## 3 Experimental Results
### Measurements of the Extrapolation Length \(L\)
In this example, \(T=40\,\mathrm{\SIUnitSymbolCelsius}\) and the expected value for \(\kappa\) at this temperature is close to \(-0.08\) according to previous measurements performed in thick samples at low voltage when the anchoring energy can be assumed to be infinite [33] or by using a full-optical method [24]. For this reason, I performed the first fit (Figure 8a,b) with this value and found \(L=0.037\,\mathrm{\SIUnitSymbolMicro m}\). In that case, the slope of the theoretical curve is correct at low voltage (below 10 Vrms), but the model fails to reproduce the increase in the capacitance at large voltage, above 10 V. Increasing the value of \(\kappa\) enables better fitting the behavior at large voltage but, this time, the slope of the theoretical curve is clearly too large at low voltage. The best global fit is obtained by taking \(\kappa=0.72\) (which is very different from the expected value \(-0.08\)) and \(L=0.05\,\mathrm{\SIUnitSymbolMicro m}\) and is shown in Figure 8c,d. This analysis shows the impossibility to reasonably fit the experimental curves in the whole range of voltages used experimentally. Something is clearly missing in the model. For this reason, I tried to fit again my curves by taking into account the dependence with the electric field of the dielectric constant \(\varepsilon_{\parallel}\) measured previously in homeotropic samples (Figure 2). This correction is here justified because the sample is almost homeotropic between 5 and 20 Vrms and was already taken into account by [PERSON] et al. [27] in a similar problem. With the aid of Mathematica 12, I realized that it was possible to considerably improve the quality of the fits. The example of the same curve as before measured at \(T=40\,\mathrm{\SIUnitSymbolCelsius}\) is shown in Figure 9. In this case the best fit as a function of \(\kappa\) and \(L\) of the whole curve gives \(\kappa=-0.08\) and \(L=0.0293\,\mathrm{\SIUnitSymbolMicro m}\) with a residual always less than \(5\times 10^{-4}\) in spite of the fact that angle \(\theta\) is pretty large at this temperature at large voltage. This is ten times better than before, as we can see by comparing the residuals of the fits shown in Figures 8 and 9. In addition, the value of \(\kappa\) found here is in very good agreement with that found by [PERSON] et al. [33] or by us (myself and [PERSON]) by using a full-optical method [24]. This good agreement was confirmed at all the other temperatures, showing the importance of taking into account the electric field dependence of \(\varepsilon_{\parallel}\) in the fit of the experimental curves at large electric field. The values of \(\kappa\) and \(L\) obtained this way are shown in Figures 10 and 11a, respectively. Finally, the last graph in Figure 11b shows the temperature dependence of the anchoring energy \(W\) calculated by using the value of \(K_{1}\) deduced from our measurements of \(V_{c}\) by taking \(\varepsilon_{a}=\varepsilon_{\parallel}(0)-\varepsilon_{\perp}\). The graph shows that \(W\) decreases when the temperature increases and approaches the melting temperature. By  
Figure 8: Capacitance curve measured at \(T=40\,\mathrm{\SIUnitSymbolCelsius}\) (the dots are the experimental points) and its fits to Equations (4) and (5) (solid lines) calculated by taking \(\kappa=-0.08\) and \(L=0.045\,\mathrm{\SIUnitSymbolMicro m}\) (**a**,**b**) or \(\kappa=0.72\) and \(L=0.05\,\mathrm{\SIUnitSymbolMicro m}\) (**c**,**d**). Here \(\varepsilon_{\parallel}\) is assumed to be constant and equal to \(\varepsilon_{\parallel}(0)\). contrast, no divergence is observed in the vicinity of the smectic A phase, contrary to what is observed for the bend and twist elastic constants. Figure 10: Values of \(\kappa\) obtained from the fit of the capacitance curves shown in Figure 7 by taken into account the electric field variation of \(\varepsilon_{\parallel}\). The agreement with the values given by [PERSON] et al. is excellent.