state ∫ π 0 dθ = 2 ∫ π 2 0 dθ . In order to deal with the azimuthal integral we introduce the kernel K ( θ, θ ′ ) via where we have renamed φ − φ ′ → φ as the difference between the azimuthal angles of the two platelets. Adding all three terms and multiplying by R 3 1 yields the Onsager contribution to the excess free energy in the dimensionless form We generalise the monodisperse functional to the case of binary mixtures using an ap- proximate term at third order in density which is based on the FMT developed in Ref. [52]. Our theory contains the exact second order Onsager term and instead of using any higher order terms from the series, such as the exact third virial level (7), an approximate term which is of third order in density is used [54]. This term is nonvanishing (and constant) for cases with common triple intersection of the three platelets involved. There are no higher order terms due to the scaled-particle roots [55] of the approach; the vanishing volume of the platelets truncates the series. Global prefactors are used to compensate for lost cases [56, 57]. We postulate the excess free energy (where the right hand side is independent of T ). The first term of the sum in Eq. (14) is equivalent to the Onsager contribution to the excess free energy and the second is the FMT βF$_{exc}$ ([ ρ$_{1}$, ρ$_{2}$ ] , V ) = ∫ d r ∫ d ω ∫ d ω ′ [ n DD 1 ( r , ω ) n D $_{2}$( r , ω $^{′}$) + 1 24 π n D 2 ( r , ω ) n DDD 2 ( r , ω , ω $^{′}$) n D 2 ( r , ω $^{′}$) ] , (14) βF (2) exc V R 3 1 = 16 π $^{2}$c2 ∫ π 2 0 dθ sin θ ∫ π 2 0 dθ $^{′}$sin θ $^{′}$K ( θ, θ $^{′}$) × [ (1 − x ) $^{2}$Ψ$_{1}$ ( θ )Ψ$_{1}$( θ $^{′}$) + x $^{2}$λ$^{3}$Ψ$_{2}$ ( θ )Ψ$_{2}$( θ $^{′}$) + x (1 − x )( λ 2 + λ )Ψ$_{1}$( θ )Ψ$_{2}$( θ $^{′}$) ] , (13) where the superscript of F ( α ) exc represents the order in density of the excess free energy. K ( θ, θ $^{′}$) = ∫ 2 π 0 dφ sin γ = ∫ 2 π 0 dφ √ 1 − ( ω · ω ′ ) 2 = ∫ 2 π 0 dφ √ 1 − (cos θ cos θ ′ + sin θ sin θ ′ cos φ ) $^{2}$, (12) D. Fundamental Measure Theory for Binary Platelet Mixtures 8