3.4 Kinetic Analysis

Having assigned the five transients bands toPy2•+, DCB^•−, and ACN2−, we now examine their time profiles (Figure III–2d–h) in more depth. The observation that Py2•+ and DCB^•− show markedly different decay kinetics (compare, e.g., Figure III–2e,f) seems counter-intuitive because it is compelling to think that those carriers with opposite charges should decay in unison, irrespective of whether geminate (first-order reaction) or non-geminate (second-order reaction) recombinations take place, in order to maintain charge balance.

Scheme III–1. Top: DCB^•− ejects an electron into the solvent ACN, resulting in the formation of neutral DCB and ACN2−. Bottom: Py2•+ captures the excess electron in ACN2−.

To account for the observed asynchronous decay kinetics in the Py–DCB system, we propose here a mechanism of the BET reaction between Py2•+ and DCB^•− in which ACN plays a pivotal role as a charge mediator (Scheme III–1). There are two possible decay channels for DCB^•−: (1) the recombination with Py2•+ and (2) ejection of the electron into the solvent leading to the formation of ACN2− (Scheme III–1, top). In case 1, DCB^•− would decay concurrently with Py2•+. The observed time profiles contradict this prediction, suggesting that case 2, i.e., the electron ejection into the solvent, is the major route of the DCB^•− decay. It follows a pseudo-first-order reaction with rate constant k1′ (= k1[ACN]²). The preference for the electron ejection into ACN over the Py2•+/DCB^•− recombination may well be due to several factors, such as orbital symmetry mismatch between the HOMO of Py and the LUMO of DCB [62,63], the high capability of ACN to solvate electrons, and the abundance of ACN as solvent molecules ([Py]:[DCB]:[ACN] = 1:10:40000). The formation of ACN2− is clearly manifested as the initial rise of the transient at 11000 cm⁻¹ with a maximum at ~6 µs (see Figure III–2d). Because DCB^•− decays preferentially via the reaction with ACN, the only available decay pathway for Py2•+ that retains charge neutrality of the system is the interaction with ACN2−, which obeys a second-order rate law with rate constant k2 (Scheme III–1, bottom).

The resulting rate equations governing the kinetics of the system are given by

d[DCB^•−]/dt = − k1′ [DCB^•−] (III–1)

d[Py2•+]/dt = − k2 [Py2•+][ACN2−] (III–2)

d[ACN2−]/dt = k1′ [DCB^•−] − k2 [Py2•+][ACN2−] (III–3)

By solving Eqs. (III–1)–(III–3) with the initial conditions [Py2•+] = [DCB^•−] = C0 and [ACN2−]

= 0 at t = 0 and with the charge neutrality condition [Py2•+] = [DCB^•−] + [ACN2−] at any given time t, we are able to obtain the time-dependent changes in the concentrations of Py2•+ and DCB^•− as follows:

[DCB^•−] = C0 exp(−k1′t) (III–4)

(III–5)

where α = C0k2/k1′ and Ei(x) represents the exponential integral defined as _{Ei( )x} ^x _{e / d}^t _{t t}

=

∫

These solutions can be verified by substitution of Eqs. (III–4) and (III–5) into Eqs. (III–1) and (III–2). The IR absorbance difference at time t and wavenumber ν!, ΔA( !ν,t) , can be expressed as a linear combination of [DCB^•−], [Py2•+], and [ACN2−]:

(III–6) where a₁(ν!) , a₂(ν!) , and a₃(ν!) represent the amplitudes of the time-dependent concentrations of DCB^•−, Py2•+, and ACN2−, respectively. Due to the charge neutrality condition, Eq. (III–6) leads to

(III–7)

with b₁(ν!) = a₁(ν!) − a₃(ν!) and b₂(ν!) = a₂(ν!) + a₃(ν!).

To test the validity of our model we performed a global curve fitting analysis of the time-resolved spectra shown in Figure III–2a–c using Eq. (III–7) together with Eqs. (III–4) and (III–5). The parameters used in the fitting are the amplitudes of the two contributions [Py2•+] and [DCB^•−] in the linear combinations (i.e., b1 and b2 in Eq. (III–7)), k1′, and α. The global analysis yields k1′ = (1.1 ± 0.2) × 10⁵ s⁻¹ and α = 0.4 ± 0.1. The smooth curves in Figure III–

2d–h are best fits obtained with these parameters, from which it is clear that the kinetic analysis based on our proposed reaction mechanism (Scheme III–1) could successfully reproduce the observed time profiles of the five transient bands. A good agreement between experiment and simulation is also found in the entire spectral window studied, as shown in Figure III–5.

