Flory model for molecules of different sizes

So far we have considered mixtures of atoms or species of similar size and shape. Now we will consider a mixture of a polymeric solute and a solvent of monomers [7, 8], The ideal entropy of mixing used until now cannot possibly hold for this polymer solution, in which the solute molecule may be thousands or more times the size of the solvent. The long chain polymer may be considered to consist of r chain segments, each of which is equal in size to the solvent molecule. Therefore r is also equal to the ratio of the molar volumes of the solute and the solvent. The solute and the solvent can be distributed in a lattice where each lattice site can contain one solvent molecule. The coordination number of a lattice site is z. 

Let V be the number of sets of r contiguous lattice sites available to each polymer molecule. If each of the NP polymer molecules added to the lattice were distinguishable from those already present, the number of ways in which all of them could be arranged in the lattice would be given by the product of the v,- for each molecule added consecutively to the lattice. The degeneracy is then given as 

If we distribute /Vp polymer molecules over NP sets of r consecutive lattice sites and then permute the polymer molecules on these fixed sets of consecutive sites, we will overestimate the degeneracy. The factor 1 /NP in eq. (9.44) takes into account that the configurations obtained by this kind of permutation are indistinguishable. 

The average probability, /, that a lattice site is occupied by a segment of one of the (i -1) preceding molecules at random is given by the number of vacant sites 1-fi =(N -ri)/N. Using this mean field approximation is replaced by /, , even though the former is somewhat smaller than the latter. The expression for v +1 then becomes 

The degeneracy factor for the arrangement of NP identical polymer molecules on the lattice consisting of N sites is now obtained by substituting eq. (9.49) in eq. (9.44) [7]  

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