Random-Network Model

The first extension of the percolation theory to address the problem of electrical transport in composite materials was conducted by Kirkpatrick rrsing a random resistor network model. Random resistor networks are created by assigning each node in the network with a random resistivity value and calculating the arrrent flow through the entire network at a fixed external voltage by solving Kirchoff s law at every node. Kirkpatrick s random resistor model provides a simple and convenient discrete model for the conductivity of a continuorrs medium if the spatial distribution of particles is known. As a... 

Mangipudi et al. [63,88] reported some initial measurements of adhesion strength between semicrystalline PE surfaces. These measurements were done using the SFA as a function of contact time. Interestingly, these data (see Fig. 22) show that the normalized pull-off energy, a measure of intrinsic adhesion strength is increased with time of contact. They suggested the amorphous domains in PE could interdiffuse across the interface and thereby increase the adhesion of the interface. Falsafi et al. [37] also used the JKR technique to study the effect of composition on the adhesion of elastomeric acrylic pressure-sensitive adhesives. The model PSA they used was a crosslinked network of random copolymers of acrylates and acrylic acid, with an acrylic acid content between 2 and 10%. 

Chapter 8 describes a number of generalized CA models, including reversible CA, coupled-map lattices, quantum CA, reaction-diffusion models, immunologically motivated CA models, random Boolean networks, sandpile models (in the context of self-organized criticality), structurally dynamic CA (in which the temporal evolution of the value of individual sites of a lattice are dynamically linked to an evolving lattice structure), and simple CA models of combat. 

Wood and Hill consider that the role of fluoride in these glasses is uncertain. Phase-separation studies suggest that the structure of the glass might relate to the crystalline species formed, in which case a microcrystallite glass model is appropriate. But other evidence cited above on the structure-breaking role of fluoride is compatible with a random network model. 

Experimental determinations of the contributions above those predicted by the reference phantom network model have been controversial. Experiments of Rennar and Oppermann [45] on end-linked PDMS networks, indicate that contributions from trapped entanglements are significant for low degrees of endlinking but are not important when the network chains are shorter. Experimental results of Erman et al. [46] on randomly cross-linked poly(ethyl acrylate)... 

The SANS experiments of Clough et al. (21) on radiation crosslinked polystyrene are presented in Figure 9, and appear to fit the phantom network model well. However, these networks were prepared by random crosslinking, and the calculations given are for end-linked networks, which are not truly applicable. 

So far, there have only been a few modeling studies to try to link local fluid flow to bed structure. Chu and Ng (1989) and later Bryant et al. (1993) and Thompson and Fogler (1997) used network models for flow in packed beds. Different beds were established using a computer simulation method for creating a random bed. The model beds were then reduced to a network of pores, and either flow/pressure drop relations or Stokes law was used to obtain a flow distribution. 

For reasons which will become clear, we examine first the case of high temperature H20(as). Two random network models relevant to our hypothesis have been described in the literature. Both are based on distortions from a single locally tetrahedral structure that is like ice Ih. Kell s model 77> is much too small to be very useful. Nevertheless, its successful construction, just as for the case of Ge(as) 78>, Si02(as) 79>, and others, shows the viability of the random network concept. 

A much more satisfactory random network model has been discussed by Alben and Boutron 82h They used a model, proposed by Polk 78> for Ge(as), scaled to fit the observed nearest neighbor 00 distance of H20(as), and with H atoms added to the OO bonds according to the Pauling ice rule that guarantees the presence of only H20 molecules 65>. In the Polk model the bond length is everywhere the same and the 000 angles are distributed with root mean square deviation of 7° about 109°. For the case of Ge(as), the observed and model radial distribution functions are in excellent agreement. 

Alben and Boutron suggest that the peak in the X-ray and neutron scattering functions at 1.7 A-1 is indicative of an anisotropic layer structure extending over at least 15 A in Polk type continuous random network models. To show this better Fig. 52 displays the radial distribution function of the Alben-Boutron modified... 

The reader should recall that the fitting of a structure to diffraction data is not unique. We have shown that both the constructed modified random network model of Polk, as well as the network simulated by allowing Gaussian distributions of atom-atom distances can fit the observed structure functions for low density H20(as), and the latter, with modification to include small OOO... 

There do not yet exist random network models with mixed lattice parentage, hence the properties of low temperature HaO(as) cannot be compared with those predicted for such a model. 

The model just described does not conform in detail to the random network model proposed earlier. In particular, the use of a continuum outside the nearest-neighbor tetrahedral structure removes some of the correlations inherent in a continuous network. Nevertheless, given the existence of a broad OOO angular distribution, coupled to an 00 distance distribution much broader than in H20(as), it is unlikely that this assumption introduces any features in serious disagreement with those characteristic of a random network model. 

A different test, one less satisfactory because the standard of comparison is simulated water not real water, is obtained by examining the functions hon(R) and dd(-R) predicted. The functions derived from the Narten model are shown in Fig. 54 they should be compared with those for simulated water, displayed in Figs. 27 and 28. Just as for the function hoo R), the curves for simulated water are narrower and higher than those in Fig. 54. In all other respects the agreement between the two sets of functions is excellent. It now remains to be shown that a full calculation, based on precisely the random network model proposed will reproduce the data as well as this (sensibly equivalent ) model. 

If the different continuous random network models of high and low temperature H20(as) are valid, the following tests are worthy of attention ... 

Clearly, any measurement that differentiates between the properties of high and low temperature forms of H20(as), and/or delineates the relationship between H20(as) and liquid H20, can be used to test the hypotheses advanced vis a vis their structures. These and the experimental tests suggested, together with the construction of continuous random network models more sophisticated than that for Ge(as), the increased use of computer simulation, and exploitation of the available experimental information to guide the choice of appproximations in a statistical mechanical theory should increase our understanding of H20(as) and, uitimately, liquid H20. 

This oversimplified random network model proved to be rather useful for understanding water fluxes and proton transport properties of PEMs in fuel cells. - - - It helped rationalize the percolation transition in proton conductivity upon water uptake as a continuous reorganization of the cluster network due to swelling and merging of individual clusters and the emergence of new necks linking them. ... 

The effective conductivity of the membrane depends on its random heterogeneous morphology—namely, the size distribution and connectivity of fhe proton-bearing aqueous pafhways. On fhe basis of the cluster network model, a random network model of microporous PEMs was developed in Eikerling ef al. If included effecfs of varying connectivity of the pore network and of swelling of pores upon water uptake. The model was applied to exploring the dependence of membrane conductivity on water content and... 

Recently, Jung et al. [42] developed two artificial neural network models to discriminate intestinal barrier-permeable heptapeptides identified by the peroral phage display experiments from randomly generated heptapeptides. There are two kinds of descriptors one is binary code of amino acid types (each position used 20 bits) and the other, which is called VHSE, is a property descriptor that characterizes the hydrophobic, steric, and electronic properties of 20 coded amino acids. Both types of descriptors produced statistically significant models and the predictive accuracy was about 70%. 

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