where γ is the angle between ω and ω ′ and ˆ n is a unit vector perpendicular to both ω and ω ′ . Since e$_{z}$ is fixed, ˆ n · e$_{z}$ = sin θ ′′ . Hence I 5 comprises two other similar terms so by symmetry, I 5 = 3 ˜ I $_{5}$. Hence I 5 = − 3 π $^{4}$/ 5 as required. We thank Chris Newton and Susanne Klein and the Liquid Crystals group of HP Labs, Bristol for useful discussions. We thank Peter Sollich for useful discussion regarding the structure of the third order term in the binary functional. Bob Evans, Paul Hopkins, Hendrik Reich, Robert Hales and Tom Smith are thanked for valuable comments on the manuscript. Financial support from the EPSRC and HP Labs, Bristol is gratefully acknowledged as well as through the SFB840/A3 of the DFG. 27 [9] G. C. Maitland, Curr. Opin. Colloid Interface Sci. 51 , 301 (2000). [8] M. C. D. Mourad, Liquid Crystal Phases of Colloidal Platelets and their Use as Nanocomposite Templates (PhD Thesis, Utrecht University, 2009). [7] M. C. D. Mourad, E. J. Devid, M. M. van Schooneveld, S. Vonk, and H. N. W. Lekkerkerker, J. Phys. Chem. B 112 , 10142 (2008). [6] E. S. H. Leach and J. S. van Duijneveldt, Langmuir 21 , 3821 (2005). [5] J. S. van Duijneveldt, S. Klein, E. Leach, C. Pizzey, and R. M. Richardson, J. Phys. Condens. Matter 17 , 2255 (2005). [4] C. Pizzey, S. Klein, E. Leach, J. S. van Duijneveldt, and R. M. Richardson, J. Phys. Condens. Matter 16 , 2479 (2004). [3] M. Dijkstra, J. P. Hansen, and P. A. Madden, Phys. Rev. E 55 , 3044 (1997). [1] F. M. van der Kooij, K. Kassapidou, and H. N. W. Lekkerkerker, Nature 406 , 868 (2000). [2] M. Dijkstra, J. P. Hansen, and P. A. Madden, Phys. Rev. Lett. 75 , 2236 (1995). Acknowledgments ˜ I 5 = ∫ d ω $^{′′}$sin θ ′′ ∫ d ω ∫ d ω $^{′}$sin γP$_{2}$ (cos θ ) P$_{2}$ (cos θ $^{′}$) = ∫ d ω $^{′′}$sin θ $^{′′}$I 4 = 4 π · ($_{−}$ π 3 10 ) = − π 4 5 . (A21)