The equilibrium of liquid crystals

According to Equation 2.19 the free energy of the system is written as 

The first term is the contribution of translation, the second term results from entropy reduction due to the orientational order of rigid rods, and the third term is the second virial term. The higher virial terms are neglected. 

By means of the Maxwell equal area rule, the dashed line in the figure is replaced by an equal pressure solid line. The regime of low pressure and low density is the isotropic phase while the regime of high pressure and high density corresponds to the anisotropic phase — nematic liquid crystal phase. The order parameter is determined by 

When the isotropic-nematic transition occurs the concentration of rods changes abruptly and the order parameter jumps from zero to a value Sc. 

The conclusions are dependent on the shape of rigid rods and hence the second virial coefficient. Thus, the concentration difference and the critical order parameter differ at the nematic — isotropic transition for different shapes of rods. 

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