Photoinduced Energy-Transfer and Electron-Transfer Processes in

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J. Phys. Chem. C 2010, 114, 16716–16725

Photoinduced Energy-Transfer and Electron-Transfer Processes in Dye-Sensitized Solar Cells: TDDFT Insights for Triphenylamine Dyes Julien Preat*,† Unite´ de Chimie Physique The´orique et Structurale, Faculte´s UniVersitaires Notre-Dame de la Paix, rue de Bruxelles, 61, 5000 Namur, Belgium ReceiVed: June 1, 2010; ReVised Manuscript ReceiVed: August 23, 2010

We have conducted a theoretical investigation to model the mechanisms of photoinduced electron injection and energy transfer for a recent organic metal-free dye derived from the triphenylamine (2TPA-R) structure. The 2TPA-R system results from the fusion between two TPA moieties connected by a vinyl group, and the rhodanine-3-acetic acid is used as the electron acceptor group. In a first step, DFT and TDDFT approaches have been exploited to calculate the key parameters controlling the intramolecular charge transfer (ICT) injection and ET transfer rate constants in the classical Marcus formalism: (i) the electronic coupling; (ii) the reorganization energies; and (iii) the variation of the Gibbs energy. In a nice agreement with the experimental trends, the results have highlighted that (i) two excited states [EE(1), lower in energy, and EE(2), higher in energy] have been calculated at 2.78 and 3.33 eV; (ii) the energy transfer (ET) between these two excited states is in competition with the electron injection from the EE(2); (iii) when 2TPA-R is excited at 3.33 eV, the ET between the resulting relaxed excited states EE(2) and EE(1) directly takes place, and the probability of injection from EE(2) is weak; (iv) the ET remains governed by the Dexter mechanism. Indeed, the ratio between the Fo¨rster and Dexter rate constants (kF/kD) is evaluated at ∼10-4. Second, we propose structural modifications improving the electron injection efficiency of the TPA-based DSSCs, and we show that using the 1-CN,2-COOH-ethylene group as the acceptor unit combined with a functionalization of the TPA moieties by -OMe groups significantly improves the key parameters related to the electron injection. 1. Introduction The present energetic and environmental crisis has stimulated interest in the design of renewable energy sources, and solar photovoltaic devices are likely to be leading technologies in a promising “low-carbon level” future.1–7 In this framework, the photosensitization of semiconductors by molecular dyes has strongly changed the photovoltaic landscape. The dyes-sensitized solar cells (DSSCs) are made of a wide band gap semiconductor (typically TiO2) deposited on a translucid conducting substrate, an anchored molecular sensitizer, and a redox electrolyte (usually the I-/I3- couple).8–12 In the DSSC field, the Ru complex photosensitizers were the most popular category of dyes over the past 20 years as they show a solar power conversion efficiency (SPCE) of about 10% in average. Nevertheless, in spite of their important SPCE, the interest in Ru-derived sensitizers tends to fall off and this decrease in popularity can mainly be interpreted as a consequence of the very expensive production costs of such dyes, as ruthenium is not abundant in nature. Consequently, metal free sensitizers such as organic dyes and natural dyes have been investigated as alternative molecules for DSSC applications. The interest in purely organic DSSCs as substitutes for Ru complexes have increased in recent years due to crucial advantages among which a high molar extinction coefficient, simple and relatively inexpensive preparation processes, and even easier compliance with environmental or health issues.13 Moreover, several solidstate DSSCs based on organic dyes appear to have higher performances than ruthenium complexes, suggesting promising * E-mail: [email protected]. † Postdoctoral Researcher of the Belgian National Fund for Scientific Research.

commercial applications.14 Therefore, metal-free dyes like coumarin,15,16 merocyanine,17 indoline,18 xanthene,19 hemicyanine,20 hydroquinones,21,22 perylene, and fluorene23,24 have been tested as good potential candidates for the next DSCCs generation. The pattern for most of the organic sensitizers is a donor (D), a bridge (B, typically a π spacer), and an acceptor (A) moieties, which are usually combined following a D-π-A rodlike configuration (see Scheme SI.1) in order to improve the efficiency of the UV/vis (UV/visible) photoinduced intramolecular charge transfer (ICT). Generally, the critical factors governing the sensitization are as follows: (i) A highest occupied molecular orbital (HOMO) must fit the iodine/iodide redox potential, combined with a lowest unoccupied molecular orbital (LUMO) that has to be higher in energy than the conduction band edge of the semiconductor. (ii) The excited state’s (EEs) redox potential has to match the energy of the conduction band (CB) edge of the semiconductor (SC). (iii) The light harvesting ability of the dye must be large enough to reach a substantial photocurrent response. (iv) The conjugation across the donor and anchoring group has to be maximized, as it determines the large photoinduced intramolecular charge transfer amplitude. (v) Obviously the electronic coupling strength between dye’s LUMO and the semiconductor CB is a decisive factor for an efficient electron injection from the dye onto the semiconductor surface. In general, items leading to the low conversion efficiency of many organic dyes in DSSC are the formation of dye aggregates or the charge recombination between the CB electrons and the positively charged dye (or the electrolyte) which is at the source of the “dark” current of the DSSC.25–27 (vi) The bridging groups steer the light absorption regions of

