1 Introduction

In a previous study [1], we reported that the slow first-order thermal Z to E isomerization of azobenzene through the inversion mechanism with a non-polar transition state was hardly influenced by the solvent properties, while the rate constant and the frequency factors of the Arrhenius plots were significantly reduced in ionic liquids for the isomerization of 4-dimethylamino-4′-nitro-azobenzene compared with the reactions in molecular solvents with similar dielectric properties. As this reaction proceeds through the rotation mechanism with a polar transition state, the observed unusually small frequency factors of 104–107 s−1 were attributed to the slow rearrangement of the cationic (1-R-3-methylimidazolium) and anionic (bis(trifluoromethylsulfonyl)amide) components of the ILs to form highly ordered solvation shells around the charge centers of the reactant in the transition state, and such motions of the cationic and anionic components of ILs was accompanied by the concerted movements of the counter ions.

To examine the behavior of ILs around polarized/charged species in more detail, we attempted to examine second-order electron transfer/electron self-exchange reactions of various redox couples in the same ILs, since theories for electron transfer reactions have been well-established for analyzing the solvent reorganization during the activation process [2,3,4,5,6,7,8,9,10]. Second-order electron transfer reactions between metal complexes may also provide further information about the solvent properties from a different view point.

However, in the preliminary attempts it was found that most metal complexes are not stable in ILs: almost all the inert metal complexes even with bidentate/tridentate ligands decomposed rather rapidly in ILs, probably because of the abundant anionic (cationic) components of ILs (such phenomena strongly indicate that ILs dissociate in the vicinity of charged metal complexes). Other factors that prevented us from making such experiments in ILs are related to the methods for investigations: rapid mixing technique such as the stopped-flow method cannot be used because of the large viscosity of ILs and the use of the NMR line broadening technique was limited to the direct observation of metal centers because of many intense signals of the nuclei on ILs. Therefore, reported investigations of homogeneous redox reactions in ILs depend mostly on the EPR method, the laser flash method and pulse radiolysis [8, 11,12,13]. On the other hand, various heterogeneous reactions in ILs on the surface of modified electrodes have been reported [14,15,16]. However, the detailed theoretical analyses of the heterogeneous rate constants are hardly possible, since the dimensions of the electrochemical electron transfer rate constant are cm·s−1 and direct comparisons with the homogeneous rate constant is almost impossible [3, 4]. Redox couples so far investigated as homogeneous reactions in ILs have been limited to small organic molecules/counter cationic radicals such as TCNE/TCNE+· [9,10,11,12,13, 17, 18]. The electron transfer/self-exchange reactions for small organic molecules are usually very rapid because of the very small activation barriers for the inner-sphere reorganization. For some reactions, electron transfer rate constants involving the photo-excited states were reported to be faster than the estimated diffusion limit in ILs [11,12,13], although the intrinsic electron transfer rate constants observed in ILs were certainly slower than those observed in molecular solvents [19, 20]. A thorough investigation of the homogeneous electron self-exchange reaction of the TCNE/TCNE+· couple in various ILs was reported by Grampp and co-workers, and their detailed analyses revealed that the observed self-exchange rate constants did not exhibit the Pekar factor dependence when the refractive indices and dielectric constants for bulk ILs were employed for the analyses [17].

In this study, we found the [Zn(TPP)]+·/0 couple and the caged [Co(sep)]3+/2+ couple [21] (TPP = 5,10,15,20-tetraphenyl porphyrin and sep = sepulchrate = 1,3,6,8,10,13,16,19-octaazabicyclo[6.6.6]eicosane) are indefinitely stable in most ILs, and investigated the redox reaction between the triplet excited state of [Zn(TPP)], [Zn(TPP)]*, and [Co(sep)]3+ and the back electron transfer reaction from [Co(sep)]2+ to the π-cation radical [Zn(TPP)]+· using the laser flash technique. The contribution of the viscosity of ILs to the rate constants was isolated using the Kramers theory, and the contribution of the outer-sphere reorganization energy to the total activation Gibbs (free) energy for the self-exchange processes of [Co(sep)]3+/2+ and [Zn(TPP)]+·/* couples was estimated on the basis of the Ratner–Levin cross relation for the cross reactions [21,22,23] and of the Marcus–Sutin model for the self-exchange reactions [2]. The differences between the experimentally isolated and theoretically calculated outer-sphere contributions to the activation Gibbs energies are compared and discussed in relation to the modification of the solvent structures and solvent properties around the charged encounter complexes.

2 Experimental Section

2.1 General

1-Chlorobutane (> 98.0%), 1-chloropentane (> 99.0%), and 1-chlorohexane (> 95.0%) were obtained from Tokyo Kasei (EP grade) and used without further purification. 1-Methylimidazole (Tokyo Kasei EP grade, > 99.0%) was distilled under reduced pressure before use. Activated charcoal and active alumina was obtained from Nakarai and Merck, respectively. Lithium bis(trifluoromethanesulfonyl)imide (LiNTf2) from Fluorochem (99%) was dried at 80–120 °C for 19 h before use.

Acetonitrile obtained from Wako Pure Chemicals Inc. (99.5%) was treated with dried molecular sieves 4A for several days followed by distillation under an argon atmosphere. This treatment reduced the content of water to less than 1 mmol·kg−1 [24]. Tetrabutylammonium tetrafluoroborate (TBABF4) from Tokyo Kasei (> 98.0%) and tetrabutylammonium perchlorate (TBAP) from Wako (> 98.0%) were recrystallized twice from ethyl acetate (99.5%)–hexane (96%) mixtures and dried under reduced pressure. All reagents and solutions were handled under an argon atmosphere.

2.2 Syntheses

All ionic liquids, 1-R-3-methylimidazolium chloride 1-R-3-methylimidazolium bis(trifluoromethanesulfonyl)imide (R = butyl, pentyl, and hexyl) were synthesized by the reported method [1, 25, 26] and dried in a vacuum oven at 60 °C for 48 h. The purity of the synthesized ILs were satisfactory as determined by the NMR analyses and the amount of water in the ionic liquids was less than 5 mmol·kg−1 for all of the ionic liquids used in this study [1].

2.2.1 BMIMNTf2

1H-NMR(chloroform-d1): δ = 8.74 ppm (s, 1H; NCHN), 7.32 ppm (m, 2H; NCHCHN), 4.17 ppm (t, J = 7.6 Hz, 2H;NCH2), 3.94 ppm (s, 3H; CH3N), 1.85 ppm (q, J = 7.5 Hz, 2H; NCH2CH2), 1.36 ppm (sext, J = 7.4 Hz, 2H; N(CH2)2CH2), 0.96 ppm (t, J = 7.6 Hz, 3H;CH3CH2). Elemental analysis calcd. (%) for C10H15N3O4S2F6: C 28.64, N 10.02, H 3.61; found: C 28.63, N 9.72, H 3.65.

