Free energy dependence of the diffusion-limited quenching rate constants in photoinduced electron transfer processes

Abstract

Electron-transfer rate constants were determined by means of lifetime measurements for the fluorescence quenching of 9,10-dicyanoanthracene by aromatic amines and methoxybenzenes as electron donors, and for the quenching of the synthetic dyeseosin Y and phenosafranine by a series of p-benzoquinones as electron acceptors. All determinations were carried out in acetonitrile at 298 K. The quenching rate constants (k_q) in the region of –1.9 eV < ΔG_et < –0.2 eV do not decrease as predicted by Marcus theory, but they show a small increase with decreasing ΔG_et. Although this behaviour is in qualitative agreement with the current theories for reactive systems in the diffusion limit region, a closer analysis of the experimental data showed that several aspects of the dependence of the k_q on ΔG_et are not entirely explained, suggesting that new, refined theoretical models may be required.

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Introduction

Since the pioneering work of Rehm and Weller¹ it was generally accepted that the electron transfer (ET) fluorescence quenching rate constants (k_q) reached a diffusion-limited plateau in the exothermic region (ΔG_et < –0.2 eV) and decreased as the ET driving force becomes positive. However, the lack of an inverted region at large negative ΔG_et, as predicted by Marcus theory,^2,3 represented for many years an intriguing dilemma which understanding demanded significant experimental and theoretical efforts.^4–9

Several explanations have been given for the absence of the inverted region in fluid media¹⁰ and among them, the dependence of the ET rate constants (k_et) on the electron donor (D)/electron acceptor (A) separation distance (R) was particularly successful. According to this interpretation the plateau reached at large negative ΔG_et is related to the solvent reorganization energy being an increasing function of R (see, for instance, eqn (17) below). This notion is illustrated in Scheme 1.

Scheme 1 Simplified 2D model for intermolecular ET. The reaction between an electron donor molecule (grey sphere) and electron acceptors (black spheres) surrounded by solvent molecules (open spheres) can occur at variable distances.

ET reactions can produce diverse ionic species that differ in their degree of solvation such as contact (CRIP) and solvent separated (SSRIP) radical ion pairs. CRIP arise from neutral A/D precursors that have diffused to the contact distance (R_AD = R_A + R_D), while SSRIP can be formed (trough a solvent-mediated superexchange coupling mechanism) from a wide distribution of A/D pairs in which the reactants are at variable distances (R).¹¹ In Marcus theory, the largest rate for ET reaction is attained when ΔG_et + λ = 0. Hence, for an ET reaction falling in the inverted region (i.e., ΔG_et + λ < 0) there is a significant probability for the process to take place from the solvent separated A/D pairs. These species are characterized by a larger A/D separation and therefore, by a larger solvent reorganization energy. Another consequence of this interpretation is that the effective A/D separation distance (R_eff) at which the ET reaction takes place should increase with decreasing ΔG_et.

One of the most frequent systems used to investigate the kinetics of ET reactions at large negative ΔG_et was the quenching of anthracenecarbonitriles by electron donors. These systems were first studied by Eriksen and Foote¹² and more recently by Jacques and Allonas^13,14 and Niwa et al.^15,16 by means of stationary fluorescence measurements. In these studies, the experimental Stern–Volmer plots show a positive deviation at high quencher concentrations. From the analysis of the curvature of the plots the ET effective distance (R_eff), the mutual A/D diffusion coefficient (D) and the intrinsic rate constant for ET (k_et) can be extracted. These studies confirmed the apparent increase of the R_eff with decreasing ΔG_et.

On the other hand, direct measurement of the quenching rate constant k_q by means of time resolved fluorescence experiments is much less frequent, in particular, in the investigation of the dependence k_q on ΔG_et in the region of high ET exergonicity.^17–19

We report here a time resolved fluorescence study of the quenching of the singlet excited state of 1,9-dicyanoanthracene (DCA) by aromatic amines and methoxybenzenes, and the quenching of the synthetic dyeseosin Y (Eos) and phenosafranine (PS) by a series of p-benzoquinones. All experiments were performed in acetonitrile at 298 K. The quenchers were chosen in order that the driving force of the ET processes were in the region of –1.9 < ΔG_et < –0.2 eV. It was found that, in all cases, the k_q increase linearly with decreasing ΔG_et. It was also observed that each fluorophore/quenchers system follows a particular relationship with the ET driving force. The experimental data are analyzed using the current theories for diffusion-mediated ET reactions.

