A Parabolic System Model for the Formation of Porous Silicon: Existence, Uniqueness, and Stability

Abstract

In this paper, a nonlinear weakly coupled parabolic system is constructed to model porous silicon formation. The model considers physical and chemical factors such as charge carriers transport, ions transport, electrochemical reactions, and material balance. In order to prove the existence of a unique solution for the proposed parabolic model with zero Neumann boundary, we use a monotone method based on the upper and lower solutions. The asymptotic stability of the steady-state solution is also demonstrated, and the behavior of this model is shown for specific cases where the analytical stationary solution is known. Finally, a discussion of the results is carried out, and we conclude that the proposed model reproduces important qualitative features which are observed experimentally. Introduction.Porous silicon is a biodegradable nanomaterial obtained by means of electrochemical anodization. Due to its physical and electrochemical properties, it is actually used in many applications such as the controlled release of drugs, bone implants, and solar cells among others [5,7,9,19,44]. Different morphological structures can be experimentally obtained in laboratories by varying conditions like impurities distribution in the silicon wafer, bulk density, electrolyte concentration, current density, and temperature [5,9,19,31]. The variability of the porous structure (quantity, size, and shape) characterizes the releasing and charge-transfer processes. Thus, silicon biocompatibility is linked to its morphology properties; see Figure 1. Moreover, the interaction mechanisms among the diverse factors involved in the structure development is not completely understood. In laboratories, the porous structures are obtained by a trial-and-error procedure. Modeling and computational tools may help to understand such complex mechanisms.In recent years, several works have been focused on the mathematical modeling of different aspects of the porous silicon process. The first proposal was the finite diffusion length (FDL) model by 37], which describes the formation mechanism of porous silicon morphology. They considered the formation of porous silicon as a diffusion limited process. In such articles, a lattice model was obtained based on the diffusion limited aggregation model [43]. Later on, a computational FDL model by means of the Monte Carlo method was performed by Vadjikar [39,40]. Afterwards, Aleksandrov and Novikov simulated the porous silicon formation by considering not only a diffusion limited process but also the drift of the charge carriers in the silicon [1,25]. Other models are focused on reproduc-

Discrete monotone method for space-fractional nonlinear reaction–diffusion equations

“…For example, from the analytical side, such iterative techniques have been applied to investigate the existence and uniqueness of solutions of a wide range of parabolic partial differential equations [1], as well as other analytical features of the solutions. In particular, this approach has been used to establish the existence of positive solutions of quasilinear parabolic systems with Dirichlet boundary conditions [2], to study quasilinear parabolic and elliptic systems with mixed quasimonotone functions [3], to analyze periodic boundary-value problems for differential equations with delay [4], to solve first-order functional-difference equations with nonlinear boundary value conditions [5], to prove the existence and asymptotic behavior of solutions for quasilinear parabolic systems [6], and, recently, to establish the existence, uniqueness, and stability of the solutions of a parabolic model in the formation of porous silicon [7], among other interesting applications.…”