Drift-diffusion theory

These two models, however, are insufficient to handle disordered organic materials, where the DOS is a Gaussian distribution, with localized carriers and discrete hopping within a distribution of energy states. Arkhipov presented an analytical model based on hopping theory [31] and Wolf preformed detailed Monte-Carlo simulations of hopping injection from a metal into an organic semiconductor [32]. Here we will present two injection models one is based on drift-diffusion theory and the other on the master equation. 

Using drift-diffusion theory, the hole current J can be written as... 

One of the robust features of the theory of polymer translocation, with the mapping to the drift-diffusion Fokker-Planck equation, is that the average translocation time (t) is proportional to the chain length N and inversely proportional to the driving force A/u,... 

The shape of the distribution function for the translocation time given by the drift-diffusion equation (Figure 10.11b) is similar to experimentally observed histograms, as long as the multiple peaks of Figure 10.1a are deconvoluted. When the reflecting boundary condition is used at the pore entry, theory yields only one peak in the distribution function of the translocation time. On the other hand, experiments show that there are two peaks, after the first peak corresponding to faster collisions between the polymer and pore is deleted. This feature is particular to the aHL protein pore and ssDNA, and is also based on the way the data are analyzed. 

Due to the low mobility in organic semiconductors, diffusion transport is also very important for the charge injection process. Therefore, an analytical diffusion controlled charge injection model particularly suited for OLEDs has been developed [33]. This model is based on drift-diffusion and multiple trapping theory. The latter can also be used to describe hopping transport in organic semiconductors [34],... 

The oxygen vacancies then diffuse to the gas interface where they are annihilated by reaction with adsorbed oxygen. The important point, however, is that metal is consumed and oxide formed in the same reaction zone. The oxide drift has thus only to accommodate the net volume difference between the metal and its equivalent amount of oxide. In theory this net volume change could represent an increase or a decrease in the volume of the system, but in practice all metal oxides in which anionic diffusion predominates have a lower metal density than that of the original metal. There is thus a net expansion and the oxide drift is away from the metal. 

This is the Einstein relation. It is probably the most important relation in the theory of the movements and drift of ions, atoms, molecules, and other submicroscopic particles. It has been daived ho e in an atomistic way. It will be recalled that in the phenomenological treatment of the diffusion coefficient (Section 4.2.3), it was shown that... 

C. Wulff. Theory of Meandering and Drifting Spiral Waves in Reaction-Diffusion Systems. PhD thesis, FU Berlin, 1996. 

While this is natural in view of the complexity of more detailed approaches to diffusion-limited reaction rates (e.g. kinetic theory), it is nevertheless a mute point as to whether the Debye—Smoluchowski equation represents an adequate description of diffusion and drift of interacting species in solution. 

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