2.2.2 PMIMNTf2

1H-NMR (chloroform-d1): δ = 8.76 ppm (s, 1H; NCHN), 7.29 ppm (m, 2H; NCHCHN), 4.16 ppm (t, J = 7.6 Hz, 2H; NCH2), 3.94 ppm (s, 3H; CH3N), 1.87 ppm (q, J = 7.6 Hz, 2H; NCH2CH2), 1.34 ppm (m, 4H; CH3(CH2)2), 0.90 ppm (t, J = 6.8 Hz, 3H; CH3CH2). Elemental analysis calcd. (%) for C11H17N3O4S2F6: C 30.49, N 9.70, H 3.95; found: C 30.45, N 9.38, H 3.99.

2.2.3 HMIMNTf2

1H-NMR (chloroform-d1): δ = 8.79 pm (s, 1H; NCHN), 7.29 ppm (m, 2H; NCHCHN), 4.17 ppm (t, J = 7.6 Hz, 2H; NCH2), 3.95 ppm (s, 3H; CH3N), 1.86 ppm (q, J = 7.3 Hz, 2H; NCH2CH2), 1.33 ppm (m, 6H; CH3(CH2)3), 0.88 ppm (t, J = 7.0 Hz, 3H; CH3CH2). Eelemental analysis calcd. (%) for C12H19N3O4S2F6: C 32.21, N 9.39, H 4.28; found: C 32.41, N 9.18, H 4.39.

2.2.4 [Co(sep)](NTf2)3

[Co(sep)]Cl3 (Strem, 95%) was recrystallized from water and treated with AgNTf2 (> 98%). White precipitates of AgCl were filtered using diatomaceous earth (from Wako) and the product crystallized. The orange crystal was recrystallized from water and dried in vacuo. Anal. (calcd.) for C, 17.8 (18.2); H, 2.81 (2.55); N, 13.2 (13.0). It has been known that results of the elemental analyses of the compound containing C–F bonds tend to report lower values for carbon and higher value for nitrogen for metal complexes because of incomplete combustion [27]. Purity of this compound was confirmed also by the UV–VIS absorption spectra [21].

2.3 Measurements

Cyclic voltammograms were recorded using a BAS 100BW. A glassy carbon electrode (1 or 3 mm·ϕ) and a Pt wire (0.5 mm·ϕ) were used as the working electrode and the counter electrode, respectively. A Ag/AgNO3 reference electrode containing 10 mmol·kg−1 AgNO3 and 0.1 mol·kg−1 TBAP in acetonitrile was used for all measurements. UV–VIS–NIR absorption spectra were recorded using a JASCO V-570 spectrophotometer. A JEOL JNM-400 NMR was used to record the NMR signals. Time-resolved absorption measurements were carried out with a Nd-YAG laser (Minilite, Continuum) equipped with a second (532 nm) harmonic generator. The energy and duration of the laser pulse were 12 mJ per pulse and 3–5 ns, respectively. The transient absorption spectra were measured using a photodiode array detector in the multi-channel spectrophotometer system, and absorbance–time traces were acquired by a photomultiplier detector with a single-wavelength monochromator system (TSP-1000, Unisoku).

Kinetic data and transient absorption spectra were stored and processed using a Unisoku TSP-1000 spectrometer. Refractive indices of the ionic liquids were measured using an Atago 3850 refractometer (PAL-RT).

3 Results and Discussion

3.1 Redox Potentials for the [Co(sep)]3+/2+ and [Zn(TPP)]+·/0 Couples in Ionic Liquids

Cyclic voltammograms of [Co(sep)]3+ and [Zn(TPP)] were examined in acetonitrile, BMIM, PMIM and HMIM (ionic liquids used in this study are abbreviated as BMIM, PMIM and HMIM for 1-R-3-methylimidazolium bis(trifluoromethylsulfonyl)amide with R = butyl, pentyl, and hexyl, respectively, since the counter anion is common, bis(trifluoromethylsulfonyl)amide, \( {\text{NTf}}_{2}^{ - } \)). Results for the [Co(sep)]3+/2+ couples are summarized in Table 1. The redox potential for the ferricinium/ferrocene couple in the same solvent was set to zero volt.

Table 1 Results of electrochemical measurements for the [Co(sep)]2+/3+ couple
Full size table

As shown in Table 1 (and also Table 2), the redox potential for the highly charged [Co(sep)]3+/2+ couple is very different in acetonitrile and in the ILs, while the redox potential is only 6–11 mV more positive with the added LiNTf2 salt in ILs, compared with the measured value without LiNTf2. Somewhat higher values of E0 in ILs with LiNTf2 indicate that the double layer structure and the solvation environment in the bulk were altered by the addition of this salt, especially by the presence of Li+ ion with its high charge density. The effect was largest in acetonitrile: when the TBAP (tetrabutylammonium perchlorate) was replaced by LiNTf2, E0 for the [Co(sep)]3+/2+ couple was shifted by + 55 mV. In contrast to this drastic shift of E0 in acetonitrile, the much smaller shift of E0 in ILs with the addition of LiNTf2 indicates that significant portions of the positive and negative components that constitute the ILs are ionized (dissociated) in the vicinity of the electrodes and the added small amount of lithium salt did not significantly alter the double layer structure. Israelachvili and co-workers reported that less than 0.1% of ILs are dissociated in the pure ILs while ILs in the double layer are significantly ionized, from the results of surface force measurements with thermodynamic arguments [28]. In addition, the voltammogram of the [Zn(TPP)]+·/0 couple is reversible under the experimental conditions described later in this section. Therefore, the electrochemical electron transfer rate constant is much larger for this couple than that for the [Co(sep)]3+/2+ couple (the signals for the [Co(sep)]3+/2+ couple are quasi-reversible and the peak to peak separation increased with increasing scan rate): the rate constant for the homogeneous electron self-exchange reaction for the [Co(sep)]3+/2+ couple is only 5.1 kg·mol−1·s−1, while that for the [Zn(TPP)]+·/0 couple was estimated as 8.39 × 109 kg·mol−1·s−1 in acetonitrile from the result of this study (see Table 4 below). It is also known that the heterogeneous electron exchange reaction at electrodes is a mirror image of the corresponding homogeneous self-exchange reaction: \( \Delta G_{\text{heterogeneous}}^{*} = \frac{{\Delta G_{\text{homogeneous}}^{*} }}{2} \) [3, 4], which indicates that the solvation environment around the metal complexes is identical to that in the vicinity of the electrodes, including the diffuse double layer, where electron transfer reactions take place. Therefore, ILs in the vicinity of charged metal complexes may also be dissociated: cationic components of solvent ILs besiege the cationic metal complexes surrounded by the anionic components of the ILs, while the ILs in the bulk are unaffected. Such a model explains the smooth diffusion of complex ions in ILs.