10.1021/jp1050035  2010 American Chemical Society Published on Web 09/13/2010

TDDFT Insights for Triphenylamine Dyes the DSSCs21,25,28 and subsequently the scale of the electron injection from the excited state of the dyes to the semiconductor surface. To further design and develop more efficient metalfree dyes for DSSCs, appropriate DBA (donor-bridge-acceptor) systems are needed with properties finely tuned by applying the adequate structural modifications. Recently, it has been found that among other molecules triphenylamine (TPA)29 derivatives and the cyanoactetic acid moiety are patterns of choice as electron donor and electron acceptor/anchoring, respectively.25,30 Indeed, TPA is expected to greatly confine the cationic charge from the semiconductor surface therefore hampering the recombination. TPA also features a steric hindrance that can then prevent unfavorable dye aggregation at the semiconductor surface.30 In recent years, many studies have been devoted to the characterization of the mechanisms of long-range charge transfer processes taking places from D to A through a molecular bridge in a rod-like DBA assembly. Theoretical investigations, based on a variety of quantum chemical (QM) techniques, have provided insights into the possible pathways for intramolecular charge transfer and the resulting electron transfer. These QM techniques remain efficient and very attractive tools for the interpretation of the experimental data. The progress in CPU resources now allows the study, at correlated levels of approximation, the absorption spectra of large molecular species such as the DSSCs dyes.31,32 These studies have underlined the crucial role carried by the electronic coupling between the reactants and the products (VRP) in the description of the electron transfer reactions. For the electronic excited states calculations, one of the most popular approaches remains the time-dependent density functional theory (TDDFT) as it commonly provides results that qualitatively agree with experimental data at reasonable computational costs, especially when hybrid functionals are used.33–54 Though, it is known that “cyanine-like” structures such as triphenylamine derivatives show a large CTcharacter electronic excitation (and sometimes a multideterminental nature), and that makes them sometimes difficult to study with DFT.55,56 The ICT process plays a crucial role in achieving higher SPCE, and though it has been largely described, a thorough understanding of the more complicated energy transfer (ET) is far less converged and remains an intense topic of research. Accordingly, we herein report the theoretical study of new molecules derived from the several organic dyes recently synthetized by Hagfeldt’s group.57 In order to reveal the working principles of organic metal-free DSSCs, we focus on a novel dye (2TPA-R, see Scheme 1) that contains two TPA units connected by a vinyl group and rhodanine-3-acetic acid as the electron acceptor. This investigation aims at helping the design of efficient organic dyes in the future by using a computational procedure that delivers a qualitatively good descriptions of the excited states of D-π-A rod-like organic dyes.58–60 The present contribution is organized as follows: in section 2, we explain the ICT injection and ET mechanisms that occur for 2TPA-R. In section 3, we first define the set of parameters required to study the rate constant related to the ICT and ET mechanisms. Second, we describe the theoretical methodology used to evaluate these parameters. The calculated ICT and ET rate constants are compared to the corresponding experimental insights in section 4, and we assess the nature of the possible involved mechanisms. Finally in section 5, we investigate three structures derived from TPA-1R (see Scheme 2), providing better injection parameters for 2TPA-R.

J. Phys. Chem. C, Vol. 114, No. 39, 2010 16717 SCHEME 1: Sketch of the D1 and D2 Systems Included in 2TPA-R

SCHEME 2: Structures of 2TPA-R1 and 2TPA-R2

2. Description of the System: Involved Mechanisms We here detail the ET and ICT processes described by Hagfeldt’s group for 2TPA-R (see Scheme 1).57 This dye shows (2) two absorption bands located at ∼350 (λ(1) max) and ∼500 nm (λmax) that could be assigned to the two vinyl-conjugated moieties, D1 and D2 (see Scheme 1). In order to support this assumption, the Hagfeldt’s group has synthesized the 2TPA and TPA-R dyes, and in a recent contribution, the authors show that both 2TPA and TPA-R UV/vis spectra completely match with the 2TPA-R one; that is, the correspondence between the 2TPA-R absorption spectrum and the 2TPA and TPA-R band is perfect. Therefore, D1 and D2 moieties were selected as two different systems to study the ICT and ET processes in the 2TPA-R dye.57 Figure 1 provides the shape of the molecular orbitals (MOs) obtained for 2TPA-R grafted to the TiO2. Accordingly to the model recently proposed by Peng et al.,61 we consider as a first approximation that the dye is connected to one Ti atom, and a set of water molecules and hydroxyl ligands are added to ensure the neutrality of the whole complex. The binding model is in perfect agreement with the recent results obtained by Tian and co-workers who have demonstrated from infrared spectroscopy analysis in CH2Cl2 solution that, for a rhodanine-TPA-based compound (similar to 2TPA-R), the preferential binding mode is a bidentate chelating interaction between the -COO- terminal group of the dye and the SC Ti atom, as depicted in Figure 1.62 (1) The electronic excitation that occurs at λmax ) 532 nm corresponds to a HOMO-1 to LUMO transition which is

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Figure 1. Molecular orbitals involved in the two electronic excitations of 2TPA-R. These orbitals have been obtained at the B3LYP/6-31G(d,p) level of theory with a contour threshold of 0.05 |e|. A detailed view of the bidentate chelating binding mode between the dye and the SC surface.

followed by the electron injection into the semiconductor (1) conduction band to form (D1-D2)+. In such a case, the λmax can be considered as the electronic transition at the origin of the injection (λICT max). After the injection, a charge transfer occurs between the HOMO and the HOMO-1 to give the relaxed geometry of the cationic dye (which formally corresponds to D 1+) which can be further reduced by the redox electrolyte in (1) the the DSSC. Moreover, Figure 1 clearly shows that for λmax LUMO is localized in the A-TiO2 region and suggests that the first excited state has a predominant DBA+-SC- configuration, very similar to the state resulting from the electron injection. When 2TPA-R is excited at higher energy (355 nm), the resulting transition can be assigned as a HOMO to LUMO+1 transition, giving D1*. Then, two processes can be foreseen: 1. An electron injection from the LUMO+1 of 2TPA-R to (2) can be give a cation (2TPA-R+) and in such a context the λmax ICT also considered as a λmax. 2. An intramolecular ET process from D1 to D2, in which the electron is injected from the dye to the SC at LUMO level and a ICT process then occurs between HOMO and HOMO-1 levels to form (D1-D2)+. This intramolecular ET certainly leads to significant improvement of the sunlight to current conversion efficiency because the injection occurs at 532 and 355 nm, respectively. On the other hand, the photoinduced ICT process is effective for intramolecular charge separation and effectively decreases the “dark” reaction. All of the above evoked possible mechanisms are characterized by rate constants and their evaluation are of importance in order to draw up the review of the probable involved mechanisms of injection onto a titanium dioxide (TiO2) surface. 3. Methodology In this section we first define the set of parameters required to study the rate constant related to the ICT injection and the

