Contact and Distant Luminescence Quenching in Solutions

Abstract

The limitations and advantages of modern encounter theories of remote transfer are discussed, as well as their application to particular transfer reactions assisted by encounter diffusion. Comparison is made with contact multiparticle theories, Brownian dynamic simulations, and the actual experimental data requiring a distant description of energy and/or electron transfer.

1. Introduction

Non-Markovian chemical kinetics started a century ago based on the famous work of Smoluchowski [1] who studied the encounter diffusion of spherical particles reacting at any contact. The model implied that the reaction is accomplished as soon as the intersphere distance 𝑟 reduces to the size of their diameter, 𝜎. A typical example of such a reaction is the irreversible quenching of excited molecule 𝐴^∗ by the impurity 𝐵, stopping abruptly the luminescence of 𝐴^∗:(1)

The time-dependent reaction constant established by Smoluchowski, 𝑘_𝐴(𝑡)=4𝜋𝜎𝐷[1+𝜎²𝜋𝐷𝑡], (2) makes the differential kinetic equation non-Markovian: 𝑁̇^∗_𝐴=−𝑘_𝐴(𝑡)𝑐𝑁^∗_𝐴−𝑁^∗_𝐴𝜏_𝐴. (3) It becomes the conventional Markovian equation of chemical kinetics only over long times when 𝑘_𝐴(𝑡) approaches the constant (stationary) value lim_𝑡→∞𝑘_𝐴(𝑡)=4𝜋𝜎𝐷≡𝑘_𝐷. This is the diffusional rate constant linear in encounter diffusion coefficient 𝐷.

In reality, not any contact of reactants results in complete quenching of 𝐴^∗. Being less efficient, such contacts are characterized by a phenomenological parameter 𝑘_𝑎, known as the kinetic rate constant. It was first introduced in the extended contact theory of Collins and Kimball [2] wherein the Markovian (stationary) limit of 𝑘_𝐴(𝑡) is𝑘=lim_𝑡→∞𝑘_𝐴𝑘(𝑡)=^𝑎𝑘_𝐷𝑘_𝑎+𝑘_𝐷. (4) However, such a generalization did not overcome the contact nature of the theory that could be scarcely applied to the distant electron transfer reactions and even more so to the long distant energy transfer.

Only an original differential encounter theory (DET) elaborated in the late 1960s overcame the contact limitations of the Smoluchowski-Collins-Kimball model [3–5]. Instead of the kinetic rate constant 𝑘_𝑎, it incorporates into the theory the noncontact, space-dependent rate 𝑊_𝐴(𝑟). The latter determines 𝑘_𝑎 as well as the whole non-Markovian kinetics of quenching which becomes exponential (Markovian) at later time. The stationary quenching constant 𝑘=4𝜋𝑅_𝑄𝐷 and its effective quenching radius 𝑅_𝑄 are also determined by 𝑊_𝐴(𝑟) and 𝐷. DET had a tremendous success in the treatment of luminescence quenching by irreversible energy or electron transfer [6]. Assuming an exponential shape of 𝑊(𝑟) [7], the theory attempted to fit the real viscosity dependence of the quenching radius 𝑅_𝑄(𝐷) [8].

Although DET covers only the irreversible transfer, it is known to be exact in the particular case of the target problem (immobile 𝐴^∗ quenched by independently moving point 𝐵's) [9]. In the simplest case the time-dependent reaction constant of DET substituted for its contact analog (2) has the following definition:𝑘_𝐴(𝑡)=𝑊_𝐴(𝑟)𝜈(𝑟,𝑡)𝑒^{𝑡/𝜏𝐴}𝑑³𝑟. (5)Here 𝜏_𝐴 is the excitation life time and 𝜈(𝑟,𝑡) is the pair correlation function of the reactants, obeying the equation for encounter diffusion accompanied by a distant reaction with the rate 𝑊_𝐴(𝑟):̇1𝜈=−(𝜏_𝐴+𝑊_𝐴)𝜈+𝐷Δ𝜈. (6)At later times, the theory becomes Markovian, that is, 𝑘_𝐴(𝑡)→𝑘=4𝜋𝑅_𝑄𝐷 which is the conventional (time independent) rate constant.

At the same time, DET failed to describe the reversible reaction between the meta-stable reactants, 𝐴^∗ and 𝐵^∗. When their lifetimes are different (𝜏_𝐴≠𝜏_𝐵), one of the DET “rate constants” diverges. To avoid such a divergency, the alternative integral encounter theory (IET) was developed [10–12] and applied to the reaction of quasiresonance energy transfer [13–15]. Since IET is a kind of memory function formalism, the time-dependent rate constants of DET give way to the memory functions (the kernels of the integral terms) of IET equations not ever turning to the Markovian ones even at later times. On the other hand, the stationary reaction characteristics such as the fluorescence yield or the yields of reaction products are even easier to calculate with IET. The Laplace transformation turns the integral equations into algebraic ones which can be solved analytically. However, IET was known from the very beginning [13–16] as a theory for the lowest concentration of reactants, 𝑁_𝐴 and 𝑁_𝐵. Some of the multiparticle effects accounted for by DET are lost in IET.

To minimize this drawback, a modified version of the integral theory (MET) was developed [17–23]. Creation of MET extended IET to higher concentrations keeping its advantages in accounting for the reaction reversibility but eliminating some of the weaknesses demerits in the asymptotic description of the long time kinetics. The contemporary matrix versions of IET [14, 15] and MET [24–26] allow considering reactions of any complexity and arbitrary number of internal electronic states of the reactants and products including their spin states.

In parallel, great progress was made in extending the contact reaction model to higher concentrations of reacting spheres. A number of the multiparticle Kernel theories, MPK1, MPK2, MPK3, were developed [27–29] as well as the superposition approximation [30–33] (SA) and some others. More recently, the relaxation time approximation (RTA) was developed which revives the archaic Smoluchowski approach by accounting for triple encounters: the pair AB with one more particle B (“bachelor” intervening in the family business). Although original, the self-consistent RTA (SCRTA) [34] does not add very much to more fundamental multiparticle theories in describing the high concentration effects. Even less warranted are the latest attempts to find better phenomenological constants for RTA, verifying the choice by how it agrees with Brownian dynamic (BD) simulations of reaction kinetics [35].

The principle difference between encounter theories and the inferior Smoluchowski-like approach is in their ability to account for the internal states of the reactants (including spin-multiplicity of the pairs) and the distant dependence of the transfer rates or corresponding Hamiltonian. To emphasize this point this review has the following structure. A brief overview of the integral theory of intra- and intermolecular transfer and later IET modifications is given in Section 2. Also the long time reaction kinetics of contact reactions studied with IET, MET and competing multiparticle theories is compared there with the latest BD simulations. Then we consider the quantum yields of the luminescence, reversible exciplex formation and electroluminescence (Section 3), focusing attention on their concentration dependence and difference between 𝛿-pulse and stationary excitation. Finally, we turn to the spin assisted reactions (Section 4) and the remote electron transfer (Section 5) using the integral and original unified theory (UT) for fitting the actual experimental data collected in these sections.

2. Integral Encounter Theories

Among numerous applications of the integral theory, the reversible reactions of intermolecular or intramolecular energy transfer are the simplest ones. To demonstrate clearly the main features of the integral (memory function) formalism it is better to start from them.

2.1. Intermolecular Energy Transfer

The transfer first studied with IET had the following reaction scheme [13]: (7) To make the problem even easier let us assume that the nonexcited partners are present in great excess ([𝐴]=𝑁_𝐴≫𝑁^∗_𝐴=[𝐴^∗] and [𝐵]=𝑁_𝐵≫𝑁^∗_𝐵=[𝐵^∗]). Then the terms in the kinetic equations which are second order in the excited state concentrations, 𝑁^∗_𝐴(𝑡) and 𝑁^∗_𝐵(𝑡), are negligible and the set of two IET equations can be linearized in the particle densities [15]:𝑁̇^∗_𝐴=𝑁_𝐴^𝑡₀⟨𝑆(𝑡−𝜏)⟩𝑁^∗_𝐵(𝜏)𝑑𝜏−𝑁_𝐵^𝑡₀⟨𝑅(𝑡−𝜏)⟩𝑁^∗_𝐴𝑁(𝜏)𝑑𝜏−^∗𝐴𝜏_𝐴,𝑁̇^∗_𝐵=−𝑁_𝐴^𝑡₀⟨𝑆(𝑡−𝜏)⟩𝑁^∗_𝐵(𝜏)𝑑𝜏+𝑁_𝐵^𝑡₀⟨𝑅(𝑡−𝜏)⟩𝑁^∗_𝐴𝑁(𝜏)𝑑𝜏−^∗𝐵𝜏_𝐵. (8)This set should be solved with the initial conditions𝑁^∗_𝐴(0)=𝑁₀,𝑁^∗_𝐵(0)=0, (9)provided that 𝐴^∗ was created by 𝛿-pulse excitation at 𝑡=0.

In any memory function formalism the main problem is how to define the kernels “memory functions” of integral equations. The great advantage of IET is the absence of such a problem. The space-averaged kernels of the integral equations (8) are defined from first principles by their Laplace transformations:⟨𝑆^{0𝑥002𝑑𝑐}1(𝑠)⟩=(𝑠+𝜏_𝐵)[𝑊_𝐵(𝑟)𝜇^{0𝑥002𝑑𝑐}₃(𝑟,𝑠)−𝑊_𝐴(𝑟)𝜈^{0𝑥002𝑑𝑐}₃(𝑟,𝑠)]𝑑³⟨𝑟,_{0𝑥002𝑑𝑐}𝑅1(𝑠)⟩=(𝑠+𝜏_𝐴)[𝑊_𝐴(𝑟)𝜈^{0𝑥002𝑑𝑐}₂(𝑟,𝑠)−𝑊_𝐵(𝑟)𝜇^{0𝑥002𝑑𝑐}₂(𝑟,𝑠)]𝑑³𝑟. (10)The auxiliary equations for the dyads of pair distributions 𝜈₃,𝜇₃, and 𝜈₂,𝜇₂ are the same:̇1𝜈=−(𝜏_𝐴+𝑊_𝐴)𝜈+𝑊_𝐵̇1𝜇+𝐷Δ𝜈,𝜇=−(𝜏_𝐵+𝑊_𝐵)𝜇+𝑊_𝐴𝜈+𝐷Δ𝜇. (11)Only the initial conditions for them are different:𝜈₂𝜇(𝑟,0)=1,₂𝜈(𝑟,0)=0;₃𝜇(𝑟,0)=0,₃(𝑟,0)=1. (12)The global structure of IET formalism is similar to that of DET. Although the kinetic equations are integral, their kernels are defined via the same transfer rates, 𝑊_𝐴(𝑟) and 𝑊_𝐵(𝑟), and the pair distribution functions obey the auxiliary equations similar to that of DET.

