Percolation theory critical volume fraction

All listed effects have an influence on the granule size and shape as well as on the distances between granules. This helps in understanding why experimental values of the critical volume fraction of metal granules xc strongly differ from the calculated ones in the framework of classical percolation theory [35], in particular, for In granules on top of an Si02 surface xc — 0.82... 

Utracki and Lyngaae-J0rgensen (78) proposed a theory based on the assumption that the critical volume fractions relate to the percolation thresholds of droplets, and phase inversion appears at the composition at which the blend... 

The percolation takes place if the critical volume fraction of secondary nanotube agglomerates Vy ggg is reached. According to classical percolation theory, the conductivity increase can be described with power law behavior (Eq. 5.9) with cr the plateau value of conductivity and the critical exponent (see also Appendix). 

For the polymers containing filler that touch each other, the percolation theory has been developed. This assumes a sharp increase in the effective conductivity of the disordered media, polymer matrix composite, at a critical volume fraction of the reinforcement known as the percolation threshold (( )percoi) which long-range connectivity of the system appears. The model that best expresses these aspects is the one created by Vysotsky (Vysotsky and Roldughin 1999), which presumes a percolation network of nanofiller particles inside the polymer matrix as shown in equation (11.10) ... 

According to the percolation theory, the percolation threshold, (j), is the critical volume fraction, where an infinite continuous conducting cluster spanning across the sample is formed. Due to the presence of a conducting or percolation path across the entire sample, a change from an insulator to a semiconductor occurs. Above the percolation threshold, the electrical conductivity is related to the content of conducting filler by a simple power law ... 

Very important is the fact that the position pc of the gel point is not universal Just like the amplitudes alone, pc is not independent of materials and models. For example, in the Stockmayer theory of f-functional gelation, pc = l/(f - 1) obviously depends on f considering lattice percolation in two dimensions for the square lattice of Fig. 1 Pc = 1/2 whereas for the triangular lattice Pc = 2 sin( r/18) = 0.34729 As a consolation for the lack of exact universality for Pc, one can offer the critical volume fraction which is not exactly the same but nearly the same for a broad class of different lattices and models with the same dimensionality. 

The initial porous texture of a catalyst pellet and the change in texture caused by metal deposition in it can be described using the percolation theory. In the percolation approach the pellet is constructed as a binary interdispersion of void space and (deposited) solid material. In this binary interdispersion, the void space can exist as (1) isolated clusters surrounded by solid material or (2) sample overspanning void space that allows mass transport from one side to the other. The total void space c can be split into the sum of the volume fraction of isolated clusters t1 and the volume fraction of accessible void space tA, If is below a critical value, called the percolation threshold all the void space is distributed as isolated clusters and transport is impossible through the pellet. 

The percolation probability has different values based on the classical theory site or bond percolation for different structures, as shown in Table 12.2. This critical percolation volume fraction, <, is calculated from the percolation threshold and the space filling factor. The volume fraction for site percolation for various structures is essentially the same as follows. In three dimensions, the site percolation threshold occurs at —16% volume. Near the percolation threshold the average cluster size diverges as does the spanning length of clusters. 

To test this theory, the room temperature conductivity of "Nafion" perfluorinated resins was measured as a function of electrolyte uptake by a standard a.c. technique for liquid electrolytes (15). The data obey the percolation prediction very well. Figure 9 is a log-log plot of the measured conductivity against the excell volume fraction of electrolyte (c-c ). The principal experimental uncertainty was in the determination of c as shown by the horizontal error bars. The dashed line is a non-linear least square law to the data points. The best fit value for the threshold c is 10% which is less than the ideal value of 15% for a completely random system. This observation is consistent with a bimodal cluster distribution required by the cluster-network model. In accord with the theoretical prediction, the critical exponent n as determined from the slope of... 

In the first case (when using butanol), the curve can be analysed using the percolation theory of conductivity [18]. In this model, the effective conductivity is practically zero as long as the volume fraction of the conductor (water) is below a critical value (the percolation threshold). Beyond this value, k suddenly takes a non-zero value and increases rapidly with further increase in In the above case (percolating microemulsions), the following equations were derived theoretically. 

The data was obtained at the shear strain of200% and frequency 2 s . There is a clear critical particle volume fraction in this suspension system, about 38 vol%. When the particle volume fraction exceeds this critical value, a sharp increment of the rheological property is observed. Percolation theory was used to explain this phenomenon, which will be discussed in detail in later on. Note that Figure 46 is consistent with Figure 47, and Figure 46 may only show the low particle volume fraction portion of Figure 47. 

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