A good and computationally efficient polynomial approximation to the Maier-Saupe nematic free energy

A new computational strategy is proposed to approximate, with a simple but accurate expression, the Maier- Saupe free energy for nematic order. Instead of the traditional approach of expanding the free energy with a truncated Taylor series, we employ a least-squares fitting to obtain the coefficients of a polynomial expression. Both methods are compared, and the fitting with at most five polynomial terms is shown to provide a satisfactory fitting, and to give much more accurate results than the traditional Taylor expansion. We perform the analysis in terms of the tensor order parameter, so the results are valid in uniaxial and biaxial states.