It is very easy to solve (11) in the contact approximation when𝑊_𝐴𝑘(𝑟)=^𝑎𝛿(𝑟−𝜎)4𝜋𝜎²,𝑊_𝐵𝑘(𝑟)=^𝑏𝛿(𝑟−𝜎)4𝜋𝜎². (13)In highly polar solvents,𝑘_𝑎=𝑊_𝐴(𝑟)𝑑³𝑟,𝑘_𝑏=𝑊_𝐵(𝑟)𝑑³𝑟 (14)are the kinetic rate constants of the forward and backward transfer. They relate both the kernels⟨𝑅^{0𝑥002𝑑𝑐}(𝑠)⟩=𝑘_𝑎Σ(𝑠),⟨𝑆^{0𝑥002𝑑𝑐}(𝑠)⟩=𝑘_𝑏Σ(𝑠), (15)to a single factor^{0𝑥002𝑑𝑐}𝑘Σ(𝑠)=[1+^𝑎/𝑘_𝐷1+(𝑠+1/𝜏_𝐴)𝜏_𝑑+𝑘_𝑏/𝑘_𝐷1+(𝑠+1/𝜏_𝐵)𝜏_𝑑]⁻¹=1,𝐹(𝑠) (16)where 𝜏_𝑑=𝜎²/𝐷 is the encounter time. With these kernels, the general IET equations (8) take the particular form specific only to the contact reactions [15]:𝑁̇^∗_𝐴=𝑘_𝑏𝑁_𝐴^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐵(𝜏)𝑑𝜏−𝑘_𝑎𝑁_𝐵^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐴𝑁(𝜏)𝑑𝜏−^∗𝐴𝜏_𝐴,𝑁̇^∗_𝐵=−𝑘_𝑏𝑁_𝐴^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐵(𝜏)𝑑𝜏+𝑘_𝑎𝑁_𝐵^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐴𝑁(𝜏)𝑑𝜏−^∗𝐵𝜏_𝐵. (17)

At a great excess of the 𝐵 molecule, one may neglect the reencounters of 𝐵^∗ with 𝐴 in the bulk. In this particular case, which attracted the most attention, there is actually only a single 𝐴 surrounded by 𝐵s. In such a situation, the general set (17) reduces to even a simpler one: 𝑁̇^∗_𝐴−𝑘_𝑎𝑁_𝐵^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐴(𝜏)𝑑𝜏−𝑁^∗_𝐴𝜏_𝐴,𝑁̇^∗_𝐵=+𝑘_𝑎𝑁_𝐵^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐴(𝜏)𝑑𝜏−𝑁^∗_𝐵𝜏_𝐵. (18a)(18b) These equations describe the quenching of the initially excited 𝐴, as well as accumulation of energy on 𝐵, its reverse transfer to 𝐴, and subsequent natural decay after the end of the encounter (Figure 1). From here on we should discriminate between the qualitatively different situations. (1)Energy quenching. This is a case when either the transfer is irreversible 𝑘_𝑏=0 or 𝜏_𝐵→0 so that nothing can be transferred backward.(2)Equal times. In a border case, 𝜏_𝐴=𝜏_𝐵=𝜏, the decay of excited states and energy redistribution between them proceed independently.(3)Energy storing. At 𝜏_𝐴<𝜏_𝐵 the initial quenching of 𝐴^∗ by the forward energy transfer results in energy storing by 𝐵's and the backward transfer from them to the short-lived 𝐴^∗ during the same encounter.

Figure 1 

The decay of the initially excited molecule, 𝑁∗𝐴(𝑡) and (A) accumulation/dissipation of transferred energy, 𝑁∗𝐵(𝑡), (B) reflected receipt and reimbursement of energy by a partner molecule [14].

To describe the energy quenching kinetics one can transform the integral equation (18a) to a differential one, (3), used in DET. This procedure is very delicate, especially since there is a power decay of the memory function. It is valid in a restricted time interval, where the main part of the quenching proceeds quasiexponentially with the time-dependent rate constant [14],𝑘_𝐴(𝑡)=𝑘_𝑎^𝑡₀Σ(𝑡^′)𝑒^{𝑡′/𝜏𝐴}𝑑𝑡^′, (19)that monotonously decreases up to the stationary (Markovian) value𝑘_𝑠=lim_𝑡→∞𝑘_𝐴=𝑘_𝑎^{0𝑥002𝑑𝑐}1Σ(−𝜏_𝐴). (20)

In fact, this value is unattainable because the reduction of IET to DET does not hold so long. The asymptotic behavior of 𝑁^∗_𝐴 in IET was shown to be the following [14]:𝑁^∗_𝐴(𝑡)𝑁^∗_𝐴=(0)exp(−𝑐^𝑡₀𝑘_𝐴(𝑡^′)𝑑𝑡^′−𝑡𝜏_𝐴(𝜏)at𝑡→0,_𝑑𝑡)^3/2𝑡exp(−𝜏_𝐴)at𝑡→∞. (21)However, the power time asymptotic quenching in IET was shown to be false at least at 𝜏_𝐵≤𝜏_𝐴 and was essentially corrected by the multiparticle theories reviewed below.

Qualitatively different are the results in the opposite, energy storing situation when 𝜏_𝐵>𝜏_𝐴. The rate constant (19) changes the sign with time and diverges while the “stationary constant” (20) becomes a complex quantity. In contrast, the effective IET analog of (19),𝑘_eff(𝑡)=−𝑑ln(𝑁^∗_𝐴𝑒^{𝑡/𝜏𝐴})𝑐𝑑𝑡, (22)can be specified using the solution to the original IET equation (18a). It does not diverge although it becomes negative at long times (Figure 2). This is just a result of changing the sign of the energy flux first directed from 𝐴 to 𝐵 but then turning back from 𝐵 to 𝐴. In other words, the initial decay of the excited state is faster than the natural one but finally becomes slower and obeys the asymptotic power time dependence over very long times [14]:𝑁^∗_𝐴𝑁₀𝜏=(^𝑑𝑡)^3/2𝑒^{−𝑡/𝜏𝐵}at𝜏_𝐴<𝜏_𝐵. (23)It should be noted that a similar power law time dependence is an exact result of the geminate reaction of reversible dissociation, 𝐴𝐵⇄𝐴+𝐵, first discovered by Berg [36] and re-derived afterwards [37]. This law was confirmed experimentally [38] and essentially generalized later on [39–41].

Figure 2 

The sign alteration of the effective reaction constant of IET (dashed line) and that of DET (solid line), in the case of energy storing (at 𝜏𝐵=∞, 𝜏𝐴=2𝜏𝑑) and 𝑘𝑎𝑐𝜏𝑑=0.1 [14].

2.2. Intramolecular Energy Transfer

Even a simpler example of a reversible reaction of the same sort provides the intersystem crossing 𝐴^∗⇄𝐵^∗ induced by an inert partner 𝐶 according to the following scheme: (24)Initially excited 𝐴^∗ as well as 𝐵^∗ are two electronic states of the same particle 𝐴 reacting with 𝐶 given in concentration 𝑐=const. In the contact approximation the IET equations take the form [26]𝑑𝑁^∗_𝐴𝑑𝑡=−𝑘_𝑎𝑐^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐴(𝜏)𝑑𝜏+𝑘_𝑏𝑐^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐵𝑁(𝜏)𝑑𝜏−^∗𝐴𝜏_𝐴,𝑑𝑁^∗_𝐵𝑑𝑡=𝑘_𝑎𝑐^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐴(𝜏)𝑑𝜏−𝑘_𝑏𝑐^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐵𝑁(𝜏)𝑑𝜏−^∗𝐵𝜏_𝐵, (25)where the Laplace transformation of Σ(𝑡) is given by (16). This set of equations is equivalent to the set (17) at 𝑁_𝐴=𝑁_𝐵=𝑐 but the reduction to (18a), (18b) is impossible and the backward transfer during reencounters is inevitable.

However, the numerical solution of (25) was an easy matter. This was done in [26] with the initial conditions 𝑁^∗_𝐴(0)=𝑁₀, 𝑁^∗_𝐵(0)=0 corresponding to the instantaneous (𝛿-pulse) excitation of state 𝐴. The results are shown in Figure 3. In the absence of 𝐶 there is only a spontaneous exponential decay of 𝐴^∗ (solid line in Figure 3(a)). The intramolecular energy transfer to 𝐵^∗ modulates this process. If the transfer is irreversible (𝑘_𝑏=0), it only facilitates the decay. This is the energy quenching proceeding exponentially but faster than the natural decay (dashed line in Figure 3(a)). At 𝑘_𝑏≠0, the energy comes back due to reencounters with 𝐶, supporting the delayed fluorescence from 𝐴^∗ if 𝜏_𝐵>𝜏_𝐴. Under such a condition, the dashed-dotted line representing 𝑁^∗_𝐴(𝑡) short after rapid quenching goes down much slower than the spontaneous decay. The smaller 𝑘_𝑏, the later, weaker and slower should be the delayed fluorescence. This energy storing is the most pronounced at 𝜏_𝐵=∞.

(a)

(b)

Figure 3 

Relaxation of state populations after instantaneous excitation of 𝐴 at 𝜏𝐵=∞ and 𝑐𝑘𝑎𝜏𝑑=1.0. (a) Spontaneous (exponential) decay of 𝐴∗ with lifetime 𝜏𝐴=2𝜏𝑑 (thick straight line) and its dissipation accompanied by intersystem crossing with the backward rates 𝑘𝑏=5.0𝑘𝐷 (dotted line), 𝑘𝑏=𝑘𝐷 (dashed-dotted line), and 𝑘𝑏=0 (dashed line). (b) Accumulation and dissipation of the initially empty state 𝐵∗, which decays through 𝐴∗ faster the higher is 𝑘𝑏, or does not decay at all if 𝑘𝑏=0 [26].

