Effective one-body potentials for orientationally anisotropic fluids

Abstract

We consider moving an axially symmetric rod, 2, about another, 1, such that (A) their relative orientation Θ₁₂ never changes and (B) they never penetrate each other. These constraints define uniquely a closed surface S inside which the second rod's centre can never be found. We determine S explicitly as a function of the rod dimensions and of the fixed angle ϑ₁₂ (Θ₁₂) between their axes. We then treat fluids of anisotropic molecules whose pair interaction energy can be written as the sum of a hard rod (h.r.) repulsion, v _h.r., and a dispersional attraction, v _attr. The effective one-body potential is given approximately by where ρ is the number density and f(Θ) is the fraction of molecules having orientation Θ. Using our results for the surface S _{ϑ12 we evaluate this mean field for typical ratios of molecular length (l) to width (w). Even when the attractive part of the pair potential consists only of the familiar Ψ(Θ) can be expressed in the form where the bars over the Pn 's denote averages over the orientational distribution f(Θ). (ϑ is the angle between a cylindrically symmetric molecule and the director of the nematic). The An 's are obtained numerically for physically reasonable values of l, w, C an and C iso; all of the ρ and T dependence of Ψ is contained in the ρ [Pbar]n 's. The terms with n ⩾ 4 are shown to be small (i.e. their contribution to Ψ is less than a few per cent) and A 2 is found to be dominated by the isotropic term (- C iso/r 12 ⁶) in the dispersive interaction. A 2/A 0 is as large as 0·3 for molecular length/width ratios of the order of 3 to 4. Our theory and results for Ψ(cos ϑ), the effective one-body potential, are contrasted with several recent discussions in the literature.}