Percolation limits

The filling factor is in good agreement with estimation from electron microscopy [6]. A filling factor of about 0.6 was obtained in all cases. The filling factor sensitively determines the position of the resonance at 0), which indeed shifts in frequency for different specimens. Moreover it is important to observe that / is already quite large and close to the boundary value for a percolation limit (which is -0.7 for spheres and -0.9 for cylinders). The realisation of such a limit would lead to a low frequency metallic Drude-like component in ai(to) for the composite. At present, this possibility seems to be... 

Figure 26 shows plots of this constriction factor verses porosity for various values of L. As observed in Figure 21 for the Michaels model of diffusion in such a cell, the percolation limits are seen where the constriction factor goes to zero at = L/(l + L). 

There is no adequate theory of the Neel temperature of a random distribution of centres in a dilute alloy, of indeed one exists. For higher concentrations of the magnetic matrix, with the assumption that only nearest neighbours interact, there is considerable theoretical work, giving a percolation limit , the concentration c0 at which long-range order disappears. The behaviour of TN is as (c —c0)12. For details see Brout (1965), Elliott and Heap (1962) and Klein and Brout (1963). 

According to the theoretical considerations, the following phase morphologies are expected for PPE/SAN blends (1) PPE contents below the percolation limit of around 19wt% (according to Utracki [46] SAN matrix, PPE fully dispersed,... 

The number of PPE particles dispersed in the SAN matrix, i.e., the potential nucleation density for foam cells, is a result of the competing mechanisms of dispersion and coalescence. Dispersion dominates only at rather small contents of the dispersed blend phase, up to the so-called percolation limit which again depends on the particular blend system. The size of the dispersed phase is controlled by the processing history and physical characteristics of the two blend phases, such as the viscosity ratio, the interfacial tension and the viscoelastic behavior. While a continuous increase in nucleation density with PPE content is found below the percolation limit, the phase size and in turn the nucleation density reduces again at elevated contents. Experimentally, it was found that the particle size of immiscible blends, d, follows the relation d --6 I Cdispersed phase and C is a material constant depending on the blend system. Subsequently, the theoretical nucleation density, N , is given by... 

In the present case, the foam density relates perfectly with the previously observed rheological properties, as a transition in the flow behavior was detected at approximately 20 wt% of PPE (Fig. 13). In the viscoelastic case (below the percolation limit), the PPE content neither significantly influences the foamability nor the blend rheology. At elevated contents (beyond percolation), however, the PPE content strongly affects the rheological response of the blend and, subsequently, degrades the foaming behavior, which is verified by a reduced expandability. 

Qualitative analysis of percolation limits. Experimental results suggest that periodicity of the growing morphologies exists only over a certain period of... 

Morphology evolutions, both solvent induced and by thermal agitation, as sketched above can be qualitatively explained in terms of Eq. (50). Percolation limits for blends of PMMA and SxMMAi x were studied in Ref. [107]. Also these results agree quite well with the outlined principles. 

In systems where the electrical properties are determined by the concentration of the component ions, the A2B04 system shows an unusual concentration dependence of resistivity. For example, delocalization of eg electrons is found in systems like LaSrAli-JNLCXf when x > 0.6. In perovskite systems such as Lai Sr Co03 and LaFei NL03, the oxides become metallic when x = 0.25-0.30. It is interesting to ponder whether such concentration limits are related to percolation limits in two-... 

The volume fraction o in Eqn. (2) defines a percolation limit for creep.69,70 Above this threshold, most particles within the slurry are part of an extended cluster that dominates the flow process. Two-dimensional experiments on slurries demonstrate that, at the threshold, 60-80% of the particles within the slurry form part of the cluster.71,72 When this happens, particle motion within the slurry is controlled by the cluster, and further deformation of the slurry is dominated by deformation of the cluster itself. 

FIGURE 10R6 Aggregate volume fraction versus time for reaction limited aggregation (RLA) and diffusion limited aggregation (DLA). When percolation limit)... 

For anisotropic particles, the percolation limit is a function of the aspect ratio. For ellipsoids of revolution, the percolation limit for a simple cubic lattice was studied by Boissonade et al. [85]. They found as the aspect ratio increases from 1 (a sphere) to 15 (a fiber), the percolation limit decreased from a volume fraction of 0.31 to 0.06 and the correlation length (i.e., aggregate size) did not change (i.e., it was the same as that of the sphere). 

FIGURE 12.6 Schematic diagram of an aggregated colloidal suspension showing a bridging network at gelation. The volume fraction of particles at which this bridging network is formed is referred to as the percolation limit. 

At the percolation limit, the rheological behavior of the suspension changes from Newtonian to either the Cross equation with a low shear limit viscosity or the Bingham plastic equation with an apparent yield... 

Beyond the percolation limit, the bridging network is more concentrated. Below the critical volume fraction, no continuously bridging networks are formed and the viscosity is low. As shown in Figure 12.7, this bridging network breaks up as the shear rate increases, giving different viscosities at different shear rates. As a result, this gives low and high shear limit viscosities observed at steady state for concentrated poljmier solutions and concentrated particulate suspensions (discussed later). 

FIGURE 12 As aggregation proceeds, the low shear viscosity increases drastically as the percolation limit is reached at the gel time, tg. 

As an illustration of the Rouse model, consider the polydisperse mixture of polymers produced by random branching with short chains between branch points. The molar mass distribution and size of the branched polymers in this critical percolation limit were discussed in Section 6.5. Close to the gel point, some very large branched polymers (with M> 10 ) are formed and the intuitive expectation is that such large branched polymers would be entangled. However, recall that hyperscaling requires... 

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