Random Network Model of Membrane Conductivity

The effective PEM conductivity depends on the random heterogeneous morphology, namely, the size distribution and connectivity of the proton-containing aqueous pathways. Random network model of PEMs was developed in Eikerling et al. (1997). It included effects of the swelling of pores and the evolving connectivity of the pore network upon water uptake. The model was applied to study the dependence of membrane conductivity on water content and temperature. It could rationalize trends in 

The number fraction of blue pores as a function of the water content A, is given by 

A rigid microporous morphology with monodisperse pores, which does not reorganize upon water uptake, corresponds to a linear law x (w) = y w. In this case, the model resembles the typical percolation problem in random porous media with rigid walls (Stauffer and Aharony, 1994). Swelling causes deviations from this law. The universal percolation exponents are, therefore, not warranted. 

Pore-size-dependent conductances are assigned to individual pores and channels. Three possible types of bonds between pores exist. The corresponding bond conductances, specifically, abb X), abri ) and cfrrW, can be established. The model was extended toward the calculation of the complex impedance of the membrane by assigning capacitances in parallel to conductances to pores. The probability distribution of bonds to have conductivity abb, br, or arr is 

The simplest method of solution of the Kirchhoff equations, corresponding to the random network of conductance elements, is the single-bond effective medium approximation (SB-EMA), wherein a single effective bond between two pores is considered in an effective medium of surrounding bonds. The conductivity ab of the effective bond is obtained as the self-consistent solution of the equation 

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Overall, the distinction of different aqueous environments is a common thread for explaining membrane operation at distinct relevant scales (microscopic mechanisms of PT, conductance in the single pore enviromnent, random network model of membrane conductivity and membrane operation in the cell). 

Efforts of polymer scientists and fuel cell developers alike are driven by one question What specific properties of the polymeric host material determine the transport properties of a PEM, especially proton conductivity The answer depends on the evaluated regime of the water content. At water content above kc, relevant structural properties are related to the porous PEM morphology, described by volumetric composition, pore size distribution and pore network connectivity. As seen in previous sections, effective parameters of interest are lEC, pKa, and the tensile modulus of polymer walls. In this regime, approaches familiar from the theory of porous media or composites (Kirkpatrick, 1973 Stauffer and Aharony, 1994), can be applied to relate the water distribution in membranes to its transport properties. Random network models and simpler models of the porous structure were employed in Eikerling et al. (1997, 2001) to study correlations between pore size distributions, pore space connectivity, pore space evolution upon water uptake, and proton conductivity, as will be discussed in the section Random Network Model of Membrane Conductivity. ... 

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