Onsager Theory of the Isotropic-Nematic Transition

Here we have lumped in the constant quantities that do not affect the phase transition, i.e., have the same value in the coexisting phases. Further c is the dimensionless concentration 

Here we have taken into account that / does not depend on the azimuthal angle (j) but only on the polar angle 9. Furthermore, the distribution function f 9) must satisfy inversion symmetry, implying the angles 9 and tc — 0 are equivalent. The Lagrange multiplier k is determined by requiring that/(0) fulfills the normalization condition 

It is easily seen that the isotropic distribution function 

An exact solution to the non-finear integral equation (6.5) for higher concentrations, where a nematic distribution minimizes the free energy, has not yet been found but ways to solve it numerically have appeared [12, 13]. For a didactic account on how to solve (6.5) numerically, see [14]. This allows the determination 

6 Phase Transitions in Suspensions of Rod-Like Colloids Plus Polymers 

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