Linear association and ring formation

More general cases of multiple association where polymer networks with multiple junctions of varied structures are formed will be treated as the gelling case in the next section. 

Let be the number of m-mer open chains, and let be the number of m-mer rings in the system. The total number of primary polymer chains (wa =n) is given by 

We follow the general strategy given in Chapter 5, and start with the free energy of the solution 

However, the equilibrium constant for the rings includes an extra factor of the probability to form a ring. This factor is proportional to (mn) / for a Gaussian chain of the length mn, but again we have to divide it by the symmetry factor or = tm for a ring, because we can close a chain at any one of m bonds to form a ring. 

The upper limit of the summation is given by the maximum possible aggregation number and should not exceed the total number N of polymers. But here we have taken the thermodynamic limit, and allow Af to go to infinity. 

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