Numerical Simulation of Non-Linear Models of Reaction—Diffusion for a DGT Sensor

Abstract

In this work, we present a novel strategy for the numerical solution of a coupled system of partial differential equations that describe reaction–diffusion processes of a mixture of metals and ligands that can be absorbed by a sensor or a microorganism, in an aqueous medium. The novelty introduced in this work consisted of an adequate database management in conjunction with a direct iterative schema, which allowed the construction of simple, fast and efficient algorithms. Except in really adverse conditions, the calculation is converging and satisfactory solutions were reached. Computing times showed to be better than those obtained with some commercial programs. Although we concentrate on the solution for a particular system (Diffusive Gradients in Thin Films [DGT] sensors), the proposed algorithm does not require major modifications to consider new theoretical or experimental configurations. Since the quality of numerical simulations of reaction–diffusion problems often faces some drawbacks as the values of reaction rate constants increase, some additional effort has been invested in obtaining proper solutions in those cases.

Mathematical Modeling of the Limiting Current Density from Diffusion-Reaction Systems

“…To describe the cathodic processes it is assumed, as in [7,8], that, locally, at the cathode in the diffusion boundary layer of the width δ X , the species distributions are not determined by thermodynamic equilibrium but by diffusion and the following single reaction:…”

“…where F is the Faraday's constant and z is the valence of the metal M and (c S , p S ) ∈ X is the solution to (7). With the representation (8) of the limiting current density j lim .…”

“…In the literature, both globalized models, see [1,3,5,6], considering the processes in a galvanic bath with descriptions of static electric fields at their core, and localized models, considering the processes directly at the electrodes concentrating on diffusion-reaction formulations, see [2,3,[7][8][9][10][11], exist and could give hints for the limiting current density. Stateof-the-art approaches to model the limiting current density, as discussed in [12], use the assumption of thermodynamic equilibrium in the bath and use the corresponding species distributions.…”

Speciation of Inorganic Compounds in Aquatic Systems Using Diffusive Gradients in Thin-Films: A Review

“…Hydrogels have distinct properties such as a high water content and controllable swelling behavior and are generally limited to two types for use in DGTs: polyacrylamide or agarose. They are FIGURE 1 | Schematic drawing of the DGT device (Adapted from Averós et al, 2020).…”

A New Forward–Backward Algorithm with Line Searchand Inertial Techniques for Convex Minimization Problems with Applications

“…The problem of finding an unknown reaction term of such an equation can be formulated as a coefficient inverse problem (CIP) for a partial differential equation (PDE). Numerical methods for solving CIPs for PDEs, as well as their applications to various subjects have been widely studied and analyzed; for more comprehensive information on this topic, see [1][2][3][4][5]. To obtain a globally convergent method for solving CIPs for PDEs, many authors have employed the convexification technique, which reformulates CIPs as convex minimization problems; for a more in-depth development and discussion, we refer to [6][7][8].…”