This picture is supplemented with the corresponding kinetics of energy accumulation in 𝐵^∗ and the subsequent decay of this state. The latter takes place even at 𝜏_𝐵=∞ due to the collision induced reverse transfer to 𝐴^∗ (Figure 3(b)). Such a decay is absent only at 𝑘_𝑏=0 and is faster, the higher is 𝑘_𝑏. At any 𝜏_𝐴<𝜏_𝐵, the energy is stored in a more stable excited state 𝐵^∗ but finally comes back to the short-lived excited state 𝐴^∗, causing the delayed fluorescence.

Between the opposite limits of energy quenching (𝜏_𝐴>𝜏_𝐵) and energy storing (𝜏_𝐴<𝜏_𝐵) there is a border case of equal times that deserves special attention. This exceptional case allows the exact multiparticle solution of the problem obtained in [42]. When 𝜏_𝐴=𝜏_𝐵=𝜏, the natural decay of both states can be excluded from consideration by a simple substitution:𝑁^∗_𝐴(𝑡)=𝒫_𝐴(𝑡)𝑒^−𝑡/𝜏,𝑁^∗_𝐵(𝑡)=𝒫_𝐵(𝑡)𝑒^−𝑡/𝜏. (26)The relative populations of excited states, 𝒫_𝐴(𝑡) and 𝒫_𝐵(𝑡), tend to their equilibrium values as 𝑡→∞:𝒫_𝐴1(∞)=1+𝐾=𝑃_𝐴,𝒫_𝐵𝐾(∞)=1+𝐾=𝑃_𝐵, (27)where 𝐾=𝑘_𝑎/𝑘_𝑏 is the equilibrium constant in a system of two excited states. The general solution of the problem can be represented as𝒫_𝐴1(𝑡)=+𝐾1+𝐾𝒫1+𝐾^eff(𝑡)=1−𝒫_𝐵(𝑡), (28)where 𝒫_eff(𝑡) was the subject of calculations in [42]. After the exact averaging over the multiparticle distribution of 𝐶 it was shown to be𝒫_eff(𝑡)=exp(−𝑐^𝑡₀𝑘(𝑡^′)𝑑𝑡^′) (29)with 𝑘(𝑡) obeying the original DET equations (5) and (6) but with substituting 𝑊_𝐴(𝑟)+𝑊_𝐵(𝑟) for 𝑊_𝐴(𝑟). In the contact approximation this leads to the conventional Smoluchowski-Collins-Kimball solution but with 𝑘_𝑎+𝑘_𝑏 substituted for 𝑘_𝑎.

2.3. Intermolecular Electron Transfer

Let us now turn to the reversible reaction of electron transfer, 𝐴^∗+𝐷⇄𝐴⁻+𝐷⁺ accompanied by charge recombination. After charge separation, the free ions recombine in the bulk to either the ground state or backward to the excited neutral products, which contribute again to the total fluorescence: (30) Such a complete scheme enables studying the luminescence quenching proceeding from left to right, as well as the electroluminescence (resulting from the recombination of injected ions) going from right to left.

The complex kinetics of the luminescence switched by 𝛿-excitation of 𝐴 in reaction (30) was studied in [43] using the general IET equations:𝑁̇^∗_𝐴=−𝑐^𝑡₀𝑅^∗(𝜏)𝑁^∗_𝐴+(𝑡−𝜏)𝑑𝜏_𝑡0𝑅^♯(𝜏)𝑃²𝑁(𝑡−𝜏)𝑑𝜏−^∗𝐴(𝑡)𝜏_𝐴+𝐼₀𝑁_𝐴,𝑃̇=𝑐^𝑡₀𝑅^†(𝜏)𝑁^∗_𝐴(𝑡−𝜏)𝑑𝜏−^𝑡₀𝑅^‡(𝜏)𝑃²(𝑡−𝜏)𝑑𝜏, (31)where 𝑃(𝑡)=[𝐴⁻]=[𝐷⁺] is the free-ion concentration and 𝑐=[𝐷]=const is the concentration of donors present in great excess. When the geminate recombination is completed and gives way to ion recombination, the latter restores either the ground or excited state of the neutral reactants. The fluorescence of the excited ones is delayed. In fact this is an electroluminescence though not from injected ions but from those which escaped the geminate recombination after photo separation. The density of restored excitations at a later stage of their decay is quadratic in the free-ion concentration that should be large enough to make the delayed fluorescence detectable. To reach this goal one has to use as strong pumping as possible and choose the fluorophors with rather long-lived excited states. Then their fluorescence at times 𝑡>𝜏_𝐴 will be stronger than in the absence of charge recombination.

The long time asymptote of the delayed decay can be described by the reduced equation (31), where the light pumping is absent and charge recombination is considered as irreversible in view of a negligible concentration of restored 𝐴^∗ [43]:𝑁̇^∗_𝐴=−𝑁^∗_𝐴𝜏_𝐴−𝑐^𝑡₀𝑅^∗(𝜏)𝑁^∗_𝐴+(𝑡−𝜏)𝑑𝜏_𝑡0𝑅^♯(𝜏)𝑃²̇(𝑡−𝜏)𝑑𝜏,𝑃=−^𝑡₀𝑅^‡(𝜏)𝑃²(𝑡−𝜏)𝑑𝜏. (32) The initial condition for free ions is given by the following equation:𝑃(0)=𝑁^∗_𝐴(0)(1−𝜂)𝜑=𝑃₀, (33)where the yield of photo-generated ions is 1−𝜂 and the yield of their separation 𝜑 is defined in (67).

Since ions are stable particles, those terms in the integral equations (32) which describe the ion recombination may be transformed at 𝑡≫𝜏_𝑑 into their differential and even Markovian analogs [13]. The same can be done with the remaining ionization term, but at a much longer time, 𝑡≫𝜏_𝐴, when the concentration of excitations levels decreases slowly following charge recombination. Hence, the delayed fluorescence can be described by the following set of differential (Markovian) equations:𝑁̇^∗_𝐴=−[1𝜏_𝐴+𝑐𝜅₀]𝑁^∗_𝐴(𝑡)+𝑘^∗𝑃²,𝑃̇=−𝑘^×𝑃², (34) where 𝑘^∗=𝑅^{0𝑥002𝑑𝑐♯}(0) and 𝑘^×=𝑅^{0𝑥002𝑑𝑐‡}(0). The delayed fluorescence can be obtained from the quasistationary solution of (34):𝑁^∗_𝐴𝑘(𝑡)=^∗𝑃²[1/𝜏_𝐴+𝑐𝜅₀]→𝐴𝑡²,𝑃𝑃=⁰1+𝑘^×𝑃₀𝑡→1𝑘^×𝑡, (35a)(35b)where𝑘𝐴=^∗[1/𝜏_𝐴+𝑐𝜅₀](𝑘^×)² (36)was calculated in the contact approximation in [43]. In the same paper the full kinetics of luminescence was shown to have a long tail approaching the delayed fluorescence asymptote (35a) (Figure 4). The latter screens the intermediate asymptote of the multiparticle quenching (21) which is known to be false in IET.

Figure 4 

The false asymptote of geminate excitation quenching (long dashed line) in comparison to the second power asymptote (dashed line) of true excitation decay (solid line) [43].

2.4. Modifications of Integral Theory

Unfortunately, the IET is the theory keeping only the lowest order terms in reactant concentrations. To account for the higher ones by MET, one only has to modify the kernels of the integral equations. In [24, 25] this was recommended to be done by the substitution of the generalized decay rates1𝜏^∗_𝐴=1𝜏_𝐴+𝑘_𝑎𝑁_𝐵^{0𝑥002𝑑𝑐}1Σ(𝑠),𝜏^∗_𝐵=1𝜏_𝐵+𝑘_𝑏𝑁Σ_𝐴^{0𝑥002𝑑𝑐}(𝑠), (37)for the inverse times 1/𝜏_𝐴 and 1/𝜏_𝐵 figuring in (16). Such an obtained MET was shown to correct IET even better than the first multiparticle kernel (MPK1) theory [27], which at first corrected the false IET asymptote of quenching. However, the authors of the MPK1 intuitively neglected some three-particle correlations in comparison with others, while MET accounts explicitly for all of them [24, 25]. An excellent analysis of the drawback in MPK1 and its consequences was recently presented by Ivanov [44]. Fortunately this drawback was later overcome by the same authors [29] making their last theory, MPK3, almost equivalent to MET [25].

MET essentially corrects IET over long times [44]. Figure 5 demonstrates the difference in time behavior between IET and MET effective quenching constants for the forward and reverse transfer,𝑘^eff_𝑓=𝑘_𝑎∫^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐴(𝜏)𝑑𝜏𝑁^∗_𝐴,𝑘(𝑡)_eff𝑟=𝑘_𝑏∫^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐵(𝜏)𝑑𝜏𝑁^∗_𝐵.(𝑡) (38)The former is the analog of (22) which is known from IET to become negative with time (Figures 2 and 5(a)) but in MET turns to be positive again at 𝑡≫𝜏_𝑑) (Figure 5(b)). At these long times the transfer proceeds not between geminate partners that have already separated but with many other 𝐵's that, being in the ground state, act as the fresh quenchers. As to the quenching constant for stable excitation 𝐵^∗, it remains positive at all times in both theories but only in MET approaches the truly stationary value 𝑘_𝑏Σ(0).

(a)

(b)

Figure 5 

The time effective rate “constants” for the forward and reverse transfer, 𝑘𝑓(𝑡)=𝑘eff𝑓/𝑘𝑎 (solid lines) and 𝑘𝑟(𝑡)=𝑘eff𝑟/𝑘𝑎 (dashed lines), calculated with IET (a) and MET (b) at 𝜏𝐵=∞, equal concentrations of 𝐴 and 𝐵 and equal kinetic constants 𝑘eff𝑓(0)=𝑘eff𝑟(0)=𝑘𝑎=10𝑘𝐷. The thin solid line shows the stationary value of 𝑘eff𝑟(∞) [44].

