# Chemical Vapor Deposition of Longitudinal Homogeneous Parylene Thin-Films inside Narrow Tubes
## 2 Theory on PPX-N CVD in Narrow Tubes
### PPX-N Deposition in the Slip-Flow Regime  
In our case of PPX-N, CVD we diluted the reactant with a residual noble gas, which is in common use to ensure a more controllable deposition [18]. By knowledge of both partial pressures, the mean free path \(\lambda\) of the monomer in the residual gas is [36]  
\[\lambda=\frac{k_{\mathrm{B}}T_{\mathrm{g}}}{\sqrt{2}\,\sigma_{\mathrm{MM}}\ p_{ \mathrm{M}}+\sqrt{1+M_{\mathrm{M}}/M_{\mathrm{R}}}\ \sigma_{\mathrm{MR}}\ p_{\mathrm{R}}}=\frac{k_{\mathrm{B}}T_{\mathrm{g}}}{ \sigma_{1}\ p_{\mathrm{M}}+\sigma_{2}\ p_{\mathrm{R}}} \tag{12}\]  
with \(T_{\mathrm{g}}\) the gas temperature and \(p_{\mathrm{M,R}}\) the partial pressure of the monomer and residual gas, respectively. In this work we use argon as diluting gas. The molar masses are \(M_{\mathrm{M}}\approx 104\) g/mol and \(M_{\mathrm{R}}\approx 40\) g/mol, and the geometric cross section \(\sigma_{\mathrm{MM}}\approx 634\) A\({}^{2}\) is approximated from the van der Waals surface area of _para_-xylylene [37]. The mixed geometric cross section \(\sigma_{\mathrm{MR}}=\pi\ (r_{\mathrm{M}}+r_{\mathrm{R}})^{2}\approx 253\) A\({}^{2}\) is calculated by assuming the monomer cross section to be circular. With 10 Pa argon base pressure and a monomer partial pressure of around 3 Pa the rarefaction parameter is \(\delta=2\ R_{\mathrm{t}}/\lambda\approx 20\ldots 40\) at the inlet of the tube which has an inner diameter of \(R_{\mathrm{t}}=1\) mm. Therefore, the gas dynamic regime is characterized to be in the slip-flow regime [35], and \(G_{p}\) and \(G_{T}\) are approximated from molecular dynamic simulations by [38; 39]  
\[G_{p}\approx 1+\frac{\delta}{4}\,\ G_{T}\approx\frac{1}{\delta} \tag{13}\]  
It follows that in the slip-flow regime, the magnitude of \(G_{p}\) is in the order of two decades above \(G_{T}\) (\(G_{p}\gg G_{T}\)) and therefore the term with the temperature slip coefficient \(G_{T}\) in (5) can be neglected if the relative temperature and pressure gradients are in the same order of magnitude. This is the case for our manipulation, with the sticking coefficient and pressure being linear proportional to the deposition rate (see Equation (1)). Equation (6) is rewritten with \(d\varepsilon_{\mathrm{g}}/dX=\left(\varepsilon_{\mathrm{g}}/2T_{\mathrm{g}} \right)\left(dT_{\mathrm{g}}/dX\right)\) as  
\[\frac{d^{2}P}{dX^{2}}+\left[\frac{1}{G_{p}}\frac{dG_{p}}{dX}-\frac{1}{2\ T_{ \mathrm{g}}}\frac{dT_{\mathrm{g}}}{dX}\right]\frac{dP}{dX}-\frac{2}{\sqrt{ \pi}}\ \left(\frac{L_{\mathrm{t}}}{R_{\mathrm{t}}}\right)^{2}\frac{S(T_{ \mathrm{s}})}{G_{p}}\ P=0,\ \frac{dP}{dX}\Bigr{|}_{1}=-\frac{1}{\sqrt{\pi}}\ \frac{L_{ \mathrm{t}}}{R_{\mathrm{t}}}\frac{S(1)}{G_{p}(1)}P(1),\ P(0)=1 \tag{14}\]  
For intermediate dilute gas mixtures, the total gas pressure will stay constant over the whole tube distance \(X\). Thus a rarefaction in the monomer partial pressure will increase the partial pressure of the residual gas [26] given by  
\[\frac{d}{dX}(p_{\mathrm{M}}+p_{\mathrm{R}})=0\to p_{\mathrm{tot}}=p_{ \mathrm{M}}(X)+p_{\mathrm{R}}(X) \tag{15}\]  
where \(p_{\mathrm{tot}}\) is the total pressure at the inlet of the tube and can be experimentally tracked during the deposition process. With this information, the Poiseuille coefficient is expressed only showing dependence in the partial pressure of the monomer and its temperature. Due to the condition from Equation (15), \(G_{p}\) and its derivative with respect to the space variable can be expressed via Equations (12) and (13) as  
\[G_{p}(X)=1+\frac{R_{\mathrm{t}}}{2k_{\mathrm{B}}T_{\mathrm{g}}}[(\sigma_{1}- \sigma_{2})\ p_{\mathrm{M}}+\sigma_{2}\ p_{\mathrm{tot}}] \tag{16}\]  
At this point it should be noted that the gas temperature inside the tube being influenced by the surface temperature should be discussed. Although the slip-flow regime is characterized by a temperature jump between the vacuum and the wall [40], radial effects are not considered in our 1D approach.