Supplemental Material Entropic Stabilization of Biaxial Nematics : a Competition with Uniaxial and Positional Order for Extremely Long Particles

To study the phase behavior of polyhedral hard rods (either cuboids or triangular prisms) we employed standard Monte Carlo (MC) simulations either in the NPT or NV T ensemble. System sizes range from N ' 2000 to N ' 5000 and several million MC steps are performed before obtaining equilibrated configurations. For NV T -MC simulations, each MC step consists on average of N/2 attempts of translating a random particle and N/2 attempts of rotating a random particle. In the case of NPT -MC simulations, an additional attempt is performed at each step in order to either scale isotropically the box volume or change only one edge of the cuboidal simulation box. Equilibrium average density, order parameters and diffraction patterns are calculated based on around one hundred equilibrated configurations. For systems of rhombic particles, cluster moves are used in NPT -MC to perform volume changes move [1, 2]. Particles interact only via a hard-core potential and overlaps are detected using algorithms based either on triangular-triangular intersection-detection, using the RAPID library [3], or based on the GJK algorithm [4, 5], depending on the particle model. In addition to MC simulations, rhombic platelets (and some selected cases of cuboids) are simulated with state-ofthe-art Event-Driven Molecular Dynamics (EDMD) [2]. The GJK overlap-detection algorithm is combined with conservative advancement and near-neighbor list to efficiently simulate around N ∼ 2 · 10 rhombic particles in the NV T ensemble. The moment of inertia and the mass of all particles is set to 1 and the system is simulated for over 10τ , where τ = v 1/3 p /v0 is the reduced time unit, with vp particle volume and v0 the initial velocity of each particle (velocities are initialized by using random unit vectors whereas angular velocities are initially set to zero). The equilibrium pressure is calculated from the particle collisions in the equilibrated configurations.