Partition function of a rigid rod solution

Flory (1956, 1984) adopted the lattice model. The Flory theory starts with the partition function of systems consisting of rigid rods and solvent molecules. 

Assume the long axis of the rigid rods makes an angle if with respect to the director of the system and the director is along one principal axis of the cubic lattice. Divide each rod into x basic units of equal width. Each basic unit occupies one cell in the lattice, x is actually the axial ratio of the rods. For simplicity, suppose that the dimension of a solvent molecule is compatible to the size of a cell lattice. In this section we adopt the same assignations as Flory. These may be different from those used in the preceding section by Onsager. 

Each sub-particle has x/y basic unit and its long axis is along the director. If a particle is perfectly aligned along the director, y is zero. As a 

Assume that the total number of cells in the system is no and (j — 1) rods have been placed in the lattice. They have occupied x(j — 1) lattice cells and hence no — x(j — 1) lattice cells remain unoccupied. In this case, there are Vj ways to put the j-th rod into the lattice 

All units of each sub-particle must be in same row of the cell. Once the first unit has been put into the lattice (the cell must be unoccupied and is allowed to put in) each of remaining units must be positioned immediately next to preceding unit (the cell must be unoccupied). There are two possibilities the cell may be unoccupied and is allowed to enter in the other possibility is that it has been occupied by the first unit of a sub-particle 

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