Polar Molecular Shape on the Flexocoefficients

Simple expressions for the flexoelectric coefficients in the nematic phase can be obtained using the generalized van der Waals approximation. In this approximation the intermolecular attraction is taken into account in the mean-field approximation while the steric repulsion is accounted for by taking into consideration short-range steric correlations via the excluded 

The first term in Eq. (1.27) is the internal energy of the system where E(xi,X2,ri2) is the pair intermolecular potential and ( 12 — r ) is the step function. Here 12 is the closest distance of approach for the two molecules 1 and 2 with fixed relative orientation. (C12 — 12) = —1 if ri2 C12, i e. if the molecules penetrate each other, and ( 12 — ri2) = 0 if ri2 Ci2- The second term in Eq. (1.27) is the so-called packing entropy and the third term is the orientational entropy. Thus in the first term the integration over ri2 is restricted to ri2 12, which takes into account that the molecules cannot penetrate each other. 

In this approximation the direct correlation function in the nematic phase can be obtained using Eq. (1.15)  

The direct correlation function can now be substituted into the general Eqs (1.24) and (1.26) to obtain the particular expressions for the flexoco-efiicients. 

The results of Straley can be obtained by neglecting the pair attraction interaction potential V(xi, X2, ri2) in the equation for the direct correlation function. Indeed, the Straley theory of flexoelectricity was developed for the system of hard polar rods, while for thermotropic liquid crystals both the molecular shape and the intermolecular attraction are important. 

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