Macroscopic network models

The expressions given in this section, which are explained in more detail in Erman and Mark [34], are general expressions. In the next section, we introduce two network models that have been used in the elementary theories of elasticity to relate the microscopic deformation to the macroscopic deformation the affine and the phantom network models. 

According to the phantom network model, the fluctuations Ar are independent of deformation and the mean f deform affinely with macroscopic strain. Squaring both sides of Equation (23) and averaging overall chains gives... 

Fluctuations are larger in networks of low functionality and they are unaffected by sample deformation. The mean squared chain dimensions in the principal directions are less anisotropic than in the macroscopic sample. This is the phantom network model. 

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

The assemblage of chains is constructed to represent the affine network model of rubber elasticity in which all network junction positions are subject to the same affine transformation that characterizes the macroscopic deformation. In the affine network model, junction fluctuations are not permitted so the model is simply equivalent to a set of chains whose end-to-end vectors are subject to the same affine transformation. All atoms are subject to nonbonded interactions in the absence of these interactions, the stress response of this model is the same as that of the ideal affine network. 

There are different ways to depict membrane operation based on proton transport in it. The oversimplified scenario is to consider the polymer as an inert porous container for the water domains, which form the active phase for proton transport. In this scenario, proton transport is primarily treated as a phenomenon in bulk water [1,8,90], perturbed to some degree by the presence of the charged pore walls, whose influence becomes increasingly important the narrower are the aqueous channels. At the moleciflar scale, transport of excess protons in liquid water is extensively studied. Expanding on this view of molecular mechanisms, straightforward geometric approaches, familiar from the theory of rigid porous media or composites [ 104,105], coifld be applied to relate the water distribution in membranes to its macroscopic transport properties. Relevant correlations between pore size distributions, pore space connectivity, pore space evolution upon water uptake and proton conductivities in PEMs were studied in [22,107]. Random network models and simpler models of the porous structure were employed. 

Effective conductivity of the membrane is related to its macroscopic morphology, viz. the random heterogeneous domain structure of polymer and solvent phases. On the basis of Gierke s cluster network model, a random network model of microporous PEMs was developed in [22]. This approach highlighted the importance of connectivity and swelling properties of pores. Random distributions of pores and channels as their interconnections were assumed. The connectivity between pores was considered as a phenomenological parameter. 

The catalyst effectiveness factor rji was calculated from the pore network model of Wood and Gladden [15] under the conditions on which capillary condensation was expected. The pore network model was solved over a range of temperatures from 553 to 580 K and for several pressures in the interval 20-40 bar to create a database of effectiveness factors for input to the macroscopic reactor model. The hydrodesulfurization of 1 mole % diethyl sulfide in an inert dodecane carrier was considered, with a molar gas oil ratio of 4. The catalyst was taken to have a connectivity of 6 and a normal distribution of pore sizes with a mean of 136 A and standard deviation of 28 A. By using the results of the pore network simulation as input to the macroscopic fixed bed reactor model, capillary condensation at the scale of the catalyst pellets was accounted for. 

This chapter discussed the art of modeling, with a few examples. Regardless of the type of model developed, a mathematical model should be validated with experimental results. Validation becomes very important in the black-box type of models such as the neural network models. Moreover, the model results are valid only within certain regimes where the model assumptions are valid. Sometimes, any model can be fit to a particular set of data by adjusting the parameter values. The techniques of parameter estimation were not presented in this chapter, the presentation was limited to lumped-parameter analysis, or macroscopic modeling. 

In the Gaylord-Douglas model [57,81] the chains are localized in a tube defined by the interactions with neighboring chains. The first term of the elastic free energy is the same as that of a phantom network model, while the second term accounts for the loss of degrees of freedom of the chains due to chain localization. In the dry network the cross-sectional dimension of the tube is of the order of the hard-core cross-sectional radius of the polymer chain, and the volume of the tube is comparable with the chain molecular volume. The tube volume is considered to be invariant with macroscopic strain, since the molecular volume of the chains is independent of the deformation. The elastic free energy is given by... 

Both deterministic and stochastic models can be defined to describe the kinetics of chemical reactions macroscopically. (Microscopic models are out of the scope of this book.) The usual deterministic model is a subclass of systems of polynomial differential equations. Qualitative dynamic behaviour of the model can be analysed knowing the structure of the reaction network. Exotic phenomena such as oscillatory, multistationary and chaotic behaviour in chemical systems have been studied very extensively in the last fifteen years. These studies certainly have modified the attitude of chemists, and exotic begins to become common . Stochastic models describe both internal and external fluctuations. In general, they are a subclass of Markovian jump processes. Two main areas are particularly emphasised, which prove the importance of stochastic aspects. First, kinetic information may be extracted from noise measurements based upon the fluctuation-dissipation theorem of chemical kinetics second, noise may change the qualitative behaviour of systems, particularly in the vicinity of instability points. 

This agrees with the result of the tetrahedra model studied by Flory and Rehner [24] to investigate the effect of junction fluctuations in a micronetwork consisting of four chains which start from the corners of the tetrahedra and are connected by one central junction. For a macroscopic network, the prefactor becomes... 

The Theory of Kuhn and Grun. The theory of birefringence of deformed elastomeric networks was developed by Kuhn and Griin and by Treloar on the basis of the same procedure as that used for the development of the classical theories of rubber-like elasticity (48,49). The pioneering theory of Kuhn and Griin is based on the affine network model that is, upon the application of a macroscopic deformation the components of the end-to-end vector for each network chain are assumed to change in the same ratio as that of the corresponding dimensions of the macroscopic sample. 

From the observed changes in the parallel and pernendicular components of the radius of gyration relative to macroscopic extension ratio, after appropriate correction for the dangling chain contributions, the chain extensive deformation is found to follow a behavior intermediate between the junction affine model and the phantom network model which allows unrestricted fluctuations of network junctions. On the other hand, the chain contractive deformation follows closely the chain affine model, indicating an asymmetry between extensive and contractive chain deformation. In either case, the deformation behavior is found to be the same for the two molecular weights. 

Key words polymer electrolyte membrane fuel cell, PEMFC, two-phase transport, porous media, pore network model, lattice Boltemarm model, direct numerical simulation, macroscopic upscaling. 

In this chapter we have shown the scope of multi-scale modeling in PEM fuel cells. The multi-scale modeling consists of pore-scale modeling and macroscopic upscaling. Pore network model and lattice Boltzmann model were specifically discussed in the context of two-phase transport in the PEMFC gas diffusion layer... 

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