Diffusion-Facilitated Direct Determination of Intrinsic Parameters for

Feb 26, 2015 - ... Photoinduced Bimolecular Electron-Transfer Reactions in. Nonpolar Solvents. Andrew D. Scully,. †. Hiroyasu Ohtaka,. ‡,§. Makot...
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Diffusion-Facilitated Direct Determination of Intrinsic Parameters for Rapid Photoinduced Bimolecular Electron-Transfer Reactions in Nonpolar Solvents Andrew D. Scully,† Hiroyasu Ohtaka,‡,§ Makoto Takezaki,‡ and Toshihiro Tominaga*,‡ †

CSIRO Manufacturing Flagship, Bayview Avenue, Clayton, Victoria 3168, Australia Department of Applied Chemistry, Okayama University of Science, 1-1 Ridai-cho, Okayama 700-0005, Japan § Department of Pharmacy, Chiba Institute of Science, 3 Shiomi-cho, Choshi, Chiba 288-0025, Japan ‡

S Supporting Information *

ABSTRACT: Bimolecular fluorescence-quenching reactions involving electron-transfer between electronically excited 5,10,15,20-tetraphenyl-21H,23Hporphine (TPP*) and 1,4-benzoquinone (BQ) or 1,4-naphthoquinone (NQ) were investigated using a set of alkane solvents that enabled the rapid reaction kinetics to be probed over a wide viscosity range, while minimizing changes in other relevant solvent parameters. Relative diffusion coefficients and reaction distances were recovered directly from analysis of fluorescence decay curves measured on a nanosecond time scale. The electron transfer from TPP* to BQ requires reactant contact, consistent with tightly associated exciplex formation in these nonpolar solvents. In contrast, electron transfer from TPP* to NQ displays a clear distance dependence, indicative of reaction via a much looser noncontact exciplex. This difference is attributed to the greater steric hindrance associated with contact between the TPP*/NQ pair. The diffusion coefficients recovered from fluorescence decay curve analysis are markedly smaller than the corresponding measured bulk relative diffusion coefficients. Classical hydrodynamics theory was found to provide a satisfactory resolution of this apparent discrepancy. The calculated hydrodynamic radii of TPP and NQ correlate very well with the van der Waals values. The hydrodynamic radius obtained for BQ is a factor of 6 times smaller than the van der Waals value, indicative of a possible tight cofacial geometry in the (TPP+/BQ−)* exciplex. The present work demonstrates the utility of a straightforward methodology, based on widely available instrumentation and data analysis, that is broadly applicable for direct determination of kinetic parameter values for a wide variety of rapid bimolecular fluorescence quenching reactions in fluid solution.

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INTRODUCTION Understanding the intrinsic parameters associated with an electron-transfer reaction between two molecular entities is of fundamental scientific interest1 and has practical importance in the field of organic electronics.2,3 Fluorescence quenching measurements of dilute solutions containing a mixture of the two reactants is the most straightforward method for extracting information about electron-transfer reaction parameters when at least one of the reactants is luminescent, and this approach has been widely utilized, such as for the study of the biologically relevant quenching of porphyrin fluorescence by quinones.4−9 However, in many reaction systems of interest the electrontransfer rate is so fast that, even in low-viscosity fluid solutions, the overall rate of the reaction is influenced by the rate of approach of the reactants,4,10−23 and this can complicate the recovery of the intrinsic electron-transfer reaction parameters. Approaches to overcoming this perceived difficulty have included tethering the reactants together by covalent linkage24−28 or immobilizing the reactants in a highly viscous or solid medium. Covalently linked systems are generally not readily available and are often not easily synthesized, and it can © 2015 American Chemical Society

be difficult to account for the influence of intervening organic linkages on the electronic properties of the reaction. Immobilizing the reactants in liquids having very high viscosity or in solid-state media, such as polymeric films or frozen solutions, typically necessitates the use of very high concentrations of at least one of the reactants.29−32 In this case, insufficient solubility may limit the reaction efficiency and can lead to the formation of molecular aggregates. The presence of luminescent molecular aggregates can cause significant complications in the determination of the intrinsic reaction parameters between the two reactants of interest. Consequently, despite the complicating effects of diffusion on the reaction kinetics, harnessing the utility of translational diffusion offers a convenient means of studying a wide variety of reaction pairs while avoiding the need for synthesis of covalently linked molecules or the difficulties associated with Received: October 15, 2014 Revised: January 31, 2015 Published: February 26, 2015 2770

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The Journal of Physical Chemistry A

(TPP*) to either 1,4-benzoquinone (BQ) or 1,4-naphthoquinone (NQ). These reactions were investigated using a set of alkane solvents as this offers a very convenient means of systematically modulating the solvent viscosity over a very wide range (around an order of magnitude in the present work) while minimizing changes in other relevant solvent parameters, such as dielectric constant. The findings of this work build on the relatively limited data available on the fluorescence quenching kinetics associated with diffusion-influenced electron-transfer reactions in nonpolar solvents.

