Percolation in one dimension

Because only one vacancy separates the train into two parts, all lattice sites must be occupied for the system to percolate. In other words, there is no postgel regime in one dimension. 

To specify the nature of the singularity near the threshold, we introduce the critical indices of percolation. The averages are 

We next introduce the connectivity correlation function (r) of the particles defined 

As p approaches Pc = I, the correlation length grows to infinity. If we introduce the critical index v by the form 

The percolation transition can be described in space of any dimension. Examples of two-dimensional percolation are deluge, forest fire, spreading of a contagious disease in an orchard, and gelation of a polymer at an air-water interface. Examples of three-dimensional percolation are substitutional alloys and bulk polymer gelation. A problem analogous to one-dimensional percolation is the condensation polymerization of bifunctional monomers described in Section 1.6.2. 

Consider the condensation polymerization of bifunctional monomers, each with two different reactive groups A and B, where A is only allowed to react with B in an intermolecular fashion. For example, if group A were 

There is exactly one unreacted A group (and one unreacted B group) per molecule. The number density of molecules n oiip) (number of molecules per monomer) is therefore equal to the fraction 1 of unreacted groups  

The number-average degree of polymerization (the number of monomers per molecule) is [see Eq. (1.55)] the reciprocal of the number density of molecules  

A linear polymer (A-mer) is a cluster of N monomers (sites) connected by N — 1 bonds and containing one unreacted A group and one unreacted B group. The number fraction distribution mole fraction ofN-mers) is given by the probability that a chosen unreacted A group is part of an N-mer. This number fraction of A-mers is the probability of A - 1 formed bonds (j) ) and one unreacted B group (1 p) [Eq. (1.52)]  

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