Exponential model 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],... 

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.

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. 

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... 

Contrary to this approximation, the exponential model, considered in Section V.A, does not assume recombination to be contact, but suggests that it takes place with a uniform backward transfer rate k-et within the reaction sphere of the volume v = 47ict3/3. As a result, Eq. (3.419) is replaced by the following one ... 

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.)...

As before, the analytical solution becomes possible only in the contact approximation (3.40). It is inappropriate for the dipole-dipole transfer rate (3.44) but can be considered as a reasonable model for the exponential exchange quenching carried out by the dipole-forbidden triplet-triplet energy transfer [204]. 

The theories of the electronic and ionic currents have some features in common. One may formulate models in which the current is limited by the injection into the film from the contacts of positively or negatively charged carriers, or one may consider an equilibrium state to exist across either or both interfaces. One may postulate space-charge limited currents, trapping, and recombination processes. One of the chief differences between the ionic and the electronic currents is that the average velocity of the ions is approximately exponentially dependent on the field for fields which produce experimentally observable ionic currents, whereas the average velocity of electrons is linearly dependent on the field at low fields with different types of nonlinearity at high fields. 

---