the use of samples at very high concentrations and, moreover, can provide valuable insight into the diffusion process itself. Time-resolved fluorescence spectroscopy has been shown to be an effective tool for studying the kinetics of fast bimolecular reactions occurring in dilute fluid solutions.33 The theoretical framework to account for the influence of diffusion on reaction kinetics is now highly developed, and a variety of commercial instrumentation and analysis software required for highprecision time-resolved fluorescence measurements on a nanosecond time scale are in use in many laboratories around the world.19,20,22,23,33,34 Recent work using ultrafast time resolution has focused on the use of complicated analytical functions that comprehensively describe the fluorescence decay kinetics.35,36 This type of approach offers the potential for more directly probing the quenching reaction kinetics, although data analysis invariably necessitates fixing the magnitude of some key parameters, such as reaction distances and/or diffusion coefficients, using estimated rather than experimentally determined values, which can undermine the usefulness of this methodology. It is commonly assumed that the magnitude of the relative diffusion coefficient, D, used in the fluorescence decay analysis is equal to the sum of the bulk diffusion coefficients of the fluorophore and quencher, (DF + DQ), obtained using classical methods. For example, for the analysis of fluorescence decay data on ultrashort time scales according to the more complex functions, the magnitude of D is usually fixed at a value calculated from the van der Waals radii of the reactants (the sum of which is also sometimes used as a fixed estimate for the encounter distance, R). However, significant discrepancies have emerged in the limited number of systems in which values of D obtained from fluorescence decay curve analysis have been compared with experimentally determined values of (DF + DQ).9,20,34,37−40 In many cases, the magnitude of D is found to be of the order of only 30−60% that of (DF + DQ); one contributing factor to this difference could be that the former includes the diffusion coefficient of the electronically excited fluorophore, whereas the latter reflects the diffusion coefficient of the fluorophore in the ground electronic state. A more significant difference is that the magnitude of D reflects the diffusion coefficient when the fluorophore and the quencher are sufficiently close to react, whereas that of (DF + DQ) reflects the bulk relative diffusion coefficient when the fluorophore− quencher separation is much larger than the reaction distance. A primary objective of the present work was to further investigate this apparent discrepancy between the magnitude of the relative diffusion coefficients recovered from fluorescence decay curve analysis and the sum of the bulk diffusion coefficients of the reactants measured independently. Fortunately, a judicious selection of measurement time-scale permits the use of a simplified fitting function,11,12,14,20,41 which means that, in principle, key parameters associated with diffusioninfluenced bimolecular electron-transfer fluorescence quenching reactions, such as D and R, can be determined directly as freely adjustable parameters.11,12,14,20,41 Additionally, we demonstrate in this work that nanosecond time-resolved measurements of bimolecular fluorescence quenching reactions, where the magnitude of the reaction rate is comparable with the rate of translational diffusion of the reactants, can offer a convenient tool for probing hydrodynamic effects. The fast bimolecular fluorescence-quenching reactions studied in the present work are the electron-transfer from electronically excited 5,10,15,20-tetraphenyl-21H,23H-porphine

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EXPERIMENTAL SECTION Materials. 5,10,15,20-Tetraphenyl-21H,23H-porphine (TPP) (Dojindo Laboratories), hexane (Nacalai Tesque, Fluorescence grade), and cyclohexane (Wako, HPLC grade) were used as received. 1,4-Benzoquinone (BQ) (Tokyo Kasei, >98%), 1,4-naphthoquinone (NQ) (Tokyo Kasei, EP grade), and ferrocene (Fc) (Tokyo Kasei, GR grade) were purified by sublimation. Decane (Tokyo Kasei, GR grade), tetradecane (Wako, GR grade), and hexadecane (Nacalai Tesque, GR grade) were passed through a silica gel column and filtered by a membrane filter (Advantec, PTFE, 0.2 μm). Benzonitrile (Tokyo Kasei, 99.9%) was used as received. Methods. Fluorescence decay profiles were measured as described previously,20 with the detector used in the present work being a Hamamatsu R3809U-50 microchannel plate photomultiplier tube cooled to −30 °C using a Hamamatsu C2773 thermoelectric cooler. Fluorescence decay curves were collected to 6.5 × 104 counts in the channel of maximum intensity, and fitting of the decay data was from the channel corresponding to 3 ns after the channel of maximum intensity of the instrument response function. The concentration of TPP in solutions used for fluorescence decay measurements was 1 μM (1 M = 1 mol dm−3). Absorption spectra were measured using spectrophotometers, Hitachi U-3310 up to 850 nm and Hitachi U-2800 up to 1100 nm. Fluorescence spectra were measured using a fluorescence spectrophotometer (Hitachi F7000) by circulating temperature-controlled water to the cell holder to keep solutions at 25 °C. Bulk diffusion coefficients were measured using the Taylor dispersion method.42,43 The concentration profiles after diffusion/dispersion were detected using a UV−visible detector (Hitachi, L-7420), and other details have been described elsewhere.20,38 Cyclic voltammograms were recorded using an electrochemical analyzer (ALS 620A, BAS Inc.). Measurements were carried out using a 3 mm ϕ glassy carbon electrode as a working electrode, platinum wire as a counter electrode, and Ag/Ag+ electrode (ALS RE-7, BAS Inc.) as a reference electrode under nitrogen atmosphere. van der Waals volumes, VvdW, were calculated according to the atom/group contributions to the van der Waals volumes listed in refs 47 and 48, and the molecular radii, r, were calculated using the relation VvdW = (4/3)πr3. Theoretical Background. According to differential encounter theory, the time-dependent quenching rate constant, k(t), for a reaction between an electronically excited fluorophore, F*, and a quenching species, Q, in which the reaction kinetics depends on their separation distance, r, is related to the pair correlation function, n(r,t), via eq 1,16−19,21,22 where R is the encounter distance between the two reactants. 2771

