= Quarter-turn nematic instability threshold
{title2=$V_c^2=4\pi^3[K_1+(K_3-2K_2)/4]/\Delta\epsilon$}
For planar strong anchoring and a quarter-turn base profile, $\phi'=\pi/(2L)$ and $\theta=0$. Expanding the <one-dimensional twisted nematic energy> through quadratic order gives $\Delta F=\frac12\int[K_1\theta'^2+\{(K_3-2K_2)(\pi/(2L))^2-\Delta\epsilon E^2/(4\pi)\}\theta^2]dz$. The lowest zero-endpoint mode is $\sin(\pi z/L)$. Its coefficient vanishes at the displayed threshold, with $V=EL$, provided the bracket and anisotropy are positive. Halving the electric coupling doubles $V_c^2$.
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