Experimental

1,9-Dicyanoanthracene (DCA), phenosafranine (PS) and eosin Y (Eos) were obtained from Aldrich and used without further purification. The electron donors N,N,N′,N′-tetramethyl-(p-phenylenediamine) (TMPD), N,N,N′,N′-tetramethylbenzidine (TMBZ), p-anisidine, p-toluidine, 1,2,4-trimethoxybenzene, 1,4-dimethoxybenzene, 1,2-dimethoxy benzene and anisole were all commercially available and were purified by standard procedures. The acceptors p-benzoquinone, chloro-p-benzoquinone, p-methylquinone, 2,6-dimethylbenzoquinone and duroquinone were obtained from Aldrich and were purified by sublimation. Acetonitrile (HPLC grade) was from Sintorgan.

Fluorescence lifetime measurements were carried out with the time correlated single photon counting technique using an Edinburgh Instruments OB-900 equipment. All measurements were carried out in air equilibrated acetonitrile solutions at 25 ± 0.1 °C.

Results

Table 1 collects some relevant electrochemical and photophysical parameters of the fluorophores involved in this study. The structure of the fluorophores is given in Chart 1 in the annexed Electronic Supporting Information, ESI.†

Table 1 Oxidation potential E_D/D+˙ (V vs.SCE), reduction potential E_A/A^–˙ (V vs.SCE), singlet excited state energy (E₀₀) and lifetimes of 1,9-dicyanoanthracene, phenosafranine and eosin Y, in acetonitrile

Fluorophore	E D/D^+˙/V	E A /A ^–˙/V	E 00/eV	τ o /ns^a
a The reported uncertainties correspond to one standard deviation calculated from five independent determinations. b Values obtained from reference12. c Ref. 20. d Ref. 21.
1,9-Dicyanoanthracene ^b	—	–0.98	2.89	12.71 ± 0.03
Phenosafranine ^c	+1.30	–0.67	2.34	3.52 ± 0.02
Eosin Y ^d	+0.80	—	2.31	3.99 ± 0.04

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In all cases, the emission of the fluorophores in air equilibrated acetonitrile solutions (at 298 K) decays monoexponentially. The experimental lifetimes (τ_o) are collected in Table 1. As expected, the emission decay times of the fluorophores decrease with increasing concentration of the quenchers. No deviation from a single exponential decay law could be detected even at the larger concentrations of the quenchers employed ([Q] ≤ 0.02 M). The χ² of the fitting were always in the range of 1.05–1.20; and applying bi-exponential or non-exponential (Föster-like) decay models to the analysis of the emission decays do not lead to significant improvements of χ² or residuals plots.

The bimolecular quenching rate constants (k_q) were obtained according to eqn (1):

where τ stands for the fluorescence decay time measured in the presence of the quencher. The plots according to eqn (1) were linear with correlation coefficients (R) better than 0.999 (even for τ_o/τ >7.0). The emission decay of DCA in the presence of TMPD 0.02 and three plots according to eqn (1) are shown in the ESI.† The quenching rate constants (k_q) measured for all the systems studied are collected in Tables 2–4. The values for the PS/D are given in Table 1 in the ESI.†

Table 2 Bimolecular rate constants (k_q) for the quenching of the singlet excited state of DCA by aromatic electron donors in acetonitrile at 298 K and exponential decay constant β calculated according eqn (11)

Quencher	E ox vs. SCE/V	ΔGet/eV	k q/10^10 M^–1 s^–1	β/nm^–1
TMPD	0.16	–1.81	2.57	8.5
TMBZ	0.32	–1.65	2.35	8.8
p-Anisidine	0.66	–1.31	2.06	9.2
p-Toluidine	0.78	–1.19	2.16	9.4
1,2,4-Trimethoxybenzene	1.12	–0.85	1.83	9.4
1,4-Dimethoxybenzene	1.30	–0.67	1.74	9.4
1,2-Dimethoxybenzene	1.47	–0.50	1.56	9.4
Anisole	1.75	–0.22	1.30	9.4

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Table 3 Bimolecular rate constants (k_q) for the singlet excited state quenching of eosin Y by electron acceptors in acetonitrile at 298 K and exponential decay constant β calculated according eqn (11)