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The Journal of Physical Chemistry A k(t ) = 4π

∫R

∞

w(r )n(r , t )r 2dr

their separation is sufficiently large for hydrodynamic effects to be negligibly small. When w(r) is given by the exponential function in eq 2 and under circumstances where long-range tunnelling is a significant reaction channel, eq 5 provides a good approximation of the relationship between R′ and D when D < W0(L × 10−9)2γ2/4 (“slow-diffusion” regime).16,17,45,46

(1)

The functional form of the distance-dependent intrinsic reaction rate, w(r), depends on the type of reaction between F* and Q. A functional form for w(r) that has been used for reactions involving the transfer or exchange of an electron via tunnelling is shown in eq 2.4,10,13,16,17,25,44,45 ⎡ −2 ⎤ w(r ) = W0 exp⎢ (r − R )⎥ ⎣ L ⎦

R′ = R + (2)

⎧ V 2π 1/2 ⎛ − (ΔG + λ)2 ⎞⎫ ⎡ −2 ⎤ w(r ) = ⎨ 0 1/2 exp⎜ (r − R )⎥ ⎟⎬ exp⎢ ⎣ ⎦ 4 k T L λ ⎝ ⎠ ( k T ) λ ⎩ B ⎭ B ⎪

⎪

⎪

(3)

Electron-transfer reactions for which −ΔG < λ are considered to lie in the so-called “normal region” and the activation energy decreases with increasing exergonicity of the reaction, whereas those for −ΔG > λ lie in the “inverted region” and the activation energy increases with increasing exergonicity. At −ΔG = λ, there is no activation energy and the reaction rate is at its fastest. The approximate form of w(r) given in eq 2 is considered to provide a satisfactory description of the reaction kinetics for reaction in the normal region provided the reaction neither requires contact nor occurs very far from contact.16,17 According to Burshtein16,17 and Rice,45 the long-time asymptote of eq 1 using the functional form of w(r) in eq 2 is given by eq 4 in which R′ is the effective reaction distance and D is the relative diffusion coefficient (being the sum of the diffusion coefficients of the reactants when separated by the effective reaction distance), and N is Avogadro’s number. ⎛ k(t ) = 4πNR′D⎜1 + ⎝

R′ ⎞ ⎟, πDt ⎠

R′2 when t > > πD

(5)

In this equation, γ = exp(Ce) where Ce is Euler’s constant (∼0.57721), and in the present work, the units used for R′, R, and L are nm, the units for W0 are s−1, and those for D are m2 s−1. From eq 5 it can be seen that values of R′ determined experimentally are expected to be substantially larger than R for sufficiently small values of D. For increasing values of D approaching W0(L × 10−9)2γ2/4, the magnitude of R′ recovered experimentally is predicted to decrease toward that of R. If the magnitude of D is increased beyond W0(L × 10−9)2γ2/ 4 (“fast-diffusion” regime) then the distance dependence of the electron-transfer reaction rate becomes more difficult to detect, and at large enough values of D, the magnitude of R′ has been shown16,44 to be equivalent to that predicted by the fixed reaction-distance Smoluchowski−Collins−Kimball model shown in eq 6, where R′ ≤ R. In this model, kact is the fixeddistance activation-limited bimolecular rate constant for electron transfer at the reaction distance R + δR, where δR is usually considered to be vanishingly small.

In this equation the pre-exponential factor, W0, is the intrinsic diabatic electron-transfer rate when the reactants are separated by a distance equal to R, and L is a constant determined by the tunnelling barrier height, which governs the distance-dependence of the electron-transfer rate. The function given in eq 2 approximates the expression shown in eq 3 derived from the Marcus model for diabatic electron-transfer rates in fluid solution16,17,19,21 such as the alkane solvents used in the present work. In eq 3, V0 is the exchange interaction matrix element, ΔG is the Gibbs free-energy change for the reaction, λ is the reorganization energy, kB is Boltzmann’s constant, and T is temperature. ⎪

⎡ W (L × 10−9)2 γ 2 ⎤ L ⎥ loge⎢ 0 2 4D ⎦ ⎣

R′ = R

kact , kact + kdiff

where kdiff = 4πNRD

(6)

17

As discussed by Gladkikh et al., kact is equivalent to k(t = 0) in the limit D → ∞ and can be expressed by the following equation upon substituting eq 2 into eq 1, and where n(r,0) = 1 in this limit. kact ≡ k(0) = 4πN

