where the kernel L ( θ, θ ′ , θ ′′ ) is The full form of the excess free energy, as used in the calculations described below, is given by the sum of Eqs. (13) and (27). In practice, along with Eqs. (12) and (28), these require numerical computation on a grid as described in the following. The minimisation principle (3) together with the FMT approximation (22) for F$_{exc}$ ([ ρ$_{1}$, ρ$_{2}$ ]) leads to two coupled Euler-Lagrange equations for the ODFs: where the constants Z$_{1}$ and Z$_{2}$ are such that the normalisation ∫ d ω Ψ$_{i}$( ω ) = 1, for i = 1 , 2. Neglecting terms of order c 2 in the exponentials, the equations for FMT reduce to the Euler-Lagrange equations of Onsager theory. We solve Eqs. (29) and (30) numerically with 11 Ψ$_{2}$( θ ) = 1 Z$_{2}$ exp [ − 8 πc ∫ π 2 0 dθ $^{′}$sin θ $^{′}$K ( θ, θ $^{′}$)[ xλ $^{3}$Ψ$_{2}$ ( θ $^{′}$) + 1 2 (1 − x )( λ 2 + λ )Ψ$_{1}$( θ $^{′}$)] − 32 πc 2 ∫ π 2 0 dθ $^{′}$sin θ ′ ∫ π 2 0 dθ $^{′′}$sin θ $^{′′}$L ( θ, θ $^{′}$, θ $^{′′}$) × [ x $^{2}$λ$^{6}$Ψ$_{2}$ ( θ $^{′}$)Ψ$_{2}$ ( θ $^{′′}$) + 2 x (1 − x ) λ $^{4}$Ψ$_{1}$ ( θ $^{′}$)Ψ$_{2}$ ( θ $^{′′}$) + (1 − x ) $^{2}$λ$^{2}$Ψ$_{1}$ ( θ $^{′}$)Ψ$_{1}$ ( θ $^{′′}$)] ] , (30) Ψ$_{1}$( θ ) = 1 Z$_{1}$ exp [ − 8 πc ∫ π 2 0 dθ $^{′}$sin θ $^{′}$K ( θ, θ $^{′}$)[(1 − x )Ψ$_{1}$( θ $^{′}$) + 1 2 x ( λ 2 + λ )Ψ$_{2}$( θ $^{′}$)] − 32 πc 2 ∫ π 2 0 dθ $^{′}$sin θ ′ ∫ π 2 0 dθ $^{′′}$sin θ $^{′′}$L ( θ, θ $^{′}$, θ $^{′′}$) × [(1 − x ) $^{2}$Ψ$_{1}$ ( θ $^{′}$)Ψ$_{1}$ ( θ $^{′′}$) + 2 x (1 − x ) λ $^{2}$Ψ$_{1}$ ( θ $^{′}$)Ψ$_{2}$ ( θ $^{′′}$) + x $^{2}$λ$^{4}$Ψ$_{2}$ ( θ $^{′}$)Ψ$_{2}$ ( θ $^{′′}$)] ] , (29) E. Self-consistency equations for the orientational distribution functions L ( θ, θ $^{′}$, θ $^{′′}$) = ∫ 2 π 0 ∫ 2 π 0 dφ $^{′}$dφ$^{′′}$| ω · ( ω ′ × ω $^{′′}$) | = ∫ 2 π 0 ∫ 2 π 0 dφ $^{′}$dφ$^{′′}$| sin θ (sin φ $^{′}$sin θ $^{′}$cos θ ′′ + cos θ $^{′}$sin φ $^{′′}$sin θ $^{′′}$) + cos θ (cos φ $^{′}$sin θ $^{′}$sin φ $^{′′}$sin θ ′′ − sin φ $^{′}$sin θ $^{′}$cos φ $^{′′}$sin θ $^{′′}$) | . (28)