 Bounded quartic nematic coexistence

For $a,c>0$, $b\le 0$ and the molecular bound $Q\in [-1/2,1]$, coexistence with $Q=0$ occurs at $b_t=-{\min }_{0<Q\le 1}(a/Q+3cQ)$. Negative $Q$ has positive free energy and cannot be the stable phase in this coefficient range. Thus $a\le 3c$ gives $b_t=-\sqrt{12ac}$ and jump $0\to \sqrt{a/(3c)}$, whereas $a>3c$ gives $b_t=-(a+3c)$ and jump $0\to 1$. The interior case is the usual small-order-parameter Landau theory result. A formal unconstrained quartic minimization can violate the molecular bounds when extrapolated too far.