Simulation of a Chemical Vapor Deposition: Four phase model

Juergen Geiser , M. Arab

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 45 pp.

MSC 2000

35K25 General theory of higher-order, parabolic equations

35K20 Boundary value problems for second-order, parabolic equations

Abstract
We are motivated to model chemical vapor deposition for metallic bipolar plates and optimization to deposit a homogeneous layer. Moreover a constraint to the deposition process is a very low pressure (nearly vacuum) and a low temperature (about 400 K). These constraints need to have a catalyst process, here in our apparatus we deal with a plasma source and precursor gases. Such a plasma have the advantage to accelerate the vaporation process and to bring the solid materials to a gaseous phase. Nevertheless there are also some drawbacks, so that a retardation and adsorption process can hinder the direct transport to the target. Here, we present a mesoscopic model, which reflects the retardation, transport and reaction of the gaseous species through a homogeneous media in the chamber. The models include immobile gaseous phases, where the transport of the mobile gaseous species are hindered. Kinetically controlled adsorption are included to taken into account the multiple species of heavier and lighter species. Such ideas are also considered in fluid dynamical models. Further, the models include the conservation of mass and the porous media, are in accordance with the Darcy`s law, which is an assumption to the flow processes of the gaseous phase. The transport through the instationary and ionized plasma field is treated as a diffusion-dominated flow with mobile and immobile zones, where the metallic deposit and the gas chamber, looking like a porous media. Numerical methods are developed to solve such multi-scale and multi-phase models and to obtain qualitative results of the delicate multi physical processes in the gas chamber. To solve such evolution models, we combine discretization methods for partial differential and ordinary differential equations. Sequentially treatment of the partial differential equations and ordinary differential equations allow to discretize with Finite volume methods for the spatial derivatives of the transport equations, while Runge-Kutta methods are used to discretize the time derivatives and the ordinary parts of the multi-phase model. With various source terms we control the required concentration at the final deposition area. Different kinetically parameters allow to simulate the different time scale of the heavier and lighter gaseous species. We present an expert system based on a multi phase model and embedded source and target controls to present an accurate computational models for the transport of gas concentrations to a plasma media. For such efficient choose of models an discussion of physically correct numerical methods are important and simulate an optimal homogeneous deposition at the target with control of the rest gaseous concentration in the plasma media. The results are discussed by means of physical experiments to give a valid model for the assumed growth.