Transport in Disordered Media

In a previous section reference was made to the random walk problem (Montroll and Schlesinger [1984], Weiss and Rubin [1983]) and its application to diffusion in solids. Implicit in these methods are the assnmptions that particles hop with a fixed jump distance (for example between neighboring sites on a lattice) and, less obviously, that jumps take place at fixed equal intervals of time (discrete time random walks). In addition, the processes are Markovian, that is the particles are without memory the probability of a given jump is independent of the previous history of the particle. These assumptions force normal or Gaussian diffusion. Thus, the diffusion coefficient and conductivity are independent of time. 

In recent years, more complex types of transport processes have been investigated and, from the point of view of solid state science, considerable interest is attached to the study of transport in disordered materials. In glasses, for example, a distribution of jump distances and activation energies are expected for ionic transport. In crystalline materials, the best ionic conductors are those that exhibit considerable disorder of the mobile ion sublattice. At interfaces, minority carrier diffusion and discharge (for example electrons and holes) will take place in a random environment of mobile ions. In polycrystalline materials the lattice structure and transport processes are expected to be strongly perturbed near a grain boundary. 

In general, the study of transport processes in disordered media has its widest application to electronic materials, such as amorphous semiconductors, and very little attention has been given to its application to ionic conductors. The purpose of this section is to discuss briefly the effect of disorder on diffusion process and to point out the principles involved in some of the newly developing approaches. One of the important conclusions to be drawn is that frequency-dependent transport properties are predicted to be of the form exhibited by the CPE if certain statistical properties of the distribution functions associated with time or distance are fulfilled. If these functions exhibit anomalously long tails, such that certain moments are not finite, then power law freqnency dispersion of the transport properties is observed. However, if these moments are finite, then Gaussian diffusion, at least as limiting behavior, is inevitable. 

There is a qualitative difference in transport properties depending on the nature of y/(f). If y/(f) is such that the time between hops has a finite first moment, that is, a mean residence time (t) can be defined, then classical diffusion is observed. An example would be 

In other words, the hopping probability is a slowly decaying function of time. Under these conditions, the dispersion of the concentration, (jc (O), becomes proportional to f , and the diffusion coefficient 

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Mohanty, K., J. Ottino, and H. Davis, Reaction and transport in disordered composite media Introduction of percolation concepts. Chemical Engineering Science, 1982, 37, 905-924. 

A question of great interest is the influence of disorder in the medium on the universality class of dissolved flexible polymers, namely ire the imiversal exponents (1) in this c ise the same as in the pure case The question of how line tr polymers behave in disordered media is not only interesting from a theoretical point of view, but is also relevant for understanding transport properties of polymer chains in porous media, such as an oil recovery, gel electrophoresis, gel permeation chromatography, etc. [14]. 

Cortis A, Chen Y, Scher H, Berkowitz B (2004) Quantitative characterization of pore-scale disorder effects on transport in homogeneous granular media. Phys Rev E 70, 041108, DOl 10.1103/PhysRevE.70.041108... 

Quintard M, Whitaker S (1993) Transport in Ordered and Disordered Porous Media Volume-Averaged Equations, Closure Problems, and Comparisons with Experiments. Chem Eng Sci 48(14) 2537-2564... 

Quintard M. and Whitaker S. 1993a. Transport in ordered and disordered porous media Volume averaged equations, closure problems, and comparison with experiment, Chem. Eng. Sci., 48, 2537-2564. 

The relaxation of this exponentiality condition, which precisely coincides with the Markovian assumption, leads to semi-Markov processes (Feller, 1964). What could be the chemical conditions of the occurence of non-exponential waiting times, and which distributions could have physico -chemical relevance Though many works have been done with the method of the continuous time random walk (CTRW) mostly in connection with transport processes in amorphous media, it seems to be very hard job to associate chemical conditions to different kinds of waiting time distributions due to temporal disorders. ... 

We add, however, that special care must be taken in determining from the experimental data which transport mechanism is likely. This is because resistivity (p) often exhibits exponential dependence on temperature T i.e. p oc exp[(7 /7 o) ], on the basis of which the transport mechanism is inferred. The exponents x= j2, 1/3, and 1/4 frequently arise in the non-crystalline or disordered media of semiconductors. When X = 1/3 happens, for instance, it is usually difficult to distinguish various conduction mechanisms including two-dimensional VRH, tunnehng of carriers, interchain conduction, etc. [97]. 

In Chap. 9 Alessandro Troisi blocks out disorder effects by focusing on the charge transport in crystalline organic media. In this way, the interplay between a compound s electrical characteristics and the chemical structure of the soft lattice dressing of the effective masses and energies by local distortions can be examined from a microscopic point of view by the means of computational chemistry. 

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