Quencher	E red vs. SCE/V	ΔGet/eV	k q/10^10 M^–1 s^–1	β/nm^–1
Chloro-p-benzoquinone	–0.34	–1.11	2.37	6.0
p-Benzoquinone	–0.50	–0.95	1.99	6.0
Methyl-p-benzoquinone	–0.58	–0.87	1.79	6.0
2,6-Dimethylbenzoquinone	–0.67	–0.78	1.54	6.0
Duroquinone	–0.84	–0.61	1.23	6.0

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Table 4 Bimolecular rate constants (k_q) for the singlet excited state quenching of phenosafranine by aromatic electron acceptors in acetonitrile at 298 K and exponential decay constant β calculated according eqn (11)

Quencher	E red vs. SCE/V	ΔGet/eV	k q/10^10 M^–1 s^–1	β/nm^–1
Chloro-p-benzoquinone	–0.34	–1.0	1.33	7.2
p-Benzoquinone	–0.50	–0.68	1.08	7.2
Methyl-p-benzoquinone	–0.58	–0.44	0.98	7.2
2,6-Dimethylbenzoquinone	–0.67	–0.35	0.79	7.2
Duroquinone	–0.84	–0.18	0.64	7.2

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Assuming that the quenching processes studied involve a single type of ET reactions,²² the free Gibb energy changes for the ET processes (ΔG_et) can be calculated according to:¹

where w = z₁z₂e²/εR_AD represents the free energy gained by bringing the formed ions (with charges z₁ and z₂) from infinite to the contact distance R_AD in a medium of static dielectric constant ε. Taking a mean value of R_AD ∼0.6 nm and +1 and –1 for the charges of the radical ions (z₁z₂ = –1) a value of ∼ –0.064 eV is calculated for the Coulombic term in acetonitrile. For the systems PS/A and Eos/A, the Coulombic terms were calculated taking into account that the products z₁z₂ are –2 and +1, respectively. The redox potentials of the quenchers were taken from several sources.^23–25 The values of ΔG_et calculated according eqn (2) are also given in Tables 2–4.

When the experimental k_q in Table 2–4, Table 1-SI† and the Rehm and Weller data are represented together vs. ΔG_et they show an acceptable correlation within the usual scatter observed in these types of free energy correlation plots (see, Fig. 3-SI, ESI †). However, a closer inspection of our data (Fig. 1) shows that the k_q increase with decreasing ΔG_et and also, that each fluorophore/quenchers system follows its own relationship with the ET driving force.

Fig. 1 Bimolecular rate constants (k_q) for the quenching of DCA (□) and PS (○) by electron donors, and acceptor PS (●) and eosin Y (■) by electron acceptors as a function of ΔG_et. The values of PS/D were taken from ref. 25.

Discussion

Let us assume a quenching process that can occur at the encounter A/D separation distance, R_AD, exclusively. In such case, the theoretical limit for a diffusional controlled quenching rate constant is given by

Eqn (3) is the Smoluchovsky's result²⁶ for the long time limit solution of the diffusion equation

where ρ(R,t) = [Q]/[Q_o] stands for the density distribution of the quencher (A or D) about the fluorophore. However, for diffusional processes where A and D can react over a distance greater than the encounter separation, eqn (4) is modified to^8,27where k_et(R) represents the distance dependent ET rate constant. In the simplest formulation possible, this can be written aswhere β stands for the exponential decay constant and R ≥ R_AD. Solving simultaneously eqn (5) and (6), the rate constant for concurrent diffusion/ET reaction, k_q(t), is obtainedwhich in the long time limit (t > 0.1 ns) simplifies to:

Since in our experimental conditions (and resolution of our instrumental) the emission decays can be fitted to a single exponential model, the results can be analyzed according to eqn (8). The values of R_eff for the DCA/donor system calculated from the experimental k_q in Table 1 and assuming D ∼ 3.3 × 10^–9 m² s^–1 (vide infra) are in good agreement with the values estimated from steady-state fluorescence measurements reported by Jacques and Allonas¹³ and Niwa et al.^15,16