Table 2 Observed redox potentials for [Co(sep)]3+/2+, [Zn(TPP)]+/0, and [Zn(TPP)]*/+
Full size table

Separation between the anodic and cathodic peaks for the [Co(sep)]3+/2+ couples became wider when supporting electrolyte (LiNTf2) was added to the IL solutions, indicating that the reversibility of the signals was reduced with the presence of excess LiNTf2. As such changes in the signals are not related to the electrochemical electron transfer rate (conditions for measurements, including the scan rate, were not changed), it is certain that the diffusion-controlled condition necessary for the observation of reversible/quasi-reversible CV signals is spoiled by the existence of added salt: it seems the added salt acted chiefly as a ‘structure breaker’. In addition, peak currents of the CV signals hardly changed with the addition of supporting electrolyte (see results in BMIM at scan rate = 20 mV·s−1 where the concentrations of [Co(sep)]3+ are identical): the migration current is not significant in ILs, even without supporting electrolyte (which is similar to the condition for aqueous solutions with > 0.1 mol·dm−3 of supporting electrolyte) [29]. This result is also consistent with the observations by Israelachvili and co-workers [28].

At this stage, we may be able to conclude that: (1) solutions of ionic liquids, BMIM, PMIM and HMIM, can be regarded as moderate electrolyte solutions in the vicinity of the electrodes and probably around the charged metal complexes; (2) introduction of Li+ to these solutions slightly alters the double layer structure at the electrodes; (3) Li+ ions in ILs also act as a structure breaker because of its very high charge density, since the observed wider peak to peak separation indicates the altered diffusion of ILs by the presence of Li+; and (4) on the other hand, small concentrations of charged metal complexes (< 1 mmol·kg−1), [Co(sep)]3+/2+ and [Zn(TPP)]+·, did not alter the double layer structure as well as the solvent structure.

Summarized in Tables 1 and 2 are the redox potentials for [Co(sep)]3+/2+ and [Zn(TPP)]+·/0 couples measured in acetonitrile and in ILs without supporting electrolyte. The redox potential between the triplet excited state and the π-cation radical [Zn(TPP)]+ was obtained by subtracting Eo([Zn(TPP)]+·/0) value (= potential difference between the ground state and π-cation radical) from the energy of the triplet excited state (1.59 eV; a phosphorescence maximum was observed at ca. 778 nm) which is 0.47 eV lower in energy than the singlet excited state (2.06 eV; fluorescence maximum 601 nm): it is known that fluorescence and phosphorescence maxima are largely independent of the solvent [30,31,32,33,34,35].

Differences in the observed redox potentials in two different solvents may roughly be dependent on the solvation energies in the two solvents and therefore it is given by the following equations: Eq. 1 corresponds to the classical Born-type equation, and Eq. 2 to the MSA (Mean Spherical Approximation) equation with the radius of the solvent being a0 [6, 36,37,38,39,40].

$$ F\left( {E_{a} {-} E_{b} } \right) = - \frac{{N_{\text{A}} e^{2} }}{{8\pi \varepsilon_{0} }}\left( {\frac{{(n - 1)_{{}}^{2} }}{{r_{B} }} - \frac{{n_{{}}^{2} }}{{r_{A} }}} \right)\left( {\frac{1}{{\varepsilon_{a} }} - \frac{1}{{\varepsilon_{b} }}} \right) $$
(1)
$$ F\left( {E_{a} {-} E_{b} } \right) = - \frac{{N_{\text{A}} e^{2} }}{{8\pi \varepsilon_{0} }}\left( {\frac{{z_{B}^{2} }}{{r_{B} }} - \frac{{z_{A}^{2} }}{{r_{A} }}} \right)\left[ {\left( {1 - \frac{1}{{\varepsilon_{b} }}} \right)\frac{1}{{1 + \delta_{2} \left( {\varpi = 0} \right)}} - \left( {1 - \frac{1}{{\varepsilon_{a} }}} \right)\frac{1}{{1 + \delta_{1} \left( {\varpi = 0} \right)}}} \right] $$
(2)

where \( \delta = \frac{{3a_{0} }}{{r[108^{{\frac{1}{3}}} \varepsilon^{{\frac{1}{6}}} - 2<Para>]}}. \)

When the ionic radii of both [Co(sep)]3+ and [Co(sep)]2+ are set to r (this assumption is valid since [Co(sep)]3+ and [Co(sep)]2+ are caged metal species) [41], these equations are given by

$$ F\left( {E_{a} {-} E_{b} } \right) = \Delta G_{b}^{\text{solv}} - \Delta G_{a}^{\text{solv}} = - \frac{{N_{A} e^{2} }}{{8\pi \varepsilon_{0} }}\left( {\frac{{(n - 1)^{2} - n^{2} }}{r}} \right)\left( {\frac{1}{{\varepsilon_{a} }} - \frac{1}{{\varepsilon_{b} }}} \right) $$
(1′)
$$ F\left( {E_{a} {-} E_{b} } \right) = - \frac{{N_{A} e^{2} }}{{8\pi \varepsilon_{0} }}\left( {\frac{{(n - 1)^{2} - n^{2} }}{r}} \right)\left( {\left( {1 - \frac{1}{{\varepsilon_{b} }}} \right)\frac{1}{{1 + \delta_{2} \left( {\varpi = 0} \right)}} - \left( {1 - \frac{1}{{\varepsilon_{a} }}} \right)\frac{1}{{1 + \delta_{1} \left( {\varpi = 0} \right)}}} \right) $$
(2′)

where ε is the relative dielectric constant and the subscript a and b refer to the two different solvents, and z1, and z2 are the formal charges of the metal complexes, respectively. By comparing the measured redox potentials for the [Co(sep)]3+/2+ couple in ILs and the redox potential measured in acetonitrile, apparent values of the relative dielectric constant for each IL were calculated. The estimated value of the apparent dielectric constant by the Born-type equation is ca. 26 for all ILs. Estimated radii of ILs by the MSA equation (Eq. 2′) are a0= 222 pm when ε is 17.5 [1] (a0= 265 pm when ε is 26). Values of a0 are small (a0= 195 pm) for PMIM (ε = 13.0), and a0= 186 pm for HMIM (ε = 12.7) [42, 43]. Although it is known that MSA estimates rather small values for a0 [6], these values estimated by Eq. 2′ may indicate that ILs in the vicinity of charged solutes behaves as the dissociated component ions.

On the other hand, the CV signals are reversible for the [Zn(TPP)]+·/0 couple, indicating that electron transfer at the electrode is very rapid. The wider peak to peak separation observed for the [Co(sep)]3+/2+ couple, under the same experimental conditions, indicates that the electron transfer process is slower for the [Co(sep)]3+/2+ couple; it is known that the self-exchange rate constant for the [Co(sep)]3+/2+ couple is much smaller than that for the [Zn(TPP)]+·/0 couple [42]. The redox potential for the [Zn(TPP)]+·/0 couple was not altered by the addition of supporting electrolyte, LiNTf2. It seems that the double layer structure was not altered by addition of LiNTf2, since the redox potential is at the positive side. The solvent environment around this redox pair seems not to be affected significantly because of the very low charge density of [Zn(TPP)]+· species. Application of Eqs. 1 and 2 to the results for the [Zn(TPP)]+·/0 couple was not successful with the same reasons. It seems that the low charge density with long conjugate system of [Zn(TPP)]+· does not induce significant dissociation of the surrounding ILs.

As the addition of supporting electrolytes to ILs significantly modified the solvent properties, kinetic experiments were carried out without using a supporting electrolyte.