In the case of irreversible quenching (𝑘_𝑏=0), the kernel modification can also be done as was recommended earlier [19–23], by the straightforward substitutionΣ(𝑡)→Σ(𝑡)exp(−𝑐𝑘𝑡), (39)where 𝑘 is the stationary (Markovian) constant of the irreversible transfer (4). The difference between the two recipes of modification originates from the way in which the “point encounter approximation” was used to simplify the three-particle terms; only in space or also in the time domain [24]. However, this choice rather insignificantly affects the results. Performing point encounter approximation in the coordinate space one implies that the reaction pair evolution proceeds at distances larger than the reaction zone which can be ignored (see [24, Equations (6.15) and (6.16)]). This allows to derive the binary kinetic equation by the simplification of three-particle terms in concentration expansion keeping only their dependence on relative mobility of the reactants.

An original matrix reformulation of IET and its modification similar to (39) were presented in [45]. Such an obtained MET is applicable to the reversible reactions of any complexity and is in fact identical to that used in [17–21]. However, the authors did not restrict themselves to the integral form of the theory but transformed their matrix integrodifferential equation into a set of two coupled differential equations which can be solved easier numerically. They also developed a general computer code (available on the Internet [45]) and demonstrated it application to the well-known Lotka-Volterra reaction which is oscillatory with time [46].

2.5. Asymptotic Excitation Decay

The pre-exponential power time dependence of 𝑁^∗_𝐴(𝑡)/𝑁^∗_𝐴(0) indicated in (21) is known to be the false IET asymptote for the long time quenching. The latter can be easily seen in the case of irreversible transfer, rather than in the reversible case.

Assuming that the transfer from 𝐴^∗ to 𝐵 is irreversible (𝑘_𝑏=0), the quenching kinetics was studied in [45] by solving the original differential equations equivalent to either IET or MET. As can be seen from Figure 6, the difference between the curves representing these theories is insignificant within the validity limits for IET established in [17, 18]: 𝑡≪ln(𝜉⁻¹)/𝑘𝑐, (40)where 𝜉=(4/3)𝜋𝜎³𝑐. The validity region for MET is known to be much wider: 𝜉𝑡≪⁻¹𝑘𝑐. (41) This was confirmed in [45] by a straightforward comparison of the MET solution with that of DET:𝑃^∗_𝐴=𝑁^∗_𝐴𝑡exp(𝜏_𝐴)=exp[−𝑐^𝑡₀𝑘_𝐴(𝑡^′)𝑑𝑡^′]. (42)The latter which is exact for the target problem is not distinguishable from MET, represented by the solid line in Figure 6. The difference appears to be less than the precision of the numerical calculations, provided 𝑃^∗(𝑡)>𝑒^−1/𝜉, where 𝜉<1. On the contrary, the dashed curve representing IET significantly deviates from the exact result when the time exceeds the border (40) indicated by the vertical line in this figure. The false asymptote of IET is to the right of it. In other words the IET is valid until 𝑃^∗(𝑡)>𝜉.

Figure 6 

The quenching kinetics obtained with IET (dashed line) and MET (solid line). The vertical strip denotes the upper boundary of the region of IET validity [45].

As has been reported, the reversible intramolecular transfer between the meta-stable states (43) results in either energy quenching, if 𝑘_𝐶>𝑘_𝐴, or in energy storing, if 𝑘_𝐴>𝑘_𝐶. However, inspecting the accuracy of different ABCD theories, one cannot use DET as a standard because this theory is not applicable to the energy storing limit (due to the divergency of the time-dependent forward rate constant) [13–16]. The only alternative is to compare the theoretical results with those obtained by the BD simulation of the transfer kinetics.

Since these simulations are always done only for contact reactions between chemically isotropic hard spheres, they can be compared only with contact IET and MET, as well as with MPK, SA, SCRTA, and others, applied to the same Smoluchowsci-like model. Even though in such a primitive model only the concentration and time dependence of 𝑁^∗_𝐴 can be inspected it is yet useful and instructive. The numerous modifications of SCRTA were the main interest of the authors of the last BD simulations but some other theories were also included into comparison shown in Figure 7 [35]. The two upper panels are represented as they were published (in black and white), while the lower ones are just the colored originals of them placed at our disposal by one of the authors.

(a)

(b)

(c)

(d)

Figure 7 

The time dependence of the survival probability, 𝑁∗𝐴(𝑡), of initially excited 𝐴∗ for several values of the unimolecular decay constants, 𝑘𝐴 and 𝑘𝐶 (in ns−1, one is zero and the other is indicated in the figure). The parameters are the same everywhere, except that the concentrations in the left panels are low (𝑐𝐵=𝑐𝐷=10−3Å−4) being large in the right ones (𝑐𝐵=𝑐𝐷=10−3Å−3). In the colored panels, the BD simulations are presented by solid lines, while all the rest are depicted by symbols indicated in these panels.

We can see from this figure that the two variants of BD simulations, depicted by the open symbols: triangles (algorithm 1) and circles (algorithm 2), are actually indistinguishable. It can also be seen from the black and white panels that they agree perfectly with MPK3 and SCRTA but not as well as with MPK1 and even worse with IET. The black and white presentations of data are suitable to give the impression that IET is the worst among the other theories. To refute this conclusion it is enough to glance at the colored version of the same data.

IET deviates from the exact results only at a high concentration (right panels) where it is known to be inapplicable. In this sense, it is actually “the inferior to the MET” as stated in [35] but only in the same sense as the theory of ideal gases is the inferior to the Van der Waals theory. Moreover, even at large concentrations IET only fails to describe the energy quenching, 𝑘_𝐶=10(𝑘_𝐴=0), but not the energy-storing case, 𝑘_𝐴=10(𝑘_𝐶=0), where the false IET asymptote is screened (as in Figure 4) by delayed fluorescence. Of course, the coincidence of the IET results with BD simulations is excellent in any case when the concentration is low. Moreover, the expected imperfection of IET at high concentration is totally corrected by implication of MET ≡ MPK3.

In short, the false asymptote of IET comes to light only at high concentrations and only in the energy quenching case. This drawback of IET is completely removed by MET which accounts for the transfer not only to the partner in a pair but also to other surrounding reactants. By this way, MET corrects the high concentration behavior of all the quantum yields considered in Section 3.

3. Concentration Phenomena

3.1. Luminescence after Pulse Excitation

The quantum yield of the fluorescence following pulse excitation is∫𝜂=^∞0𝑁^∗_𝐴(𝑡)𝑑𝑡𝑁^∗_𝐴(0)𝜏_𝐴 (44)provided that the light excites instantaneously (at 𝑡=0) only 𝐴 and that the luminescence comes only from this particle. The concentration dependence of the yield is always represented by the Stern-Volmer law𝜂=^{0𝑥002𝑑𝑐}𝑁^∗_𝐴(0)𝑁₀𝜏_𝐴=11+𝑁_𝐵𝑘_𝑞𝜏_𝐴, (45)where 𝑘_𝑞 is the quenching constant which is to be investigated.

Let us start doing this from the reversible energy transfer studied in Section 2.1. Using the Laplace transformation of the IET equations (17), one can easily find from (45) that𝑘_𝑞=𝐾_𝐴𝐵1+𝐾_𝐵𝐴𝑁_𝐴𝜏_𝐵. (46)Here 𝐾_𝐴𝐵=𝑘Σ_𝑎^{0𝑥002𝑑𝑐}(0) and 𝐾_𝐵𝐴=𝑘Σ_𝑏^{0𝑥002𝑑𝑐}(0), that is,𝐾_𝐴𝐵𝑘_𝑎=11+(𝑘_𝑎/𝑘_𝐷)/(1+𝜏_𝑑/𝜏_𝐴)+(𝑘_𝑏/𝑘_𝐷)/(1+𝜏_𝑑/𝜏_𝐵)=𝐾_𝐵𝐴𝑘_𝑏. (47)This result obtained by means of IET is concentration-independent but known to be valid at only the lowest concentrations of 𝐴 and 𝐵 [13].

3.2. Irreversible Geminate Reactions

To demonstrate how DET and MET, as well as other multiparticle theories, correct this result at higher concentrations, let us focus upon the irreversible transfer setting𝐾_𝐵𝐴=𝑘_𝑏=0,𝑁_𝐵=𝑐. (48)In such a case, the theory becomes universal that is equally good for any irreversible quenching, including the parallel transfer of energy and electron [47] (49) or the double-channel electron transfer studied in [48, 49]: (50) All of them can be briefly represented by the unified scheme of the irreversible energy quenching: 𝐴^∗+𝐵→𝐴+𝐵.

The quantum yield of the irreversible quenching obeys the Stern-Volmer dependence on concentration 𝑐=𝑁_𝐵 represented in (45). As in the original Stern-Volmer law, the quenching constant obtained from (46) and (47) is concentration-independent:𝑘_𝑞=𝑘_𝑎1+𝑘_𝑎/𝑘_𝐷(1+𝜏_𝑑/𝜏_𝐴)=𝜅_𝑖. (51)However, contrary to 𝜅_𝑖, the experimentally found 𝑘_𝑞≡𝜅 is concentration-dependent and this is a challenge for the theory to find out how the true Stern-Volmer constant 𝜅(𝑐) differs from its IET analog 𝜅_𝑖=lim_𝑐→0𝜅(𝑐).