∫R

∞

⎛ 2R ⎞ = 4πNW0 exp⎜ ⎟ ⎝ L ⎠

w(r )r 2dr

∫R

∞

⎛ −2r ⎞ 2 ⎟r d r exp⎜ ⎝ L ⎠

⎛ L L2 ⎞ = 2πNW0R2L⎜1 + + ⎟ R ⎝ 2R2 ⎠

(7)

For the limiting case of a completely diffusion-controlled electron-transfer reaction, where kact ≫ kdiff, evidence for the distance-dependence of the reaction rate cannot be detected, and it can be seen from eq 6 that the magnitude of R′ will be independent of D, with R′ = R. The time dependence of [F*] in the presence of Q is given by eq 8

(4)

Using values of 1 nm and 1 × 10−9 m2 s−1 for R′ and D, respectively, as exemplary values for a rapid reaction in a low viscosity solvent, the function shown in eq 4 is expected to provide a good approximation for times greater than around 0.3 ns. In other words, the appropriate measurement time-scale for applicability of analysis using eq 4 is on the order of nanoseconds. The diffusion coefficient of the fluorophore is usually assumed to be independent of whether it is in its ground or excited electronic state and so the value of parameter D recovered from the analysis is the relative diffusion coefficient of F and Q at separation distances sufficiently close for a reaction to occur. At these small intermolecular separation distances (typically less than a few nanometers) hydrodynamic effects can be expected to influence the translational diffusion of the reactants. In contrast, the bulk relative diffusion coefficient, DFQ, is the sum of the diffusion coefficients of F and Q when

d[F*]t = −[F*]t (τ0−1 + k(t )[Q]) dt

(8)

where τ0 is the unquenched fluorescence lifetime of F*, [F*]t is the concentration of F* at time t, and [Q] is the concentration of the quencher. The function that is obtained for the timedependent decay of fluorescence intensity, I(t), upon substitution of eq 4 for k(t) in eq 8 and then integration is given by eq 9. This fitting function is included in numerous commercially available software packages for fluorescence decay analysis and has been shown to recover reliable values for R′ 2772

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The Journal of Physical Chemistry A and D on the time-scales used in the present work provided that the first 1−2 ns of the decay curve is not included in the curve fitting analysis.41 I(t ) = I0 exp( −At − Bt 1/2)

(9)

where

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A = τ0−1 + 4πDR′N[Q ]

(10)

B = 8(πD)1/2 R′2 N[Q ]

(11)

RESULTS AND DISCUSSION Kinetics of TPP* Fluorescence Quenching by Quinones. In the absence of quinone quenching species, the decay of fluorescence from TPP* in all the solvents used in this work was found to be exponential, with the fluorescence lifetime being essentially independent of the alkane solvent used. Figure 1 shows the fluorescence decay curves measured for TPP in

Figure 2. Distributions of reduced residuals obtained from analysis of the decay curves shown in Figure 1.Top: results from analysis using a monoexponential decay function, with associated χ2 values being (a) 1.11, (b) 1.42, (c) 1.34, and (d) 1.61. Bottom: results from analysis using eq 9, with associated χ2 values being (a) 1.09, (b) 1.14, (c) 1.06, and (d) 1.12. Figure 1. Fluorescence decay curves measured for solutions of TPP and BQ in hexadecane at 298.2 K. Concentrations of BQ (M): (a) 0, (b) 13.7 × 10−3, (c) 19.4 × 10−3, and (d) 25.2 × 10−3. Curve (e) is the instrument response function.

hexadecane solutions containing various concentrations of BQ. The distributions of reduced residuals that correspond to the best-fit curves calculated using either an exponential function or the function given by eq 9 are shown in Figure 2. For all solutions in which BQ was present it was found that the decay could not be described by a simple monoexponential decay function, as demonstrated by the nonrandom residuals and larger χ2 values. However, the fluorescence decay kinetics in these cases was found to be described very well by the function given by eq 9. Analysis of the nonexponential decay curves according to eq 9 provides A and B values as a function of the BQ concentration. Plots of the recovered values of A and B as a function of BQ concentration are shown in Figure 3 for hexadecane solutions of TPP at 298.2 K and are typical of the results obtained for the other solvents used in this work. The good linear relationship in each case is compelling evidence that, for sufficiently long times, the expression for k(t) shown in eq 4 provides a satisfactory description of the reaction kinetics for this bimolecular fluorescence quenching reaction in solution. The R′ and D values presented in Table 1 were calculated using the slopes of the straight lines according to eqs 10 and 11 (see Supporting Information). The recovered values of R′ are plotted as a function of D in Figure 4. It is clear from this graph that the magnitude of R′ is independent of the magnitude of D for the electron-transfer quenching of TPP* by BQ for the conditions used in this work. If the reaction kinetics were influenced by that of the electron-

Figure 3. Results of analysis of fluorescence decay of hexadecane solutions of TPP containing BQ (□) or NQ (○) at 298.2 K, according to eqs 10 and 11.

transfer process, then the dependencies of R′ on D shown in either eq 5 or 6 would be expected. The finding that R′ is independent of D is compelling evidence for the validity of the fixed reaction-distance approximation model for the fluorescence quenching reaction between TPP* and BQ.17 From eq 6 it can be seen that R′ will be independent of D if kact ≫ kdiff, in which case R′ = R. Evidently this is the situation for the range of D values accessible in the present work. 2773

