The Quasi-ballistic Model

Equation (10.6) for the mobility in the two-state model implicitly assumes that the electron lifetime in the quasi-free state is much greater than the velocity relaxation (or autocorrelation) time, so that a stationary drift velocity can occur in the quasi-free state in the presence of an external field. This point was first raised by Schmidt (1977), but no modification of the two-state model was proposed until recently. Mozumder (1993) introduced the quasi-ballistic model to correct for the competition between trapping and velocity randomization in the quasi-free state. 

The new model is called quasi-ballistic because the electron motion in the quasi-free state is partly ballistic—that is, not fully diffusive, due to fast trapping. It is intended to be applied to low- and intermediate-mobility liquids, where the mobility in the trapped state is negligible. According to this, the mean 

Since there are (Tf + Tt) 1 cycles of trapping and detrapping per unit time, the drift velocity is (Ax)/(xf + tt), from which the effective mobility is derived to be 

Here (g)T = (e/m)Tf2/(r( + Tt) is called the ballistic mobility and (/t)H = + Tt) is the usual trap-controlled mobility. (q)F is the applicable mobility when the velocity autocorrelation time ( 1) is much less than the trapping time scale in the quasi-free state (fTf l). In the converse limit, (jj)t applies, that is—trapping effectively controls the mobility and a finite mobility results due to random trapping and detrapping even if the quasi-free mobility is infinite (see Eq. 10.8). 

FIGURE 10.2 Variation of log(telf at 300 K with binding energy e0 for n, = 1019cm Jand/u ( = 100 cn v s-1. Notice the sensitivity of the movility to the binding energy. Reproduced from Mozumder (1993), with permission of Elsevier . 

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In comparing the results of the quasi-ballistic model with experiment, generally pq[ = 100 cn v s-1 has been used (Mozumder, 1995a) except in a case such as isooctane (Itoh et al, 1989) where a lower Hall mobility has been determined when that value is used for the quasi-free mobility. There is no obvious reason that the quasi-free mobility should be the same in all liquids, and in fact values in the range 30-400 cmV -1 have been indicated (Berlin et al, 1978). However, in the indicated range, the computed mobility depends sensitively on the trap density and the binding energy, and not so much on the quasi-free mobility if the effective mobility is less than 10 crr v s-1. A partial theoretical justification of 100 cm2 v 1s 1 for the quasi-free mobility has been advanced by Davis and Brown (1975). Experimentally, it is the measured mobility in TMS, which is considered to be trap-free (vide supra). 

Table 10.4 lists the values of trap density and binding energy obtained in the quasi-ballistic model for different hydrocarbon liquids by matching the calculated mobility with experimental determination at one temperature. The experimental data have been taken from Allen (1976) and Tabata et ah, (1991). In all cases, the computed activation energy slightly exceeds the experimental value, and typically for n-hexane, 0/Eac = 0.89. Some other details of calculation will be found in Mozumder (1995a). It is noteworthy that in low-mobility liquids ballistic motion predominates. Its effect on the mobility in n-hexane is 1.74 times greater than that of diffusive trap-controlled motion. As yet, there has been no calculation of the field dependence of electron mobility in the quasi-ballistic model. 

In applying the quasi-ballistic model to electron scavenging, Mozumder (1995b) makes the plausible assumption that the electron reacts with the scavenger only in the quasi-free state with a specific rate fesf. Denoting the existence... 

TABLE 10.4 Electron Mobility, Trap Density, Binding Energy, and Activation Energy in the Quasi-ballistic Model... 

Mozumder (1996) has discussed the thermodynamics of electron trapping and solvation, as well as that of reversible attachment-detachment reactions, within the context of the quasi-ballistic model of electron transport. In this model, as in the usual trapping model, the electron reacts with the solute mostly in the quasi-free state, in which it has an overwhelmingly high rate of reaction, even though it resides mostly in the trapped state (Allen and Holroyd, 1974 Allen et ah, 1975 Mozumder, 1995b). Overall equilibrium for the reversible reaction with a solute A is then represented as... 

FIGURE 10.5 Standard free energy change, in various liquid hydrocarbons, versus temperature upon electron trapping from the quasi-free state according to the quasi-ballistic model. Reproduced from Mozumder, (1996), with the permission of Am. Chem. Soc. ... 

Various theoretical explanations have been proposed for the diffusive behaviour of the electron, such as quantum tunnelling [64] between solvent sites and the quasi-ballistic model [65],... 

The main experimental elfects are accounted for with this model. Some approximations have been made a higher-level calculation is needed which takes into account the fact that the charge distribution of the trapped electron may extend outside the cavity into the liquid. A significant unknown is the value of the quasi-free mobility in low mobility liquids. In principle, Hall mobility measurements (see Sec. 6.3) could provide an answer but so far have not. Berlin et al. [144] estimated a value of = 27 cm /Vs for hexane. Recently, terahertz (THz) time-domain spectroscopy has been utilized which is sensitive to the transport of quasi-free electrons [161]. For hexane, this technique gave a value of qf = 470 cm /Vs. Mozumder [162] introduced the modification that motion of the electron in the quasi-free state may be in part ballistic that is, there is very little scattering of the electron while in the quasi-free state. 

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