In turns, R_eff and k_q can be related to the parameters in eqn (6) by ²⁷

where γ is the Euler's constant (0.5772). In principle, eqn (9) could be used to rationalize the dependence of k_q on ΔG_et, provided that D and the dependence of β and k_o on ΔG_et were known. Considering that the quenchers chosen are similar in sizes and shapes, it seems reasonable to assume that D is nearly constant throughout each particular correlation in Fig. 1. Hence, the slopes of the plots must be related to the variation of β or k_o (or both). The exponential factor β in eqn (6) is given by the Gamow's factor ^28,29where m_e stands for the electronic mass and ΔG_eff represents the effective tunneling barrier. ΔG_eff is related to the adiabatic energy gap for the electron transfer (or hole transfer) from the reactants to the solvent molecules. Approximated values ofΔG_eff can be estimated from the ionization potentials and electronic affinities of the reactants and solvent,³⁰ or directly from their redox potentials. Recently, Kobori et al.³¹ proposed an alternative model for explaining the dependence of β on ΔG_eff. According to these authors, β is given bywhere d_ss is the center-to-center distance between the nearest neighbor solvent molecules in the condense medium, λ_ex is the total (solvent plus internal) reorganization energy of the superexchange ET process and ν_s represents the solute–solvent and/or the solvent–solvent electronic coupling which, by simplicity, were assumed to be the same. Taking a value of λ_ex = 1 eV mol^–1, Kobori et al. used eqn (11) to fit a series of β values measured for several A/D systems in fluid and frozen media using ν_s as an adjustable parameter. The best value of ν_s was found to be ∼850 cm^–1; which as pointed out by the same authors, appears to be excessively large considering that it represents the electronic coupling between randomly oriented solvent molecules. We used eqn (10) and (11) to estimate approximated values of β for the systems studied. Details of these calculations are given in the ESI.† The values of β calculated from Kobori's equation are collected in Tables 2–4. The values of β are ∼6–7 nm^–1 for the Eos and PS/A systems and ∼10 nm^–1 for DCA and PS/D systems. These values are in good agreement with previously reported values of β ≈ (3–10) nm^–1.³² The β calculated from Gamow's equation are just about twice those estimated from eqn (11).

As it may be concluded from the analysis above, the estimated β are constant within each particular system in Fig. 1, and therefore, β cannot be the main responsible for the observed dependence of k_q on ΔG_et. Consequently, everything points to k_o as being the factor controlling the variation of k_q. This result is not unexpected taking into consideration that k_o involves the ET Franck–Condon factor; which in turns, depends on ΔG_et and on the reorganization energy of the system. Assuming values of ∼1 eV and ∼0.4 eV for the solvent and A/D internal reorganization energies (vide infra), most of the A/D systems studied should be in the normal Marcus's region and therefore, an increase of k_o with decreasing ΔG_etis expected.

An alternative explanation for the dependence of k_q on ΔG_et may be given in terms of the model developed by Tachiya and Murata.⁹ Hug and Marciniak¹⁰ analyzed Rehm and Weller¹ data using this model and found that it satisfactorily explains the dependence of k_q on ΔG_et in the diffusion-controlled limit. According to Tachiya and Murata, the second-order quenching rate constant k_q(R) can be expressed as

where k_a(R) represents the second-order ET rate constant³³and k_d(R) is the diffusion-limited rate constant. In eqn (13), k_et(R)represents the distance dependent rate constant for the ET as given by the semi-classical version of the ET theories:whereis the distance dependent electronic coupling matrix element. The reorganization energy has two contributions:

In eqn (16), λ_s(R) accounts for the solvent reorganization energy and λ_ν is the internal reorganization energy of the reactants. λ_s(R) can be calculated according to

where ε_ois the permittivity of vacuum, and n²and ε represents the refraction index and the static dielectric constant of the solvent, respectively.

According to Tachiya and Murata,⁹ the diffusion-limited rate constant, k_d(R), is given by

The value of k_d(R)can be calculated numerically using the Simpson's rule algorithm, but such computation requires the knowledge of many parameters that characterize the A/D systems. Approximated values for R_A and R_D, required for the calculation of λ_s(R), were estimated following Zissimos et al.³⁴ The calculated values are: DCA 0.37 nm, PS 0.41 nm, Eos 0.48 nm, and 0.31 and 0.27 nm as the average values for the aromatic donors and quinine quenchers, respectively. Approximated values for λ_ν were obtained from molecular orbital calculations.^35,36 The estimated contributions of the donors (λ_νD) and acceptors (λ_νA) to λ_ν are summarized in Table 1-SI.† The value λ_ν for each fluorophore/quencher system was obtained by addition of the donor and acceptor contributions. The free Gibbs ET energy change ΔG_et depends on R as shown in eqn (2). The molecular radii were also used to estimate mutual A/D diffusion coefficients (D) from the Stokes–Einstein equation:

Taking η = 0.346 cp for the viscosity of the solvent at 298 K, values of ∼3.3 × 10^–9 m² s^–1 (DCA/D), 3.2 × 10^–9 m² s^–1 (PS/D), 3.5 × 10^–9 m² s^–1 (PS/A) and 3.3 × 10^–9 m² s^–1 (Eos/A), were calculated. The values estimated for the DCA/D systems are in good agreement with the average of the experimental D reported by Jacques and Allonas¹³ for the same systems.