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Table 1. Results of Analysis of Quenching Kinetics According to Equation 9 for Solutions Comprising TPP and Quencher at 298.2 K quencher

solvent

slope of A vs [Q] (109 M−1 s−1)

slope of B vs [Q] (104.5 M−1 s−0.5)

D (10−9 m2 s−1)

R′ (nm)

DFQa (10−9 m2 s−1)

BQ

hexadecane tetradecane cyclohexane decane hexane hexadecane tetradecane decane cyclohexane hexane

2.92 3.71 7.05 6.91 15.1 1.31 1.91 4.08 5.91 9.95

2.24 2.33 4.01 3.08 4.08 4.64 4.68 4.70 7.50 8.11

0.69 0.92 1.51 1.75 4.09 0.14 0.24 0.65 0.78 1.49

0.56 0.53 0.62 0.52 0.49 1.19 1.06 0.83 1.00 0.88

0.93 1.29 2.09 2.42 5.52 0.73 0.97 1.97 1.74 4.62

NQ

a

Sums of the bulk diffusion coefficients obtained using the Taylor dispersion method for TPP and BQ or TPP and NQ as reported previously,38 except for cyclohexane where DTPP = (0.427 ± 0.005) × 10−9 m2 s−1, DBQ = (1.66 ± 0.04) × 10−9 m2 s−1, and DNQ = (1.31 ± 0.03) × 10−9 m2 s−1.

kinetics in all cases was found to be described very well by the function given by eq 9. Analysis of the nonexponential decay curves according to eq 9 provides A and B values as a function of the NQ concentration. Like the TPP*/BQ pair, the good linear relationship in each case (see Supporting Information) is, again, strong evidence that the expression for k(t) shown in eq 4 provides a satisfactory description of the reaction kinetics for this reaction. The slopes of the straight lines obtained from the plots of the recovered values of A and B as a function of NQ concentration yielded the R′ and D values given in Table 1. The recovered values of R′ for the TPP*/NQ system are plotted as a function of D in Figure 4. The magnitude of R′ is clearly dependent on the magnitude of D for the electrontransfer quenching of TPP* by NQ for the conditions used in this work. The increase observed in the magnitude of R′ with decreasing D is consistent with eq 5 and implies that the magnitude of D is in the “slow-diffusion” range (i.e., D < W0(L × 10−9)2γ2/4). This results in the reaction rate having a sufficiently large distance dependence to be detectable for the alkane solvents used in this work. In this case, the relationship between R′ and D given in eq 5 is applicable. Hubig et al.49−51 have reported an extensive number of examples of electron-transfer reactions involving sterically “unhindered” quinone electron acceptors where the kinetics of inner-sphere electron-transfer via encounter complex formation is highly favored over the (distance-dependent) outer-sphere electron-transfer reaction. While the extent of this dominance is affected to some extent by ΔG values, these authors showed that steric effects have a major bearing on the relative importance of these two mechanisms. Therefore, one explanation for the apparent absence of distance dependence for the electron-transfer reaction between TPP* and BQ, in contrast with the distance-dependent quenching reaction between TPP* and NQ, is that the electron-transfer in the former reaction pair occurs predominantly via formation of an encounter complex (such as an exciplex) and is consequently extremely rapid and occurs effectively at contact. In contrast, the steric hindrance associated with the extra benzenoid ring in the NQ quencher inhibits the necessary close approach to the central tetrapyrrole core of TPP, rendering the distancedependent outer-sphere reaction dominant. The values of R′ as a function of D for the TPP*/NQ system were analyzed according to the linearized form of eq 5 shown in eq 12.

Figure 4. Results for TPP/BQ (□) and TPP/NQ (○) in alkane solvents at 298.2 K from Table 1. The dashed line through the squares is the average R′ value for the TPP/BQ data. The solid line through the circles is the best-fit line according to eq 12 for the TPP/NQ data, with the best-fit parameter values being a = −1.82 nm and b = 0.13 nm.

The magnitude of R, the sum of the radii of the external reaction layers around TPP* and BQ, estimated from the mean values of R′ shown in Figure 4 is 0.54 ± 0.02 nm. This value of R′ is in good agreement with that reported previously of 0.51 nm for the quenching of TPP fluorescence by BQ in acetone.6 The van der Waals end-to-end distances calculated for TPP and BQ are 0.51 and 0.28 nm, respectively,47,48 the sum of which is substantially larger than the value of R obtained in the present work. Given that the geometries of both TPP and BQ are far from spherical and that the reactivity between the reactants may well be anisotropic, R is likely to represent an average reaction radius, the value of which may well be substantially smaller than simply the sum of the van der Waals radii if the reaction requires a cofacial geometry. This result strongly suggests that in these nonpolar solvents TPP* and BQ form a tightly associated encounter complex prior to the electron-transfer quenching reaction. In practice, for a reaction to appear to be completely diffusion controlled, kact must exceed kdiff by at least an order of magnitude. Since the largest value found for kdiff in the present work is in the quenching of TPP fluorescence by BQ in hexane and has a value of 1.5 × 1010 M−1 s−1, a lower limit for the magnitude of kact is estimated to be around 2 × 1011 M−1 s−1. As found for the case of the TPP*/BQ reaction pair, for all solutions in which NQ was present the decay of fluorescence from TPP* could not be described by a simple monoexponential decay function. However, the fluorescence decay 2774

