# 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  
The film thickness growth of PPX is generally understood as surface polymerization of di-radical monomers from the gas phase upon contact with a surface [19; 22; 25; 29]. Initial models are based on the adsorption of monomers on the surface with Flory-type [30] or Langmuir-type [22] adsorption isotherms. Subsequently, a reaction kinetic separation with partial temperature-dependent diffusion of monomers to the next reactant in the forming film (chain end) as well as an additional surface polymerization was considered [22]. Due to the complex modeling, there are different proportionalities in the growth rate of the film thickness, such as those from the monomer partial pressure in the adsorption isotherms or the substrate temperature, which exert influence on the diffusion coefficient as well as the vapor pressure of the monomer during Flory adsorption [22]. However, these models have the disadvantage of consisting of many complex terms with free parameters that may contain effective terms and are difficult to study separately [10]. At this point, a chemisorption model was proposed by [PERSON] et al. [23] based on a work investigating the sticking probability of chemisorbed gases on surfaces [31]. Here, the equilibrium rate between migration, desorption, and chemisorption of adsorbed molecules was combined into a generalized sticking coefficient [10; 23]. Temperature dependence was addressed by the Boltzmann factor within the reaction probabilities, introducing a dependence on surface temperature within this model. By defining a sticking coefficient for the di-radical parylene monomer, the film-growth rate of PPX-N is given by the product of molecules from the gas phase that chemisorb after collision with the surface, (with \(j_{\text{M}}\) the flux density and \(S(T_{\text{s}})\) the sticking coefficient in dependence of the substrate surface temperature \(T_{\text{s}}\)) and the molecular volume \(V_{\text{M}}\) of the reactant. This relation is in dependence of the partial pressure of the film forming monomer \(p_{\text{M}}\) written as  
\[\dot{d}_{\text{f}}=j_{\text{M}}\;V_{\text{M}}\;S(T_{\text{s}})=\frac{1}{\sqrt{ \pi}\;\hat{c}_{\text{g}}\;\rho_{\text{f}}}\;S(T_{\text{s}})\;p_{\text{M}} \tag{1}\]  
Here \(\hat{c}_{\text{g}}\) is the most probable velocity of the di-radical monomer from kinetic gas theory and \(\rho_{\text{f}}\) the film density. Herewith, a mathematical expression for the film thickness, in dependence of a spatial manipulated sticking coefficient, along the interior wall of a one-sided closed tube can be derived. We assume the CVD conditions as steady state, which means that the timescale of the change on monomer pressure at the inlet of the tube (due to possible effects of process-related pressure changes) is much greater than the diffusion time of the monomer from the inlet of the tube to its end. In addition, enough particle collisions of initial hot monomers and the residual gas or the reactors walls, both at room temperature, are given between cracker and substrate, so that the molecules will be in thermal equilibrium with the room temperature at the deposition site [32; 33; 34]. Due to the choice of a rotation-symmetrical tube, the problem may be reduced towards one dimension (parallel to tube axis).