## 3 Experimental Results
### Measurements of the Extrapolation Length \(L\)  
As the anchoring is expected to be strong with Brilliant Yellow, a pretty thin sample must be used in order for the tilt angle \(\theta\) on the plates to deviate from 0 in a measurable way under the action of the electric field. For this reason, I used a 8.3 \(\upmu\)m-thick sample and I measured the capacitance between 0.1 V and 0.5 Vrms to obtain \(C_{\perp}\) and between 5 and 20 Vrms to measure the extrapolation length \(L\). At a voltage V = 5 Vrms the condition \(V>6V_{c}\) is always satisfied, so the basic Equations (4) and (5) apply. Typical curves \(\overline{C}(\overline{V})\) measured between 5 and 20 Vrms are shown in Figure 7. These curves are almost linear, suggesting that the angle \(\theta\) remains small and Equation (6) applies. In that case, fitting the curves with the linear law \(1+\gamma_{app}-p\,\overline{V}\) in any voltage range \((V_{1},V_{2})\) with \(V_{1}>5\) Vrms and \(V_{2}<20\) Vrms should allow me to measure \(L\) by using the equation  
\[\frac{L}{d}=\frac{\gamma_{app}-\gamma}{2\gamma(1+\gamma_{app})}, \tag{7}\]  
derived from Equation (6) and the value of \(\gamma\) given in Figure 4. To test this method, I fitted the curves within different intervals of voltage \((V_{1},V_{2})\). For each fit, I calculated \(L\) and \(V_{\text{max}}\) by using the formula given in the introduction to check whether \(V_{2}<V_{\text{max}}\) in order that the model applies. In principe, all the measurements should coincide if the model applies. In practice, it is not at all the case, the value of \(L\) crucially depending on the choice of the interval \((V_{1},V_{2})\), even when the model is _a priori_ valid, as the reader can see in Table 1 obtained by fitting the curve at \(T=35\,^{\circ}\)C. To summarize, I met the same difficulties as [PERSON] et al. when they used the YyS or RV method. In a similar way, I also noted that the larger \(V_{2}\), the larger the value of \(L\) found from the linear fit. In their paper, [PERSON] et al. [17] suggest that this problem could be due to inhomogeneities of the anchoring conditions. I do not share that view, at least in my experiments, and suggest that the problem is rather due to the approximations performed by using Equation (6) with \(\varepsilon_{\parallel}\) constant. For this reason, I tried in a first attempt to fit my experimental curves with the full Equations (4) and (5), while still keeping \(\varepsilon_{\parallel}\) constant, equal to \(\varepsilon_{\parallel}(0)\). In that case, there are two fit parameters: the elastic anisotropy \(\kappa\) that essentially fixes the slope of the curve and the ratio \(L/d\) that fixes the \"height\" of the curve. Surprisingly, I noted that it was \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Voltage Range \((V_{1},V_{2})\)** & **Fitted Value of \(L\) (in \(\upmu\)m)** & \(V_{2}\)** & **Is the Model Applicable?** \\ \hline (5 V, 9 V) & 0.0104 & 29 V & Yes \\ (5 V, 13 V) & 0.0144 & 21 V & Yes \\ (5 V, 17 V) & 0.0178 & 17 V & Yes \\ (5 V, 20 V) & 0.0202 & 15 V & No \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Value of \(L\) obtained from fitting in the voltage range \((V_{1},V_{2})\). Figure 7: Reduced capacitance curves measured in a planar sample of thickness \(d=8.2\)\(\upmu\)m at different temperatures. From top to bottom, \(T=34,35,36,37,38,39,40\) and \(40.5\,^{\circ}\)C.  
not possible to fit correctly my experimental curves at large voltages even with the full equations. The graphical comparison between the experimental curve and the calculated ones for two particular values of \(\kappa\) is shown in Figure 8. For each fit, the root-mean-square deviation between the experimental curve and the fit curve is minimized as a function of \(L\).