accounts also for imperfect relative orientations and structural heterogeneity that reduce κ2 and thus R0. The emerging picture is that the 6.3 ps lifetime is actually a time for establishing a dynamic equilibrium through exchange of energy between the excited state populations of BOD* and DPP*. Depending on the molecular conformation, the triad then forms the CT state in 52 ps involving DPP (Figure 9), or undergoes the 0.5 ns internal conversion of BOD. If neither of these nonradiative paths is available, the equilibrated (DPPBOD)* population relaxes within 2.2 ns due to the faster excited state lifetime of isolated DPP*. Since the reduction and oxidation potential determined of TPA and DPP do not change significantly when going from the TPA-DPP-TPA reference compound to the TPA-DPP-BOD triad, it is surprising that the (TPA•+, DPP•−) CT state is only observed in the triad. Following the Rehm−Weller equation,64 the CT formation process is exergonic for both compounds. However, according to the well-established Marcus formula,65 the rate of ET depends on the reorganization energy λ with both the interand intramolecular component. In the present situation ΔG ≪ λ, which is typically on the order of 1 eV for THF. We suspect small variations in the intramolecular contribution of λ to slow down the ET in TPA-DPP-TPA such that its rate becomes similar to or smaller than the radiative rate. Note that the fluorescence quantum yield of TPA-DPP-TPA in THF is only 0.32 (Table 1). An evaluation of the branching ratios is possible for the processes occurring after BOD/DPP excited state equilibration (Figure 11). Since this equilibration occurs roughly 1 order of magnitude faster than any quenching process, the fluorescence quantum yield of the triad does not depend on excitation wavelength (Table 1). The following branching ratio thus reflects the molecular conformations. Both the TA DADS and the fluorescence decay amplitude indicate that the reactive paths for DPP and BOD are more efficient than the nonreactive

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CONCLUSION We studied with several complementary spectroscopic setups the separated TPA, DPP, and BOD components of our triad, extracting the characteristic spectra of their ground, excited, or charged states. Picosecond fluorescence determines the time scales of excited state quenching for lifetimes >10 ps, and indicates the existence of minority nonreactive populations that decay radiatively. With these, we were able to analyze the five DADS obtained by pump−probe spectroscopy on the triad, and deduce a reaction scheme. Using the correct species spectra, we were able to reproduce the features of the DADS, and thus extract the information about which species were involved in each steps of the triad’s dynamics. The use of the full features of each species allowed us to disentangle the contributions of DPP and BOD, whose signatures would otherwise have been mixed. From the ground state TPA-DPPBOD, three states are created by the 320 nm excitation: TPA*DPP-BOD, TPA-DPP*-BOD, and TPA-DPP-BOD*. FRET establishes a dynamic equilibrium between DPP* and BOD* in 6.3 ps. Within 52 ps, a charge transfer state with the J



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TPA radical cation and DPP radical anion is formed, that recombines within 500 ps. When the triad is in the “reactive” TPA-DPP-BOD* excited state, internal conversion is observed on the same 0.5 ns time scale. A minority “nonreactive” fraction of the remaining TPA-DPP*-BOD decays radiatively in 2.2 ns, along with a nonreactive fraction of TPA-DPP-BOD*.

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ASSOCIATED CONTENT

S Supporting Information *

Experimental procedures, 1H,13C{1H},11B{1H} NMR traces, and characterization data for a selection of the new compounds can be found in SI. This materialis available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Dr. G. Ulrich, Dr. O. Crégut, and Dr. J. Léonard for their expertise and helpful comments, and B. Omiecienski for technical assistance with the spectroelectrochemical measurements. We thank the Région Alsace for a fellowship to E.H. and T.R. Financial support came from the Centre National de la Recherche Scientifique (CNRS), the CPER NanoMat and the FP7 Interreg Program Rhin-Solar. A.R. and S.L. acknowledge financial support within the framework of the Emmy-Noether program by the DFG.

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REFERENCES

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