ET mechanisms. Second, we describe the theoretical methodology used to evaluate these parameters. Key Parameter Evaluation. Intramolecular Charge Transfer: Electron Injection. As suggested by Figure SI.4, for a photoinduced ICT injection, the electron transfer process is initiated by photoexcitation of the grafted dye. This vertical electronic transition from the ground state (denoted as DBASC) in its equilibrium geometry (QS0) to the excited state DBA*SC is followed by a nuclear relaxation on the excited state potential curve, leading to an equilibrium energy generalized coordinate (QS1). Then, the injection from the relaxed DBA*SC (QS1 as reactant can also be noted Qr) to the charge-separated state DBA+-SC- (Qp, as the product of the injection) can take place, and the geometric relaxation of DBA* proceeds faster than the electron injection. In the classical description of the electronic transfer, the rate of the electron injection can be derived from the Marcus theory for an activationless process63–67

kinject )

(

π p χkBT 2

)

0.5

|VRP | 2e[-(∆Ginject

+ χ)2/4χkBT]

(1)

In eq 1, kinject is the rate constant (in s-1) of the electron injection form the dye excited state to the semiconductor CB, VRP is the coupling constant between the reagents and the products potential curves, kBT is the Boltzmann thermal energy, p is related to the Planck constant (h/2π), ∆Ginject is the free energy of injection, and χ is the reorganization energy of the system. Equation 1 is practical if one assumes that the injection is restricted to energy levels close to the conduction band edge and that the density of acceptor states in this energy range remains constant.63 Of course, the higher kinject, the best SPCE factor and most efficient the DSSC.

TDDFT Insights for Triphenylamine Dyes

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Since the electron injection is studied in solvent-phase, the inner reorganization energy includes a solute part (the dye) and a solvent part. The parameter χ in eq 1 is therefore expressed in two components63,68

formalism (GMH) allows to evaluate |VRP| for a photoinduced charge transfer64,65

|VRP | ) χ ) χdye + χsolvent )

χdye 1

+ 2

χdye 2

+ χsolvent

(2)

µRP |∆ERP |

(7)

√(µR - µP)2 + 4µRP2

where χdye corresponds to the energy required to accommodate the nuclear rearrangements occurring upon the injection when going from DBA*-SC (Qr) to DBA+-SC- (Qp). χdye can be estimated as the average value of the χ1dye and χ2dye associated to the two states represented by the two energy curves in Figure SI.5. Moreover, χ1dye and χ2dye can be defined to a very good approximation as

where µRP is the transition dipole moment, ∆ERP is the energy difference, and µR (µP) is the permanent dipole of DBA*-SC (DBA+-SC-). The transition dipole moment has been evaluated using the transition densities by means of the PCM-TDDFT framework. The transitions density parameter has been evaluated in the nuclear arrangement of the ith excited state (QSi, Figure SI.4).69 In eq 7, the energy difference that is easily obtained by the following:

DBA* DBA* χdye 1 ) E(Qp) - E(Qr)

DBA* ER ) E(Q + ESC r)

(3)

+

DBA EP ) E(Q + ESC p)

and +

DBA DBA χdye - E(Q 2 ) E(Qr) p)

+

(4)

(

χsolvent ) ∆q

)(

1 1 1 1 1 + 2RD 2RA R ε∞ εs

)

(5)

where RD, RA, R, ε∞, and εs are the radii of the donor and acceptor, the distance between their centers (i. e., between the 2TPA and the rhodanine geometric centers), and the optical frequency, and static relative dielectric constants of the solvent, respectively. ∆q is the amount of charge transferred (i.e., +1 for the injection of one electron from the dye to the semiconductor CB). As the MOs analysis suggests, for 2TPA-R, the donor is the TPA moiety (Scheme SI.1) whereas the rhodanine (1) and function is the acceptor unit (Scheme SI.1), for both λmax (2) λmax. The coupling constant in eq 1 is provided by

|VRP | ) 〈ΨR |H|ΨP〉

(6)

where ΨR and ΨP are respectively associated with the reagents and products states, whereas H is the operator describing all the interactions between the particles composing the investigated system. The value of VRP defines the adiabatic or non adiabatic character of the electron transfer, these two descriptions conveniently setting the limiting cases of a transition state formalism. We refer the reader to the seminal papers dealing with this topic.63,68 The use of the Generalized Mulliken-Hush

-

(9)

and +

DBA* corresponds to the relaxed excited state of DBA where E(Q r) DBA+ and E(Qp) corresponds to the cationic dye resulting from the electron injection. Note that the SC reorganization energy has been neglected; this can be justified by the large size of SC crystal lattice that remains unperturbed when the electron is SC SC SC) E(Q and E(Q injected in the CB, and we consider that E(Q p) r) p) SC ) E(Q . χsolvent describes the solvent reorganization energy and r) a standard estimate for the latter was obtained by Marcus by modeling reactants and products as spheres, and the solvent as a dielectric continuum. The reorganization energy is simply

(8)

-

+

DBA DBA* DBA ∆ERP ) E(Q - E(Q + ESC - ESC ) E(Q p) r) p) DBA* SC + ECB E(Q r)

(10)