3.2 Flash Photolysis Experiments

The Marcus–Sutin theory [2] predicts contributions of inner- and outer-sphere reorganization energies for the self-exchange process, and the rate constant kex is given by Eq. 3.

$$ k_{\text{ex}} = K_{\text{OS}} \kappa_{\text{el}} \nu_{\text{n}} \kappa_{\text{n}} $$
(3)

where \( \kappa_{\text{el}} \), \( \nu_{\text{n}} \), and \( \kappa_{\text{n}} \) are the electronic transmission coefficient, the nuclear vibrational frequency given generally by kBT/h, and the nuclear factor given by exp(− ∆G*/RT), respectively [2]. ∆G* includes the sum of the inner- and outer-sphere contributions to the activation Gibbs energy in the case of electron self-exchange reaction: when the energy to form the encountered complex (precursor complex) is included, ∆G* includes \( \Delta G_{\text{Fuoss}}^{*} = \, - RT\,{ \ln }\,K_{\text{OS}} \).

The reactions of photo-activated [Zn(TPP)]0 in molecular solvents have been well established [44,45,46,47] and the photo-processes are shown in Scheme 1. It is known that the lifetime of the singlet photo-excitation product of [Zn(TPP)] is ca. 3 ns [33] and it rapidly relaxes to the triplet state by the inter-system crossing. The triplet state [Zn(TPP)]* readily reacts with the oxidizing reagents (metal complexes such as [Co(sep)]3+) to form a π-cation radical, [Zn(TPP)]+·, followed by the back reaction with reducing reagents (such as [Co(sep)]2+) to the ground state [Zn(TPP)]0.

Scheme 1
scheme 1

Scheme of the overall photo reaction examined in this study

Full size image

The lifetime of the triplet excited state is long (44 μs in toluene and 230 μs in methanol) [48] and the relatively slow electron transfer to [Co(sep)]3+ is expected to be observed by monitoring the decrease in the absorption in the ca. 460 nm region that corresponds to the absorption maximum of the triplet excited state or by monitoring the increase in the absorption at 405 nm region for the absorption maximum of the π-cation radical. The overall reaction is described as follows:

$$ \left[ {{\text{Zn}}\left( {{\text{TPP}}} \right)} \right]\xrightarrow{{h\nu }}\left[ {{\text{Zn}}\left( {{\text{TPP}}} \right)} \right]^{*} \left( {{\text{singlet}}} \right) $$
$$ \left[ {{\text{Zn}}\left( {\text{TPP}} \right)} \right]^{*} \left( {\text{singlet}} \right) \to \left[ {{\text{Zn}}\left( {\text{TPP}} \right)} \right]^{*} \left( {\text{triplet}} \right) \,{\text{intersystem crossing }}\left( {\sim 3{\text{ ns}}} \right) $$
$$ \left[ {{\text{Zn}}\left( {{\text{TPP}}} \right)} \right]^{*} \left( {{\text{triplet}}} \right){\mkern 1mu} + {\mkern 1mu} \left[ {{\text{Co}}\left( {{\text{sep}}} \right)} \right]^{{3 + }} \xrightarrow{{k_{1} }}\left[ {{\text{Zn}}\left( {{\text{TPP}}} \right)} \right]^{{ + \bullet }} \left( {\pi {\text{-radical}}} \right){\mkern 1mu} + {\mkern 1mu} \left[ {{\text{Co}}\left( {{\text{sep}}} \right)} \right]^{{2 + }} $$
(4)
$$ \left[ {{\text{Zn}}\left( {{\text{TPP}}} \right)} \right]^{{ + \cdot}} \left( {\pi {\text{-radical}}} \right){\mkern 1mu} + {\mkern 1mu} \left[ {{\text{Co}}\left( {{\text{sep}}} \right)} \right]^{{2 + }} \xrightarrow{{k_{2} }}\left[ {{\text{Zn}}\left( {{\text{TPP}}} \right)} \right]{\mkern 1mu} + {\mkern 1mu} \left[ {{\text{Co}}\left( {{\text{sep}}} \right)} \right]^{{3 + }} $$
(5)

Typical transient absorption spectra observed in the range of 350–650 nm region are shown in Figs. 1 and 2, both of which were observed at 1 and 250 μs after 532 nm pulse of 5 ns duration. The triplet excited state [Zn(TPP)]* was observed after 1 μs of the flash at ca. 460 nm in Fig. 1, which decreased by the slow reaction with [Co(sep)]3+. As the reaction of the triplet excited state with [Co(sep)]3+ produces the π-cation radical, [Zn(TPP)]+·, the absorption at ca 405 nm region increased after 250 μs.

Fig. 1
figure 1

Typical transient absorption spectrum observed in the range of 350–650 nm in BMIM, after 1 μs of laser irradiation. [Zn(TPP)] = 9.2 × 10−6 mol·kg−1, [Co(sep)3+] = 4.0 × 10−4 mol·kg−1, and T = 298 K. Differential spectrum in the rgion from 415 to 430 nm was unreliable and therefore deleted because of the very strong Soret band

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Fig. 2
figure 2

Typical transient absorption spectrum observed in the range of 350–650 nm in BMIM, after 250 μs of laser irradiation. [Zn(TPP)] = 9.2 × 10−6 mol·kg−1, [Co(sep)3+] = 4.0 × 10−4 mol·kg−1, and T = 298 K. Differential spectrum in the region from 415 to 430 nm was unreliable and therefore deleted because of the very strong Soret band

Full size image

Typical kinetic traces observed at 405 and 460 nm are shown in Figs. 3 and 4. The kinetic trace at 460 nm shows that the triplet excited state of [Zn(TPP)]* immediately produced after irradiation was consumed by the reaction with [Co(sep)]3+. The kinetic trace observed at 405 nm clearly shows the increase of the absorption by [Zn(TPP)]+· at short times, followed by the decrease of the absorption due to the reaction with [Co(sep)]2+. The increase in the absorption at short times corresponds to the creation of the π-cation radical by the reaction of the triplet state with [Co(sep)]3+, while it is consumed somewhat more slowly by the reaction with [Co(sep)]2+. The concentration of [Co(sep)]3+ is sufficiently greater than that of [Zn(TPP)], indicating that the pseudo-first-order conditions can be applied to the analysis. All observed kinetic traces were analyzed by double-exponential functions to isolate rate constants for the two consecutive processes and/or to isolate the natural deactivation process of the π-cation radical (see footnote 1). Some of the data in BMIM wereFootnote 1 analyzed by combining two data sets observed at 405 and 460 nm, and the rate constants were examined to check the consistency with the independent analyses using data observed at two different wavelengths. In Figs. 5 and 6, the observed first-order rate constants, k1 and k2, thus obtained in BMIM are plotted against the concentration of [Co(sep)]3+. The rate constant k1 depends linearly on the concentration of [Co(sep)]3+. The intercept of the linear line of Fig. 5 corresponds to the decay of the triplet excited state of [Zn(TPP)]* to the ground state by intersystem crossing. The second-order rate constant for reaction (4) was obtained from the slope of the plot in Fig. 5. Note that the linear dependence of k1 on the concentration of [Co(sep)]3+ indicates the activity coefficients of ionic species in solutions were hardly changed although the ionic strength of the solution was not adjusted: it seems that charged species such as [Co(sep)]3+ are associated with anions in solution and the charge on this species was effectively cancelled.

