## 2 Basic Equations  
Let us consider a planar sample. The \(x\)-axis is taken parallel to the anchoring direction on the electrodes and the \(z\)-axis is perpendicular to the electrodes, with \(z=0\) at the bottom electrode and \(z=d\) at the top electrode. I denote by \(\phi(z)\) the tilt angle of the director with respect to the \(x\)-axis and by \(\theta\) the tilt angle on the electrodes (\(\theta=\phi(0)=\phi(d)\)). I suppose that there is no pretilt angle, which means that \(\theta=0\) when no electric field is applied. For now, the anchoring potential is assumed to be of the Rapini-Papoular form [16]  
\[W(\theta)=\frac{1}{2}W\sin^{2}\theta, \tag{1}\]  
and I neglect the flexoelectric effect. When an electric field is applied and the anchoring is very strong the sample destabilizes above the Freedericksz critical voltage \(V_{c}\), given by [1; 21]  
\[V_{c}=\pi\sqrt{\frac{K_{1}}{\varepsilon_{0}\varepsilon_{a}}}. \tag{2}\]  
In this equation, \(\varepsilon_{0}\) is the vacuum permittivity, \(\varepsilon_{a}=\varepsilon_{\parallel}-\varepsilon_{\perp}\) is the dielectric anisotropy of the nematic phase and \(K_{1}\) is the splay constant. The actual critical voltage \(V_{c}^{\prime}\) is slightly smaller if the anchoring energy is taken into account and it is the solution of the following equation [1; 16]:  
\[\frac{d}{L}=\pi\frac{V_{c}^{\prime}}{V_{c}}\tan\left(\frac{\pi V_{c}^{\prime}} {2V_{c}}\right), \tag{3}\]  
where \(L=K_{1}/W\) is the anchoring extrapolation length. This equation shows that the ratio \(d/L\) must be larger than \(\approx 200\) for \(V_{c}^{\prime}\approx V_{c}\) to better than \(1\%\). The formulas that give the capacitance as a function of the applied voltage are well known [22; 23] and are recalled in [24]. They considerably simplify at large voltage, when the maximum tilt angle \(\phi_{m}\) in the middle of the cell is very close to \(\pi/2\). Numerical calculations show that this condition is typically satisfied when \(V\gtrsim 6V_{c}\). In this limit,\(\phi_{m}\approx\pi/2\) to better than \(10^{-3}\) rad [5]. If the flexoelectric effects are screened out by the ions contained in the LC-and this is usually the case experimentally-the capacitance can be calculated by using the formula [25]:  
\[\frac{C}{C_{\perp}}=1+\gamma-\frac{2}{\pi}\gamma\sqrt{1+\gamma}\frac{V_{c}}{V} \int_{\sin\theta}^{1}\sqrt{\frac{1+\kappa x^{2}}{1+\gamma x^{2}}}\mathrm{d}x, \tag{4}\]  
where \(\gamma=\varepsilon_{\parallel}/\varepsilon_{\perp}-1\) and \(\kappa=K_{3}/K_{1}-1\) (with \(K_{3}\) the bend constant) and \(C_{\perp}=\varepsilon_{0}\varepsilon_{\perp}S/d\) is the capacitance measured below the onset of instability (with \(S\) the surface area of the electrodes). The surface angle \(\theta\) is obtained by solving the surface torque equation which reads by using the Rapini-Papoular potential [26]:  
\[\sin\theta=\sqrt{\frac{1+\kappa\sin^{2}\theta}{1+\gamma\sin^{2}\theta}}\frac{ \pi}{\sqrt{1+\gamma}}\frac{L}{d}\frac{V}{V_{c}}\frac{C}{C_{\perp}}. \tag{5}\]  
Solving these two equations can determine the ratio \(C/C_{\perp}\) as a function of \(V/V_{c}\) provided that \(\kappa\), \(\gamma\) and the ratio \(L/d\) are known. If \(\theta\) is small, expanding the two previous equations in power series of \(\sin\theta\) yields [27]  
\[\frac{C}{C_{\perp}}=\frac{1+\gamma}{1-2\gamma\frac{L}{d}}-\frac{2}{\pi}\frac{ \gamma\sqrt{1+\gamma}}{1-2\gamma\frac{L}{d}}I\frac{V_{c}}{V}, \tag{6}\]  
where \(I\equiv\int_{0}^{1}\sqrt{\frac{1+\kappa x^{2}}{1+\gamma x^{2}}}\mathrm{d}x\). This equation is valid as long as \(\frac{2}{3}(\kappa-\gamma)\sin^{3}\theta\ll I\), i.e., as long as \(\sin^{3}\theta\ll 1\) since \(I\) and \(\kappa-\gamma\) are usually measured to be of order unity. This imposes that the voltage is not too large, typically less than voltage \(V_{\mathrm{max}}\) defined in the introduction for \(\theta<0.2\) rad.