where µ$_{i}$ is the chemical potential of species i = 1 , 2 and T is absolute temperature. The grand potential functional can be decomposed as where the spatial integral (over r ) is over the system volume V and the angular integral (over ω ) is over the unit sphere; V i $_{ext}$( r , ω ) is an external potential acting on species i ; F$_{exc}$ ([ ρ$_{1}$, ρ$_{2}$ ] , V, T ) is the excess (over ideal gas) contribution to the Helmholtz free energy and describes the inter-particle interactions. The free energy functional for a binary ideal gas of uniaxial rotators is given by where Λ$_{i}$ is the (irrelevant) thermal wavelength of species i and β = 1 / ( k$_{B}$T ), where k$_{B}$ is the Boltzmann constant. We let Λ$_{i}$ = R$_{1}$ , which is equivalent to fixing an additive constant to the chemical potential and hence does not affect any observable properties of the system. One systematic way to express the excess free energy functional is to expand it in a virial series in density [16]. Onsager theory is based entirely on the second-virial level. FMT (as described in Sec. II D below) approximates higher order terms using single particle geometries. Nevertheless we find it useful to give the terms in the virial expansion up to third order in density explicitly: the second and third order contributions to the (exact) virial series for the excess free energy βF$_{exc}$ ([ ρ$_{1}$, ρ$_{2}$ ] , V, T ) are given respectively by where each line in the diagrams represents a Mayer function f$_{ij}$ ( r − r ′ , ω , ω ′ ) = exp ( − βu$_{ij}$ ( r − r ′ , ω , ω ′ )) − 1, which equals − 1 if the two particles overlap and zero oth- erwise. The shaded circles, field points , indicate multiplication by the one-body density − 1 2 [ GLYPH<a116> 1 1 GLYPH<a116> + GLYPH<a116> 2 2 GLYPH<a116> + 2 GLYPH<a116> 1 2 $^{GLYPH<a116>}$] , (6) − 1 6 [ GLYPH<a116> 1 1 1 GLYPH<a116> GLYPH<a116> ❅ + GLYPH<a116> 2 2 2 GLYPH<a116> GLYPH<a116> ❅ + 3 GLYPH<a116> 1 2 1 GLYPH<a116> GLYPH<a116> ❅ + 3 GLYPH<a116> 1 2 2 GLYPH<a116> GLYPH<a116> ❅ ] , (7) 6 βF$_{id}$ ([ ρ$_{1}$, ρ$_{2}$ ] , V, T ) = 2 ∑ i =1 ∫ d r ∫ d ω ρ$_{i}$ ( r , ω ) × [ln( ρ$_{i}$ ( r , ω )Λ 3 $_{i}$) − 1] , (5) Ω([ ρ$_{1}$,ρ$_{2}$ ] , µ$_{1}$, µ$_{2}$, V, T ) = F$_{id}$ ([ ρ$_{1}$, ρ$_{2}$ ] , V, T ) + F$_{exc}$ ([ ρ$_{1}$, ρ$_{2}$ ] , V, T ) + 2 ∑ i =1 ∫ d r ∫ d ω ρ$_{i}$ ( V i $_{ext}$( r , ω ) − µ$_{i}$ ) , (4)