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 reduces to the size of their diameter, . A typical example of such a reaction is the irreversible quenching of excited molecule by the impurity , stopping abruptly the luminescence of :(1)

The time-dependent reaction constant established by Smoluchowski, makes the differential kinetic equation non-Markovian: It becomes the conventional Markovian equation of chemical kinetics only over long times when approaches the constant (stationary) value . This is the diffusional rate constant linear in encounter diffusion coefficient .

In reality, not any contact of reactants results in complete quenching of . Being less efficient, such contacts are characterized by a phenomenological parameter , 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 is 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 , it incorporates into the theory the noncontact, space-dependent rate . The latter determines as well as the whole non-Markovian kinetics of quenching which becomes exponential (Markovian) at later time. The stationary quenching constant and its effective quenching radius are also determined by and . DET had a tremendous success in the treatment of luminescence quenching by irreversible energy or electron transfer [6]. Assuming an exponential shape of [7], the theory attempted to fit the real viscosity dependence of the quenching radius [8].

Although DET covers only the irreversible transfer, it is known to be exact in the particular case of the target problem (immobile quenched by independently moving point 's) [9]. In the simplest case the time-dependent reaction constant of DET substituted for its contact analog (2) has the following definition:Here is the excitation life time and is the pair correlation function of the reactants, obeying the equation for encounter diffusion accompanied by a distant reaction with the rate :At later times, the theory becomes Markovian, that is, which is the conventional (time independent) rate constant.

At the same time, DET failed to describe the reversible reaction between the meta-stable reactants, and . When their lifetimes are different (), 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, and . 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 -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 ( and ). Then the terms in the kinetic equations which are second order in the excited state concentrations, and , are negligible and the set of two IET equations can be linearized in the particle densities [15]:This set should be solved with the initial conditionsprovided that was created by -pulse excitation at .

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:The auxiliary equations for the dyads of pair distributions , and are the same:Only the initial conditions for them are different: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, and , 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 whenIn highly polar solvents,are the kinetic rate constants of the forward and backward transfer. They relate both the kernelsto a single factorwhere is the encounter time. With these kernels, the general IET equations (8) take the particular form specific only to the contact reactions [15]:

At a great excess of the molecule, one may neglect the reencounters of with in the bulk. In this particular case, which attracted the most attention, there is actually only a single surrounded by s. In such a situation, the general set (17) reduces to even a simpler one: These equations describe the quenching of the initially excited , as well as accumulation of energy on , its reverse transfer to , 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 or so that nothing can be transferred backward.(2)Equal times. In a border case, , the decay of excited states and energy redistribution between them proceed independently.(3)Energy storing. At the initial quenching of by the forward energy transfer results in energy storing by 's and the backward transfer from them to the short-lived during the same encounter.

Figure 1: The decay of the initially excited molecule, and (A) accumulation/dissipation of transferred energy, , (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],that monotonously decreases up to the stationary (Markovian) value

In fact, this value is unattainable because the reduction of IET to DET does not hold so long. The asymptotic behavior of in IET was shown to be the following [14]:However, the power time asymptotic quenching in IET was shown to be false at least at and was essentially corrected by the multiparticle theories reviewed below.

Qualitatively different are the results in the opposite, energy storing situation when . 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),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 to but then turning back from to . 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]:It should be noted that a similar power law time dependence is an exact result of the geminate reaction of reversible dissociation, , 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 , ) and [14].

2.2. Intramolecular Energy Transfer

Even a simpler example of a reversible reaction of the same sort provides the intersystem crossing induced by an inert partner according to the following scheme: (24)Initially excited as well as are two electronic states of the same particle reacting with given in concentration . In the contact approximation the IET equations take the form [26]where the Laplace transformation of is given by (16). This set of equations is equivalent to the set (17) at 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 , corresponding to the instantaneous (-pulse) excitation of state . The results are shown in Figure 3. In the absence of there is only a spontaneous exponential decay of (solid line in Figure 3(a)). The intramolecular energy transfer to modulates this process. If the transfer is irreversible (), 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 , the energy comes back due to reencounters with , supporting the delayed fluorescence from if . Under such a condition, the dashed-dotted line representing short after rapid quenching goes down much slower than the spontaneous decay. The smaller , the later, weaker and slower should be the delayed fluorescence. This energy storing is the most pronounced at .

Figure 3: Relaxation of state populations after instantaneous excitation of at and . (a) Spontaneous (exponential) decay of with lifetime (thick straight line) and its dissipation accompanied by intersystem crossing with the backward rates (dotted line), (dashed-dotted line), and (dashed line). (b) Accumulation and dissipation of the initially empty state , which decays through faster the higher is , or does not decay at all if [26].

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

Between the opposite limits of energy quenching () and energy storing () 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 , the natural decay of both states can be excluded from consideration by a simple substitution:The relative populations of excited states, and , tend to their equilibrium values as :where is the equilibrium constant in a system of two excited states. The general solution of the problem can be represented aswhere was the subject of calculations in [42]. After the exact averaging over the multiparticle distribution of it was shown to bewith obeying the original DET equations (5) and (6) but with substituting for . In the contact approximation this leads to the conventional Smoluchowski-Collins-Kimball solution but with substituted for .

