ρ$_{i}$ ( r , ω ) and integration over the coordinates r and ω [16]. The number alongside each field point represents the species i . Here we consider only spatially homogeneous states, such that the one-body density does not depend on position r and, for the case of uniaxial nematic states considered in this paper, depends only on the polar angle θ of ω with respect to the nematic director. Hence the one-body densities factorise as ρ$_{i}$ ( r , ω ) = ρ$_{i}$ Ψ$_{i}$( θ ) where Ψ$_{i}$( θ ) is the orientational distribution function (ODF) and ρ$_{i}$ the number density of species i . Before laying out the FMT we focus on the second-virial level. The diagrams in Eq. (6), using the fact that ρ$_{i}$ ( r , ω ) = ρ$_{i}$ Ψ$_{i}$( θ ) for spatially homogeneous states, become where we have renamed r − r ′ → r . The spatial integral over the Mayer bond yields (minus) the excluded volume −E $_{ij}$( ω , ω ′ ) between two particles of species i and j , as a function of the angle γ between ω and ω ′ . Hence Therefore The integrals in Eq. (10) (omitting the prefactor) can be written as where θ and θ ′ are the polar angles of two platelets with respect to the nematic director and φ and φ ′ are the azimuthal angles. Due to the inversion symmetry of the nematic ∫ 2 π 0 dφ ∫ π 0 dθ sin θ ∫ 2 π 0 dφ ′ ∫ π 0 dθ sin θ ′ × Ψ$_{1}$( θ )Ψ$_{1}$( θ $^{′}$) sin γ, (11) − 1 V GLYPH<a116> i j GLYPH<a116> = 16 π $^{2}$( R 2 i R$_{j}$ + R 2 $_{j}$R$_{i}$ ) ρ$_{i}$ρ$_{j}$ ∫ π 2 0 dθ sin θ (10) × ∫ π 2 dθ $^{′}$sin θ $^{′}$K ( θ, θ $^{′}$)Ψ$_{i}$ ( θ )Ψ$_{j}$( θ $^{′}$) . E $_{ij}$( ω , ω $^{′}$) = − ∫ d r f$_{11}$ ( r , ω , ω $^{′}$) = 2 π ( R 2 i R$_{j}$ + R 2 $_{j}$R$_{i}$ ) sin γ, (9) 7 − 1 2 GLYPH<a116> i j GLYPH<a116> = − 1 2 ρ$_{i}$ρ$_{j}$ ∫ d ω ∫ d ω ′ ∫ d r f$_{ij}$ ( r , ω , ω $^{′}$)Ψ$_{i}$ ( θ )Ψ$_{j}$( θ $^{′}$) , (8) C. Onsager Second-Virial Theory for Binary Platelets