# Chemical Vapor Deposition of Longitudinal Homogeneous Parylene Thin-Films inside Narrow Tubes
## 2 Theory on PPX-N CVD in Narrow Tubes
### General Model for the Deposition in Narrow Tubes
Neglecting the transient change of the tube's radius \(R_{\text{t}}\) caused by the film growth, the change of monomer mass flow per distance interval inside the tube is in combination with Equation (1) given by  
\[-\frac{\dot{m}_{\text{M}}}{dx}=2\;\pi\;R_{\text{t}}\;\rho_{\text{f}}\;\dot{d} _{\text{f}}=\frac{2\;\sqrt{\pi}\;R_{\text{t}}}{\hat{c}_{\text{M}}}\;S(T_{ \text{s}})\;p_{\text{M}} \tag{2}\]On the other hand, a pressure or temperature gradient of a gas, with the local gas temperature \(T_{\rm g}\)--here the gaseous monomer, in a tube will induce a local mass transfer, which is for the investigated cylindrical geometry given by [35]  
\[\dot{m}_{\rm M}=\frac{\pi R_{\rm t}^{3}}{\dot{\varepsilon}_{\rm M}}\ p_{\rm M} \left(\frac{G_{\rm T}}{T_{\rm g}}\frac{dT_{\rm g}}{dx}-\frac{G_{p}}{p_{\rm M}} \frac{dp_{\rm M}}{dx}\right) \tag{3}\]  
Here \(G_{p}\) is referred to the Poiseuille coefficient and the term containing \(G_{T}\) describes the induced gas flow by a relative temperature gradient. With Equations (2) and (3), the ordinary differential equation for our problem is given as  
\[\frac{S(T_{\rm s})}{\dot{\varepsilon}_{\rm M}}\ p_{\rm M}=\frac{\sqrt{\pi}\ R_{ \rm t}^{2}}{2}\frac{d}{dx}\left[\frac{p_{\rm M}}{\dot{\varepsilon}_{\rm g}} \left(\frac{G_{p}}{p_{\rm M}}\frac{dp_{\rm M}}{dx}-\frac{G_{\rm T}}{T_{\rm g} }\frac{dT_{\rm g}}{dx}\right)\right] \tag{4}\]  
The boundary condition at the closed end of the tube \(x=L\) is given by the film thickness growth on the perpendicular closing tube surface \(A_{\rm t}\) induced by the mass flow of the reactant at this position \(\dot{m}_{\rm M}(L)=\rho_{\rm f}\,\dot{d}_{\rm f}(L)\ A_{\rm t}\), and is (with Equations (1) and (3))  
\[\frac{dp_{\rm M}}{dx}\bigg{|}_{L}=-\frac{1}{\sqrt{\pi}\ R_{\rm t}}\frac{S(L)} {G_{p}(L)}\ p_{\rm M}(L) \tag{5}\]  
Thereby, the temperature gradient at the end of the tube is set to zero. At the inlet, the boundary value is given by the partial pressure of the reactant in the gas phase \(p_{\rm M}(0)\). By introducing the relative variables \(P=p_{M}/p_{M}(0)\) and \(X=x/L\), the boundary value problem is rewritten in a more general form  
\[\frac{S(T_{\rm s})}{\dot{\varepsilon}_{\rm g}}\ p=\frac{\sqrt{\pi}}{2}\ \left(\frac{R_{\rm t}}{L_{\rm t}}\right)^{2}\frac{d}{dX}\bigg{[}\frac{P}{ \dot{\varepsilon}_{\rm g}}\left(\frac{G_{p}}{p_{\rm M}}\frac{dP}{dX}-\frac{G_ {\rm T}}{T_{\rm g}}\frac{dT_{\rm g}}{dX}\right)\bigg{]},\ \ \frac{dP}{dX}\bigg{|}_{1}=-\frac{1}{\sqrt{\pi}}\ \frac{L_{\rm t}}{R_{\rm t}}\frac{S(1)}{G_{p}(1)}P(1),\ P(0)=1 \tag{6}\]  
In order determine the film thickness profile, the solution of Equation (6) is inserted back into Equation (1) and integrated over the process duration \(\tau\) taking into account a slowly varying (much less than the diffusion time) Parylene partial pressure at the inlet  
\[d_{F}(X,\tau)=\frac{S(X)}{\sqrt{\pi}\ \rho_{\rm f}\,\dot{\varepsilon}_{\rm g}(X)} \ \int_{0}^{\tau}dt\ p_{\rm M}(0,t)\ P(X,t) \tag{7}\]  
where for comparison of experiment and theory the relative film-thickness with respect to the inlet may be used [5; 6]  
\[D(X)=d_{\rm f}(X,\tau)/d_{\rm f}(0,\tau) \tag{8}\]  
It is apparent that the complex form of Equation (6) is not analytically solvable for a spatial arbitrary sticking coefficient. The coefficients \(G_{p}\) and \(G_{T}\) are also dependent on the gas rarefaction parameter \(\delta=2\ R_{\rm t}/\lambda\)[35] and the monomer partial pressure \(p_{\rm M}\) and respectively the \(X\) coordinate. Therefore, an approximation for our case will be presented in the following subsection. However, prior a brief comparison to existing literature models is carried out. A common thread among these models is that solely steady-state conditions and a spatial constant sticking coefficient (\(p_{M}(0,t)=p_{M}(0)\) and \(S(X)=S\)) are taken into account. In addition, common approximations are constant Poiseuille coefficients or diffusion coefficients, respectively.