Effective Medium Models for a Porous Membrane

We apply simple effective medium models in an attempt to understand the diffusion process in the complex pore network of a porous SiC sample. There is an analogy between the quantities involved in the electrostatics problem and the steady state diffusion problem for a uniform external diffusion flux impinging on a coated sphere. Kalnin etal. [17] provide the details of such a calculation for the Maxwell Garnett (MG) model [18]. The quantity involved in the averaging is the product of the diffusion constant and the porosity for each component of the composite medium. The effective medium approach does not take into account possible effects due to charges on the molecules and/or pore surfaces, details in the size and shape of the protein molecules, fouling (shown to be negligible in porous SiC), and potentially important features of the microstructure such as bottlenecks. 

We consider two effective medium models, corresponding to distinct morphologies on the micro- or nanoscale. The resulting quantity (pD)e replaces D in Equation (12.1). Subscripts 1 and 2 refer to the two 

This expression clearly is not symmetric in the two components. References [17,19-21] contain representative examples of applications of the MG model to diffusion in porous media. 

For application to porous SiC membranes, let medium 1 be the solution and medium 2 the SiC, so that D = D, the diffusion constant of the molecule in the solution, pi = 1, D2 = 0 and p2 = 0. The expressions for the two effective medium models simplify considerably  

In these expressions, p is the porosity of the SiC, i.e. the volume fraction of empty space. In the asymmetric MG model, we have chosen the coating to be the solution, since the opposite choice of SiC-encapsulated liquid spheres will not permit diffusion through the medium. With this choice, the SiC does not percolate and hence there is no structural support. The selectivity of the membrane is based in part on the size and shape of the protein molecules. The expressions for (pD)eff in the effective medium models [Equations (12.2) and (12.3)] do not contain a size scale, but it is necessary to introduce a scale in order to account for the size of a protein molecule. For simplicity, we assume that the proteins are spherical with effective (hydration) radius r. The excluded volume within the pores due to nonzero size is taken into account by replacing the porosity p with an effective porosity p. For the columnar 

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