Onsager distance

While it is not clear how the constant frequency low field dielectric relaxation measurements mentioned above should be applied to reactions in liquids, save for a complete time-dependent theory of liquids, these effects are very significant. At short times (<10ps) the effective Onsager distance may be 20 nm, even in methanol or ethanol, but over the next two or three decades of time reduce to more nearly 2 nm. Such a change can reduce the rate of reaction much more rapidly than that which occurs by decay of the transient time dependence discussed in the previous sub-section. 

This connection is further discussed in Sect. 2.3. (The prescribed diffusion method of solution of this problem has been used by Mozumder [76], but it was shown to be unsatisfactory by Hong and Noolandi [72].) Recently, Clifford et al. [322] have shown that the diffusion and drift equation can be re-cast in a form which is symmetric to the sign of the coulomb potential. Consequently, the care necessary to define the sign of the Onsager distance, rc, is no longer required and it is sufficient to solve for an attractive potential. This is particularly valuable when performing numerical studies, as only attractive potentials need be considered (and such situations are more easily solved numerically than repulsive cases). 

However, when the Onsager distance, rc, is much larger than the radii of either ion, hydrodynamic repulsion is unimportant (see Fig. 45, p. 268). The tensorial form of the diffusion equation is [cf. eqn. (141)]... 

Fig. 29. Escape probability for an ion-pair as a function of inverse separation distance, r0, in a solvent with Onsager distance = 15R (e.g. 7.5nm), permittivity of 8, and with an electron jump distance X R (e.g. 0.5 nm). The three curves represent the...

Fig. 45. The rate coefficient for a diffusion-limited reaction between ions of charges Zje and z2e in a medium of relative permittivity e. The Onsager distance is rc = ZjZ2e2/...

Considering first, however, polar solvents where only the coulomb and screened coulomb potentials are signficant by comparison with thermal energies, it should be emphasized that theory is quite successful in explaining the experimental results. The Debye [68] expression for the rate of reaction has been discussed in Chap. 3, Sect. 1. With an Onsager distance 0.7—1.0 nm, the effective reaction radius is 0.9—1.2 nm or... 

Figure 5.4 Recombination of ions in low-polar solvents change with time of the calculated distribution function for the ion-pair separation. The plots show the relative numbers F(r) of ion pairs at various distances r apart (up to about half the Onsager radius fc) at a series of times in the picosecond range rc is the Onsager distance, at which the chance of escape is fifty-fifty. As the time from the start increases, the peak moves towards higher values of r and becomes flatter, indicating that the average distance increases and that the distribution becomes more uniform. The three curves are calculated for times 1.3, 2.4 and 6.7 ps. Data from Ref. [16]. The diagram is after J.M. Warman, Ref. [l,a, p. 454].

The free ion yield at zero field strength refers to the number of electron/ion pairs which achieved an initial separation distance larger than the Onsager distance, r (see Section 4.4). Theoretically, electron escape from spurs (spherical geometry) leads to a finite value of G /E = 0). From the track of a high LET particle (cylindrical geometry), escape at E = 0 is zero. The observed small Gfi(E = 0) values are probably due to escape via 5-rays. 

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