MET solves this problem substituting 𝜏_𝐴 in (51) by 𝜏^∗_𝐴(0) from (37):𝑘𝜅=^𝑎1+𝑘_𝑎/𝑘_𝐷(1+(𝜏_𝑑/𝜏_𝐴)[1+𝑐𝜅_𝑖𝜏_𝐴]). (52)In general, this 𝜅 is concentration-dependent unless the transfer is strongly under kinetic control when 𝜅=𝑘_𝑎≪𝑘_𝐷. In the alternative limit of diffusional transfer, MET gives𝜅=𝑘_𝐷(1+𝜏_𝑑𝜏_𝐴[1+𝑐𝜅_𝑖𝜏_𝐴]). (53)In [50] this result was compared with that of IET and other theories: SA [30–33] and DET (Figure 8). The Stern-Volmer constant of MET and SA differs a bit from the exact result represented by DET but all of them except IET increase with the dimensionless concentration 𝜉=𝑐𝑣 (𝑣=(4/3)𝜋𝜎³). At moderate concentrations this dependence is always linear,𝜅𝑘_𝐷=𝜅𝜅_𝑖+𝐴𝜉=1+𝜏_𝑑𝜏_𝐴+𝐴𝜉, (54)but the slope of it appears to be different. For MET obtained by modification (37), 𝐴=(3/2)𝜏_𝐴/𝜏_𝑑+3/2 while in the old MET originated from another modification, (39), 𝐴=(3/2)𝜏_𝐴/𝜏_𝑑. The former coincides with that of SA while the latter (shown in figure) is a bit smaller. The true DET value is in between 𝐴=(3/2)𝜏_𝐴/𝜏_𝑑+3(1−2/𝜋) [50]. Since the concentration correction in (54) is only actual for large 𝜏_𝐴≫𝜏_𝑑, the constant term in 𝐴 is insignificant. In the opposite case, the whole correction is negligible compared to the preceding IET term. Hence the difference between the concentration corrections at small 𝜉 is not essential.

Figure 8 

The Stern-Volmer constants as functions of the dimensionless concentration 𝜉=4𝜋𝜎𝑐3/3 obtained in the contact approximation and under diffusional control at 𝜏𝑑/𝜏𝐴=0.01. The thick line represents DET which is exact for the target problem (immobile donors and independently moving acceptors). The rest of the curves are obtained with SA, MET, and IET [50].

However, the comparison of all the theories (of contact multiparticle quenching by point particles 𝐵) was done later for any 𝜉 (Figure 9) [47]. Theoretically, 𝜅 increases with concentration from the IET value 𝜅_𝑖 up to the kinetic rate constant 𝑘₀≡𝑘_𝑎, though experimentally available is only the lower (left) part of the graph, 𝑐𝑣<1. Contact MET, which is identical to MPK3 developed later [29], underestimates the exact 𝜅(𝑐) represented by DET. The latter is equivalent to the irreversible version of the first multiparticle kernel theory, MPK1 [27]. The intermediate version of this theory [28], MPK2, as well as the latest model theory SCRTA [34], almost coincide with DET = MPK1 (irreversible), unlike the linearized extended superposition approximation [51] (LESA) which overestimates 𝜅(𝑐).

Figure 9 

The concentration dependence of the irreversible quenching Stern-Volmer constant 𝜅 in units of 𝑘0≡𝑘𝑎 for a number of contact theories, provided 𝑘𝑎=3.7⋅1010M−1s−1 is the same for all of them [47] and [48, Figure 3.88].

3.3. IET of Reversible Geminate Reaction

Let us now turn to the geminate reaction similar to that included in scheme (49) but carried out by reversible electron transfer: (55) Here 𝐷 is the donor of an electron, while 𝐴 is its acceptor. The excitation of 𝐴 by the short light pulse, 𝐼(𝑡), resulting in charge separation, produces the free ions, 𝐷⁺ and 𝐴⁻, with the yield 𝜙. This is the fraction 𝜑 of ions initially born in amount 1−𝜂 but separated escaping recombination to the ground state with the rate 𝑊_𝑅 and the backward electron transfer to excited products with the rate 𝑊_𝐵.

If 𝐴's are present in very low concentration, the density of the free ions is small as well, so that their recombination in the bulk is negligible during a bounded time domain comparable with encounter time 𝜏_𝑑∼𝜎²/𝐷. The IET equations (31) can be reduced for this case by omitting the bulk terms quadratic in 𝑃, as well as the pumping term:𝑁̇^∗_𝐴=−𝑐^𝑡₀𝑅^∗(𝜏)𝑁^∗_𝐴(𝑡−𝜏)𝑑𝜏−𝑁^∗_𝐴(𝑡)𝜏_𝐴,𝑃̇=𝑐^𝑡₀𝑅^†(𝜏)𝑁^∗_𝐴(𝑡−𝜏)𝑑𝜏. (56) Instead of the pumping term one has to add the initial conditions for these equations which represent the instantaneous excitation of 𝐴:𝑁^∗_𝐴(0)=𝑁₀,𝑃(0)=0. (57)

The luminescence of the pulse excited 𝐴^∗ quenched in a limited time has the yield specified by (44) which completely neglects the subsequent delayed fluorescence resulting from the bulk recombination. Such a yield obeys the Stern-Volmer law (45) but with the geminate quenching constant 𝑘_𝑞≡𝜅_𝑔. The latter is concentration-independent unlike its analog in (46) accounting for the backward energy transfer during bulk encounters.

Hence, the luminescence quenched by reversible ionization after instantaneous excitation has the yield calculated from (45) and (56) [52]:1𝜂=1+𝑐𝜅_𝑔𝜏_𝐴, (58)where the geminate Stern-Volmer constant is [43, 52]𝜅_𝑔=𝑅^{0𝑥002𝑑𝑐∗}𝜅(0)=^𝑖1+𝜅_𝑖/𝐾(𝑘_𝐷+𝑘_𝑐). (59)Here 𝜅_𝑖 is the Stern-Volmer constant for irreversible transfer (51) while𝑘𝐾=^𝑎𝑘_𝑏=exp(−Δ𝐺_𝑖𝑇) (60)is an equilibrium constant for ionization, with free energy Δ𝐺_𝑖 (here and further on the Boltzmann constant 𝑘_𝐵=1). The kinetic reaction constants of the outer-sphere electron transfer𝑘_𝑎=𝑘^′_𝑎𝑒^{−𝑢𝑎/𝑇},𝑘_𝑏=𝑘^′_𝑏𝑒^{−𝑢𝑏/𝑇} (61)obey the Arrhenius law with activation energies satisfying the free energy gap (FEG) law [9]:𝑢_𝑎=𝑢_𝑏+Δ𝐺_𝑖=(Δ𝐺_𝑖+𝜆_𝑐)²4𝜆_𝑐, (62)where 𝜆_𝑐 is the contact reorganization energy of the polar media. The charge recombination constant also obeys the FEG law:𝑘_𝑐=𝑘^′_𝑐𝑒^{−𝑢𝑐/𝑇}with𝑢_𝑐=(Δ𝐺_𝑟+𝜆_𝑐)²4𝜆_𝑐 (63)but with different free energy −Δ𝐺_𝑟=ℰ+Δ𝐺_𝑖 where ℰ is the excitation energy of 𝐴^∗.

For highly exergonic charge separation (−Δ𝐺_𝑖>𝑇) when 𝐾→∞ the reaction (55) becomes irreversible, that is, 𝜅_𝑔→𝜅_𝑖 which is the same for the irreversible transfer of either energy or electron. In the opposite limit (𝐾≲1), the transfer can also be irreversible provided the charge separation or their geminate recombination is fast: either 𝑘_𝐷→∞ or 𝑘_𝑐→∞. Otherwise, the distribution between neutral and charged reactants is equilibrated and the luminescence having the Stern-Volmer constant 𝜅_𝑔≈𝐾(𝑘_𝐷+𝑘_𝑐) disappears simultaneously with the ion pair that either separates or recombines.

3.4. Stationary Luminescence

The pumping light intensity is 𝐼=(𝑁₀/𝑁_𝐴)𝛿(𝑡) in the case of pulse excitation but 𝐼=𝐼₀=const when the fluorescence is an induced and studied stationary. In the former case, the quantum yield 𝜂 has to be calculated from formula (45) while in the latter case it is defined in another way [53, 54]:𝑁𝜂=^𝑠𝐴𝐼₀𝑁_𝐴𝜏_𝐴=11+𝑐𝜅₀𝜏_𝐴, (64)where 𝑁^𝑠_𝐴 is the stationary density of the excited states. If the fluorescence is quenched by reversible intramolecular transfer according to scheme (24), then both recipes were shown to give the same result [42]. A different situation arises when the quenching is performed by the intermolecular transfer whose charged products recombine in the bulk restoring the excitation according to scheme (30). In such a case, 𝑁^𝑠_𝐴 can be found from the set of corresponding IET equations (31) setting 𝑁̇^∗_𝐴=𝑃̇=0. Substituting the stationary solution thus obtained into (64) we get the corresponding Stern-Volmer constant𝜅₀=𝑅^{0𝑥002𝑑𝑐∗}(0)[1−^{0𝑥002𝑑𝑐}𝑅^♯(0)_{0𝑥002𝑑𝑐}𝑅^‡×(0)^{0𝑥002𝑑𝑐}𝑅^†(0)_{0𝑥002𝑑𝑐}𝑅^∗(0)]=𝜅_𝑔[1−𝜒⋅𝜑]. (65)It is smaller than the geminate one (𝜅_𝑔=𝑅^{0𝑥002𝑑𝑐∗}(0)) because not all the excitations are quenched forever at first encounter. Some of them are restored with an efficiency𝜒=^{0𝑥002𝑑𝑐}𝑅^♯(0)_{0𝑥002𝑑𝑐}𝑅^‡(0) (66)in the subsequent bulk encounters of the free ions, which are separated with the yield𝜑=^{0𝑥002𝑑𝑐}𝑅(0)_{0𝑥002𝑑𝑐}𝑅^∗(0). (67)All the components of (65) are well defined via the IET kernels.

Calculated in the contact approximation, they reduce expression (65) to the following one [43, 54]:𝜅₀=𝜅_𝑖1+𝜅_𝑖/𝐾𝑘_𝑐. (68)The principle difference between this result and the geminate one, (59), is the absence of 𝑘_𝐷 in the denominator. Diffusional ion pair separation cannot make the stationary energy quenching irreversible. The charge separation does not put an end to the reversible reaction, though interrupts it for a while. Only the irreversible recombination to the ground state of the neutral products proceeding with rate constant 𝑘_𝑐 causes this to happen. If 𝑘_𝑐→∞, the ionization is fully irreversible, that is, 𝜅₀≡𝜅_𝑖 Figure 12.

