Equivalent network model

Past work on electrical measurements of columns consisting of spheres of ion-exchange resins and electrolyte solutions, and on the interpretation of the results by the simple equivalent-network model described above is then reviewed. The experimental results covered are conductivity measurements at low frequencies (60 and 1000 Hz), electrical potential differences between two different NaCl solutions separated by such columns, and the variation of the dielectric constant and conductivity of the columns above 20 MHz, all over a range of sulution concentrations. It will be shown that not only does the proposed network describe the variation of the measured parameters with solution concentration and/or frequency, but that the geometrical parameters of the model are roughly the same for all these measurements. 

Equation (32a) has been very successful in modelling the development of birefringence with extension ratio (or equivalently draw ratio) in a rubber, and this is of a different shape from the predictions of the pseudo-affine deformation scheme (Eq. (30a)). There are also very significant differences between the predictions of the two schemes for P400- In particular, the development of P400 with extension ratio is much slower for the network model than for the pseudo-affine scheme. 

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. 

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. 

It can be shown using Eq. (1-20) that the upper-convected Maxwell equation is equivalent to the Lodge integral equation, Eq. (3-24), with a single relaxation time. This is shown for the case of start-up of uniaxial extension in Worked Example 3.2. Thus, the simplest temporary network model with one relaxation time leads to the same constitutive equation for the polymer contribution to the stress as does the elastic dumbbell model. 

A great number of studies have been published to deal with relation of transport properties to structural characteristics. Pore network models [12,13,14] are engaged in determination of pore network connectivity that is known to have a crucial influence on the transport properties of a porous material. McGreavy and co-workers [15] developed model based on the equivalent pore network conceptualisation to account for diffusion and reaction processes in catalytic pore structures. Percolation models [16,17] are based on the use of percolation theory to analyse sorption hysteresis also the application of the effective medium approximation (EMA) [18,19,20] is widely used. 

Gierke also considered that these clusters are interconnected by short, narrow channels in the fluorocarbon backbone network. The diameter of these channels is about 1 mm estimated from hydraulic permeability data. He further considered that the Bragg spacing ( 5nm from SAXS data) can represent the distance between clusters. The cluster-network model is a phenomenological description. Recently, Hsu and Gierke " have derived a semi-phenomenological expression to correlate the variation of cluster diameter with water content, equivalent weight, and cation form of the membrane. They have shown that the short channels are thermodynamically stable. 

In the next section we will present the data and arguments on which the cluster-network model is based. We will also discuss the effects of equivalent weight, ion form, and water content on the dimensions and composition of the clusters. In the third section we will present a formalism, which follows from the cluster-network model, based on absolute reaction rate theory (2) and hydroxyl rejection in "Nation perfluorinated membranes. Finally we will outline the concepts of percolation theory and demonstrate that ion transport trough "Nation" is well described by percolation. 

Experiment 1 compares Hebbian trained Hopfield networks with their equivalent HEDA models. The aim is to discover whether or not the HEDA model can achieve the capacity of the Hopfield network. Hopfield networks were trained on patterns using standard Hebbian learning, with one pattern at a time being added until the network s capacity was exceeded. At this point, the learned patterns were set to be the targets for the HEDA search using Eqs. 21 and 22 and the network s weights were reset. 

In both cases an expression for the free energy results which is identical with tq. (37) derived by the replica method From the congruent results obtained by different theoretical methods for physically equivalent models, it can be concluded that Eq. (37) is representative for network models with harmonic constraining potentials. Equation (37) will be used extensively in all further discussions concerning the properties of the tube model of polymer networks. 

Figure 12.5 The equivalent circuit of a porous carbon electrode. It consists of a single vertical ladder network in series with an flC parallel network. The ladder network models the response of pores in the body of...

Based on the concept of equivalent geotechnical model, back propagation neural networks was applied to determinesome geotechnical parameters, which are difficult to be ascertained. Verification through comparing detected deformation and calculated deformation qualifies equivalent geotechnical model for FEM assessment of slope stabihty. 

The greatest physiological application for the equivalent circuit model is where myocardial cells are represented as a ladder network. In the case of the heart, myocytes are represented as being connected end to end (which they are within the heart) with an internal resistance (/ ) between two cells. Within the context of the heart, it must be realised that myocytes are connected to adjacent cells and therefore the network most accurately reflects the physiological scenario when considered in two or even three dimensions. 

Bourouina T, Grandchamp JP (1996) Modeling micropumps with electrical equivalent networks. J Micromech Microeng 6(4) 398-404... 

An agreement with experimental results was obtained by taking into account the increased effective fraction of the filler, Veff, due to the glassy interphase of the bound epoxide layer and assuming a co-continuous morphology of the epoxy-silica hybrid network. Mechanical properties in dependence on the phase continuity are treated by parallel and series models for bicontinuous morphology and discontinuous phases, respectively. The equivalent box model (EBM) developed by Takayanagi (13) (eqs 2-5) and Davies model (14) (eq. 6) were used to compare the experimental data with the theory (9). 

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