Solution of Onsager Equations in a Simplified Case

In practice, Eq. [13] is a nonlinear integral equation and its exact solution in the general case is difficult. Therefore, we will solve the Onsager equations in a very simplified case. Let us constrain the hard rods to point in only three orthogonal directions, say oj, and z. Choose z as the preferred direction. If the fraction of hard rods pointing in the oj-direction is r, the same fraction should point in the /-direction, since x and y directions are equivalent by uniaxial symmetry of the nematic phase. The fraction pointing in the 2 -direction is then 1 — 2r. The order parameter 

Here and y = are the excluded volumes of two hard rods when they are respectively, perpendicular and parallel to each other. 

To maximize (N, V, kT) /N, we merely differentiate with respect to S and set the result equal to zero. This gives 

In Figs. 4 and 5, reduced pressure is plotted versus reduced 

Zwanzig, First-Order Phase Transitions in a Gas of Long Thin Rods, J. Chem. Phys., Vol. 39, p. 1714 (1963). 

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