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The Journal of Physical Chemistry A ⎤ ⎡ −9 2 L ⎧ ⎛ W (L × 10 )γ ⎞⎫⎥ L ⎟⎬ − loge D R′ = ⎢R + ⎨loge⎜ 0 ⎥ ⎢⎣ 2⎩ ⎝ 4 2 ⎠⎭⎦ ⎪

⎪

⎪

⎪

= a − b loge D

diffusion coefficients of the reactants, DFQ. For example, the ratio D/DFQ is predicted to be 0.25 for the case of a contact reaction occurring between reactants having equal radius. The slope of the best-fit straight line in Figure 5 is less than unity, providing evidence that hydrodynamic effects play a role in the diffusion of the reactants when they are sufficiently close to react. Previous work has shown that the magnitude of the bulk diffusion coefficient for TPP in alkane solvents spanning a wide range of viscosity is described very well by the Stokes−Einstein equation with stick boundary conditions.38 In contrast, in the same set of solvents, the bulk diffusion coefficients of BQ and NQ deviate markedly from Stokes−Einstein behavior38 which precludes any clear conclusion about whether the stick or slip boundary condition is more appropriate for these molecules. Nevertheless, for the TPP/NQ system, the ratio D/DFQ is substantially smaller than the value of 0.5 (Table 1) predicted for the slip boundary condition at the contact distance. Furthermore, the linearity of the UV absorbance of NQ in hexadecane both at 404 nm (measured using a 10 mm opticalpass cell) and at 330 nm (measured using a 0.1 mm optical-pass cell), and the linearity of the A and B values as a function of the NQ concentration (Figure S1 in Supporting Information), render the possibility that the subunity slope might be an artifact arising from quinone aggregation highly unlikely. On the basis of these factors, eq 13 is used in the present work. The reaction between TPP* and BQ occurs at the effective reaction radius R′, and the validity of the fixed-distance reaction model, in which R′=R, was demonstrated in the previous Section. In this case, and with R ≈ rF + rQ, eq 13 can be reduced to the following expression:

(12)

The line of best-fit is shown in Figure 4 and the recovered value for L is 0.26 nm (L = 2b). This value for L is consistent with the value of ∼0.2 nm usually considered typical for electron-transfer reactions.17,22,25,52 Although the values for R and W0 cannot be extracted individually from this analysis because they are combined within the best-fit value of parameter a in eq 12, values for these parameters are estimated in the following section. Hydrodynamic Effects. As mentioned above, a particular point of merit of the analysis of the quenching kinetics on a nanosecond time scale according to eq 9 is that values for D can be obtained directly. Interestingly, when compared with the independently measured or estimated sum of the bulk diffusion coefficients of the reactant pair, DFQ (= DF + DQ), values of D obtained in this way are found to be smaller by as much as 40− 70%. The same trend was found in the present work. The origin of this apparent discrepancy has not been fully resolved and warrants further investigation. Of particular interest is the question as to whether this difference can be explained on the basis of hydrodynamic effects. The recovered values of D plotted as a function of measured values of DFQ (see Table 1) are shown in Figure 5. According

⎛ 3rFrQ ⎞ ⎛ l ⎞ ⎟ ≈ D D ≡ D(r = R′) = DFQ ⎜1 − ⎟ FQ ⎜1 − ⎝ ⎝ R′ ⎠ R′2 ⎠ (14)

From the slope of the line in Figure 5, and with R′ = 0.54 nm, the best-fit value for the product rFrQ is 0.025 nm2. With rF + rQ = 0.54 nm, and given rF > rQ, it is then possible to solve for rF and rQ to yield values of 0.49 and 0.05 nm, respectively. The calculated hydrodynamic radius for TPP of 0.49 nm is in very good agreement with the calculated van der Waals radius of 0.51 nm. However, the value calculated for the BQ hydrodynamic radius is smaller than the van der Waals radius by a factor of around 6. This suggests that the reaction rate may be highly anisotropic with respect to the relative orientations of the reactants and/or possibly highlights the likely inadequacies of classical hydrodynamic model in describing the local solvent structure experienced by the reactants when they are essentially at contact.55 The fluorescence quenching reaction kinetics for a reaction having a distance-dependent reaction rate probes the microenvironment of the reactants when they are at some distance remote from each other. Burshtein et al.44 showed that when the distance dependence of electron transfer follows eq 2, the value of R′ determined from experiment can be considered to be a good estimate of the root-mean-square distance between the charged species formed upon electron transfer. Since in this case R′ represents the most probable separation distance for electron transfer, it is reasonable to conclude that the most probable intermolecular separation distance at which the value of D is determined from analysis of the fluorescence decay quenching kinetics is r = R′. An assumption implicit in the use

