Generalized van der Waals theory of nematic liquid crystals: Requirements for self-consistency

Abstract

In the generalized van der Waals (GVDW) theory of nematic liquid crystals, nematogenic molecules are assumed to interact via hard core repulsions and longer ranged attractive forces, and the latter are treated in the mean field approximation. The resulting expression for the configurational Helmholtz free energy Ac as a functional of the orientational distribution function f (Ω) is N−1Ac[ρ,T;f (Ω)]=N−1Ac[ρ,T;f (Ω)] + N−1V(a)N(ρ) =N−1Ac[ρ,T;f (Ω)] + 1/2ℱf (Ω) ? (ρ,Ω) dΩ, where ρ is the number density, Ac[f (Ω)] is the free energy functional for a system of molecules with only hard core repulsions, ?(a)N is the GVDW average of the total potential energy of attraction V(a)N, and ? (ρ,Ω) is the pseudopotential describing the interaction between a molecule with orientation Ω and the mean field produced by intermolecular attractions. In this paper, it is shown that the theory exhibits statistical mechanical self-consistency if and only if both N−1?(A)Nand?(Ω)are directly proportional to the first power of the density ρ. This result is independent of the shape of the molecular hard cores and of the assumed orientation dependence of the pseudopotential ?.