Gelation concepts and definitions

Using Eq. (6.35) relates the maximum generation for a perfect dendrimer to geometric properties of a monomer  

To make a large generation dendrimer, monomers with large aspect ratio are needed, so that f vq. For example, to make a perfect tetrafunctional (n=f 4) seventh generation dendrimer requires / /vq 3. 

The regular lattice constructed in this way is called a Bethe lattice (see Fig. 6.13). The mean-field model of gelation corresponds to percolation on a Bethe lattice (see Section 6.4). The infinite Bethe lattice does not fit into the space of any finite dimension. Construction of progressively larger randomly branched polymers on such a lattice would eventually lead to a congestion crisis in three-dimensional space similar to the one encountered here for dendrimers. 

Gelation is a connectivity transition that can be described by a bond percolation model. Imagine that we start with a container full of monomers, which occupy the sites of a lattice (as sketched in Fig. 6.14). In a simple bond percolation model, all sites of the lattice are assumed to be occupied by monomers. The chemical reaction between monomers is modelled by randomly connecting monomers on neighbouring sites by bonds. The fraction of all possible bonds that are formed at any point in the reaction is called the extent of reaction p, which increases from zero to unity as the reaction proceeds. A polymer in this model is represented by a cluster of monomers (sites) connected by bonds. When all possible bonds are formed (all monomers are connected into one macroscopic polymer) the reaction is completed (/ = 1) and the polymer is a fully developed network. Such fully developed networks will be the subject of Chapter 7, while in this chapter we focus on the gelation transition. 

At the percolation threshold or gel point pc, the system undergoes a connectivity transition. Slightly below the gel point, the system is a poly-disperse mixture of branched polymers shown in Fig. 6.14(a). Slightly 

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