Figure 5. Values of D plotted as a function of the bulk relative diffusion coefficients, DFQ, using the TPP/BQ data in Table 1. The line through the data points is the line of best-fit calculated using eq 14 where the best-fit gradient is 0.74.

to classical hydrodynamics theory, for the case where the solute molecules are larger than the surrounding solvent (the so-called “stick” boundary conditions), the distance dependence of the relative diffusion coefficient is given by eq 13, where r is the intermolecular separation distance, rF and rQ are the hydrodynamic radii of the reactants, and rF + rQ ≤ R.15,53 3rFrQ ⎛ l⎞ D ≡ D(r ) = DFQ ⎜1 − ⎟ , with l = ⎝ r⎠ rF + rQ (13) A different mathematical expression for D has been proposed for cases in which solute and solvent molecules are of comparable size (the so-called “slip” boundary condition).45,54 It can be seen from eq 13 that at sufficiently small reaction distances, hydrodynamic effects are predicted to result in the magnitude of the experimentally determined diffusion coefficient D being appreciably smaller than the sum of the bulk 2775

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The Journal of Physical Chemistry A of eq 9 for the analysis of the fluorescence decay kinetics associated with a reaction having a distance-dependent rate is that the magnitude of the reaction rate is finite over a sufficiently narrow distribution of distances around the rootmean-square reaction distance R′ that the distance dependence of D can be considered to be negligibly small. The full-width at half-maxima of charge distribution as a function of distance for diffusion-influenced electron-transfer reactions in solution have been reported to be of the order of 0.2 nm.16,19,21,44 On the basis of eq 13, and using the van der Waals values radii of TPP and NQ, the variation in D(r) within a distance of ±0.1 nm around r = R′ is ≤13% (depending on the magnitude of R′), which is considered to be sufficiently small to be within the experimental uncertainty of the results obtained from the decay curve analysis in the present work. In the case of the fluorescence quenching reaction kinetics for the reaction between TPP* and NQ, the expected diffusion coefficient values taking into account hydrodynamic effects, D(r = R′), for a given value of rQ can be calculated by substituting the following values into eq 15; rF = 0.49 nm (the value obtained for TPP/BQ), r = R′ (using the R′ values given in Table 1), and the DFQ values given in Table 1. Values of D(r = R′) calculated in this way for various values of rQ are shown in Figure 6. ⎞ ⎛ 3rFrQ ⎟⎟ D ≡ D(r = R′) = DFQ ⎜⎜1 − (rF + rQ )R′ ⎠ ⎝

the magnitude of this parameter reported for other electrontransfer reactions.4,17,23 Substitution of the values for W0, R, and L obtained for the TPP*/NQ system into eq 7 yields a value for kact of 3.2 × 1010 M−1 s−1, which is around an order of magnitude smaller than the lower-limit of kact estimated for TPP*/BQ reflecting the much closer proximity achieved by the latter reaction pair. Energetic and Steric Effects on Mechanism. The freeenergy for diabatic electron-transfer is given by the following equation TPP TPP Q ΔG = Eox − E00 − Ered +

ETPP ox

z1z 2e 2 4πε0εsR

(16)

EQred

where and are the oxidation potential of TPP and the reduction potential of the quinone quencher, respectively, ETPP 00 is the energy of the 0−0 transition from ground-state to the singlet excited-state of TPP, z1 and z2 are the charges on the product ions, e is the elementary charge, ε0 is the vacuum permittivity, εs is the solvent dielectric constant, and R is the distance at which electron transfer occurs. The experimentally BQ NQ TPP determined values for ETPP ox , Ered , Ered , and E00 in benzonitrile (BN) are 0.56 V, −0.93 V, −1.15 V (vs Fc/Fc+), and 1.9 eV, respectively, and the dielectric constant of BN is 26. Using the values of R given above, the calculated values of the free energy for electron transfer for TPP/BQ and TPP/NQ in benzonitrile, ΔGBN, are −0.51 and −0.26 eV, respectively. Following the work of Murata and Tachiya,56,57 the free energy for electron transfer in nonpolar solvents, ΔGNP, can be estimated from ΔGBN using eq 17, in which εNP and εBN are the dielectric s s constants for the nonpolar solvent and BN, respectively, and σF and σQ are taken as the van der Waals radii of the fluorophore and quencher, respectively.

(15)

ΔGNP = ΔGBN +

e2 ⎛ 1 1 ⎞⎛ 1 1 2⎞ ⎜ NP − BN ⎟⎜⎜ + − ⎟⎟ σQ 8πε0 ⎝ εs R⎠ εs ⎠⎝ σF (17)