Fig. 3
figure 3

Typical kinetic trace observed at λ = 405 nm in BMIM at 298 K. [Zn(TPP)] = 4.3 × 10−6 mol·kg−1, [Co(sep)3+] = 1.0 × 10−4 mol·kg−1

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Fig. 4
figure 4

Typical kinetic trace observed at λ = 460 nm in BMIM at 298 K: [Zn(TPP)] = 4.3 × 10−6 mol·kg−1, [Co(sep)3+] = 1.0 × 10−4 mol·kg−1

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Fig. 5
figure 5

Dependence of k1 on [Co(sep)3+] at 298 K in BMIM. [Zn(TPP)] = 4.3 × 10−6 – 1.30 × 10−5 mol·kg−1 and [Zn(TPP)] ≪ [Co(sep)3+]. The slope of the plot corresponds to the second-order rate constant for reaction (4), and the intercept to the decay of the triplet excited state, 3ZnTPP*, via the intersystem crossing to S0

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Fig. 6
figure 6

Dependence of k2 on the concentration of [Co(sep)]3+ at 298 K in BMIM: [Zn(TPP)] = 4.3 × 10−6 – 1.30 × 10−5 mol·kg−1 and [Zn(TPP)] ≪ [Co(sep)3+]

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Time dependence of the absorption change for reaction (5) was expected to be second-order, because the concentration of [Co(sep)]2+ is identical to that of π-cation radical. However, the observed rate constants could not be described by a simple second-order reaction: it was close to a first-order process and the plot of this first-order rate constants against the concentration of [Co(sep)]3+ is quasi-linear, as shown in Fig. 6, followed by the other first-order reaction. The trace was, as a whole, better described by using a double exponential function.

There are two possibilities to explain the observed kinetic results for reaction (5). One is given by a catalytic reaction involving a trace amount of contaminant, and the other relates to the relative energies of the [Zn(TPP)]+· and [Co(sep)]2+energy levels.

3.2.1 Case 1

A small amount of a reactive contaminant, X, catalyzes the reaction as described in reactions (6) and (7).

$$ \left[ {{\text{Zn}}\left( {{\text{TPP}}} \right)} \right]^{{ + \cdot}} + {\text{ X}}\xrightarrow{{k_{3} }}\left[ {{\text{Zn}}\left( {{\text{TPP}}} \right)} \right]{\mkern 1mu} + {\text{ X}}^{ + } $$
(6)
$$ {\text{X}}^{ + } + {\mkern 1mu} \left[ {{\text{Co}}\left( {{\text{sep}}} \right)} \right]^{{2 + }} \xrightarrow{{k_{4} }}{\text{X }} + {\mkern 1mu} \left[ {{\text{Co}}\left( {{\text{sep}}} \right)} \right]^{{3 + }} $$
(7)

In this case, the kinetic trace of the reaction should be first-order and the observed first-order rate constant may be linearly dependent on the concentration of [Co(sep)]3+ when k4 is large.

3.2.2 Case 2

Another explanation is possible if the π-carion radical [Zn(TPP)]+· produced immediately after the reaction with [Co(sep)]3+ is in its excited state: it is possible that the immediate product after reaction (4) is in the structurally excited state since the driving force of reaction (4) is very large. In such a case, there are three possibilities for the deactivation of excited [Zn(TPP)]+·: (a) [Zn(TPP)]+· is in its ground state which is lower in energy than [Co(sep)]2+ (Scheme 2), (b) [Zn(TPP)]+· is in its ground state which is higher in energy than [Co(sep)]2+ (Scheme 3), (c) [Zn(TPP)]+· is in the excited state which is higher in energy than [Co(sep)]2+ while it relaxes to its ground state having lower energy than[Co(sep)]2+, with a rate constant k (Scheme 4). In the case of (a), a seocnd-order reaction with [Co(sep)]2+ should be observed, while in (b) there is no reaction of [Zn(TPP)]+· with [Co(sep)]2+, and we will observe a simple first-order decay of [Zn(TPP)]+· without dependence on [Co(sep)]3+. In case (c) as depicted by Scheme 4, the decrease of [Zn(TPP)]+· with time is first-order with the rate constant k when k <<  k2[Co(sep)2+]. When k >> k2[Co(sep)2+], we will observe a second-order reaction, while the reaction is expected to be a consecutive process described by a double-exponential function when k ~ k2[Co(sep)2+].

Scheme 2
scheme 2

[Zn(TPP)] triplet excited state reduces Co(III) and [Zn(TPP)]+· π-cation radical oxidizes Co(II)

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Scheme 3
scheme 3

[Zn(TPP)] triplet excited state reduces Co(III) while [Zn(TPP)]+· π-cation radical cannot oxidize Co(II)

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Scheme 4
scheme 4

[Zn(TPP)] triplet excited state reduces Co(III) and an excited state of [Zn(TPP)]+· π-cation radical is formed. However, it cannot oxidize Co(II) unless it relaxes to the ground state of [Zn(TPP)]+· π-cation radical

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In Table 3, the second-order rate constants for reaction (4) in three ILs and in acetonitrile are summarized. As it was not possible to obtain accurate second-order rate constants for reaction (5), analyses on the basis of the Marcus theory are carried out essentially for reaction (4) in the following section.

Table 3 Second-order rate constants for reaction (1) in three IL’s and in acetonitrile
Full size table

As the second-order rate constants estimated in acetonitrile and ILs are close to the diffusion limits calculated using Eq. 8, it was necessary to isolate the contribution of diffusion from the obtained second-order rate constants. By assuming that the diffusion limit for second-order reactions is given by the Stokes–Einstein equation (Eq. 8) [23], the diffusion limits of the second-order rate constants in acetonitrile and ILs were calculated.

$$ k_{\text{diff}} = \frac{{2RT\left( {r_{A} + r_{B} } \right)^{2} }}{{3000\eta r_{A} r_{B} }}\left( {\frac{U}{{{ \exp }(U) - 1}}} \right),\quad U = \frac{{z_{A} z_{B} e^{2} }}{{4\pi \varepsilon_{0} \varepsilon \left( {r_{A} + r_{B} } \right)kT}} $$
(8)

where rA (= 570 pm), rB (= 400 pm), and zA (= 0), zB (= + 3) are the ionic/molecular radii and formal charges of reactants, respectively [50, 52]. The estimated diffusion-controlled rate constants are 1.56 × 1010, 1.89 × 108, 1.68 × 108, and 1.56 × 108 kg·mol−1·s−1 (density of each solvent was accounted for) for the reaction between [Zn(TPP)]* and [Co(sep)]3+ in acetonitrile, BMIM, PMIM and HMIM, respectively. The rate constants in the last column of Table 3 were calculated using Eq. 9 to remove the contribution of diffusion from the observed second-order rate constants [50,51,52,53,54].

$$ \frac{1}{{k_{\text{obsd}} }} = \frac{1}{{k_{\text{diff}} }} + \frac{1}{{k_{\text{reaction}} }} $$
(9)

Logarithmic values of the rate constants for reaction (4) thus corrected for the diffusion contribution are plotted against the logarithm of the viscosity of each solvent in Fig. 7. The plot appears to be linear with a slope of − 0.834 (average slope, including the rate constant observed in acetonitrile) indicating that the dependence of the second order rate constant k1 on the viscosity of the solvents roughly obeys the predictions of either the Sumi–Marcus theory \( \left( {\log_{10} k \propto \eta^{ - \alpha } , \, 0 \, < \alpha < \, 1} \right) \) [55, 56] or Kramers theory \( \left( {{ \log }_{ 10} k \propto \eta^{\alpha } , \, \alpha = - 1} \right) \) [57, 58].

