# Chemical Vapor Deposition of Longitudinal Homogeneous Parylene Thin-Films inside Narrow Tubes
## 4 Results and Discussion
### Defining the Required Temperature Gradient and Boundary Value Problem Analysis
Here, the blue dashed line depicts the solution with the assumption of a constant Poiseuille coefficient as well as constant gas temperature, according to Equation (9). The red curve shows the pressure solution for the boundary value problem from Equation (18), where the variable coefficient \(G_{p}\) and the local temperature change of the gas along the tube (\(T_{g}=T_{s}\)) were taken into account. The latter was determined from the model of [PERSON] et al. via the relative sticking coefficient. This was necessary since the absolute values at the reference temperature of 30 \({}^{\circ}\)C differ from [PERSON] et al. and our work. A comparison of the two solutions (red and blue line in Figure 3b) does not reveal any differences at first glance. However, for both calculations, the pressure gradient turns out to be lower in comparison to a deposition process at constant temperature (black dashed line, corresponding to solution from Figure 2a). In order to investigate the virtually non-existent mismatch between the simple and complex form of the boundary value problem, the corresponding terms of the differential equation (Equation (18)) are shown in Figure 2c. By definition, the sum of these must equal zero, which corresponds to the black dashed curve. The increased complexity of Equation (18) is caused by the local temperature dependence of the gas as well as a spatially variable Poiseuille coefficient, which are part of the term containing the first pressure derivative \(dP/dX\). This contribution is shown as a magenta line in Figure 3b and obviously yields the smallest influence on the overall solution. The terms with the second pressure derivative \(d^{2}P/dX^{2}\) (red line) as well as the pressure \(P\) itself (blue line) are thus approximately axially symmetric around the zero line (corresponding to the difference of the magenta curve). This explains the negligible deviation of the solutions of Equations (9) and (18) in Figure 3b and supports the assumption that, for the derivation of the required sticking coefficient, it is sufficient to consider the simple form of the differential equation. Finally, the pressures from Figure 3b for the simple as well as complex differential equation can be multiplied by the relative sticking coefficient in order to obtain the relative layer thickness \(D_{\text{r}}(X)\). The results are presented in Figure 3d as difference to the ideal value according to the following expression  
\[\frac{\Delta D_{\text{r}}(X)}{D_{\text{r}}(0)}=\frac{D_{\text{r}}(X)-S_{\text{ r}}(0)}{S_{\text{r}}(0)} \tag{23}\]  
Here, the black dashed line corresponds to the pressure solution from Equation (9) and, as required by theory, gives a perfect match with the zero line. By choosing the representation with the relative layer thickness change, individual contributions in the differential equation can be examined more precisely. Especially those which may not be clearly evident in the pressure course. Therefore, individual contributions were systematically exhibited in Equation (18), namely the temperature dependence of the gas (blue line) and the local dependence of the Poiseuille coefficient (red line). The solution for considering all terms (red pressure plot from Figure 3b) is shown in magenta, and the dashed lines refer to results derived from neglecting the deposition at the closed end of the tube. At first glance, it can be seen that, by forcing the pressure gradient to zero at the end of the tube, the monomer partial pressure in this region increases slightly, which exerts a high effect on the film thickness profile due to the high relative sticking coefficient at this position (above two, see Figure 3a). Furthermore, neglecting the temperature gradient as well as the Poiseuille coefficient gradient, ergo the term in Equation (18) which contains the pressure derivative, leads to similar results as considering the entire terms, especially in the last third.