SC is the reduction potential of the semiconductor where ECB conduction band. Though it is often difficult to accurately SC experimentally, because it is highly sensitive to determine ECB SC values of the conditions (e.g., the pH of the solution), the ECB SC ) -4.0 several materials have been reported, and we use ECB eV for TiO2.70 This experimental value refers to conditions where the semiconductor is in contact with aqueous redox DBA+ electrolytes of fixed pH 7.0.71,72 In eq 10, the difference E(Q p) DBA* - E(Q corresponds to the oxidation potential of the dye in r) dye* ) and when the entropic component can the excited state (EOX be neglected in the calculation of the injection energetic balance, ∆ERP corresponds to the injection free energy change (∆Ginject). Commonly, two empirical models can be used for the dye* 63,68 . The first implies that the electron evaluation of EOX injection occurs from the unrelaxed excited state. For this reaction path, the excited state oxidation potential can be dye and extracted from the redox potential of the ground state EOX the vertical transition energy (ICT transition)71

dye* dye EOX ) EOX - λmax

(11)

Note that this relation is valid only if the entropy change during the light absorption process can be neglected. For the second model, we assume that electron injection occurs after relaxation. Given this condition, the “relaxed” Edye* OX is expressed as71,73 dye* dye dye EOX-relax ) EOX - E0-0

(12)

dye where E0-0 is the 0-0 transition energy between the ground state and the excited state related to λmax. Though electron injection from unrelaxed excited states has been observed in

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TiO274 and SnO2,75,76 the relative contribution of the ultrafast injection path is not yet clear, and the experimental groups commonly assume that electron injection dominantly occurs after relaxation.71,73 Consistently, here we strive for the relaxed path of injection for computing the ∆Ginject of novel dyes, following eq 12.60 To calculate the 0-0 “absorption” line, we need both the S0 (singlet ground state) and the Si (ith singlet excited state) equilibrium geometries (see Figure SI.4). More precisely, the 0-0 transition energy is calculated as

E0-0 ) λmax - ESreorg i

(13)

ESreorg ) ESi(QS0) - ESi(QSi) i

(14)

ESi(QS0) ) ES0(QS0) + ∆ESi

(15)

where

and

QS0 and QSi are the equilibrium geometries for the S0 and Si states, respectively. ESi(QS0) and ESi(QSi) denote the internal energies for the Si state calculated at QS0 and QSi, respectively, whereas ∆ESi is the S0 f Si excitation energy. Intramolecular Energy Transfer. General OVerView. During the energy transfer between molecule units, the excited donor unit (D*) transfers its energy to the acceptor (A) which is in turn promoted into an excited state (Figure SI.6)

D* + A f D + A*

(16)

and as a starting point, eq 1 can be used to evaluate the rate constant of this energy transfer between the excited states of 2TPA-R. The methodology to compute the key parameters remains unchanged with respect to the previous section, and in the case of the energy transfer, the R and P states correspond to the first and second excited states of 2TPA-R. On the other hand, it is generally accepted that two limiting cases can be used to describe the intramolecular energy transfer reactions. Dexter and Fo¨rster Mechanism. On the one hand, the Fo¨rster mechanism describes a non radiative resonance excitation energy transfer occurring when D* can transfer its excitation energy to an acceptor over long distances (R > 15 Å).78,80 In such a context, the Coulomb term (βC) represents the interaction of the charge distributions that can be expanded into multipole (M) terms (dipole-dipole, dipole-quadrupole, ...). At small distance (R) between D and A, the dominant term remains limited to the dipole-dipole interactions

βC )

MDMA R3

(17)

The oscillating dipole of D* causes electrostatic forces acting on the electronic system of the acceptor and the resulting Fo¨rster rate constant is expressed at not too small R as

F kET )

2π|VDA | 2 J(ν) p

(18)

where VDA is the coulomb coupling constant between D (EE(2)) and A (EE(1)). 2

VDA )

1 µDA κ 4πεs R3

(19)

in which appear the transition dipolar moment related to the D (EE(2)) f A (EE(1)) conversion which can be easily obtained from the transition densities computed for the first excited state within the TDDFT framework, where J(ν) measures the overlap between the emission spectrum of D and absorption spectrum of A at a given frequency ν. κ takes into account the effect of the relative orientation of the two transition dipole moment vectors and it is common to assume a random orientational average value of (2/3)0.5, because this parameter is not easily extracted from most experiments.81 In such a context, the donor is D1 and the acceptor site is defined by D2 (see Scheme and Figure SI.6). On the other hand, the Dexter mechanism considers a non radiative electron exchange energy transfer that occurs when D* and A are close enough (R < 15 Å).77,78 The exchange term represents the interaction of the exchange charge distributions and the Dexter’s rate constant is given by a formalism similar to eq 18,

D kET

2π|VEx | 2 ) J(ν) p

(20)

where VEx is Dexter’s coupling constant. The evaluation of this exchange coupling factor are seen in work using parametrized Hamiltonians or hydrogen-like orbitals.65 These two last approaches can be considered as relatively crude approximations and in this contribution, the short-range coupling has been characterized by taking the difference between the GMH [eq 7] and coulomb (VDA) couplings, as suggested in the recent approach elaborated by Hsu.65 Computational Level. All calculations have been performed with the Gaussian 03 and Gaussian 09 packages,82,83 following a three-step procedure: (i) the optimization of the ground state geometry with DFT using a tight threshold that corresponds to rms (residual mean square) forces smaller than 10-5 a.u., (ii) the determination of the vertical electronic excitation energies, their corresponding oscillator strengths (and the related transition dipole moments), and (iii) the optimization of the excited states .53,84 by means of TDDFT and the evaluation of ESreorg i For the first and third step, we have used the popular threeparameter B3LYP functional,85 in which the exchange is a combination of Hartree-Fock exchange, Slater functional, and Becke’s generalized gradient approximation (GGA) correction,86 whereas the correlation part combines local and Lee-Yang-Parr (LYP) functionals.87 For the vertical excitation calculations as well as for f, we used CAM-B3LYP which is the Handy and co-workers’ long-range corrected version of B3LYP using the Coulomb-attenuating method.41,88 We have selected the 6-31G(d,p)89 basis set (BS) for the ground-state and excited-state optimizations and 6-311G+ (2d,2p)89 for the vertical TDDFT calculations, as these BS have been shown to return converged λmax for a series of TPA, while a smaller BS would give too short λmax (in nm).60 The evaluation of the oxidation potential was performed by using the restricted and unrestricted formalisms of the more refined CAM-B3LYP with 6-31G(d,p) and taking into account the solvent effects.