Fig. 7
figure 7

Plot of logarithmic value of the rate constant k12 for cross reactions (T = 298 K, and the rate constants were corrected for the diffusion contribution) against the logarithmic value of the apparent (reported) viscosity of solvents (acetonitrile, BMIM, PMIM and HMIM)

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However, the slope of the plot for the data observed in ionic liquids only was much larger (− 4.10) as shown by the right three circles in Fig. 7, indicating that the rate constants in ILs are governed by other factors than the macroscopic viscosity.

In the following discussion, the influence of viscosity η was assumed to follow the Kramers theory (= the rate constant depends on η−1 through Eqs. 8 and 9, and the effect of the Pekar factor, \( \frac{1}{{n^{2} }} - \frac{1}{{\varepsilon_{s} }} \)) (see also Eq. 15 below), on the observed and calculated activation Gibbs energies for the electron self-exchange processes will be discussed.

The Marcus cross relation is generally expressed by Eq. 10 [2], with additional work terms introduced by Espenson [59].

$$ k_{AB} = \left( {k_{AA} k_{BB} K_{AB} f_{AB} } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} W_{AB} ,\,\ln f_{AB} = \frac{{\left( {\ln K_{AB} + \frac{{\left( {w_{AB} - w_{BA} } \right)}}{RT}} \right)^{2} }}{{4\ln \left( {\left( {\frac{{k_{AA} k_{BB} }}{{z^{2} }} + w_{AA} + w_{BB} } \right)\frac{1}{RT}} \right)}} $$
(10)

where Z is the collision frequency (~ 1012, which is close to kBT/h in water and acetonitrile), \( W_{AB} = \exp \left( {\frac{{ - \,\left( {w_{AB} \, + \,w_{BA} \, - \,w_{AA} \, - \,w_{BB} } \right)}}{2RT}} \right) \), and \( w_{ij } \) is the coulombic work given by \( w_{ij} = \frac{{z_{A} z_{B} e^{2} }}{{4\pi \varepsilon \varepsilon_{0} r\left( {1 + Br\sqrt \mu } \right)}} \): r is the distance between two reactants, μ is the ionic strength and B is a parameter related to the Debye–Huckel theory for electrolyte solutions (μ is unknown for IL solutions). Use of this relation was not successful for the electron transfer reactions involving small hydrated molecules/ions or large molecules/ions such as enzymes [23]. It was also shown that Eq. 10 does not correctly explain the cross reactions with large KAB values such as the reaction of [Fe(OH2)]3+with [Co(sep)]2+ in aqueous acidic solution [60].

In 1980, Ratner and Levin derived a similar cross relation given by Eqs. 11 or 12 solely on the basis of the thermodynamic relations [22, 23]. They used only two assumptions to derive the relation: (a) the activation process for each reactant is independent of the other reactant, and (b) the activated species are the same for the self-exchange and the cross reactions.

$$ \Delta G_{AB}^{*} = \frac{1}{2}\left( {\Delta G_{AA}^{*} + \Delta G_{BB}^{*} + \Delta G_{AB}^{0} } \right) $$
(11)
$$ k_{AB} = \left( {k_{AA} k_{BB} K_{AB} \frac{{Z_{AB}^{2} }}{{Z_{AA} Z_{BB} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} $$
(12)

where Z′s are the collision frequencies for the second-order reaction (given by Eq. 13). Zij values are ~ 1012 for water and ordinary non-viscous fluids and \( {Z_{ 1 2}^{2}} \sim {Z_{ 1 1} Z_{ 2 2} }\) when the reaction has charge symmetry [23], while the effect of charges on the reactants to Zij is related to Eq. 8.

$$ k_{ij} = Z_{ij} \exp \left( {\frac{{\Delta G_{ij}^{*} }}{RT}} \right) $$
(13)

Use of the cross relation given by Eqs. 11–13 is more appropriate than the use of Eq. 10 for the analyses of the results obtained in ILs, since it is not possible to correctly estimate the work terms of the reactants (we do not know the ionic strengths of the solutions nor the exact properties of the solvents in the vicinity of solutes necessary for the use of Eq. 10).

However, Eq. 10 is the standard theoretical equation and is reliable for analyses of the reactions in conventional molecular liquids. Therefore, we used Eq. 10 for the estimation of activation parameters of the reactions in acetonitrile. Use of Eqs. 11 and 12 seems the more appropriate for the analyses of the reactions observed in ILs, since the contributions of \( \Delta G_{\text{Fuoss}}^{*} \) and the frequency factor of reactions are included in Zij, and some parameters necessary for the calculations of \( \Delta G_{\text{Fuoss}}^{*} \) are unavailable for ILs. The quotient \( \frac{{Z_{AB}^{2} }}{{Z_{AA} Z_{BB} }} \) was taken to be ca. 1 for the estimation of the kinetic parameters of the reactions in ILs since effective charge cancellation of Co(III)/(II) complexes by counter anions is expected from the small bulk dielectric constants of the solvents used in this study [see Appendix 1 for the justification of this assumption].

The total activation Gibbs energy for a self-exchange reaction in acetonitrile is given by the following equation [2, 23].

