# 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  
Next the temperature gradient for the generation of homogeneous layer thicknesses was investigated. For this, according to Equation (1), the product of local sticking coefficient and monomer partial pressure must remain constant within the entire tube. \(S(X)P(X)=S(0)P(0)=S(0)\) applies. A natural consequence is that the resulting pressure gradient compared to a non-manipulated process is reduced, and therefore a homogeneous layer thickness may only be realized by a partial suppression of deposition on the first section of the tube and vice versa. Owing to the complexities of Equation (14) and Equation (18), Equation (9) was used to approach this problem. The disregard of complex and spatially varying terms has already been discussed in the previous section, with the result that a simplified mathematical description of the problem yields sufficiently accurate results--at least when dealing with constant temperatures. With the given domain condition \(S(X)P(X)=S(0)\), Equation (9) yields a parabolic pressure profile after twofold integration  
\[P(X)=\alpha\ S_{\rm r}(0)\ \bigg{[}\frac{X^{2}}{2}-\left(1+\frac{R_{\rm t}}{2 \ L_{\rm t}}\right)\ X\bigg{]}+1 \tag{20}\]  
\(\alpha\), determined by fitting Equation (11), already includes the absolute sticking coefficient for the constant reference temperature \(T_{\rm ref}\). This makes it sufficient, as mentioned in the previous section, to calculate the sticking coefficient relative to the temperature present in the original experiment. The parabolic course of the pressure gradient follows with \(P(X)>0\) for \(0<X<1\) an upper bound for the applicable sticking coefficient within this model,  
\[S_{\rm r}(0)<\frac{2}{\alpha\ (1+R_{\rm t}/L_{\rm t})} \tag{21}\]  
where, for long aspect ratios, this is simplified to \(S_{\rm r}(0)<2/\alpha\). From the measurement of the sticking coefficient, we further know that the minimum relative sticking coefficient (just below the ceiling temperature) is \(\min\{S_{\rm r}\}=0.14\). By defining a new relative and ideal film thickness via \(D_{\rm r}(X)=D_{\rm manipulated}/D_{\rm unmanipulated}=S_{\rm r}(X)\ P(X)=S_{ \rm r}(0)\) we find a small theoretical bandwidth for achieving homogeneous film thicknesses with values between 0.14 and 0.24 \(d(0)\). Here \(d(0)\) is the film thickness at constant reference temperature at the inlet of the tube, which is 2.55(6) \(\upmu\)m (see above). The required sticking coefficient profile along the tube is then finally given with  
\[S_{\rm r}(X)=\bigg{[}\alpha\bigg{[}\frac{X^{2}}{2}-\left(1+\frac{R}{2\ L} \right)X\bigg{]}+1/S_{\rm r}(0)\bigg{]}^{-1} \tag{22}\]In particular, Equation (22) is valid for manipulated CVD processes in tubes with linear deposition kinetics, and the corresponding temperature gradient is derived from the specific dependence of the sticking coefficient. In our case, we will use for the final experiments the numerical smoothing spline from our measured relative sticking coefficient. At this point, the employed approximation shall be discussed. In this regard Equations (9) and (18) were solved numerically using the defined sticking coefficient from Equation (22). Since we know \(L_{\text{t}}\gg R_{\text{t}}\) the aspect ratio in the equation was neglected. The calculation was carried out in Comsol Multiphysics 5.5, and the required sticking coefficient was defined for a constant film thickness corresponding to 0.9 times the maximum achievable homogeneous film thickness (Equation 21) (see Figure 3a). With \(\alpha=8.29\) for our tube geometry at a reference temperature of 30 \({}^{\circ}\)C, the relative film thickness corresponds to 21.7 % of the film thickness at the beginning of the tube of a non-manipulated process. This ratio was calculated as \(D_{\text{r}}(X)=D_{\text{manipulated}}/D_{\text{unmanipulated}}=S_{\text{r}}(X)\ P(X)\). Figure 3b shows the determined pressure curves \(P(X)\) in comparison.