Size of ideal randomly branched polymers

So far we have discussed the distribution of degrees of polymerization of molecules both below and above the gel point. In the present section, we will describe their spacial sizes in the polymerization reactor. The Bethe lattice introduced above for the mean-field gelation model properly describes the connectivity of monomers into trcc-like branched molecules, but is not 

Here we calculate the size of ideal randomly branched polymers, ignoring excluded volume interactions and allowing each molgcule to achieve the state of maximum entropy (recall the discussion of ideal chains in Chapter 2). Since branched molecules have many ends, the mean-square end-to-end distance used to characterize the size of linear chains is not appropriate for them. The simplest quantity describing the size of branched molecules is their mean-square radius of gyration j g [see Eq. (2.44) fonthe definition]. 

In Chapter 2 we have presented a proof of the Kramers theorem for branched molecules containing N monomers of size b, but no loops (Eq. 2.65). The mean-square radius of gyration of these molecules is 

The Kramers theorem relates the ideal size of molecules to a purely structural property—the number of ways of dividing a molecule into two 

The probability that a chosen bond divides a molecule into two branches—the first one with N monomers and the second one with N-Ni monomers is jv, (p)Wjv-a/, (p). Therefore, the Kramers theorem can be rewritten  

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