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TABLE 1: Key Parameters Obtained for the Electron Injection Processes from the First and Second Excited States of 2TPA-Ra dye E OX

unrelaxed

(1) λ max

dye E OX

exp.

theory

exp.

theory

exp.

theory

4.90

5.39 0.49

2.78

2.56 0.22

2.12

2.84 0.72

-1.88

*(1)

(1) E0-0

theory

MAE

(1) ∆G inject

theory

MAE

relaxed

*(1)

exp.

dye EOX-relax

(2) λ max

exp. -1.16

theory

exp.

theory

exp.

theory

2.31

2.30 0.01

2.59

3.09 0.50

-1.41

exp. -0.91 0.50

*(2)

(2) ∆G inject

theory

exp.

theory

exp.

theory

3.33

3.24 0.09

1.54

2.15 0.61

-2.46

0.72 (1) ∆Ginject-relax

dye E OX

*(2)

(2) E0-0

dye EOX-relax

exp. -1.85 0.61

(2) ∆Ginject-relax

theory

exp.

theory

exp.

theory

2.87

2.95 0.08

2.03

2.44 0.41

-1.97

exp. -1.56 0.41

a

(1) (2) (1) (2) λmax and λmax (in eV) provided by PCM(CH2Cl2)-TDCAM-B3LYP//6-311+G(2d,2p), E0-0 and E0-0 (in eV) that are obtained at the dye PCM(CH2Cl2)-B3LYP/6-31G(d,p) level. EOX (in eV) has been using the same framework but with CAM-B3LYP using the restricted and dye* unrestricted formalism of this hybrid, and the resulting EOX and ∆Ginject (in eV) calculated in the unrelaxed and relaxed framework using eqs 11 and 12, respectively. We also provide the corresponding deviations to experimental data (MAE in eV).57

Indeed, the iodine/iodide couple is used as regenerator of the DSSC, implying that the solar cells work in solvent phase. This is why the oxidation potentials as well as the UV/vis experimental data for triphenylamine-based dyes are reported in solution and the polarizable continuum model (PCM)36,90,91 has been used for evaluating bulk solvent effects on both the cationic and neutral dyes’ geometry and the UV/vis spectrum. In PCM, one usually divides the problem into a solute part (the dye) lying inside a cavity and a solvent part. By solving the Poisson’s equation at the interface, PCM gives a valid approximation of solvent effects. For the vertical excitation energies, we have selected the so-called nonequilibrium PCM solutions36 and according to the experimental set up, we selected dichloromethane (CH2Cl2). All of the theoretical λmax reported in the following correspond to the singlet excited state with a huge dipole-allowed transition from the ground state. 4. Results In this Section, we first present the excited states’ optical properties obtained for EE(1) and EE(2). Second, we compare the theoretical to the experimental key parameters that have been used to estimate the rate constant related to the injection. The ICT dye dye* , EOX , EOX , and ∆Ginject have also been compared to the λmax corresponding experimental insights in order to validate the methodology. Then, we present the calculated ICT and ET rate constants and we assess the nature of the possible involved mechanisms. Excited State Optical Properties. The excited state transitions computed at the TDCAM-B3LYP level on the basis of EE(1) can be classified according to the occupied and unoccupied MOs combinations depicted in the Supporting Information. We find among them two excitations of interest: (i) the first one corresponds to a HOMO f LUMO transition in which the electron is transferred from the donor to the acceptor (D+BA-). Indeed, as shown in SI.7, the LUMO(EE(1)) is essentially localized at the A unit, and this confirms that this state has a predominant DBA+-SC- configuration, i.e., very close to the state configuration obtained after the electron injection (P state). The transition electric dipole moment between EE(1) and the P state values 14.62 D (D) and is related to the ICT electron injection from EE(1) of 2TPA-R; (ii) the second excitation of interest is related to a HOMO f LUMO+1 transition, and from the MOS topolgy analysis (Figure SI.7), it is clear that the final state associated to this excitation is similar to EE(2). The TDDFT provides a transition dielectric dipole moment of 11.39 D, and can be related to the EE(2) f EE(1) ET.

