content_image/374715.jpg content_image/374713.jpg content_image/374712.jpg content_image/374711.jpg
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. 

C. Onsager Second-Virial Theory for Binary Platelets 

The diagrams in Eq. (6), using the fact that ρ$_{i}$ ( r , ω ) = ρ$_{i}$ Ψ$_{i}$( θ ) for spatially homogeneous states, become 

<img src='content_image/374715.jpg'>

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 

<img src='content_image/374713.jpg'>

Therefore 

The integrals in Eq. (10) (omitting the prefactor) can be written as 

<img src='content_image/374712.jpg'>

<img src='content_image/374711.jpg'>

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 