On the contrary, at 𝑘_𝑐=0 the quenching is reversible, that is, the fluorescence is not quenched at all (𝜅₀=0). Almost the same is true when 𝐾≫𝜅_𝑖/𝑘_𝑐, where 𝐾 and 𝑘_𝑐 were specified in (60) and (63). The quasireversible ionization is controlled by RIP recombination which proceeds with the rate constant𝜅₀≈𝐾𝑘_𝑐=𝑘^′_𝑐𝑒^−𝑈/𝑇. (69)The total activation energy of such a reaction,𝑈=Δ𝐺_𝑖+𝑢_𝑐, (70)becomes negative at highly exergonic transfer (when |Δ𝐺_𝑖|>𝑢_𝑐).

3.5. Association/Dissociation of the Exciplex

A different situation arises when the luminescence is interrupted by the reversible association of 𝐴^∗ with impurity (𝐵) present in great excess, resulting in exciplex (𝐶^∗) formation [55]: (71) Here 𝜏_𝐶 and 𝜏_𝐴 are the decay times of bound and unbound excitation that may be either equal or different. The densities of the excited particles obey the set of IET equations:𝑑𝑁^∗_𝐴𝑑𝑡=−𝑐𝑘_𝑎^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐴(𝜏)𝑑𝜏+𝑘_𝑐^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐶𝑁(𝜏)𝑑𝜏−^∗𝐴𝜏_𝐴,𝑑𝑁^∗_𝐶𝑑𝑡=𝑐𝑘_𝑎^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐴(𝜏)𝑑𝜏−𝑘_𝑐^𝑡₀Σ(𝑡−𝜏)𝑁^∗_𝐶𝑁(𝜏)𝑑𝜏−^∗𝐶𝜏_𝐶. (72) The luminescence quantum yields of 𝐴^∗ and 𝐶^∗ are𝜂_𝐴𝐴=1𝜏_𝐴^∞₀𝑁^∗_𝐴(𝑡)𝑑𝑡/𝑁₀,𝜂_𝐶𝐴=1−𝜂_𝐴𝐴, (73)provided that only 𝐴 is subjected to instantaneous light excitation: 𝑁^∗_𝐴(0)=𝑁₀, 𝑁^∗_𝐶(0)=0.

The conventional Stern-Volmer law,1𝜂_𝐴𝐴=1+𝑐𝜅𝜏_𝐴𝑘𝜅=^{𝑎0𝑥002𝑑𝑐}Σ(0)1+𝑘_𝑐𝜏Σ_𝐶^{0𝑥002𝑑𝑐}(0), (74)has the Stern-Volmer constant 𝜅, which depends on the concentration via ^0𝑥0Σ^02𝑑𝑐(0,𝑐). Only its minimal value ^0𝑥0Σ^02𝑑𝑐₀=Σ^{0𝑥002𝑑𝑐}(0,0) calculated with IET from (73) and (72) is concentration-independent:𝜅₀=lim_𝑐→0𝑘𝜅(𝑐)=^{𝑎0𝑥002𝑑𝑐}Σ₀1+𝑘_𝑐𝜏Σ_𝐶^{0𝑥002𝑑𝑐}_{00𝑥002𝑑𝑐}Σ₀𝑘=[1+^𝑎𝑘_𝐷(1+𝜏_𝑑/𝜏_𝐴)]⁻¹. (75)For the irreversible binding (𝑘_𝑐=0) this 𝜅₀=𝑘Σ_𝑎^{0𝑥002𝑑𝑐}₀ coincides with 𝜅_𝑖 given in (51).

Equations (72) as well as the general 𝜅 definition given in (74) are common for all multiparticle theories. They differ only in ^0𝑥0Σ^02𝑑𝑐(𝑠) that were collected in [55, Tables 1 and 2] for the target and trap problem (only 𝐴 is moving between immobile traps). All ^0𝑥0Σ^02𝑑𝑐(0) monotonously increase with 𝑐 approaching 1. Simultaneously the upper limit of the Stern-Volmer constant is achieved:𝜅_𝑚=𝑘_𝑎1+𝑘_𝑐𝜏_𝐶. (76)Since only a few theories deal with 𝜏_𝐴≠𝜏_𝐶, in Figure 10, only the equal lifetimes case is examined. It is easy to see that all MPK theories and CA give similar results, unlike SCRTA and LESA which deviate from them into opposite sides. LESA was independently shown to give an inappropriate description of reversible transfer at equal times [42]. SCRTA in its turn strongly overestimates the difference between the target and trap problems as compared to MPK3≡ MET. As to the concentration-independent IET result, it is always reproduced but only as 𝑐→0. At higher concentration it is better to replace IET with MET.

(a)

(b)

Figure 10 

The Stern-Volmer constant of the reversible exciplex formation at 𝜏𝐴=𝜏𝐶=𝜏 as a function of the dimensionless concentration of 𝐵's. In a wide range of concentration (a) all curves increase from the minimal (IET) value 𝜅0 up to the maximal one, 𝜅𝑚. At low concentrations (b) the difference between them is more pronounced [55].

Quite recently the kinetics of energy quenching by exciplex formation and resulting free energy dependence of Stern-Volmer constant were thoroughly investigated with IET and compared with available experimental data [56].

3.6. Electroluminescence

The ions injected from electrodes recombine to either the ground or excited state of the neutral products. The latter can be detected by their luminescence and the quantum yield of such an electroluminescence is 𝜙_ecl=𝜙_e𝜙_es. (77)The quantum yield of excitations, 𝜙_es, can be extracted from this relationship since the emission quantum yield from the excited state, 𝜙_e, is usually known. In [57–59] the 𝜙_es dependence on the free energy of ionization, Δ𝐺_𝑖=−Δ𝐺_𝑏, was measured for a number of systems. To specify this dependence we have to calculate𝜙_es=∫^∞₀𝑁^∗_𝐴(𝑡)𝑑𝑡𝑃(0)𝜏_𝐴, (78)borrowing 𝑁^∗_𝐴(𝑡) from the solution of (31) where we set 𝐼₀=0 and use the appropriate initial conditions created by the external injection of ions into solution:𝑁^∗_𝐴(0)=0,𝑃(0)=𝑃₀. (79)

Making the Laplace transformation of (31) we obtain from (78)𝜙_es=^{0𝑥002𝑑𝑐}𝑁^∗_𝐴(0)𝑃₀𝜏_𝐴=_{0𝑥002𝑑𝑐}1𝑅^‡(0)/𝑅^{0𝑥002𝑑𝑐♯}(0)+𝑐𝜏_𝐴[𝑅^{0𝑥002𝑑𝑐∗}(0)𝑅^{0𝑥002𝑑𝑐‡}(0)/𝑅^{0𝑥002𝑑𝑐♯}(0)−𝑅^{0𝑥002𝑑𝑐†}.(0)] (80)Using the expressions for all the kernels obtained in contact approximation in [53] we get the contact analog of this equation [43]:𝜙_es=11+(1+𝑐𝜏_𝐴𝜅_𝑖)𝑘_𝑐𝐾/𝜅_𝑖. (81)This expression reduces to a much simpler one provided the recombination into the excited state is irreversible as was assumed in [57–59]. If this is really the case, then 𝜅_𝑖=𝑘_𝑎=0 and 𝐾/𝜅_𝑖=𝐾/𝑘_𝑎=1/𝑘_𝑏 so that (81) takes the form used in these works:𝜙_es=11+𝑘_𝑐/𝑘_𝑏. (82)Using the contact estimates of the kinetic rate constants given in (61) and (63) we obtain the following final result:𝜙_es𝑘=[1+^′𝑐𝑘^′_𝑏exp(−ℰ(Δ𝐺_𝑖−𝜆_𝑐+ℰ/2)2𝜆_𝑐𝑇)]⁻¹. (83) This is the stepwise function approaching unity when Δ𝐺_𝑖 increases making the recombination to the excited state more favorable than to the ground one. Finally, the excitation becomes the unique reaction product since the recombination to the ground state is switched off.

Being calculated with (81), which accounts for the transfer reversibility (ionization of excited state), this function appears to be different from the simplest ones, (82) and (83), suited for irreversible recombination. Shown in Figure 11 these functions, although they are different, resemble the experimental results obtained in [57–59]. The correct accounting for the reaction reversibility is the main but not only advantage of IET, compared to DET and Markovian chemical kinetics. Taking into account the space dependence of the rates one should use (80) without contact simplifications and it was really employed in [43]. Moreover, in the next work the spin states of the free ions and radical ion pairs (RIPs) formed from them were also taken into account, as well as the spin conversion in the RIP and recombination to triplet products [60]. Even after that, the full correspondence with the experimental findings was not reached: the height of the true plateau remains lower than 1 for unknown reasons.

Figure 11 

The quantum yield of the excited states, 𝜙es, calculated in the contact approximation with (dashed line) and without (dashed-dotted line), taking into account their ionization [43].

(a)

(b)

Figure 12 

The Rehm-Weller plot for a few systems which differ by their triplet RIP recombination rate. (a) The theoretical curves for 𝑘′𝑡/𝑘𝐷=0;10−8;10−5;0.1;20 at 𝑘′𝑐=𝑘′𝑏=20𝑘𝐷. (b) Interpolation through experimental points from [61, Figure 2].

4. Spin-Assisted Complex Reactions

Until now we considered only the simplest reactions, which are sometimes termed as ABC when dealing with exciplex formation (71) or ABCD, when addressing either 𝐴^∗𝐵𝐴𝐵^∗, or 𝐴^∗𝐵𝐴⁺𝐵⁻ reactions (7), (49), and (50). It is rather easy to study the spinless reactants especially when transfer is contact and irreversible. However, the real chemistry deals with much more complex reactions than ABC, ABCD, and so on. They include the reactants with a few internal states and radical ions subjected to spin conversion and reverse recombination to their precursors and/or neutral products. Here we confine ourselves to spin-assisted contact reactions leaving noncontact effect for the next section.

Any realistic theory should discriminate between the singlet and triplet states of the radical-ion pair (RIP) and account for reversible transitions between them (Figure 13), either coherent (Hamiltonian) or incoherent with the model rate 𝑘_𝑠. In general, the same is true for excited states of 𝐴: singlet (¹𝐴^∗) and triplet (³𝐴^∗), but the internal conversion is neglected here. If there are no triplet quenchers and the triplets are generated in a low concentration making their annihilation negligible, then the reaction scheme of reversible triplet production is the following one: (84) where 𝜏_𝐴 and 𝜏_𝑇 are the singlet and triplet life times. The forward and reverse electron transfer, to and from the excited triplet acceptor ³𝐴^∗, has the rate constants 𝑘_𝑡 and 𝑘_−𝑡, respectively, which fit the detailed balance principle 𝑘_−𝑡=𝑘_𝑡exp(−Δ𝐺_𝑡𝑇), (85) where Δ𝐺_𝑡 is the free energy of triplet ionization. There are two parallel ways of spin conversion: either within the geminate RIP or through the bulk where 1/4 of the meeting-free ions associate into the singlet RIP and 3/4 into the triplet one.