Second, the same procedure can be performed on the basis of EE(2) optimized geometry and the excitation of interest is a HOMO f LUMO transition showing a strong D+ BAcharacter with a transition electric dipole moment that values 2.32 D. This value is in complete agreement with the MOs (2) , however, the LUMO+1 analysis which shows that, for λmax remains localized on D1, and consequently the transition dipole moment for an electron injection from EE(2) is weak. These values of µRP are used in the next section to evaluate (i) the ICT electron injection rates from EE(1) and EE(2) and (ii) the EE(2) f EE(1) ET rate constant. ICT Electron Injection. Table 1 lists the experimental and dye dye* , EOX ) obtained for the theoretical key parameters (λmax, EOX first and second electronic excitation of 2TPA-R and that have been used to estimate the free energy change related to the electron injection (∆Ginject.), using the relaxed and unrelaxed formalisms. These results show that (i) the procedure underestimates the oxidation potential; (ii) for the unrelaxed scheme, TDDFT provides a better description of second excited state with an experiment-theory absolute deviation of 0.09 vs 0.22 eV for the first absorption peak. This results in a relatively better dye*(2) (2) , and consequently in a good ∆Ginject. ; evaluation of the EOX (iii) the TDCAM-B3LYP oscillator strengths are also in a good qualitative agreement with the molar absorption coefficients of 2TPA-R: f (1) and f (2) values of 1.8761 and 0.9998, respectively, whereas experiment provides ε(1) and ε(2) values of 39 000 and 35 000 M-1 cm-1, respectively;57 (iv) the TDDFT results are much more accurate for the relaxed scheme, and we notice that the first excited state energy is well-reproduced by theory with a mean average error (MAE) of 0.01 eV. Note that the error on dye * and ∆Ginject-relax is less important for the second excited EOX state; this can be explained by a compensation of errors, i.e., (2) (MAE of 0.08 eV) compenthe slight underestimation of E0-0 sates the 0.49 eV error in the oxidation potential evaluation; dye deviation could be related to the activation of (v) this EOX bending modes that induce distortion of the 2TPA-R structure: the lack of such “twisted” geometries in our model possibly explains the difficulty to fully describe the electrochemical properties of these derivatives. Moreover, it is also well-known that for extended conjugated organic systems, the DFT framework tends to overestimate the delocalization and therefore underestimate the ionization potential as well as the excitation energies; (vi) we here followed the relaxed path of injection dye dye* , EOX , for computing the ∆Ginject and while the evaluated EOX and ∆Ginject-relax values present a fairly good accuracy ((0.40 to 0.50 eV), the computed shifts are in agreement with the dye* or experimental data, e. g. the measured variation of EOX

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TABLE 2: χdye (in eV), χCH2Cl2 (in eV), VRP (in eV), ∆Ginject-relax (in eV), and kinject (in s-1) Obtained for the Electron Injection Processes from the First and Second Excited States (EE(1) and EE(2)) of 2TPA-R at 298 K eq 1 EE(i) i)1 i)2

χdye

a

b

χCH2Cl2

0.35 0.20

0.09 0.09

c V RP

∆Ginject-relax

(i) k inject

0.56 0.24

-1.41 -1.97

6.84 × 10 1.14 × 10-26

eq 21 (1) (i) k inject /kinject

6

(i) k inject

(1) (i) k inject /kinject

4.73 × 10 3.3 × 1015

15

1.00 ∼1033

1.00 1.43

a DBA* E(Qp) is obtained by performing a SCF calculation on the optimized cation geometry using the TDDFT gradient at the PCM-B3LYP/ DBA+ is evaluated via a PCM-UB3LYP/6-31G(d,p) SCF calculation on the optimized geometry of the ith excited 6-31G(d,p) level. Similarly, E(Q r) state. b RD ) 4.31 Å, RA ) 3.63 Å, R ) 14.00 Å, ε∞ ) 2.02, and εs ) 8.93.82 c µR (µP) corresponds to the first and second excited states (ground states cationic form) dipole moment and values 8.56 (EE(1)) and 11.55 (EE(2)) (30.34) D, respectively.

∆Ginject-relax between the first and second EE (-0.65 eV) is well reproduced by theory (-0.56 eV). Table 2 compares the electron injection rate constants obtained for the two excited states of 2TPA-R using the Marcus formalism defined in eq 1. The results show that the injection rate from the first excited state is on the order of 10-8 s. For the second excited state, eq 1 provides a very low injection rate (the time of injection is evaluated at 1026 s). Note that these values have been obtained by considering that the systems have time enough to relax after the electronic excitation, and this assumption is justified by the rather slow transfer rates in comparison to the typical vibrational relaxation rate.57 However, while the Marcus theory gives a good fit at low potentials, the rate constant is underestimated at high potentials, and as one enters the inverted region, the predicted rate rapidly decreases instead of slowly increasing.63 Undoubtedly eq 1 underestimates (2) . Indeed, the the rate constant of the electron injection and kinject probable time scale for an injected electron to enter into the semiconductor CB from the dye excited state is of the order of pico to femtoseconds.57 Moreover, from a more general point of view, eq 1 deviations can be attributed to (i) the large uncertainty concerning the conformational space of the systems; (ii) the error related to the different key parameters combined to the high numerical sensitivity of the exponential function; and (iii) for EE(2), µP(µDBA+-TiO2) has been evaluated for the cationic dye (DBA+) and its has therefore probably been overestimated by our DFT procedure. Therefore, a more refined relationship which deals with a square root function of the form

kinject )

( ) π p kBT 2

0.5

|VRP |[-∆Ginject - (EC - EF - χ)]0.5

(21)

where EC - EF is the difference between the energy of the bottom of the conduction band and the energy of the Fermi level for TiO2, and this difference is 1.6 eV.63 This formula derives from the Helmholtz potential function which delivers, at high potential, a almost linear dependence of rate constant on free energy of injection. This behavior is due to the SC density of states function and the date may be therefore fitted by a square root function.92 By using eq 21, one calculates rate constant which is more realistic for an injection from the first and second excited states: 4.73 × 1015 for the first EE and 3.3 × 1015 for the second EE which corresponds to very close injection time scale of 2.1 × 10-16 and 3.0 × 10-16 s, respectively. As shown by the results, the electron injection from EE(2) is characterized by an electronic coupling weaker than that computed for the injection from the excited state of lowest

TABLE 3: χdye (in eV), χCH2Cl2 (in eV), VRP (in eV), ∆GET (in eV), and kET (in s-1) Obtained for Energy Transfer Process between the Excited States of 2TPA-R at 298 K χdye

χCH2Cl2a

V RPb

∆GET

kET

0.33

0.01

0.27

-0.56

5.2×1014

a RD ) RD1 ) 5.48 Å, RA ) RD2 ) 5.00 Å, R ) 6.02 Å, ε∞ ) 2.02, and εs ) 8.93.82 b µP (µR) corresponds to the first (second) excited state dipole moment and values 11.55 (8.56) D. The transition electric dipole has been evaluated at 11.39 D.

