Models of competitive hydration and gelation

The model we smdy is the polymer solution in which each primary polymer chain carries / reactive groups attached along the chain backbone. Each reactive group is assumed to be capable of forming pairwise hydrogen bond with either solvent molecule (water molecule in an aqueous solution) or other reactive group on a different chain. 

The total free energy of the solution is given by the sum of the general form (5.13). The chemical potential of an (/, r)-cluster is 

as usual, we split the free energy of cluster formation into three parts  

m)-cluster has Z - 1 cross-links connecting / constituent polymer chains, and fl — 2/ -t-2 unreacted functional groups, m of which are hydrated by solvent molecules. The combinatorial factor is given by f toi for the connection of the /-functional polymers into a tree, where a i = fl—Z) // (/Z—2/ -b 2) is the Stockmayer factor (3.19). For each type of the polymer tree, m attaching sites must be chosen from f I —21+2 open sites for hydration. This selection gives a binomial factor fi-ii+iCm- The total combinatorial free energy change then takes the form 

To find the configurational entropy, we employ the conventional lattice-theoretical formula for the entropy of disorientation (2.90) 

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