Resolvability of boundary problems describing film atom diffusion in underlying surface at formation of thin-film structures

Abstract

Relevance of work. Extraction of geo assets requires the development of new technological solutions for their production, for example, the coating of drill screws with anticorrosion agents. A solution to this problem is possible using diffusion lacquer coating (corrosion inhibitor) on underlying surface (auger). The mathematization of such physical process as diffusion growth of thin films on the underlying surface is currently unexplored. In mathematical models the question on the existence and uniqueness of the solution of boundary-value problems describing the specified physical process often arises. Many domestic and foreign scientists have studied the analytical and numerical methods for solving the initial-boundary value problems, in which it is originally explicitly or implicitly assumed that the solution of the problem exists and it is unique. As a rule, the authors of publications devoted to various problems of mathematical modeling of diffusion, either do not address this question at all (about the existence and uniqueness of the solution) or refer to the classic works without good reason. Therefore, the studies on solvability of boundary value problems carried out in the paper are relevant. The aim of the research is to develop the criteria of resolvability (existence and uniqueness) of the boundary problems arising at mathematical modeling of the thin-film structure growth on underlying surface in various spaces. Research methods. The achievement of a goal is based on correct use of results and methods of the equations of mathematical physics, the integrated equations, the mathematical analysis, the equations in private derivatives, physics of a solid body, a crystallography. Results. The authors have studied the resolvability of the boundary problems describing the diffusive growth of thin films on substrates; developed the criteria of existence and uniqueness of the solution of the specified tasks in various spaces. Conclusions. At mathematical modeling of diffusive growth of a thin film on underlying surface the authors developed the theorems (criteria) providing resolvability (existence and uniqueness of the decision) of initial-boundary tasks. The paper considers the boundary problems for cases of full reflection and absorption of atoms of a film by the underlying surface. The present article is of considerable interest in applied research, and allows answering a question: is it possible to proceed immediately to a numerical (or possibly analytic) solution of the specific boundary value problem describing the diffusion growth of thin-film structures on substrates, or further carry out research on its regularization