energy. Clearly, for 2TPA-R, the evolution of ∆ERP from EE(1) to EE(2) cannot be considered as the key parameter controlling the magnitude of VRP. clearly, the evolution of the coupling term between these two states is mainly driven by the changes in µRP. This conclusion is completely supported by the MOs (2) however, the LUMO+1 analysis which shows that, for λmax remains localized on D1, and consequently coupling factor (1) is related to an ICT injection from EE(2) is weak. Note that VRP close to |∆ERP|/2 (∼0.15 eV difference), confirming an “adiabatic” character of the ICT injection from EE(1). Moreover, as underlined in the introduction, for 2TPA-R, the injection from the second excited state is in competition with the energy transfer process from this state to EE(1), and we present in the next subsection a theoretical investigation of the ET mechanism that we compare to the ICT. Energy Transfer Mechanism. General Marcus Approach. Table 3 provides the ET key parameters and the resulting rate constant. Though this ET presents a reorganization energy similar to those obtained for EE(1) and EE(2), the EE(2) f EE(1) energy trasfer shows a very weak ∆Ginject (-0.56 eV) and it results a small coupling factor, which is very close to |∆ERP|/2 (0.28 eV) and confirms the “adiabatic” character of the energy transfer. This “adiabatic” character of the ET is supported by the fact that EE(1) and EE(2) are rather similar (8.56 vs 11.55 D). By combining the parameters of Table 3 with eq 1, we calculate a energy transfer time scale of 10-15 s (1.92 × 10-15 s) that is 1 order of magnitude slower than the ICT injection process. Moreover, as depicted in Figure 1, the LUMO+1 of 2TPA-R is essentially centered on the D1 part of the dye and remains isolated from the -COOH anchoring group and consequently, the excited electronic cloud is far from being injected into the semiconductor conducting band via the carboxyl group. We would like to underline that, if eq 21 delivers a more realistic evaluation of the injection rate constants, the square root function probably tends to overestimate these rate constants. On the other hand, if the model based upon the Marcus theory provide a quite realistic kET value, we cannot ensure that eq 1 does not underestimate the EE(2) f EE(1) rate constant. Therefore, we can reasonably conclude that when 2TPA-R is excited at 355 nm (HOMO f LUMO+1 transition), the ET between the resulting relaxed excited states EE(2) and EE(1)

TDDFT Insights for Triphenylamine Dyes directly takes place, and the probability of injection from the high energetic EE(2) is weak. This interpretation is in perfect agreement with the experimental trends published by Tian and Hagfeldt, who have shown that for 2TPA-R the same oxidized state of the dye has been obtained after the electron injection from the excited state of the dye to the TiO2 CB, independent of the excitation wavelength.57 Moreover, we would like to underline that, though eq 1 is indeed weak at correctly predicting the rate constant of injection at high potential, it is certainly appropriate for the description of an electron transfer from the LUMO+1 to the LUMO in 2TPA-R.32,63–67 Dexter and Fo¨rster Formalisms. It is common to describe ET processes by using two limiting theories: (i) the Dexter mechanism based upon a nonradiative electron exchange ET and (ii) the Fo¨rster mechanism dealing with a non radiative resonance excitation ET based upon dipolar interactions. We propose to evaluate the relative weight of these two components for the 2TPA-R EE(2) f EE(1) transfer. Table SI.1 provides the parameters that have been introduced into eqs 18 and 20, and the resulting ratio between the Fo¨rster and Dexter rate constants (kF/kD) is evaluated at 2.61 × 10-4. This result lets us conclude that, as expected at short D-A distances, the energy transfer is governed by the Dexter mechanism. Since we can suspect that eq 1 underestimates the ET rate constant, we have examined the possibility to evaluate kET within the Dexter formalism by using eq 17. The evaluation of the overlap function and its integration is detailed in the Supporting Informations. Therefore, we have to underline that the value of J(ν) amounts 5.20 × 1023 for 1 mol of dye (i.e., 8.67 × 10-1 for one molecule) and the resulting kD is estimated at 5.86 × 1014 and corresponds to a injection time scale of 1.7 × 10-15 s. Obviously, this formalism is in better agreement with the reality and delivers a ET rate constant in the same order of magnitude than the one obtained within the Marcus framework. 5. Optimal Structures In the present section, we propose structural modifications improving the electron injection efficiency in TPA-based DSSCs. We impose that the dyes possess at least a terminal -COOH group on the acceptor side, as the carboxylic group is necessary to anchor the dye to the semiconductor surface.30 We then consider three criteria: (i) a high exergonic free energy of injection ∆Ginject in TiO2. Indeed, the larger -∆Ginject, the faster the electron injection from the valence excited state. Moreover, it is also well-established that the more exergonic the injection, slower the recombination reaction;63 (ii) the oxidation potential of the dyes must be more positive than the I-/I3- redox couple, ensuring that there is enough driving force for a fast and efficient regeneration of the dye cation radical; (iii) the light harvesting efficiency (LHE) of the dye has to be as large as possible to maximize the photocurrent response. In a previous investigation, we have shown that using the 1-CN,2-COOH-ethylene group as the acceptor unit combined with a functionalization of the TPA moieties by -OMe and -CN groups significantly improves the key parameters related to the electron injection.60 The transposition of these conclusions to 2TPA-R leads to 2TPA-R1 and 2TPA-R2 (Scheme 2) for which the ground and excited state parameters are provided in Table 4. Of course, for these two new compounds, we have checked that they indeed have a oxidation potential higher than the I-/I3- redox potentials (4.8 eV ( 0.1 eV).57 From 4, one can enumerate the following trends: (i) the LHE factor remains similar for both the three compounds, and we calculate TDCAM-B3LYP f values of 0.9867, 0.9874, and 0.9881 for

J. Phys. Chem. C, Vol. 114, No. 39, 2010 16723 dye dye* TABLE 4: E OX , E0-0, E OX-relax , ∆Ginject-relax and the LHE Factor Obtained for the Electron Injection Processes for 2TPA-R, -R1, and -R2a dye dye* compounds E OX E0-0 E OX-relax ∆Ginject-relax (LHE)/(LHE2TPA-R)b

2TPA-R 2TPA-R1 2TPA-R2

4.90 2.31 4.93 2.81 5.06 2.92

2.59 2.12 2.14

-1.41 -1.88 -1.86

1.00 1.00 1.00

a Energies are expressed in eV. b The light harvesting efficiency factor (LHE) of a dye is related to its oscillator strength and is equal to 1-10-f. This factor has to be as high as possible to maximize the photocurrent response of the cell.

