Hopping distance

The hopping model This model assumes that molecules can hop over the surface. The surface flux is calculated by the mean hopping distance and the velocity, with which the molecules leave their site. Weaver and Metzner (1966) developed a detailed model to calculate the mean hopping distance. Ponzi et al. (1977) developed a simpler way of estimating the mean hopping distance. 

Usually it is assumed that tc is the only temperature-dependent variable in Eq. 9. This might be the case for an order-disorder type rigid lattice model, where the only motion is the intra-bond hopping of the protons, since the hopping distance is assumed to be constant and therefore also A and A2 are constant. This holds, however, only for symmetric bonds. Below Tc the hydrogen bonds become asymmetric and the mean square fluctuation amplitudes are reduced by the so-called depopulation factor (l - and become in this way temperature-dependent also. The temperature dependence of tc in this model is given by Eq. 8, i.e. r would be zero at Tc, proportional to (T - Tc) above Tc and proportional to (Tc - T) below Tc. 

Amino acid residue models such as a tyrosine residue model (p-cresol) lengthen remarkably the charge hopping distance, a phenomenon which can solve i he problem in the electrocatalysis mentioned in the above item 5) and enhance remarkably the catalytic activity. 

Regarding item 6) above on electrocatalysis, the coexistence of tyrosine residue model, p-cresol (p-Crej, enhanced remarkably the catalytic activity of Ru-red confined in a Nafion membrane coated on an electrode (Fig. 19.3).20) This was attributed to the nearly twofold lengthening of the charge hopping distance by p-cresol from 1.28 nm to 2.25nm). 

It appears that the small amounts of Co2+ and Co3+ which occur, probably on equivalent lattice sites, have little effect upon conductivity because of the large hopping distances. 

This form of the conductivity is termed variable range hopping because the average hopping distance is not constant, but decreases as the temperature is raised. The conductivity follows a characteristic 7 law rather than an Arrhenius behavior. More detailed analysis of the hopping mechanism confirms Eq. (7.79), but gives a substantially different value for the prefactor... 

A key result of the analysis of Van der Auweraer et al. is that the drift velocity increases with increasing field at low fields, then decreases with increasing field at high fields. This gives rise to a negative field dependence of the mobility at high fields. The results are illustrated in Fig. 30. For these calculations, the temperature was 298 K. the rale constant kQ 10-6 s-f and the hopping distance... 

In the lattice gas model, the hopping frequency of a carrier decreases exponentially with the hopping distance p as v = vQ exp(-2p/p0). Here, pQ is a wavefunction decay constant and vQ a frequency factor. From the Einstein relationship, the zero-field mobility is... 

The lattice gas model is based on the assumption that the hopping distance can be described as p = (M/Ach) 9 where M is the molecular weight of the dopant... 

The mobility is concentration dependent, see Table 8.2, since increasing the density of conjugated sites reduces the mean hopping distance. Data for DEH loaded in polycarbonate is shown in Fig. 8.28. The solid line is a fit to the empirical relationship ... 

Mobility Scales with Average Inter site Hopping Distance, Gill identified an exponential dependence of electron drift mobility on average TNF intersite hopping distance and noted an associated decrease in hole drift mobility (35). Gilfs subsidiary observation that TNF addition also decreased hole mobility precisely because complexed carbazole is removed as a hole transport site provided evidence for the key role of the discrete carbazole groups in hole transport. 

Figure 34. Dependence of the overall-magnitude parameter on separation between NIPC molecules (p) in NIPC-doped bisphenol-A pwlycarbonate. The slowly varying factor compensates for variation of the average hopping distance. Data taken from Ref. [51].

The diffusion coefficient is Da = (1 /xa)R2, where l/xa is the phonon-assisted fracton-hopping time and R is the mean hopping distance for fracton modes with frequency near CD,. The rate of phonon-assisted fracton hopping is calculated with Eqs. (34) and (35) specifically for the cases where one fracton decays into another fracton and a phonon, as well as for the reverse process. 

The mean hopping distance must depend on frequency because of Drude s condition. There are therefore, all the insignia of a variable range hopping, but with the hopping range determined by the a.c. frequency. Thus,... 

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