2.3. Intermolecular Electron Transfer

Let us now turn to the reversible reaction of electron transfer, 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 -excitation of in reaction (30) was studied in [43] using the general IET equations:where is the free-ion concentration and 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 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 [43]: The initial condition for free ions is given by the following equation:where the yield of photo-generated ions is and the yield of their separation 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 into their differential and even Markovian analogs [13]. The same can be done with the remaining ionization term, but at a much longer time, , 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: where and . The delayed fluorescence can be obtained from the quasistationary solution of (34):wherewas 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 ratesfor the inverse times and 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,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 ) (Figure 5(b)). At these long times the transfer proceeds not between geminate partners that have already separated but with many other 's that, being in the ground state, act as the fresh quenchers. As to the quenching constant for stable excitation , it remains positive at all times in both theories but only in MET approaches the truly stationary value .

Figure 5: The time effective rate “constants” for the forward and reverse transfer, (solid lines) and (dashed lines), calculated with IET (a) and MET (b) at , equal concentrations of and and equal kinetic constants . The thin solid line shows the stationary value of [44].

In the case of irreversible quenching (), the kernel modification can also be done as was recommended earlier [19–23], by the straightforward substitutionwhere 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 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 to is irreversible (), 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]: where . The validity region for MET is known to be much wider: This was confirmed in [45] by a straightforward comparison of the MET solution with that of DET: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 , where . 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 .

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 , or in energy storing, if . 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 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.

Figure 7: The time dependence of the survival probability, , of initially excited for several values of the unimolecular decay constants, and (in , 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 () being large in the right ones (). 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, , but not the energy-storing case, , 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 isprovided that the light excites instantaneously (at ) only and that the luminescence comes only from this particle. The concentration dependence of the yield is always represented by the Stern-Volmer lawwhere 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) thatHere and , that is,This result obtained by means of IET is concentration-independent but known to be valid at only the lowest concentrations of and [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 settingIn 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: .

The quantum yield of the irreversible quenching obeys the Stern-Volmer dependence on concentration represented in (45). As in the original Stern-Volmer law, the quenching constant obtained from (46) and (47) is concentration-independent:However, contrary to , the experimentally found is concentration-dependent and this is a challenge for the theory to find out how the true Stern-Volmer constant differs from its IET analog .

MET solves this problem substituting in (51) by from (37):In general, this is concentration-dependent unless the transfer is strongly under kinetic control when . In the alternative limit of diffusional transfer, MET givesIn [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 (). At moderate concentrations this dependence is always linear,but the slope of it appears to be different. For MET obtained by modification (37), while in the old MET originated from another modification, (39), . The former coincides with that of SA while the latter (shown in figure) is a bit smaller. The true DET value is in between [50]. Since the concentration correction in (54) is only actual for large , the constant term in 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 is not essential.

Figure 8: The Stern-Volmer constants as functions of the dimensionless concentration obtained in the contact approximation and under diffusional control at . 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 ) was done later for any (Figure 9) [47]. Theoretically, increases with concentration from the IET value up to the kinetic rate constant , though experimentally available is only the lower (left) part of the graph, . Contact MET, which is identical to MPK3 developed later [29], underestimates the exact 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 .

Figure 9: The concentration dependence of the irreversible quenching Stern-Volmer constant in units of for a number of contact theories, provided 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 is the donor of an electron, while is its acceptor. The excitation of by the short light pulse, , resulting in charge separation, produces the free ions, and , with the yield . This is the fraction of ions initially born in amount but separated escaping recombination to the ground state with the rate and the backward electron transfer to excited products with the rate .

If '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 . The IET equations (31) can be reduced for this case by omitting the bulk terms quadratic in , as well as the pumping term: Instead of the pumping term one has to add the initial conditions for these equations which represent the instantaneous excitation of :

The luminescence of the pulse excited 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 . 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]:where the geminate Stern-Volmer constant is [43, 52]Here is the Stern-Volmer constant for irreversible transfer (51) whileis an equilibrium constant for ionization, with free energy (here and further on the Boltzmann constant ). The kinetic reaction constants of the outer-sphere electron transferobey the Arrhenius law with activation energies satisfying the free energy gap (FEG) law [9]:where is the contact reorganization energy of the polar media. The charge recombination constant also obeys the FEG law:but with different free energy where is the excitation energy of .

For highly exergonic charge separation () when the reaction (55) becomes irreversible, that is, which is the same for the irreversible transfer of either energy or electron. In the opposite limit (), the transfer can also be irreversible provided the charge separation or their geminate recombination is fast: either or . Otherwise, the distribution between neutral and charged reactants is equilibrated and the luminescence having the Stern-Volmer constant disappears simultaneously with the ion pair that either separates or recombines.

