Refinement beyond the simple mean field approximation

4 Refinement beyond the simple mean field approximation 

Many refinements of the Flory-Huggins theory (FH theory) of polymer solutions have been attempted ever since it was proposed in the 1940s. The first one is the detailed study of the mixing entropy and its improvement. The second is to take the semiflexibility of the polymer chains into the theoretical framework. The third is to consolidate its interaction term to accommodate specific interactions such as hydrogen bonds, dipole interaction, hydrophobic force, etc. 

The mixing entropy (2.91) of FH theory is known to be widely applicable under the condition that the free volume of each component in the mixture is proportional to its molecular volume, as was pointed out by Huggins [9] and Hildebrand [5,16]. We try to derive its refined form by employing the detailed probability (2.85) of the monomer-solvent contact in the counting problem of the number of possible arrangements of polymers on the lattice. 

Let us start from the number of possible arrangements W Nq,Ni) (2.83), and use the detailed form (2.85) of the monomer-solvent contact probability (surface contact) for n +i. 

Guggenheim generalized this result to the mixtures of two components with arbitrary molecular volumes, and found that the mixing entropy is given by [17] 

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