Percolation point

As polymer molecular weight grows, the polyurea hard segment reaches a percolation point, where the driving force for interchain hydrogen bonding exceeds... 

Graphites with larger surface areas or greater porosities have a distinctly lower percolation threshold. It is assumed that the conductivity of a compound depends upon the structured agglomerates being sufficiently close to each other, or in direct contact above the percolation point, and on the continuous current pathways created thereby (14-15). 

Similarly, one can study the growth of the elastic constants (say the rigidity modulus) of a randomly formed elastic network, near the percolation point. The central force elastic problem (for networks formed out of linear springs only) belongs however to a different class of percolation problem, known as elastic percolation or central force percolation, and is discussed separately later (see Section 1.2.1(f)). 

As we will see in the appropriate sections of the next two chapters, the precise ranges of the validity of the Weibull or Gumbel distributions for the breakdown strength of disordered solids are not well established yet. However, analysis of the results of detailed experimental and numerical studies of breakdown in disordered solids suggests that the fluctuations of the extreme statistics dominate for the entire range of disorder, even very close to the percolation point. 

Phase A percolation point. Pc Phase B Compositon of phase A / B composite... 

An attempt to use a highly structured PAc (burrlike) to create blends by means of solvent suspensions on the lines of conventional percolation theory (low percolation point, but no further increase in conductivity as the concentration continues to rise) [64],... 

In adding conductive filler to an insulating resin, the volume resistivity changes slowly until a critical level of filler is reached, called the percolation point. The percolation point occurs when the resistivity drops abruptly, then continues to drop slowly (Fig. 2.11). " Almost continuous linkage of metal particles occurs at the percolation point where typical filler volumes for silver flakes are 25-30%. According to the percolation theory, there is a minimum critical volume of filler required for electrical conductivity in a polymer at which each filler particle must contact two other particles. A misconception in the use of silver flakes is that increasing the number of contacts lowers volume resistivity. Actually, the converse is true because, once the percolation point has been reached, each additional contact adds resistance. Thus, increasing the particle size can increase conductivity since the total number of contacts for a fixed volume decreases." ... 

Figure 2.11 Percolation point (Fc) for loading conductive fillers in epoxy resin.

In adding conductive filler to an insulating resin, the volume resistivity changes slowly until a critical level of filler is reached, called the percolation point. The percolation point occurs when the resistivity drops abruptly, then continues to drop slowly (Fig. 2.12). Almost continuous linkage... 

Consider the behavior of the m-component spin model on the disordered lattice with dilution near the percolation threshold cuid at low temperatures. As p decreases, also the critical temperature Tc(p) decreases and reaches zero at the percolation point Pc- At T = 0 the critical properties are determined by the properties of the connected spin clusters, i.e. this is a percolation problem. It has been argued [78] that the point p = Pc,T = 0 should be viewed as an multicritical point, here the thermal correlation length and correlation length of the percolation cluster diverge simultaneously. The studies of Ising model [91]... 

The percolation models discussed so far undergo piuely geometrical transitions because the objects treated have no center of mass translational motion. They are only randomly placed either on the lattices or in the continuum space. Therefore, they don t reveal any thermodynamic singularities. If particles are moving in a space, however, the entropy associated with the translational motion may partly vanish at the percolation point since the mass center of the infinite cluster (gel) ceases to move. If its derivative with respect to the concentration across the percolation point has a discontinuity, the transition becomes a real thermodynamic one. 

To define the threshold of pc statistically, the probability of path continuation is introduced. It is the probability that a randomly chosen site has at least one p>ath emanating from it that continues to infinity. This probability increases as the increase in p, and becomes equal to unity at p. Now, we will consider the path continuation from the randomly chosen site to its neighbour. There are (z - 1) sites available around the neighbour except for the first site chosen randomly the neighbour can form a bond to each of these sites with the probability of p. Therefore, the randomly chosen site can form bonds of (z - l)xp with its neighbours on average. At the critical percolation point, we have (z - l)xpc =1 and the gel point cein be given as. 

This behavior has been called percolation (Figs. 19.2 and 19.3). The percolation behavior of carbon blacks shows a dependence on the specific surface area. Carbon blacks with a large surface area display a much lower percolation point (which can be expressed in percentage by weight or volume, wt % or vol 9f) 9J. 

For a given type of carbon black the percolation point should be independent of the properties (polarity, viscosity) of the polymer matrix. 

Any dependence on viscosity should mean that overprocessing with highly viscous polymers results in a higher percolation point than with low viscosity polymers ... 

It is not possible on the basis of percolation theory to explain the percolation behavior of the spherical ICP primary particles that exist in totally disperse form below the percolation point. The percolation theory is based on the assumption that particles that do not enter into any interaction with the matrix are statistically distributed within it. This enabled the theoreticians to arrive at a mathematical description of the percolation behavior from a purely statistical and mechanistic point... 

---