$$ \Delta G_{\text{Total}}^{*} = \Delta G_{\text{OS}}^{*} + \Delta G_{\text{IS}}^{*} + \Delta G_{\text{Fuoss}}^{*} $$
(14)

where \( \Delta G_{\text{OS}}^{*} \), \( \Delta G_{\text{IS}}^{*} \), and \( \Delta G_{\text{Fuoss}}^{*} \) are the outer-sphere activation Gibbs energy, the inner-sphere activation Gibbs energy and the energy to bring two reactants to form a precursor complex \( \left( {\Delta G_{\text{Fuoss}}^{*} = - RT\ln K_{\text{OS}} } \right) \). \( \Delta G_{\text{OS}}^{*} \) and \( \Delta G_{\text{Fuoss}}^{*} \) terms are given by Eqs. 15 and 16 [2,3,4,5,6,7,8, 23]:

$$ \Delta G_{\text{os}}^{*} = \frac{{Ne^{2} }}{{16\pi \varepsilon_{0} }}\left( {\frac{1}{{\varepsilon_{op} }} - \frac{1}{{\varepsilon_{s} }}} \right)\left( {\frac{1}{{2r_{1} }} + \frac{1}{{2r_{2} }} - \frac{1}{{r_{12} }}} \right) $$
(15)
$$ \begin{aligned} \Delta G_{\text{Fuoss}}^{*} & = - RT \ln K_{\text{OS}} , \,K_{\text{OS}} = \frac{{4\pi N_{\text{A}} a^{3} }}{3000}\exp \left( { - \frac{U}{{k_{\text{B}} T}}} \right) \,{\text{and}}\, \\ U & = \frac{{z_{1} z_{2} e^{2} }}{{4\pi \varepsilon_{0} \varepsilon rk_{B} T(1 + Br\sqrt \mu )}} \\ \end{aligned} $$
(16)

where r’s are the radii of the reactants, μ is the ionic strength and B is a parameter related to the Debye–Huckel theory for electrolyte solutions. By using these equations, the inner-sphere contribution to the activation Gibbs energy, \( \Delta G_{\text{IS}}^{*} \), for the self-exchange reaction of the [Co(sep)]3+/2+ couple in acetonitrile was estimated as 39.1 kJ·mol−1 (a reported value of kex = 5.1 kg·mol−1·s−1 was used for this calculation [52]).

Table 4 lists the kinetic parameters calculated for the self-exchange reaction of the [Zn(TPP)]+·/* couple in acetonitrile by applying the cross relation (Eq. 10) to the rate constant for the cross reaction of [Zn(TPP)]* with [Co(sep)]3+ in acetonitrile observed in this study. Application of cross relation 12 yields kex = 4.9 × 109 kg·mol−1·s−1 as the self-exchange rate constant for the [Zn(TPP)]+·/* couple in acetonitrile when \( {{Z_{12}^{2} } \mathord{\left/ {\vphantom {{Z_{12}^{2} } {Z_{11} Z_{22} }}} \right. \kern-0pt} {Z_{11} Z_{22} }} = 1 \) was assumed: this value is consistent with that obtained by using Eq. 10 (8.39 × 109 kg·mol−1·s−1), upon consideration of the accuracy inherent in these cross relations. Therefore, the assumption of \( {{Z_{12}^{2} } \mathord{\left/ {\vphantom {{Z_{12}^{2} } {Z_{11} Z_{22} }}} \right. \kern-0pt} {Z_{11} Z_{22} }} = 1 \) seems valid for the reactions even in acetonitrile and the analyses using Eqs. 10–16 are quite accurate: charges of the metal complexes such as [Co(sep)]3+ and [Co(sep)]2+ are effectively cancelled by the counter ions in acetonitrile, indicating that the same situation is also expected for the same reaction in ILs since the dielectric constants of the ILs used in this study are much smaller than that of acetonitrile (ε = 37.5 at 293 K, see also Appendix 1).

Table 4 Kinetic parameters calculated for the self-exchange reaction of the [Zn(TPP)]+·/* couple in acetonitrile by applying the cross relation to the rate constant for the cross reaction of [Zn(TPP)]* with [Co(sep)]3+
Full size table

Equation 11 is expressed by Eq. 17 by using the corresponding activation Gibbs energies for each self-exchange reaction:

$$ \begin{aligned} 2\Delta G^{*} \left( {{\text{Co}}^{3 + } + {\text{Zn}}\left( {\text{TPP}} \right)^{*0} } \right) = & \,\Delta G_{\text{OS}}^{*} \left( {\text{solv}} \right)_{\text{Total}} + \left[ {\Delta G_{\text{IS}}^{*} \left( {{\text{Co}}^{{{{3 + } \mathord{\left/ {\vphantom {{3 + } {2 + }}} \right. \kern-0pt} {2 + }}}} } \right) + \Delta G_{\text{IS}}^{*} \left( {{\text{Zn}}\left( {\text{TPP}} \right)^{ + \cdot /*} } \right)} \right] \\ & + \,\Delta G^{0} \left( {{\text{Co}}^{3 + } + {\text{Zn}}\left( {\text{TPP}} \right)^{*0} } \right) \\ \end{aligned} $$
(17)

where \( \Delta G_{\text{OS}}^{*} \left( {\text{solv}} \right)_{\text{Total}} = \Delta G_{\text{OS}}^{*} \left( {{\text{Co}}^{{{{3 + } \mathord{\left/ {\vphantom {{3 + } {2 + }}} \right. \kern-0pt} {2 + }}}} } \right) + \Delta G_{\text{OS}}^{*} \left( {{\text{Zn}}\left( {\text{TPP}} \right)^{ + \cdot /*} } \right) \). \( \Delta G_{\text{OS}}^{*} \left( {\text{solv}} \right)_{\text{Total}} \) estimated by this equation, \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-1}} \right)_{\text{Total}} \), reflects the exact nature of the solvent ILs in the vicinity of the precursor and successor complexes. Estimated values of \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-1}} \right)_{\text{Total}} \) are listed in Table 5 for the reactions in BMIM, PMIM and HMIM: \( \Delta G_{\text{IS}}^{*} \left( {{\text{Zn}}\left( {\text{TPP}} \right)^{ + \cdot /*} } \right) \) and \( \Delta G_{\text{IS}}^{*} \left( {{\text{Co}}^{{{{3 + } \mathord{\left/ {\vphantom {{3 + } {2 + }}} \right. \kern-0pt} {2 + }}}} } \right) \) in ILs were taken to be identical to those estimated for the self-exchange reactions in acetonitrile since the inner-sphere contribution to the activation Gibbs energy is not dependent on the solvent [5].

Table 5 Estimated values of \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-1}} \right)_{\text{Total}} \) and \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-2}} \right)_{\text{Total}} \) for the reactions in BMIM, PMIM and HMIM
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Values of \( \Delta G_{\text{OS}}^{*} \left( {\text{solv}} \right)_{\text{Total}} \), \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-2}} \right)_{\text{Total}} \), calculated using Eq. 15 are 30.9, 30.5 and 30.3 kJ·mol−1 (Table 5) in BMIM, PMIM and HMIM, when measured/reported refractive indices and dielectric constants for the bulk ILs are used in the calculations. The values of \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-1}} \right)_{\text{Total}} \) and \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-2}} \right)_{\text{Total}} \) in Table 4 seem close to each other, which indicates that the analyses employed in this study is reasonable. However, when we look into the details, the estimated \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-1}} \right)_{\text{Total}} \) values increase in the order, BMIM < PMIM < HMIM, while theoretically calculated \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-2}} \right)_{\text{Total}} \) increase in the opposite order, BMIM > PMIM > HMIM. Similar results were reported by Fawcett et al. for the adiabatic electron transfer process at the electrode for the Ferrocene/Ferricinium couple in ILs [61], indicating that the observation by Israelachvili and co-workers [28] together with the explanations by Nakamuta and Shikata [43] are valid even for the ILs in the vicinity of charged/polarized ions (see details in Appendix 2). It seems certain that the solvent macroscopic properties related to the Pekar factor, the refractive index and dielectric constant, are different from those for the bulk ILs in the vicinity of the charged encounter (precursor/successor) complexes: as discussed in the section of electrochemical studies and also indicated by the authors of references [28, 43, 61], ILs seem dissociated, or at least the distance between the cationic and anionic components are elongated, in the vicinity of the precursor/successor complexes for electron transfer reactions. As the component ions of ILs are positively and negatively charged, it is certain that the “dissociated” component ions of ILs exist as pairs, and probably changing partner ions in a concerted manner [1].

