# Multiparameter Determination of Thin Liquid Urea-Water Films
## 3 Evaluation Strategy  
### Film Thickness Evaluation  
The film thickness \(\delta\) scales linearly with the absorbance spectra, as  
\[\alpha_{v}=\kappa_{v}(c,\ T)\cdot\delta \tag{2}\]  
where regions within the absorbance spectra are required, for which the dependencies on \(c\) and \(T\) are negligible, such that \(\alpha_{v}\approx f(\delta)\). In Figure 2, the absorbance spectra for all calibration measurements are shown in the wavenumber region of 3500 cm\({}^{-1}\) to 8000 cm\({}^{-1}\). The depicted spectra are color coded with respect to the film thickness. Outside of this wavenumber region, the detected absorbances showed either a high amount of superposed noise or a low absolute value and were therefore not further investigated.  
\begin{table}
\begin{tabular}{c c c} \hline
**Film Thickness \(\delta\) (\(\upmu\)m)** & **Temperature T (\({}^{\circ}\)C)** & **Urea Concentration \(c\) (\%\({}_{\text{meas}}\))** \\ \hline
91 & 25 & 32.5 \\
115 & 30 & 39.4 \\
185 & 40 & 48.6 \\
230 & 50 & 57.7 \\
277 & 60 & \\
310 & 70 & \\  & 80 & \\  & 90 & \\ \hline \end{tabular}
\end{table}
Table 1: Parameter variations for calibration data set.  
In the region above 4100 cm\({}^{-1}\) and below 4400 cm\({}^{-1}\) the spectra for common film thicknesses overlay become clearly visible, as shown in Figure 2a. This suggests that the absorbance spectra within this region are predominantly a function of \(\delta\), whereas the influences of \(c\) and \(T\) can be neglected. In turn, a linear dependency of the absorbance data to \(\delta\) can be assumed such that a first-order polynomial is suited for calibration purposes. In Figure 3, \(\alpha\) values at the exemplary discrete wavenumber \(\
u_{1}=\) 4260 cm\({}^{-1}\) are plotted in an \(\alpha_{\
u_{1}}\)-\(\delta\)-coordinate system. The corresponding linear fit is shown as a red line.  
To assess the quality of the calibration, the root mean square error (RMSE) between the known \(\delta\) and the calculated film thicknesses \(\delta_{calc}\) is introduced. Firstly, the fit coefficients \(b_{0}\) and \(b_{1}\) of the calibration polynomial are needed:  
\[\delta\ =\ b_{1}\cdot\alpha_{\
u_{1}}+b_{0} \tag{3}\]  
where a best fit is yielded in a least-squares sense. Secondly, \(\delta_{calc}\) is calculated by evaluating \(\alpha_{\
u_{1}}\) from the calibration set with known fit coefficients with Equation (3). The optimized wavelength for film thickness determination can then be assessed by calculating the sRMSE\({}_{\delta}\) within the promising wavenumber region. The resulting RMSE\({}_{\delta}\) curve over \(\
u_{1}\) is shown in Figure 4. RMSE\({}_{\delta}\) < 2 \(\upmu\)m are achieved in the wavenumber region between 4245 cm\({}^{-1}\) and 4271 cm\({}^{-1}\)  
Figure 3: Known film thicknesses \(\delta\) of the calibration set over respective absorbance values \(\alpha\) at \(\
u_{1}=\) 4260 cm\({}^{-1}\) (black dots) and corresponding first-order polynomial fit of the data points (red line).  
Figure 2: Absorbance spectra of all calibration measurements over the wavenumber. Spectra are color coded with respect to the film thickness \(\delta\) and contain all T and c variations. (a) Enlarged section between 4100 cm\({}^{-1}\) and 4450 cm\({}^{-1}\).