Figure 2. HOMO (down) and LUMO (up) involved in the λICT of 2TPA-R1. These orbitals have been obtained at the B3LYP/6-31G(d,p) level of theory with a threshold of 0.05 |e|.

2TPA-R, -R1, and -R2, respectively; (ii) for both 2TPA-R1 and -R2, we show an improvement of the exergonic character of the injection that can be related to the huge increase of the excitation energy combined to an almost unchanged oxidation potential. It therefore results a decrease of the excited state oxidation potential and a higher - ∆Ginject-relax value (1.88 and 1.86 eV, for 2TPA-R1 and -R2, respectively); (iii) the -CN groups do not affect the Edye* OX-relax of 2TPA-R1 and are therefore useless; (iv) certainly, the -OMe functionalization of the TPA moiety combined to the cyanoactetic acid moiety as acceptor leads to a remarkable improvement of the thermodynamic properties related to the injection. To get further insights into the molecular structures and electronic distributions of these organic dyes, we have performed a molecular orbital analysis (MOA) at the B3LYP/6-31G(d,p) level of theory. The electron distributions for the HOMO and LUMO of 2TPA-R1 linked to the semiconductor are depicted in Figure 2. It is clearly shown that the HOMO is essentially localized on the central nitrogen atom whereas the LUMO is located on both the anchoring group and the SC through the π-bridge. The MOA also reveals that the cyanoacrylic acid group is essentially coplanar to the phenyl acceptor group. It is also important to highlight that, in view of the frontier orbitals topology of 2TPA-R1, the HOMO f LUMO excitation induced by light irradiation could move the electron distribution from the TPA moiety to the anchoring group, and should therefore favor the injection into the TiO2 CB. Furthermore, the LUMO of 2TPA-R (see Figure 2) is mainly centered on the rhodanine, especially on the carbonyl and thiocarbonyl groups, leaving the LUMO isolated from the -COOH anchoring group. Conse-

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quently in 2TPA-R, electrons are far from being efficiently injected into the TiO2 conducting band via the carboxyl group.

Dexter and Fo¨rtser rate constants ratio for the ET. This material is available free of charge via the Internet at http://pubs.acs.org.

6. Conclusions

References and Notes

We have managed not only to gain insights into the geometrical and electronic structures of triphenylamine (2TPAR) organic dyes but also to bring out the adequate structural modifications optimizing the properties of the TPA-based DSSCs. In particular, DFT and TDDFT approaches have been exploited to calculate the key parameters controlling the intramolecular charge transfer (ICT) injection and ET transfer rate constants in the classical Marcus theory. In complete agreement with the experimental trends, we have shown that: (i) two excited states (EE(1) and EE(2)) have been calculated at 2.78 and 3.33 eV, respectively; (ii) the energy transfer (ET) between these two states is in competition with the electron injection from the EE of higher energy (EE(2)); (iii) we have clearly underlined that the standard Marcus model is certainly unsuitable to delivers realistic rate constants for electron injections at high potential; (iv) on the other hand, the use of a square root model helps to calculate valuable injection time scales; (v) when 2TPA-R is excited at 3.33 eV, the ET between the resulting relaxed excited states EE(2) and EE(1) directly takes place, and the probability of injection from the high energetic EE(2) is weak; and (vi) this ET is mainly governed by the Dexter mechanism. We also aimed at improving the electron injection process. Consequently, we have proposed structural modifications that lead to an improvement of the electron injection efficiency of the 2TPA-R DSSCs. We predict that by using the 1-CN,2COOH-ethylene group as the acceptor unit, combined to a functionalization of the TPA moieties by -OMe and -CN groups should significantly improve the key parameters related to the electron injection. Indisputably, the future of organic solar cells relies on their economic potential that depends upon several critical factors like the efficiency, manufacturing costs and sustainability, weight, scalability and lifetime. At this point, two ways of conditioning techniques can be foreseen, solvent or solventfree processing, and the fundamental-level understanding of the electron excitation and injection processes allows a smart optimization strategy.

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Acknowledgment. The author thanks the Belgian National Fund for Scientific Research (FRS-FNRS) for their respective positions. All calculations have been performed on the Interuniversity Scientific Computing Facility (ISCF), installed at the Faculte´s Universitaires Notre-Dame de la Paix (Namur, Belgium), for which the authors gratefully acknowledge the financial support of the FNRS-FRFC and the “Loterie Nationale” for the convention number 2.4578.02 and of the FUNDP. The author also thank Dr. Eric Perpete and Dr. Denis Jacquemin for their help and exciting discussions on the topic. Supporting Information Available: Geometries and frequencies of the neutral and cationic ground state of 2TPA-R, -R1, and -R2 and the geometries of the 2TPA-R excited state EE(1) and EE(2); procedure used for the evaluation of the overlap function and the related J(ν) value appearing in eqs 18-20 for the ET evaluation following the Dexter and Fo¨rster formalisms; (iii) several graphics and figures have been included in order to make easier the comprehension of the discussion in this manuscript; (iv) MOA on the basis of the optimized EE(1) geometry; and (v) table including the calculation details of the

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