Functional Theory of Flexoelectricity

A molecular-statistical theory of the flexoelectric effect in the nematic phase can be derived in a general way using the density-functional approach to the theory of liquid crystals. In this approach, the free energy of a liquid crystal, F, is a functional of the density po(a ) = Po/(w) where /(w) is the orientational distribution fimction. The general structure of the functional F p) is not known, but the functional derivatives are known and are related to the direct correlation functions of the nematic phase. 

The free energy of the distorted nematic liquid crystal can generally be written as a sum of two terms  

The reduced free energy H p) is the generating functional for the direct correlation functions of the system  

In this equation the first term describes the so-called direct correlations between the molecules 1 and 2 while the second term describes the correlations via a third particle. The direct correlation fimction C 2(o i, W2) generally describes short-range intermolecular correlations and decays rapidly with increasing intermolecular separation while the full pair correlation function 32( 1,012) normally has a long-range tail. Note that from a mathematical point of view Ci(w) and C2(wi,w2) are functions of w and, simultaneously, can be considered as functionals of the density p(w). For example, the functional derivative 5Ci oj )/5p(uj2) = C2(wi, W2)- Thus it is more precise to use the forms Ci(w, [p(w)]) and C2(wi, W2, [p(wi)], [ ( 2)]), which will be used below. 

The one-particle distribution function of the distorted liquid crystal can be determined by minimization of the free energy with respect to /(w) taking into account the normalization condition. We obtain 

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