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The Onsager versions of these equations are given by the same expressions but without the final bracketed term in c 3 for βF$_{iso}$/V and p ∗ iso and c 2 for µ ∗ i, $_{iso}$. The authors of Ref. [61] claim that inclusion of the exact third virial term along with the Onsager term, at least for monodisperse platelets, would give a worse equation of state in the isotropic phase than the Onsager term alone. 

## H. Isotropic-Nematic Bifurcation Analysis 

On increasing the density in the isotropic state, a point is reached known as the bifurcation density, where an infinitesimal nematic perturbation destabilises the system. The first I - N bifurcation analysis for a liquid crystalline system was performed in Ref. [62]. This was extended to a class of liquid crystal models in Ref. [63]. The bifurcation concentration lies inside the coexistence region for the monodisperse case of platelets [26] but as we will see, this is not always true in the binary case. We insert Ψ$_{1}$( θ ) = [1+ ϵ$_{1}$P$_{2}$ (cos θ )] / 4 π and Ψ$_{2}$( θ ) = [1 + ϵ$_{2}$P$_{2}$ (cos θ )] / 4 π into the free energy (5) and (22), where ϵ$_{1}$ and ϵ$_{2}$ are small parameters measuring the strengths of the nematic perturbation and P$_{2}$ (cos θ ) = (3 cos $^{2}$( θ ) − 1) / 2 is the second degree Legendre polynomial in cos θ . We then extract all second order terms, i.e. those proportional to ϵ 2 $_{1}$, ϵ 2 and ϵ$_{1}$ϵ$_{2}$ , respectively. The coefficients of these terms are denoted by a$_{1}$ ( c ), a$_{2}$ ( c ) and a$_{12}$ ( c ), respectively. These involve integrals of Legendre polynomials and are obtained using the integrals I $_{3}$, I $_{4}$, I 5 defined in Appendix A. One then solves det M = 0 [53] where 

Here 

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