Figure 13 

The energetic scheme of reversible ionization of singlet and triplet excitations, 1𝐴∗ and 3𝐴∗.

The corresponding set of integral equations for the singlet and triplet populations, 𝑁_𝑆=[¹𝐴^∗] and 𝑁_𝑇=[³𝐴^∗], and the concentration of charges, 𝑃=[𝐴⁻]=[𝐷⁺], is the most complex one: 𝑁̇_𝑆=−𝑁_𝑆𝜏_𝐴−𝑐^𝑡₀𝑅^∗(𝑡−𝜏)𝑁_𝑆+(𝜏)𝑑𝜏_𝑡0𝑅^♯(𝑡−𝜏)𝑃²(𝜏)𝑑𝜏+𝑐^𝑡₀𝑅^♣(𝑡−𝜏)𝑁_𝑇̇(𝜏)𝑑𝜏,𝑃=𝑐^𝑡₀𝑅^†(𝑡−𝜏)𝑁_𝑆(𝜏)𝑑𝜏−^𝑡₀𝑅^‡(𝑡−𝜏)𝑃²(𝜏)𝑑𝜏+𝑐^𝑡₀𝑅^⋄(𝑡−𝜏)𝑁_𝑇𝑁̇(𝜏)𝑑𝜏,_𝑇𝑁=−^𝑇𝜏_𝑇+𝑐^𝑡₀𝑅^♡(𝑡−𝜏)𝑁_𝑆(𝜏)𝑑𝜏+^𝑡₀𝑅^♠(𝑡−𝜏)𝑃²(𝜏)𝑑𝜏−𝑐^𝑡₀𝑅^⋆(𝑡−𝜏)𝑁_𝑇(𝜏)𝑑𝜏. (86) There are 9 different kernels (memory functions), having separate IET definitions via different auxiliary functions obeying equations similar to (11). All of them were solved in the contact approximations. The kernels obtained were published in the appendix of [54] and later on in [62], where the general solution of the problem is given for both singlet and triplet luminescence.

4.1. Zero Spin Conversion

When the spin conversion in geminate pairs is rather slow, it can be neglected compared to the parallel track of triplet production; via bulk recombination of free ions into triplet RIPs: (87)

Under permanent illumination 𝐼(𝑡)=𝐼₀=const, there are stationary populations of all species, 𝑁^𝑠_𝑆, 𝑃_𝑠, 𝑁^𝑠_𝑇, which obey the set of equations following from (86):𝑁^𝑠_𝑆𝜏_𝐴−𝐼₀𝑁_𝐴=−𝑐^𝑡₀𝑅^∗(𝑡−𝜏)𝑁^𝑠_𝑆+(𝜏)𝑑𝜏_𝑡0𝑅^♯(𝑡−𝜏)𝑃²_𝑠(𝜏)𝑑𝜏+𝑐^𝑡₀𝑅^♣(𝑡−𝜏)𝑁^𝑠_𝑇(𝜏)𝑑𝜏,0=𝑐^𝑡₀𝑅^†(𝑡−𝜏)𝑁^𝑠_𝑆(𝜏)𝑑𝜏−^𝑡₀𝑅^‡(𝑡−𝜏)𝑃²_𝑠(𝜏)𝑑𝜏+𝑐^𝑡₀𝑅^⋄(𝑡−𝜏)𝑁^𝑠_𝑇𝑁(𝜏)𝑑𝜏,_𝑠𝑇𝜏_𝑇=+𝑐^𝑡₀𝑅^♡(𝑡−𝜏)𝑁^𝑠_𝑆(𝜏)𝑑𝜏+^𝑡₀𝑅^♠(𝑡−𝜏)𝑃²_𝑠(𝜏)𝑑𝜏−𝑐^𝑡₀𝑅^⋆(𝑡−𝜏)𝑁^𝑠_𝑇(𝜏)𝑑𝜏. (88) After Laplace transformation, they become the algebraic equations for 𝑁^𝑠_𝑆 and other populations that can be easily found. Using the former in (64) instead of 𝑁^𝑠_𝐴, one can reproduce not only the general Stern-Volmer law but specify its constant (65) as well. The components of the latter are well defined through the Laplace transformations of the memory functions (kernels of (86)) [62].

At zero spin conversion (𝑘_𝑠=0), they are𝜅_𝑔=𝑅^{0𝑥002𝑑𝑐∗}(0),𝜒=^{0𝑥002𝑑𝑐}𝑅^♯(0)_{0𝑥002𝑑𝑐}𝑅^‡⋅(0)^{0𝑥002𝑑𝑐}𝑅^⋆(0)𝑅^{0𝑥002𝑑𝑐‡}(0)_{0𝑥002𝑑𝑐}𝑅^⋆(0)𝑅^{0𝑥002𝑑𝑐‡}(0)−𝑅^{0𝑥002𝑑𝑐♠}(0)𝑅^{0𝑥002𝑑𝑐⋄},(0)𝜑=^{0𝑥002𝑑𝑐}𝑅^†(0)_{0𝑥002𝑑𝑐}𝑅^∗.(0) (89) Calculating them in contact approximation, neglecting triplet decay, the following Stern-Volmer constant was obtained [62]:𝜅₀=𝜅_𝑖1+𝜅_𝑖/𝐾𝑘_𝑐=𝜅_𝑖=𝑘_𝑎1+𝑘_𝑎/𝑘^∗_𝐷as𝑘_𝑐𝑘→∞(ionizationcontrol),_𝑐𝐾as𝑘_𝑐→0(recombinationcontrol), (90)where 𝑘^∗_𝐷=𝑘_𝐷/(1+𝜏_𝑑/𝜏_𝐴). The reversible production of stable triplets does not affect this dependence identical to the spinless theory result (68).

As always one should discriminate between the ionization and recombination-controlled quenching. Under ionization control, the free energy dependence of ln𝜅₀ reproduces that of ln𝜅_𝑖, which coincides with the bell-shaped FEG curve, ln𝑘_𝑎=−𝑢_𝑎(Δ𝐺_𝑖)/𝑇, the top of which is cut by the diffusional plateau, ln𝜅_𝑖=ln𝑘^∗_𝐷. With increasing Δ𝐺_𝑖, the irreversible singlet ionization becomes quasireversible and gives a way to recombination control, which turns down the free energy dependence of the Stern-Volmer constant and, the earlier, the slower is singlet RIP recombination (Figure 14). After the permanent illumination is suddenly switched off (as well as after 𝛿-pulse excitation), the luminescence goes out together with singlet excitations, 𝑁_𝑆(𝑡). However, their population temporarily restores by recombination of ions into singlet: until the latter are reproduced by reverse electron transfer from the long-lived triplets. In this time domain, the reaction (87) proceeds from the right to left backing delayed luminescence.

Figure 14 

The experimental results from [63] for the Stern-Volmer constants 𝜅0(▴,•) and triplet quenching constants 𝑘𝑄(Δ,∘) fitted in [62]. The black lines are 𝜅0 and red ones are 𝑘𝑄. The triangles and upper curves, calculated for strong electron transfer (𝑘𝑎=𝑘𝑡=30𝑘𝐷), are related to Lumicrome (LC) quenched by aromatic donors in methanol. The circles and the lower curves, obtained for weak kinetic controlled transfer (𝑘𝑎=𝑘𝑡=0.5𝑘𝐷), belong to the LC quenched by aliphatic amines in the same solvent. The blue line shows the Stern-Volmer constant for irreversible ionization of singlet LC, 𝜅𝑖, at the very same parameters.

Since the decay of singlets is the fastest process, the recombination of triplets proceeds quasistationary, so that, 𝑁_𝑆(𝑡)∝𝑃²(𝑡)∝𝑁_𝑇(𝑡) and𝑁̇_𝑇=−𝑁_𝑇𝜏_𝑇−𝑐𝑘_𝑄𝑁_𝑇,where𝑘_𝑄=𝑅^{0𝑥002𝑑𝑐⋆}(0)[1−^{0𝑥002𝑑𝑐}𝑅^♠(0)_{0𝑥002𝑑𝑐}𝑅^‡(0)]. (91)In the contact approximation [62],𝑘_𝑄=𝑘_−𝑡1+𝜅_𝑖/𝑘_𝑐𝐾1+𝜅_𝑖/𝑘_𝑐𝐾+(𝑘_−𝑡+4𝑘_𝑡)/𝑘_𝐷+3𝑘_𝑡/𝑘_𝑐. (92)The quasi-exponential quenching of triplets was actually detected recently in line with the luminescence quenching [63]. Some of the data obtained are shown in Figure 14. The circles and the lower curves obtained for quenching of Lumicrome (LC) by aliphatic amines relate to the kinetic-controlled transfer following classical FEG law which is the same for the Stern-Volmer and triplet quenching constants. The triangles and upper curves, obtained for LC quenched by aromatic donors relate to stronger electron transfer represented by diffusion by diffusional plateau which is higher for singlet quenching subjected to transient effect. Unfortunately, both plateaus are too long extending even into endothermic region. This paradox was solved only recently assuming that the quenching is due to exciplex formation [56].

