In the absence of surface anchoring realignment ($\bar{W} = 0$) the colloids undergo a conventional isotropic-uniaxial nematic transition. This is the classic Onsager scenario where an isotropic (I) phase ($c_{I} = 4.19$, $S_{I} =0$) coexists with a (uniaxial) nematic phase ($c_{N} = 5.34$, $S_{N} =0.792$). For $\bar{W} >0$ the $O(3)$ symmetry of the isotropic phase and the uniaxial $D_{\infty h}$ symmetry of the nematic will both be broken in favour of a biaxial, orthorhombic symmetry ($D_{2h}$), and a coexistence between two orthorhombic phases with different overall colloidal concentrations is expected. Resolving the coexistence conditions by imposing equality of chemical potential $\mu$ and pressure $\Pi$ in both phases we may explore phase diagrams in the surface anchoring amplitude - colloid concentration ($\bar{W} - c$) plane. The results are shown in \fig{phdiag} and demonstrate that a phase coexistence between two orthorhombic nematic phases is indeed possible in the weak coupling regime ($\bar{W} <1$). A critical point beyond which no phase transition is possible is located at a surface anchoring energy equivalent to a few times the thermal energy.
% TODO: rearrange diagrams figure to fit 1-column
\begin{figure}
\includegraphics[width = .9\columnwidth]{figures/chapter-4/diagrams}
\caption{(a) $\bar{W} - c $ phase diagram for discs with planar surface anchoring, $qD = 1$. (b) $\bar{W} - c $ phase diagram for rods with homeotropic or tangential surface anchoring, $qL = 1$.}
https://github.com/mtorresl-works/TORRES_thesis_manuscript/blob/1d341cdeb3ab1b758f02ec20b5e65a6348ac53d6/chapters/chapter-4.tex#L91