Models of electron transport

The physical models describing the electronic states of excess electrons in dielectric fluids can be classified according to their mobility Qortner and Gaathon, 1977). Generally, three regions can be distinguished  

In region (1), the electron can be considered as quasifree. (An electron is regarded as completely free only in the vacuum.) The structure of the fluid is unperturbed by the presence of the excess electron. The wave function is extended. The basic electron/liquid interaction may be treated as single scattering of an electron on a molecule or atom modified by the structure factor of the liquid, S(q) (Lekner, 1967), or it is considered as multiple scattering off density fluctuations in the framework of the deformation potential theory (Basak and Cohen, 1979). 

In region (2), the excess electron can be viewed as being localized in preexisting traps (vacancies, holes) or in potential fluctuations which develop into deeper traps. The configurational fluctuations of the liquid are not affected by the excess electron and the trapping site represents an attractive potential for the electron. The state of the electron is described by a localized wave function. Transport occurs by multiphonon absorption of the electron and transfer to a new trapping site or by photodiffusion. An exception are the electron bubbles in IHe, INe, and IH2 which are discussed in Section 7.2. 

In the following sections we discuss briefly some of the classical models which have been proposed for the explanation of the electron mobility, electron/ion recombination, and electron attachment in nonpolar liquids. Recently, Feynman path integral methods have been introduced in the description of excess electrons in fluids (Chandler and Leung, 1994). 

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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... 

Emberly EG, Kirczenow G (2001) Models of electron transport through organic molecular monolayers self-assembled on nanoscale metallic contacts. Phys Rev B 64(23) 235412... 

For certain liquids like cyclohexene [158], o-xylene, and m-xylene [159], the mobility increases with increasing pressure (see Fig. 11). These results provided the key to understand the two-state model of electron transport. In terms of the model, AFtr is positive for example, for o-xylene, AFtr is +21 cm /mol. Since electrostriction can only contribute a negative term, it follows that there must be a positive volume term which is the cavity volume, Fcav(e). The observed volume changes, AFtr, are the volume changes for reaction (23). These can be identified with the partial molar volume, V, of the trapped electron since the partial molar volume of the quasi-free electron, which does not perturb the liquid, is assumed to be zero. Then the partial molar volume is taken to be the sum of two terms, the cavity volume and the volume of electrostriction of the trapped electron ... 

Thus considerable support exists to support the two-state model of electron transport. The magnitude of the mobility is dependent on many factors including Fq, qf, AGsoin(e), temperature, pressure, and other factors. Presumably, differences in these factors can... 

Fig. 20.8 Schematic models of electron transport in mesoporous Ti02 electrodes.

Nanocrystalline systems display a number of unusual features that are not fully understood at present. In particular, further work is needed to clarify the relationship between carrier transport, trapping, inter-particle tunnelling and electron-electrolyte interactions in three dimensional nan-oporous systems. The photocurrent response of nanocrystalline electrodes is nonlinear, and the measured properties such as electron lifetime and diffusion coefficient are intensity dependent quantities. Intensity dependent trap occupation may provide an explanation for this behaviour, and methods for distinguishing between trapped and mobile electrons, for example optically, are needed. Most models of electron transport make a priori assumptions that diffusion dominates because the internal electric fields are small. However, field assisted electron transport may also contribute to the measured photocurrent response, and this question needs to be addressed in future work. 

To avoid the account of the edge effects let us consider rather long structures (L > 50 nm), i.e. we will consider the armchair single-wall carbon nanotubes with the length greater than electron mean free path [2-6]. To describe the electron-phonon transport in nanotubes like that the semiclassical approach and the kinetic Boltzmann equation for one-dimensional electron-phonon gas can be used [4,6]. In this connection the purpose of the present study is to develop a model of electron transport based on a numerical solution of the Boltzmann transport equation. 

Electron transfer during photosynthesis, according to the tunnel mechanism, was examined in detail in Refs. 116-121. The most interesting of all for the modeling of electron transport in biological membranes is the case when the redox reaction at the membrane/electrolyte interface leads to ion permeability, and not only to electron permeability. Let us assume that ion B is insoluble in the membrane, and, therefore the membrane is impermeable to it. However, if at the interface the ion undergoes redox transformations ... 

The model of electron transport along pendant viologen vanadium(II) groups to colloidal platinum is presented below. The dotted line illustrates the direction of the flow of electrons (see scheme p. 132). 

J. Nelson, Continuous-time random-walk model of electron transport in nanocrystalline Ti02 electrodes, Phys. Rev. B Con-dens. Matter 1999, 59(23), 15374-15380. 

These three main models of electron transport are outlined in the following section. The discussion is presented in the context of organic solid materials because conductive biofilms are organic solids. These processes can occur simultaneously, and each may dominate at different values of the applied electric field or different temperature range [18, 19]. Figure 7.2 depicts the differences between these mechanism in the context of microbial nanowires and cytochromes aligned along the nanowires [12]. 

In this section we shall discuss simple models of electron transport. Some more detailed considerations will be presented in Chapter 7. Basically, three cases will be described here (1) electrons moving in delocalized states (motion in the conduction band) (2) trap-modulated motion in delocalized states and (3) hopping mobility. The three modes of transport are depicted schematically in Figure 35. 

In Section 3.11.2 we discussed the phenomenological model of electron transport in the conduction band of tetramethylsilane modulated by trapping by biphenyl. Such a model can be generalized for the electron transport in low mobility hydrocarbons, as, for instance, n-hexane. Localized electrons have been detected by their optical absorption. The traps in hydrocarbons are assumed to be structural voids which upon occupation by an electron increase further in depth. The electrons are only partially localized in these traps (Schiller et al., 1973). By thermal activation they... 

Zhang JF, Hughes T, Steigerwald M, Bms LA, Friesner R (2012) Realistic cluster modeling of electron transport and trapping in solvated Ti02 nanoparticles. J Am Chem Soc 134(29) 12028-12042... 

In this chapter I will review random walk Monte Carlo, MC, simulation models of electron transport in DSSC. In Sect. 2,1 will place these studies in the context of DSSC transport models. MC methods and results are covered in Sect. 3 The concluding section. Sect. 4, looks at future applications of these methods and the related MC models for polymer blend cells that are covered in [7]. 

Electrons in these liquids spend most of their time in localized states. One model of electron transport, derived from semiconductor theory, is that each electron is from time to time thermally excited into the delocalized state (conduction band), where it migrates relatively freely until it becomes de-excited into a localized state again. 

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