[Py₂ⁱ⁺] = C₀exp[α(1− exp(−k₁!t))]

1+αexp(α)[Ei(−α) − Ei(−αexp(−k₁!t))]

ΔA( !ν,t) = a₁(ν!^)[DCBi−]+ a₂(ν!^)[Py₂ⁱ⁺]+ a₃(ν!^)[ACN₂^−]

ΔA( !ν,t) = a₁(ν!)[DCBⁱ⁻]+ a₂(ν!)[Py₂ⁱ⁺]+ a₃(ν!)([Py₂ⁱ⁺]−[DCBⁱ⁻])

= b₁(ν!^)[DCBi−]+ b₂(ν!^)[Py₂^i+]

Figure III–5. Two-dimensional (2D) plots of the observed time-resolved spectra (a), the fitted result (b), and the residue (c). Each of the three 2D plots is represented in a rainbow pseudo color scale: the highest ΔA value appears red and the lowest appears purple. Note that the maximum ΔA value is 5 × 10⁻⁵ in (a) and (b) and 5 × 10⁻⁶ in (c).

The above global fitting alone does not allow us to determine C0 and k2 independently, because they appear together as parameter α in the fitting functions (see Eqs. (III–4) and (III–

5)). However, we are able to provide estimates for the values of C0 and k2. Approximately 1%

of Py molecules are calculated to be photoexcited under the present excitation conditions, so

the upper limit of the initial concentration C0 is 5 × 10⁻⁶ M. Given the reported quantum yield of 0.38 of the ion-pair (Py^•+ and DCB^•−) formation in ACN [54], the value of C0 is most likely of the order of 10⁻⁶ M, resulting in k2 ≈ 10¹⁰ M⁻¹ s⁻¹.

III-3.5 Concentration Dependence Studies of the BET Reaction between Py and DCB in ACN Solution

In this section, concentration dependence of the BET reaction between Py and DCB is examined. As the formation of Py2•+ and recombination reaction between Py2•+ and ACN2− are both second-order reactions, diffusion plays a prominent role. Thus it will be useful to test the proposed reaction scheme (refer to Section III-3.4) at various different concentrations. We measured TRNIR/MIR spectra of 5.0 mM Py and 5.0 mM DCB, where the concentration of Py was increased by one-fold and that of DCB was kept the same. Lowering the concentration of Py or DCB was not feasible as the intensity of the vibrational bands recorded at 0.5 mM Py and 5.0 mM DCB are close to the detection limit of the present apparatus.

Figure III–6 Left: Time-resolved IR spectra in the regions 3800–12000 cm⁻¹ (a), 2000–2200 cm⁻¹ (b), and 1120–1260 cm⁻¹ (c), of Py and DCB dissolved in ACN (5.0 and 5.0 mM, respectively) excited at 355 nm with Ar bubbling. Each time-resolved spectrum is offset by 2

× 10⁻⁴ for clarity of display. Right: Time profiles at 11000 (d), 6800 (e), 2100 (f), 1216 (g), and 1136 (h) cm⁻¹. The smooth black curves are the result of the kinetic analysis based on the reaction mechanism shown in Scheme III–1.

The TRNIR/MIR spectra recorded at the higher concentrations of Py and DCB (Figure

11000 6800 2100 1216

concentrations (Figure III–2). However, there are appreciable differences in terms of band intensity and shape. The intensity ratio of the 2100 cm⁻¹ band to the 1216, 1136, and 11000 cm⁻¹ bands increases compared to the lower Py-concentration experiment. This increase could be due to the increase in the concentration of Py2•+ relative to ACN2–. In addition, the NIR transient at ~7000 cm⁻¹ exhibits two peaks at ~7400 and ~6000 cm⁻¹. This change in the band shape of the electronic band of Py2•+ could be due to the occurrence of a side reaction that is not taken into account in Scheme III–1. The concentration dependence is more evident in the kinetic behavior of the transients (Figure III–6 d, e, f, g and h). The 6800 cm⁻¹ band (Figure III–6e) shows a slower rise compared with the other transients (Figure III–6 d, f, g and h) at the higher Py concentration than at the lower concentration (Figure III–2e). The synchronous decay seen for the 6800, 1216, and 1136 cm⁻¹ bands at the lower concentration (Figure III–2e, g and h) is no longer observed at the higher concentration (Figure III–6e, g and h). At the higher Py concentration, BET appears to proceed through a different scheme. To confirm this, a global fitting analysis is performed on the kinetic traces of the transients using Scheme III–

1. The reaction scheme is found to fail to reproduce the experimental data, strongly suggesting that a different scheme should apply at the higher Py concentration. The BET reaction mechanism that can account for the TRNIR/MIR spectra at 5.0 mM Py and 5.0 mM DCB needs further investigation.