Random resistor network

For typical catalyst layers impregnated with ionomer, sizes of hydrated ionomer domains that form during self-organization are of the order of 10 nm. The random distribution and tortuosity of ionomer domains and pores in catalyst layers require more complex approaches to account properly for bulk water transport and interfacial vaporization exchange. A useful approach for studying vaporization exchange in catalyst layers could be to exploit the analogy to electrical random resistor networks of... 

Reference 70 provides the first quantitative test of the random resistor network model. In Ref. 121 the authors employed the random resistor network model to determine the behavior of the low-field Hall effect in a 3D metal-nonmetal composite near the percolation threshold. For the following power laws of effective values of ohmic conductivity a, Hall coefficient R, and Hall conductivity a 12, Bergman et al. 121 have obtained the critical exponents ... 

We have given four examples of connectivity transitions described by percolation models. Other examples include the conductor-insulator transition for a random resistor network, oil recovery through a porous... 

Let us consider Young s modulus and electrical conductivity in a more general context. There is some evidence of Young s modulus and conductivity of a random resistor network scales with the... 

We have already pointed out (see equations 18 and 19) the analogy of the elastic and conductive properties so that the following remarks will apply to the conduction of a random resistor network near p as well (for a recent extended discussion of random conducting systems see ref. 82). 

M. Siekierski, K. Nadara, Modeling of conductivity in composites with random resistor networks, Electrochim. Acta, 2005,50, pp. 3796-3804. 

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... 

The model developed by Yuan et al. [82] is close in concept to the preceding one but differs from it in a computational approach used. The authors employ a random resistor-network model in which FM metallic particles with the number density p randomly fill the space of a sample. The computations are carried out with the use of the Monte Carlo method. The authors obtained the results close to those we described in this chapter, but they did not carry out the detailed analysis of the data. 

However, the picture is hardly an exact theory moreover, it was recently questioned whether the elasticity of the gel really varies with the conductivity of random resistor networks instead, the elasticity exponent was defined as y + 2j8 (which happens to be again 3 in the classical theory, but is about 2.6 in the percolation theory). Then, also the identification of viscosity with superconductor mixtures may be questionable. Even if this is not the case, entanglement effects may lead to a change in the viscosity exponent as compared to the conductivity exponent. Therefore, we use question marks instead of giving numerical predictions for k in Table 1. But Table 5 summarizes, with increasing order of reliability, the viscosity exponents determined by means of these three approximations, for both the percolation and classical theory. 

Above the gel point, the system becomes elastic and the shear modulus E increases as E Ap, where t = 1.7, according to the percolation theory where an analogy is made between E and the electric conductance of a random resistor network . ... 

We suppose now that our sample is a random lattice resistor network with 1 — p fraction of missing resistors, where p > Pc such that the lattice is still conducting. We shall now show that there is a relation between the failure current 7f of such a network and its third harmonic coefficient B (Yagil et al 1993). 

The simulations of conductivity in random resistor-capacitor (RC) networks confirmed freqnency dependence of Eqnation (65) (Panteny et al., 2005). In random networks percolated resistors lead to the DC plateau at low frequencies. In contrast, non-percolated resistors generate negative deviation of conductivity from DC plateau and non-monotonons decrease of conductivity at low frequencies. These regular or irregular deviations from DC plateau are in polymer-salt systems due to electrode polarization. At high frequencies random network simulations recovered the power-law dependency of conductivity as in Equation (65). Generally, one may say conductivity in random RC networks will be preferably determined by resistors at low frequencies and by capacitors at high frequencies. 

The first model of porous space as a 2D lattice of interconnected pores with a variation of randomness and branchness was offered by Fatt [220], He used a network of resistors as an analog PS. Further, similar approaches were applied in a number of publications (see, e.g., Refs. [221-223]). Later Ksenjheck [224] used a 3D variant of such a model (simple cubic lattice with coordination number 6, formed from crossed cylindrical capillaries of different radii) for modeling MP with randomized psd. The plausible results were obtained in these works, but the quantitative consent with the experiment has not been achieved. 

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