Diffuse interface method for simulation of the chemical vapor deposition of pyrolytic carbon: Aspects of the mathematical formulation

The present work is inspired by Anderson et al. (Phys. D Nonlinear Phenom. 2000; 135(1¯2):175¯194) and Noll (http://www.math.cmu.edu/wn0g/noll) and falls in the conceptual line of the Ginzburg-Landau class of first-order phase-transition models based on the concept of phase-field parameter. Trying to keep the exposition as much general as possible, we develop below a thermodynamically consistent rationalization of the physical process of (anisotropic) deposition of pyrolytic carbon from a gas phase. The derivation line we follow is well established in the field of the modern continuum physics. From the covariance of the first principle of thermodynamics, the second Newton's law and the Liouville's theorem with respect to the one-dimensional Lie groups of transformations, the balance laws for the temperature, linear momentum and density are formulated. This system of partial differential equations is comprehended further by the constitutive laws for the phase field, the stress, and the heat and entropy fluxes obtained in a form consistent with the Clausius-Duhem understanding of the second law. The result referred to as a local, strongly coupled initial boundary value problem of chemical vapor deposition (IBVP-CVD) constitutes the general mathematical description of the CVD process. The weak form of isotropic IBVP-CVD is then derived and discretized by means of the discontinuous Galerkin method. At the end of the paper, we also derive the weak formulations for the local lifting operators that provide the stabilization mechanism for the discontinuous Galerkin discretization scheme.