Contact approximation

The theory of geminate recombination experienced a similar evolution from primitive exponential model and contact approximation [19,20], to distant recombination carried out by backward electron transfer [21], However, all these theories have an arbitrary parameter initial separation of reactants in a pair, / o. This uncertainty was eliminated by unified theory (UT) proposed in two articles published almost simultaneously [22,23], UT considers jointly the forward bimolecular electron transfer and subsequent geminate recombination of charged products carried out by backward electron or proton transfer. The forward transfer creates the initial condition for the backward one. This is the distribution of initial separations in the geminate ion pair/(ro), closely analyzed theoretically [24,25] and inspected experimentally [26,27], It was used to specify the geminate recombination kinetics accompanied by spin conversion and exciplex formation [28-31], These and other applications of UT have been covered in a review published in 2000 [32],... 

The diffusional dependence of Rq in the GCK model is qualitatively the same as in the original contact approximation (except that R is a bit larger than ct) ... 

Figure 3.10. The ideal Stern-Volmer constants Ko (solid lines) as functions of diffusion in the contact approximation (a) and for the exponential transfer rate with different tunnelling lengths / = 1.6A (b) and / = 2.5 A (c) (From Ref. 46.) The contact stationary constant k (dashed line) is shown for comparison with contact Kq (a).

Figure 3.22. The schematic representation of (a) the exponential model (EM) and (b) contact approximation. In EM the reaction sphere of radius a is transparent for particles, which leave it by a single jump with the rate ksep. In contact calculations, the same sphere surrounds an excluded volume and recombination takes place only at its surface, or more precisely in a narrow spherical layer around it.

In the contact approximation (CA) the electron transfer proceeds in the thin reaction layer adjacent to the nontransparent sphere of radius a [Fig. 3.22(a)], However, the ions do not necessarily start from there as in EM. When their initial separation exceeds a, the ions do not recombine until they are delivered to the contact by encounter diffusion. Their distribution obeys the diffusional equation similar to (3.19) but for ions in the Coulomb well ... 

The geminate recombination is actually controlled by diffusion, if the initial separation of ions is so large that their transport from there to the contact takes more time than the reaction itself. The exponential model excludes such a situation from the very beginning, assuming that ions are bom in the same place where they recombine. Thus, EM confines itself to the kinetic limit only and fixes Z = z = const. The kinetic recombination in the contact approximation does not imply that the starts are taken from the very contact. If they are removed a bit and diffusion is fast, the recombination is also controlled by the reaction and its efficiency Z = qz is constant although smaller than in EM. 

Although the contact approximation is a good alternative to EM for low-viscosity solvents, at higher viscosities and especially in the static limit (D = 0), CA becomes inappropriate. In this limit both cp0 and cpc turn to zero, but cp(r(l) does not, if ro / cr. Particles that are separated by the distance ro > ct survive because there is no diffusion to bring them in contact where the reaction takes place. Of course, this is nonsense if electron transfer is remote. Sooner or later it involves in the reaction all immobile partners wherever they are, turning cp(/ o) to zero. 

In this limit geminate recombination becomes quasistatic and can be accomplished before the particles leave their initial positions. This is the static limit missed in the contact approximation ... 

Figure 3.25. The dependence of the quantum yield cp(ro) on the contact quantum yield

If recombination of ions not only into the exciplex but also to the ground state is contact, then the latter can be accounted for by omitting W(r)G in Eq. (3.235) and including term kcG (k, = J WR(r)cfr) in the rhs of Eq. (3.236). This is the total contact approximation, which is better suited for the transfer of protons than electrons. It allows easy expression of the Green function G(c>, t) through the Green function of the free diffusion Go(c>, f) ... 

Using the contact approximation and polar solvents (rc = 0), one can find for the irreversible ionization [38] ... 

This maximum separates the branch where Z increases with diffusion from that where Z decreases with it. As we have seen in Figure 3.4067 ), the latter can never be reached if the forward and backward transfer rates are the same. In such a case the initial charge distribution is always more remote than the reaction zone, so that ions enter it mainly from outside, only in this case the popular contact approximation for recombination is a reasonable alternative provided... 

Figure 3.41. The diffusional dependence of the recombination efficiency Z in the contact approximation (dotted line) at starting distance ro — 1.124 a and the same for the remote recombination in a normal (solid line) and inverted (dashed line) Marcus region, in highly polar solvents. The horizontal dashed-dotted line represents the exponential model result, Z — z — const. (From Ref. 152.)...

At larger AGr the difference between the IET and contact approximation is more pronounced and not only at slow diffusion, but also in the opposite limit where the IET curve passes through the maximum. This maximum cannot be reproduced either with contact or with the exponential model of the rates. The variation of free energies does not change the exponential shape of the rate, affecting only the preexponential factor chosen from the relationship... 

Figure 3.50. The partial quantum yields

Below this boundary the recombination to the ground state is preferable above it, the excitation dominates. If = 1.6 eV, then at large Xc = 1.2 eV the border free energy, AG = 0.4 eV, is positive, while at small hc = 0.5 eV the border free energy is negative AG = —0.3 eV. These are cases (a) and (h) compared in Figure 3.54. Beyond the contact approximation the relative positions of the excitation and ionization layers in these cases are the opposite. 

Figure 3.57. The ion survival probability as a function of time at To = 0.5 ns with a great excess of acceptors. In line with UT and IET (above) and Markovian theory (below) (dashed curve), the contact approximation (dashed-dotted line in the middle) and exponential model with fcjep = A et = 1.0 ns-1 (dotted line) are also shown. The horizontal thick lines indicate the free-ion quantum yield ((). The concentrations and ionization parameters are the same as in Figure 3.56, while wy = 3.4ns-1, D = D = 1.2 X 10-6 cm2/s, k1 — 7S4 A3/ns, and kr — 4S6 A3/ns. (From Ref. 195.)...

In Figure 3.57 we show almost the full identity of the results obtained with IET and UT. They demonstrate the well pronounced maximum appearing when i > is shorter than the characteristic time of the subsequent geminate recombination [22]. In the contact approximation the results are qualitatively the same but the ionization quantum yield / is half as much as in distant theories (see Table III). This was expected because at such a short To a significant fraction of ions are produced during the initial static ionization that is missed in the contact approximation. 

Figure 3.62. The light dependence of the Stem—Volmer constant K, i (/, i for diffusional quenching with given exponential rate (Wq — 103 exp[—2(r — cr)/L] ns-1), but at different diffusion in pairs containing the excited molecules (a) D =0.1 Dd, (6) D = D= 10 5cm2/s (dashed line—the same, but in contact approximation), (c) D = WD. Other parameters a = 5A, L = 1.0 A, i = 10ns, ko = f Wq(r)d3r = 1.9 x 105 A3/ns. (From Ref. 200.)...

In the same figure we also demonstrate the sensitivity of the results to the ratio of the diffusion coefficients in the stable and excited pairs, D Q and D Q, which increases from (a) to (c). The difference in the diffusion coefficients ignored in the convolution recipe affects the field dependence no less than the contact approximation. This is the unique advantage of IET, that it is free of any limitations of this sort and can be used for quantitative investigations of field effects, at any space dispersion of quenching rate and at arbitrary diffusion coefficients. In fact, this is diffusion of the excited molecule that affects the field dependence of k0. By slowing down this diffusion, one increases the encounter time x,/ and thus enhances the field acceleration of quenching. 

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