3.4. Stationary Luminescence

The pumping light intensity is in the case of pulse excitation but when the fluorescence is an induced and studied stationary. In the former case, the quantum yield has to be calculated from formula (45) while in the latter case it is defined in another way [53, 54]:where 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, can be found from the set of corresponding IET equations (31) setting . Substituting the stationary solution thus obtained into (64) we get the corresponding Stern-Volmer constantIt is smaller than the geminate one () because not all the excitations are quenched forever at first encounter. Some of them are restored with an efficiencyin the subsequent bulk encounters of the free ions, which are separated with the yieldAll 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]:The principle difference between this result and the geminate one, (59), is the absence of 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 causes this to happen. If , the ionization is fully irreversible, that is, Figure 12.

On the contrary, at the quenching is reversible, that is, the fluorescence is not quenched at all (). Almost the same is true when , where and were specified in (60) and (63). The quasireversible ionization is controlled by RIP recombination which proceeds with the rate constantThe total activation energy of such a reaction,becomes negative at highly exergonic transfer (when ).

3.5. Association/Dissociation of the Exciplex

A different situation arises when the luminescence is interrupted by the reversible association of with impurity () present in great excess, resulting in exciplex () formation [55]: (71) Here and 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: The luminescence quantum yields of and areprovided that only is subjected to instantaneous light excitation: , .

The conventional Stern-Volmer law,has the Stern-Volmer constant , which depends on the concentration via . Only its minimal value calculated with IET from (73) and (72) is concentration-independent:For the irreversible binding () this coincides with given in (51).

Equations (72) as well as the general definition given in (74) are common for all multiparticle theories. They differ only in that were collected in [55, Tables 1 and 2] for the target and trap problem (only is moving between immobile traps). All monotonously increase with approaching 1. Simultaneously the upper limit of the Stern-Volmer constant is achieved:Since only a few theories deal with , 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 MPK MET. As to the concentration-independent IET result, it is always reproduced but only as . At higher concentration it is better to replace IET with MET.

Figure 10: The Stern-Volmer constant of the reversible exciplex formation at as a function of the dimensionless concentration of 's. In a wide range of concentration (a) all curves increase from the minimal (IET) value up to the maximal one, . 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 The quantum yield of excitations, , can be extracted from this relationship since the emission quantum yield from the excited state, , is usually known. In [57–59] the dependence on the free energy of ionization, , was measured for a number of systems. To specify this dependence we have to calculateborrowing from the solution of (31) where we set and use the appropriate initial conditions created by the external injection of ions into solution:

Making the Laplace transformation of (31) we obtain from (78)Using the expressions for all the kernels obtained in contact approximation in [53] we get the contact analog of this equation [43]: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 and so that (81) takes the form used in these works:Using the contact estimates of the kinetic rate constants given in (61) and (63) we obtain the following final result: This is the stepwise function approaching unity when 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, , calculated in the contact approximation with (dashed line) and without (dashed-dotted line), taking into account their ionization [43].

Figure 12: The Rehm-Weller plot for a few systems which differ by their triplet RIP recombination rate. (a) The theoretical curves for at . (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 , or 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 . In general, the same is true for excited states of : singlet () and triplet (), 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 and are the singlet and triplet life times. The forward and reverse electron transfer, to and from the excited triplet acceptor , has the rate constants and , respectively, which fit the detailed balance principle where 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, and .

The corresponding set of integral equations for the singlet and triplet populations, and , and the concentration of charges, , is the most complex one: 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 , there are stationary populations of all species, , , , which obey the set of equations following from (86): After Laplace transformation, they become the algebraic equations for and other populations that can be easily found. Using the former in (64) instead of , 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 (), they are Calculating them in contact approximation, neglecting triplet decay, the following Stern-Volmer constant was obtained [62]:where . 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 reproduces that of , which coincides with the bell-shaped FEG curve, , the top of which is cut by the diffusional plateau, . With increasing , 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 -pulse excitation), the luminescence goes out together with singlet excitations, . 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 and triplet quenching constants fitted in [62]. The black lines are and red ones are . The triangles and upper curves, calculated for strong electron transfer , are related to Lumicrome (LC) quenched by aromatic donors in methanol. The circles and the lower curves, obtained for weak kinetic controlled transfer (), 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, , at the very same parameters.

Since the decay of singlets is the fastest process, the recombination of triplets proceeds quasistationary, so that, andIn the contact approximation [62],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 and charge separation yield are two measurable quantities, while the yield of the ground state recombination products, , can be easily calculated from them: [64]. All yields are obtained from the limiting values of excitation, ions and triplet populations, which obey the following set of reduced equations: In reality, 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) thatwhere the total yield of ionsHere is the same as in (89) while [62]If the spin conversion is negligible,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 . 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 . If the highly positioned singlet excitation is out of game (), the set of (94) is reduced to the following one: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 (), 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, and :The rate equations for these densities account for spin conversion in ion pair born at :whereis an operator of the encounter diffusion (with coefficient ) in the interparticle potential . For an ion-radical pair this is the Coulomb interaction , with Onsager radius (at temperature and dielectric constant ). This is the simplest but widely used elementary spin model (ESM) of incoherent spin-conversion in RIP with a single phenomenological parameter (rate) [9, 16].