Exponents conductivity

Here, R0 is the resistance of the occupied lattice units and CQ is the capacitance of the non-occupied lattice units. In a first attempt, not considering anomalous diffusion effects, the exponent has been evaluated as q=/u+s 2.7, where s 0.7 and /t 2.0 are the conductivity exponents (in three dimensions) below and above the percolation threshold, respectively. However, by including anomalous diffusion effects one obtains q=m=vdww3.3 [136, 137]. 

If the estimated fitting parameters are compared to the predicted values of percolation theory, one finds that all three exponents are much larger than expected. The value of the conductivity exponent ji=7A is in line with the data obtained in Sect. 3.3.2, confirming the non-universal percolation behavior of the conductivity of carbon black filled rubber composites. However, the values of the critical exponents q=m= 10.1 also seem to be influenced by the same mechanism, i.e., the superimposed kinetic aggregation process considered above (Eq. 16). This is not surprising, since both characteristic time scales of the system depend on the diffusion of the charge carriers characterized by the conductivity. 

Conductivity exponent 38, 43 Connectivity 34-35, 46, 52 Contact area 107 Contact mechanics 107 Copper 129... 

Several attempts have been made to relate these conductivity exponents tc or Sc with the percolation cluster statistical exponents discussed earlier. For example, in the node-link-blob model, the conductivity E of the network is proportional to the number of parallel links, while the... 

Again several attempts have been made to relate the elastic exponents Tg and Sg with the conductivity exponents tc and Sc and other percolation exponents. Identifying the elastic energy E F /Y of the... 

The other feature that emerges from the experimental data is the functional dependence of the observed o x) that is manifested in particular by the conductivity exponent t. Having the framework given in Sections 5.2-S.4, we can understand the various observed values of t as follows. In the cases where the onset of percolation is associated with the coalescence of particles (as in granular metals [42]) or coalescence-like (as in semiconductor-dielectric composites [41] shown in Figure 5.6), the values of t [35, 39, 40, 42, 59] are, within the experimental uncertainty, very close to the classical critical values of 1.7 < t < 2 that were derived from calculations or computations [1, 3, 15]. This observation is well understood by following two considerations. Either, as suggested above, we approach the S Z picture, or we can virtually divide the continuous phase network into small elements (say, spheres or... 

The conductivity exponent of about 1.23 indicates 2D character of the percolation transition. Similar values of the conductivity exponent were obtained for the hydration dependence of the conductivity of embryo and endosperm of maize seeds [595, 596], where the percolation threshold is /t = 0.082 and 0.127 g/g, respectively. In hydrated bakers yeast, protonic conductivity evidences 2D percolation transition of water at h = 0.163 g/g, and the value of the conductivity exponent is about 1.08 [597]. In this system, increase in conductivity due to 3D water percolation is observed at essentially higher hydration level h= 1.47 g/g, where conductivity exponent is about 1.94, i.e., close to the 3D value t = 2.0. Conductivity measurements of Anemia cysts at various hydrations show strong increase in conductivity starting from the threshold hydration h = 0.35 g/g [598] (see Fig. 97). The conductivity exponent in this system is 1.635, which is in between the values expected for 2D and 3D systems. DC conductivity of lichens, evaluated from the dielectric studies at frequencies between 100 Hz and 1 MHz [599], shows strong enhancement at some hydration level. Fit of the conductance-hydration dependence to equation (24) gave the following parameters he = 0.0990 g/g, t = 1.46 for Himantormia lugubris and he = 0.0926 g/g, t = 1.18 for Cladonia... 

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. 

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