The solid curve in Fig. 2 corresponds to the fitting of the experimental k_q measured for DCA/donors system to eqn (12), taking r_A = 0.37 nm, r_D = 0.31 nm, λ_ν = 0.37 eV, V_o = 100 cm^–1, β′ = 10 nm^–1 and D as an adjustable parameter. The value of D ∼ 4.0 × 10^–9 m² s^–1, which provided the best fit of the experimental data, is similar to that calculated for the DCA/donors system from the Stokes–Einstein equation. As it may be concluded from Fig. 2, although the theoretical curve predicts correctly the increase of k_q with decreasing ΔG_et, the fit is just fair. Similar results were found for the PS/D system (Fig. 4-SI).† However, for the systems Eos/A and PS/A we were not able to find a realistic set of parameters (V_o and D) to fit the experimental k_q.³⁷

Fig. 2 Experimental k_q for the quenching of DCA emission by electron donors (□) and eosin Y by electron acceptors (■). The solid line corresponds to the simulation of the second order diffusion-mediated electron transfer rate constant according to the model developed by Tachiya and Murata.⁹ The parameters used for the simulation are: R_A = 0.37 nm, R_D = 0.31 nm, λ_ν = 0.37 eV, V_o = 100 cm^–1, β′ = 10 nm^–1 and D = 4.0 × 10^–9 m² s^–1.

The main inconvenient found for fitting the experimental k_q to eqn (12) is related to the uncertainties on R_AD. Although, this can be considered just as a “technical” problem, it has clear mechanistic implications. For instance, an inspection of the values of R_eff (and D) reported for the DCA/donor systems by Jacques and Allonas¹³ and Niwa et al.^15,16 shows that the experimental R_eff measured for the less exothermic reactions approach to those expected for exciplex (CRIP) formation, i.e. ∼ 0.35–0.42 nm. These R_effvalues are consistent with the expected inter-planar aromatic-aromatic A/D separation of the systems studied and they are considerably smaller than the “contact” separation distance (R_AD) calculated from the reactant's volumes. Niwa et al.¹⁶ proposed that this behavior obeys to a change in the ET reaction mechanism occurring around ΔG_et ∼ –0.5 eV. In the strongly exothermic region (ΔG_et < –0.5 eV) the ET reaction produces SSRIPs, while the reaction goes via exciplex formation at ΔG_et > –0.5 eV. Similar conclusions were reached by Gould et al.³⁸ in the study of the quenching of tetracyanoanthracene and DCA by alkylbenzenes, although these authors proposed a branched mechanisms in which the relative importance of the SSRIP/exciplex reaction routes is a continuous function of ΔG_et. These mechanistic complications are not being taken into account by the models used here for interpreting the dependence of k_qon the ET driving force.⁹ In Tachiya and Murata models^9,11 the CRIP species are treated as weakly interacting radical ion pairs at a contact distance (assumed to be the sum of the reactants radii, typically 0.6 nm). Although these parameters may be appropriated for describing the A/D systems discussed by these authors,¹¹ they can hardly be used for characterizing exciplexes. In addition, it seems clear that alternative models, particularly those which consider the reactants and solvent molecules as discrete entities,³⁹ may have better chances to explain this discontinuity (or branch) in the ET mechanism.

Summary

ET quenching rate constants obtained by direct measurement of the decay of the excited fluorophore, produced results in line with the predictions of current theories for reactive systems in the diffusion limit region. The rate constants increase with decreasing ΔG_et in all the range explored. However, a detailed analysis showed that some aspects of the dependence of the rate constants on the driving force are not fully explained, suggesting that new, refined theoretical models may be required.

Acknowledgements

The authors thank to Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET-Argentina), Agencia Nacional de Promoción Científica (FONCYT-Argentina) and Secretaría de Ciencia y Técnica de la Universidad Nacional de Río Cuarto for the financial support.

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Footnote

† Electronic supplementary information (ESI) available: Chart 1, Fig. 1–4SI, and Tables 1–6SI. See DOI: 10.1039/b708797g

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