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The fundamental measures of a platelet of species i are the integral mean curvature ξ IMC i = πR$_{i}$/ 4 and the surface ξ S i = 2 πR 2 $_{i}$. The first term of Eq. (22) may be expressed as 

<img src='content_image/335088.jpg'>

remembering that Ψ$_{i}$( ω ) = Ψ$_{i}$( θ ) for uniaxial nematics. Eq. (24) is simply the fundamental measures interpretation [52] of the Onsager contribution to the free energy. Note that the term which represents the excluded volume between two particles of different species is given by ξ IMC 2 ξ S 1 + ξ IMC 1 ξ S $_{2}$. This leads to a scaling of the excluded volume by λ $^{2}$+ λ , which is quite different from the scaling that occurs for binary mixtures of rods. The second term in Eq. (22) is the FMT contribution to the excess free energy. This is given by 

where 

<img src='content_image/335087.jpg'>

<img src='content_image/335086.jpg'>

We choose coordinates such that ω = (sin θ, 0 , cos θ ), ω ′ = (cos θ ′ sin θ ′ , sin φ ′ sin θ ′ , cos θ ′ ) and ω ′′ = (cos θ ′′ sin θ ′′ , sin φ ′′ sin θ ′′ , cos θ ′′ ) where θ, θ ′ and θ ′′ are the polar angles of three platelets. The third order contribution in density to the FMT excess free energy in these coordinates is given by 

<img src='content_image/335085.jpg'>