4.2. Double-Channel Geminate Recombination/Separation

If the spin conversion is efficient, the geminate reaction followed pulse excitation produces a number of ions and triplets. The first experimental study of such a double-channel reaction [64] raised a few questions about the accumulation kinetics and quantum yields of its products in solvents of different viscosities. The reaction scheme for the double-channel geminate reaction is easy to get from a general one, (84), by omitting all the bulk reactions: (93)

The yields of triplet products 𝜑_𝑇 and charge separation yield 𝜑 are two measurable quantities, while the yield of the ground state recombination products, 𝜑_𝑆, can be easily calculated from them: 𝜑_𝑆=1−𝜑_𝑇−𝜑 [64]. All yields are obtained from the limiting values of excitation, ions and triplet populations, which obey the following set of reduced equations:𝑁̇_𝑆=−𝑁_𝑆𝜏_𝐴−𝑐^𝑡₀𝑅^∗(𝑡−𝜏)𝑁_𝑆(𝜏)𝑑𝜏,𝑁_𝑆(0)=𝑁₀,𝑃̇=𝑐^𝑡₀𝑅^†(𝑡−𝜏)𝑁_𝑆𝑁̇(𝜏)𝑑𝜏,𝑃(0)=0,_𝑇𝑁=−^𝑇𝜏_𝑇+𝑐^𝑡₀𝑅^♡(𝑡−𝜏)𝑁_𝑆(𝜏)𝑑𝜏,𝑁_𝑇(0)=0. (94) In reality, 𝜏_𝑇 is the longest time that may be set infinite, when the geminate reaction is studied. In such a case, one can easily find from the Laplace transformed equations (88) that^0𝑥0𝑁^02𝑑𝑐_𝑆𝑁(0)=⁰𝜏_𝐴1+𝑐𝜏𝑅_𝐴^{0𝑥002𝑑𝑐∗},(0)𝑃(∞)=lim_𝑠→0𝑠𝑃^{0𝑥002𝑑𝑐}(𝑠)=𝑐𝑅^{0𝑥002𝑑𝑐†}(0)𝑁^{0𝑥002𝑑𝑐}_𝑆(0)=𝑁₀𝜓𝑁𝜑,_𝑇(∞)=lim_𝑠→0𝑠𝑁^{0𝑥002𝑑𝑐}_𝑇(𝑠)=𝑐𝑅^{0𝑥002𝑑𝑐}𝑁^{♡0𝑥002𝑑𝑐}_𝑆(0)=𝑁₀𝜓𝜑_𝑇, (95)where the total yield of ions𝜓=1−𝜂=𝑐𝜅_𝑔𝜏_𝐴1+𝑐𝜅_𝑔𝜏_𝐴. (96)Here 𝜅_𝑔=𝑅^{0𝑥002𝑑𝑐∗}(0) is the same as in (89) while [62]𝜑=^{0𝑥002𝑑𝑐}𝑅^†(0)_{0𝑥002𝑑𝑐}𝑅^∗(0),𝜑_𝑇=^{0𝑥002𝑑𝑐}𝑅^♡(0)_{0𝑥002𝑑𝑐}𝑅^∗.(0) (97)If the spin conversion is negligible,1𝜑=1+𝑘_𝑐/𝑘_𝐷,𝜑_𝑇=0 (98)are exactly the same as in the spinless theory [9]. How these quantities are affected by spin-conversion, either coherent or incoherent, will be discussed later in the frame of the unified theory (Section 5.5).

Equations (88) describe the whole geminate reaction followed instantaneous creation of singlet excitation at 𝑡=0. This reaction is composed from two sequential stages: accumulation and dissipation of ions. Another situation appears when the ions are created by a straightforward pulse excitation at the moment 𝑡=0. If the highly positioned singlet excitation is out of game (𝑘_𝑎=𝑘_𝑏=0), the set of (94) is reduced to the following one:𝑃̇=𝑐^𝑡₀𝑅^†(𝑡−𝜏)𝑁_𝑆(𝜏)𝑑𝜏,𝑃(0)=𝑃₀,𝑁̇_𝑇=−𝑁_𝑇𝜏_𝑇+𝑐^𝑡₀𝑅^♡(𝑡−𝜏)𝑁_𝑆(𝜏)𝑑𝜏,𝑁_𝑇(0)=0. (99)The natural triplet decay is accompanied by their quenching by ionization and subsequent recombination of ions to either ground or triplet states.

4.3. Magnetic Field Effect in Double-Channel RIP Recombination

If besides that the ionization of triplet is also negligible (𝑘_−𝑡=0), then the geminate reaction scheme (93) reduces to the simplest one: (100) Since any electron transfer in this scheme is irreversible there is an alternative to IET how to find the solution of the problem by conventional methods of quantum chemistry [65]. In contact approximation, the total populations of charged and triplet products may be expressed via pair correlation functions of singlet and triplet RIPs, 𝑚_𝑆(𝑟,𝑡) and 𝑚_𝑇(𝑟,𝑡):𝑃(𝑡)=[𝑚_𝑆(𝑟,𝑡)+𝑚_𝑇(𝑟,𝑡)]𝑑³𝑁̇𝑟,_𝑇=𝑘_𝑡𝑚_𝑇(0,𝑡). (101)The rate equations for these densities account for spin conversion in ion pair born at 𝑟=𝑟₀:𝑚̇_𝑆=−3𝑘_𝑠𝑚_𝑆+𝑘_𝑠𝑚_𝑇+𝐿𝑚_𝑆,𝑚_𝑆(𝑟,0)=𝛿(𝑟−𝑟₀)4𝜋𝑟²,𝑚̇_𝑇=3𝑘_𝑠𝑚_𝑆−𝑘_𝑠𝑚_𝑇+𝐿𝑚_𝑇,𝑚_𝑇(𝑟,0)=0, (102)where𝐿=𝐷^{0𝑥002𝑑𝑐}1𝑟²𝜕𝑟𝜕𝑟²𝑒^{−𝑈(𝑟)/𝑘𝐵𝑇}𝜕𝑒𝜕𝑟^{𝑈(𝑟)/𝑘𝐵𝑇} (103)is an operator of the encounter diffusion (with coefficient (𝐷^{0𝑥002𝑑𝑐})) in the interparticle potential 𝑈(𝑟). For an ion-radical pair this is the Coulomb interaction 𝑈(𝑟)=−𝑟_𝑐/𝑟, with Onsager radius 𝑟_𝑐=𝑒²/𝜖𝑘_𝐵𝑇 (at temperature 𝑇 and dielectric constant 𝜖). This is the simplest but widely used elementary spin model (ESM) of incoherent spin-conversion in RIP with a single phenomenological parameter (rate) 𝑘_𝑠 [9, 16].

The ion recombination into ground and triplet states can be taken into account via boundary conditions to (102):4𝜋𝜎𝐷^{20𝑥002𝑑𝑐}𝑒^{𝑟𝑐/𝑟}𝜕𝑒𝜕𝑟^{−𝑟𝑐/𝑟}𝑚_𝑆|_𝜎=𝑘_𝑐𝑚_𝑆,4𝜋𝜎𝐷^{20𝑥002𝑑𝑐}𝑒^{𝑟𝑐/𝑟}𝜕𝑒𝜕𝑟^{−𝑟𝑐/𝑟}𝑚_𝑇|_𝜎=𝑘_𝑡𝑚_𝑇. (104) They represent the double-channel irreversible recombination at contact.

Equations (102) may be also written in operator form for two-component vector (^𝑚𝑆_𝑚𝑇) where 𝑚_𝑇=𝑚₋+𝑚₀+𝑚₊ is actually a sum of three triplet components populations. For any particular mechanism of incoherent spin-conversion, the true rate equations should be written for either four-component vector𝑚_𝑆𝑚₀𝑚₋𝑚₊𝑚ordensitymatrix𝜌=^𝑆∗∗∗∗∗𝑚₀∗∗∗∗∗𝑚₋∗∗∗∗∗𝑚₊ (105) with account of real transitions between the components peculiar to the chosen mechanism. For instance, the stochastic Liouville equation for the density matrix of Ruthenium complex was specified in [66, 67] for the Δ𝑔-mechanism of spin conversion governed by the spin Hamiltonian1𝐇=2𝜔₊0000−𝜔₊00000𝜔₋00𝜔₋0. (106) Here𝜔_±=(𝑔₊±𝑔₋)𝛽ℏ𝐻, (107)𝛽 is the Bohr magneton, 𝑔₊ and 𝑔₋ are 𝑔-factors of positive and negative ions and 𝐻 is the magnetic field. The relaxation superoperator that has the rank 16×16 in the Liouville space was specified in [66, 67] as well.

For moderate magnetic field and relatively fast transversal relaxation rate𝜔₋=Δ𝑔𝛽𝐻ℏ≪1𝑇₂, (108) the quasistationary solution for all off-diagonal elements is allowed [9, Section VIII A]. It reduces the coherent description of density matrix evolution to the incoherent one which substitutes the phenomenological ESM set (102) by the more appropriate one [9, 68, 69]:𝑚̇_𝑆=(𝑘₀+1𝑇₂−12𝑇₁)𝑚₀−(𝑘₀+1𝑇₂+12𝑇₁)𝑚_𝑆+𝑚₊+𝑚₋2𝑇₁+𝐿𝑚_𝑆,𝑚̇₀=(𝑘₀+1𝑇₂−12𝑇₁)𝑚_𝑆−(𝑘₀+1𝑇₂+12𝑇₁)𝑚₀+𝑚₊+𝑚₋2𝑇₁+𝐿𝑚₀,𝑚̇₊=𝑚_𝑆+𝑚₀2𝑇₁−𝑚₊𝑇₁+𝐿𝑚₊,𝑚̇₋=𝑚_𝑆+𝑚₀2𝑇₁−𝑚₋𝑇₁+𝐿𝑚₋, (109)where the ab-initio-derived rate of spin-conversion is𝑘₀=𝜔²₋𝑇₂. (110)

The rest depends on the relationship between the transversal (𝑇₂) and longitudinal (𝑇₁) relaxation times. At 𝑇₁=∞ the general set (109) reduces to only two equations for 𝑆 and 𝑇₀ components [68] which are converting with the rate𝑘_𝑠=𝑘₀+1𝑇₂. (111)On the other hand, for 𝑇₂=𝑇₁=𝑇₀ the ESM equations (102) are approved but with another conversion rate [68]:𝑘_𝑠=12𝑇₀+13𝑘₀. (112)It was proven by the exact analytic solution of the general set (109) for a contact born pair (𝑟₀=𝜎). The comparison of exact double-channel solution from [70] with that of ESM obtained in [64] showed that they are exactly the same only at zero magnetic field (𝑘₀=0) when conversion is carried on by the transversal relaxation but differ a bit even in the lowest order approximation with respect to𝑘₀𝑇₀=(𝜔₋