Tortuous path theory

One must be very careful in reviewing the older, and some more recent, literature in consideration of the tortuosity and constriction factors some work has attempted to separate these two factors however, more modem developments show that they cannot be strictly decoupled. This aspect will be particularly important when reviewing the barrier and tortuous-path theories of electrophoresis, as discussed later. 

This geometrical approach may reduce the deviations between experimental data and model predictions but it cannot resolve the main limitation of the tortuous path theory. This model is based on the assumption that the presence of nano-particles does not affect the diffusivity of a polymer matrix. 

The tortuous-path and barrier theories consider the effects of the media on the electrophoretic mobility in a way similar to the effect of media on diffusion coefficients discussed in a previous section of this chapter. The tortuons-path theory seeks to determine the effect of increased path length on electrophoretic mobility. The barrier theory considers the effects of the barrier or media conductivity on the electrophoretic mobility. 

Values of r2 of 2-6 are common. Occasionally, much higher values are quoted which are more likely to indicate shortcomings in the theory rather than highly tortuous paths. There is some confusion in the literature between r2 and t, as discussed by Epstein(35). 

Because the disks use the same packing material that has been described throughout the book, the theory of operation of the disks is identical for all of the methods described. The difference with the disks lies in the use of smaller particle size and in faster flow rates for large-volume samples. Figure 11.7 shows the relative difference in particle size for 40 to 60-pm particles and for the lO-pm particle sizes that are used in disks. For the same bed height as a cartridge, the disk has a much more tortuous path of flow, which means that there is considerably more surface area available, and the kinetics of sorption will be substantially quicker. This result is shown in the Empore trade literature, which shows that dye analytes are tightly bound near the surface of the disk. 

Percolation theory [53] is also used to calculate the effective properties such as the ionic conductivity in the SOFC electrodes. The effective conductivity of a composite electrode is less than that of the pure material due to the composite structure and porosity of the electrode. Percolation theory calculates an effective ionic conductivity that accounts for the tortuous path of the electrolyte phase in the electrodes and is based on the probability of finding a percolated chain of the electrolyte phase through the electrode [53]. 

Equation 25.14 can be used to determine the electrical potential in the electrolyte phase. Each term in this equation has been volume averaged, where the last term is a source term describing the volumetric electrochemical reaction. Here, a is the specific interfacial area described as the interfacial area per unit of electrode volume. The effective conductivities and x are used to account for the actual tortuous path length of charge transport and local porosity. Two empirical relations are often used in the porous electrode theory, namely ... 

The theory of permeation through microporous membranes in ultrafiltration and microfiltration is much less developed and it is difficult to see a clear path forward. Permeation through these membranes is affected by a variety of hard-to-compute effects and is also very much a function of membrane structure and composition. Measurements of permeation through ideal uniform-pore-diameter membranes made by the nucleation track method are in good agreement with theory. Unfortunately, industrially useful membranes have nonuniform tortuous pores and are often anisotropic as well. Current theories cannot predict the permeation properties of these membranes. 

A Bethe-tree is a particular case of more general networks considered in percolation theory. Sahimi and Tsotsis [1985] applied percolation theory and Monte Carlo simulation to deactivation in zeolites, approximated by a simple cubic lattice. Beyne and Froment [1990, 1993] applied percolation theory to reaction, diffusion and deactivation in the real ZSM-5 lattice. The finite rate of growth was described in terms of a polymerization mechanism. Pore blockage was reached in this small pore zeolite. It also affects the path followed by the diffusing molecules that becomes more tortuous, so that the effective diffusivity has to be expressed in terms of the blockage probability. 

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