Mean Field Approach for the Nematic Phase

Consider the nematic phase. It has cylindrical symmetry and the orientational order parameter 2 = V2(3cos i9 — l) with angle 9 between a molecular long axis and the symmetry axis (the director n). The tasks of the molecular theory is to use the symmetry arguments and properties of molecules and (a) to find the temperature dependence of P2 (T), (b) to calculate thermodynamic and other properties in terms of P2 , (c) to discuss the phase transition from finite P2 to zero (N-Iso transition), and (d) to discuss the role of the higher order parameters P4 , Pe etc. 

The key problem is a form of the interaction potential. The two-pair potential (6.59) is too complicated and we would like to substitute it by a single molecule potential  

At first, the pair potential W12 is expanded into two series of spherical harmonics 

polar coordinate frame was introduced based on the director n as a polar axis. To obtain the single molecular potential as a function of the first molecule orientation with respect to n, one has to take three successive averages of VF12 (a) over all orientations of intermolecular vector r, (b) over all orientations of molecule 2, and (c) over all intermolecular separations Irl. 

Finally, the single-molecule potential has been found in the form of expansion over Legendre polynomials  

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