Pore network modelling steady state

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. 

Randrianalisoa and Baillis (2008) used Monte Carlo simulation to model heat conduction in porous Si. In their method, an original 3D pore network that reproduces the morphology of mesoporous Si was developed. The nonlinear phonon dispersion curves of Si and the phonon mean-free path dependent on temperature, frequency, and polarization were also considered. The model of steady-state phonon transport through the pore network was simulated. Their results were compared with experimental results of porous Si thin films on a p" -type Si substrate. 

Before proceeding into any detailed calculations, we immediately recognize that our model network (5.1) topologically coincides with the network (4.38) of the model for the enzyme-catalyzed reaction of Section 2.2. This means that we can formally transfer all results from Section 2.2 to the present problem and only have to interpret those results according to the physical and biological context of the pore model. In this way, we derive the steady state flux 1 across the pores from (2.6)... 

Determine the relaxation behaviour of the networks for the pore and carrier models and for active transport. Prove that under any external conditions the state of the network exponentially tends into the steady state by solving the full equations of motion or that for the differential fluctuations as in Sections 2.6 and 2.7. 

Construct a network for different substrates S and S competing for the same carrier. Show that the interaction between the steady state fluxes and J2 of S and S2 is nonlinear as in the case of the pore model in Section 5.2. 

In the language of network topology, feedback manifests itself as a closed loop of bonds, junctions and elements. The reverse conclusion, however, does not hold the networks of the pore and carrier models and that of active transport in Sections 5.1 to 5.5 actually involve closed loops, but nevertheless show a unique globally and asymptotically stable steady state. The simplest case of a nontrivial feedback loop, i.e., a feedback loop which really leads to multiple steady states or limit cycles, is an autocatalytic reaction... 

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