εNP s

The value of is ∼2 for the nonpolar solvents used in this work. Using the value for εNP of 1.88 for hexane in this s equation, the reaction distances, R, at which electron transfer will occur (i.e., −ΔGNP = 0 eV) for the TPP*/BQ and TPP*/ NQ systems are 0.48 and 0.45 nm, respectively. Murata and Tachiya showed56 that the energy stabilization resulting from strong coupling with the D*A state when the reactants are in close proximity means that charge transfer to form an exciplex can actually occur at separation distances somewhat greater than these. Our finding that the electron-transfer reaction in the TPP*/BQ system occurs at a distance of 0.54 nm, which, as mentioned earlier, is considerably closer than the sum of their van der Waals radii and indicative of a requirement for cofacial orientation, is strong evidence that the origin of the observed fluorescence quenching in this system is exciplex formation. Fluorescence from the exciplex was not detected, suggesting that its lifetime is very short. In contrast to the TPP*/BQ system, the distance-dependence of the electron-transfer between TPP* and NQ in nonpolar solvents is consistent with the reactants being prevented from approaching sufficiently close for the inner-sphere mechanism to be operative. This results in a weakly coupled outer-sphere electron-transfer mechanism presumably involving a loosely bound or solvent-separated exciplex, although it is noted that evidence has been reported57 for full long-range electron transfer in nonpolar solvents under certain conditions. As

Figure 6. Values of D (●) plotted as a function of the bulk relative diffusion coefficients, DFQ, using the data for the TPP*/NQ system shown in Table 1. Also shown are the values for D(r = R′) calculated according to eq 15 using rF = 0.49 nm, DFQ and R′ values from Table 1, and the following rQ values: 0.22 nm (□; 0.29), 0.32 nm (△; 0.91), and 0.42 nm (◊; 0.68), where the numbers inside the parentheses are the respective correlation coefficients.

Solving graphically, the best correlation between the values for D(r = R′) and D occurs at a value for rQ of 0.32 nm, which is in excellent agreement with the van der Waals radius of 0.32 nm calculated for NQ.47,48 Also evident from the data in Figure 6 is the poor correlation obtained between the values for D(r = R′) and D when values for rQ differing appreciably from the van der Waals radius are used in eq 15. On the basis of the very good correlation between the values for D(r = R′) and D evident in Figure 6, the reaction radius for TPP and NQ is R = rF + rQ = 0.81 nm. It is then possible to calculate the value of W0 for the TPP/NQ reaction pair using the best-fit values for parameter a shown in the caption of Figure 4. The value of W0 calculated in this way is 3.6 × 1010 s−1 and is comparable with 2776

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mentioned earlier, this dominance of the remote electrontransfer path in the TPP*/NQ system is consistent with the greater steric hindrance associated with NQ compared with BQ.

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Article

ASSOCIATED CONTENT

S Supporting Information *

Absorption and fluorescence spectra; steady-state fluorescence quenching results; exciplex formation computation. This material is available free of charge via the Internet at http:// pubs.acs.org.

CONCLUSION

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It has been demonstrated in this work that key reaction parameters for the electron-transfer reactions between TPP* and BQ or NQ, including the local relative diffusion coefficients and reaction radii, can be obtained directly by using widely available nanosecond time-resolved fluorescence instrumentation and data analysis software, and by harnessing the utility of translational diffusion. The absence of any discernible viscosity dependence for the electron-transfer reaction distance between TPP* and BQ provides compelling evidence for a contact reaction, consistent with the formation a tightly associated exciplex in the set of alkane solvents used in this work. In contrast, the recovered values from the fluorescence decay analysis for the reaction between TPP* to NQ display a clear distance dependence, which implies reaction proceeds via a much looser noncontact exciplex. Analysis of this distance dependence using the so-called “slow-diffusion” approximation yielded values for the intrinsic diabatic electron-transfer rate (W0) and distance-attenuation factor (2/L) of 3.6 × 1010 s−1 and 7.7 nm−1, respectively, which are in very good agreement with the generally accepted range of values for this type of reaction. The noncontact nature of the reaction between TPP* and NQ reaction is attributed to the greater steric hindrance associated with very close approach of the TPP*/NQ pair compared with TPP*/BQ. The previously reported trend of the magnitude of the diffusion coefficients recovered from fluorescence decay curve analysis being substantially smaller than the corresponding measured bulk relative diffusion coefficients was, again, observed in the present work. Classical hydrodynamics theory was found to provide a satisfactory explanation for this apparent discrepancy, consistent with fluorescence decay analysis providing a probe for the local diffusion of the reactants when they are in sufficiently close proximity to react. The values for the hydrodynamic radii of TPP and NQ obtained are 0.49 and 0.32 nm and correlate very well with the respective calculated van der Waals values. However, the hydrodynamic radius obtained for BQ of 0.05 nm is some 6 times smaller than the calculated van der Waals value. The origin of this discrepancy is unclear at present, but possible contributing factors could include the large deviation of the shape of BQ from the spherical geometry assumed in classical theory or a highly anisotropic reaction rate arising from the cofacial geometrical arrangement in the tightly associated (TPP+/ BQ−)* exciplex. The approach described in the present work conveniently avoids pitfalls associated with adopting more complex analytical strategies that necessitate the fixing of parameter values used in the decay curve analysis and the complications associated with using covalently linked reactants or highly concentrated solid samples. The approach illustrated here is expected to be broadly applicable to direct determination of kinetic parameter values for a wide variety of rapid bimolecular fluorescence quenching reactions in fluid solution.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Mr. Takashi Nishihara for his help in CV measurements and Ms. Airi Yoshida for her help in the measurements of absorption and fluorescence spectra. This work was partially supported by Matching Fund Subsidy for Private Universities from MEXT.

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