Thermodynamics of rodlike polymer solutions

Floryi has presented a theory of rodlike polymers in solution. The solvent is chosen to be a small molecule that defines the lattice geometry. The rod is represented as a linear sequence of x lattice cells where, as usual, x is the ratio of the molar volumes of the polymer and the solvent. The lattice is characterized by a coordination number 2. An initial theoretical effort focused on the athermal case. For a mixture of solvent molecules and N2 polymer molecules, the canonical ensemble partition fxmction for the isotropic liquid mixture can be expressed as  

It is known empirically that solutions of rodlike polymers are highly ordered at higher concentrations. The partition function for a completely orientationally ordered but otherwise random solution is  

The free energy of mixing for an athermal anisotropic solution can then 

The value of y for a given value of x and 2 is obtained by minimizing the free energy of mixing. The condition of equilibrium demands tiiat  

For large values of x, it can be shown that stable anisotropic solutions exist only for volume fractions higher than  

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