It is known that the refractive index is related to the polarizability α of a molecule by the Lorentz–Lorenz formula (Eq. 18).

$$ \frac{{n^{2} - 1}}{{n^{2} + 2}} = \frac{N\alpha }{{3\varepsilon_{0} }} $$
(18)

where N is the number of molecules per unit volume. As the cationic and anionic components of dissociated ILs are expected to act as pairs even after dissociation (though they are not the tightly-bound pairs like bulk ILs), the polarizability and therefore the refractive index of such a pair may not be significantly different from those for the bulk ILs (polarizability is proportional to the volume of a molecule): it is expected that the polarizabilities for each cationic and anionic component of ILs are not largely different from those in the bulk ILs and therefore the averaged polarizability of dissociated ILs may not be very different from those of the bulk ILs.

By assuming that the refractive index is not affected by the dissociation of ILs, values of 23.6, 47.0 and 145 were calculated as the relative dielectric constants of BMIM, PMIM and HMIM, in the vicinity of a charged precursor/successor complex. Although such a large value as 145 may not be acceptable, this order of relative dielectric constants, BMIM < PMIM < HMIM, in the vicinity of the charged precursor/successor complexes explains the larger \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-1}} \right)_{\text{Total}} \) observed for HMIM. However, this result also indicates that the refractive indices of the ILs around the charged species may also be slightly decreased from the values for bulk ILs in order to reproduce acceptable values for dielectric constants.

When a slow relaxation of solvent is involved in the activation process, the following solvent friction model may be invoked [61].

$$ k = K_{\text{OS}} \kappa_{el} \tau_{\text{L}}^{ - 1} (\Delta G_{\text{OS}}^{*} /4\pi RT)^{{\frac{1}{2}}} { \exp }\left[ { - \frac{{\Delta G_{\text{OS}}^{*} + \Delta G_{\text{iS}}^{*} }}{RT}} \right] $$
(19)

where \( \tau_{\text{L}} \) is the longitudinal relaxation time of the solvent, defined by the product of the Debye relaxation time, τD, and (n2/ε): \( \tau_{\text{L}} = \left( {\frac{{n^{2} }}{\varepsilon }} \right)\tau_{\text{D}} = 3\,V_{\text{M}} \eta n^{2} /\varepsilon RT \). For viscous solvents with large molar volume \( V_{\text{M}} \) like ILs, \( \tau_{\text{L}} \) may be longer compared with an ordinary molecular solvent. This relation accounts for the order of the observed rate constants for the cross reactions and \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-1}} \right)_{\text{Total}} \): Grampp and co-workers discarded the possibility of this mechanism because the observed rate constants for their experiment did not exhibit the Pekar factor dependence [17]. However, as indicated by the results of this study, the electron self-exchange reactions in ILs may not exhibit the Pekar factor dependence since the solvent properties are altered around the precursor/successor complexes, by the dissociation of ILs. The expected decrease of \( \tau_{\text{L}} \) was also confirmed by the experiments by Nakamura and Shikata [43] for the ILs in the vicinity of the electrode [see Appendix 2].

Dissociation of ILs is also expected around the polar transition state of the isomerization reaction of 4-dimethylamino-4′-nitro-azobenzene [1], and the intramolecular electron transfer/charge separation processes can be treated in the same way (Eq. 19). In the thermal isomerization reaction of 4-dimethylamino-4′-nitro-azobenzene, increased dielectric constant of ILs in the vicinity of the molecules stabilizes the polarized transition state and a smaller activation energy is expected for the reaction in HMIM than in BMIM (Ea = 41.0 kJ·mol−1 in HMIM and Ea = 45.6 kJ·mol−1 in BMIM). The observed smaller frequency factors for the reaction in HMIM (1.2 × 104 s−1 for HMIM and 1.4 × 106 s−1 for BMIM) may be attributed to the slower longitudinal relaxation times, \( \tau_{\text{L}} \), for HMIM than that of BMIM [43], although the activation energy and the frequency factor for the same reaction in PMIM (Ea = 56.9 kJ·mol−1 and 1.2 × 107 s−1) cannot be explained by this model unless we assume some specific interaction between the solute and solvent molecules.

4 Conclusion

Results of the electrochemical measurements and the kinetic analyses, together with the observations by Israelachvili and co-workers [28], Nakamura and Shikata [43] and Fawcett et al. [62], indicate that ILs around charged metal complexes (including precursor/successor complexes for electron transfer reactions) dissociate and form local microstructures around the metal complexes [63]: dissociated anionic and cationic components of ILs seem to act as pairs even after dissociation and exchange the partner ions in a concerted manner [1, 43]. However, the thickness of such a modified solvent layer probably extends only to a couple of IL layers around the charged metal complexes (since the charges of the cobalt complex is only 3+), which is very much like the structure of the double layers at electrodes [28, 29].

The outer-sphere reorganization energies, \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-1}} \right)_{\text{Total}} \), estimated from the experimental results are in the opposite order, BMIM < PMIM < HMIM to those calculated by using Marcus theory with bulk dielectric constants for ILs, \( \Delta G_{\text{OS}}^{*} \left( {\text{solv-2}} \right)_{\text{Total}} \), BMIM > PMIM > HMIM, which is attributed to the effect of the slightly decreased refractive index and increased dielectric constant of the medium caused by the induced dissociation of ionic liquids around charged metal/encountered complexes.

Dependence of the rate constants on the Pekar factor may not be observed when the dielectric constant and refractive index for the bulk IL are used for the calculation [17], since the dielectric constant and refractive index of the medium in the vicinity of the precursor/successor complexes are different from those for the bulk ILs.

It was shown that the methods employed in this study can be used to estimate the exact values of Pekar factors of ILs in the vicinity of polarized/charged precursor/successor complexes, although it may not be possible to evaluate the isolated values of the refractive indices and dielectric constants.

Previously reported activation energies and frequency factors for the isomerization reactions of 4-dimethylamino-4′-nitro-azobenzene in HMIM and BMIM [1] may be explained by the solvent friction model because of the decreased \( \tau_{\text{L}} \) value as a result of the dissociation of ILs or elongation of the distance between the cationic and anionic components of ILs [43]. However, it was not possible to explain the kinetic parameters for the same reaction in PMIM.

Conventional approaches using a Born-type equation (dielectric continuum model) such as the Marcus theory) seems valid for the analyses of the reactions of relatively large metal complexes in ILs, while the analyses of the reactions of small organic molecules may not be successful [6] (see Appendix 2).