An Interface Equilibria-Triggered Time-Dependent Diffusion Model of the Boundary Potential and Its Application for the Numerical Simulation of the Ion-Selective Electrode Response in Real Systems

Abstract

A simple dynamic model of the phase boundary potential of ion-selective electrodes is presented. The model is based on the calculations of the concentration profiles of the components in membrane and sample solution phases by means of the finite difference method. The fundamental idea behind the discussed model is that the concentration gradients in both membrane and sample solution phases determine only the diffusion of the components inside the corresponding phases but not the transfer across the interface. The transfer of the components across the interface at any time is determined by the corresponding local interphase equilibria. According to the presented model, each new calculation cycle begins with the correction of the components' concentrations in the near-boundary (first) layers of the membrane and solution, based on the constants of the interphase equilibria and the concentrations established at a given time as a result of diffusion. The corrected concentrations of the components in the boundary layers indicate the start of a new cycle every time with respect to the calculations of diffusion processes inside each phase from the first layer to the second one, and so on. In contrast to the well-known Morf's model, the above-mentioned layers do not comprise an imaginary part and are entirely localized in the corresponding phases, and this allows performing the calculations of the equilibrium concentrations by taking into account material balance for each component. The model remains operational for any realistic scenarios of the electrode functioning. The efficiency and predictive ability of the proposed model are confirmed by comparing the results of calculations with the experimental data on the dynamics of the potential change of a picrate-selective electrode in nitrate solutions when determining the selectivity coefficients using the methods recommended by IUPAC.

Evaluation, Pitfalls and Recommendations for the “Water Layer Test” for Solid Contact Ion‐selective Electrodes

“…7-9) [7]. The equations which have been used to describe the interface have been criticized [14], and possible modifications were recommended [15][16][17]. Here we describe the responses of a SC-ISE with the presence of a water layer between the ion-selective membrane and the solid-contact to the concentration changes used in the water layer test.…”

Potentiometric Sensing

“…The obvious advantage of this material is its hydrophobicity, large capacitance and good solubility in the membrane matrix. [47][48][49] It was found that in some cases the earlier model shows principal limitations that may lead to failure of the calculations. 47 The authors studied the influence of the boundary conditions and simulation parameters to find a solution for this problem.…”

“…This group proposed to modify the established calculation protocol and termed it an interface equilibria-triggered time-dependent diffusion model. 49 52 It included Nikolsky-Eisenman equation and its permutated form, the most rigorous self-consistent model, which is however mathematically complex, and two different approximations valid for cases of low level of interference. One of these approximations was introduced for the first time.…”

Description of the Effects of Non‐ion‐exchange Extraction and Intra‐membrane Interactions on the Ion‐selective Electrodes Response within the Interface Equilibria‐triggered Model

“…The calculations were carried out for "pure" solutions of the corresponding amine in the absence of foreign ions. The diffusion of components inside the corresponding phases was calculated using equations identical to those presented in the original model [12].…”

“…A simple and, in our opinion, universal solution to this problem was proposed in the framework of the interface equilibria‐triggered (IET) dynamic diffusion model, according to which each calculation cycle for an instant of time t begins with recalculation of equilibrium concentrations of each component in rather thin near‐boundary layers. To perform this recalculation, the concentrations of the corresponding components in near‐boundary layers attained at the previous instant of time t–δt as the result of diffusion are used and interphase and intra‐phase equilibrium constants together with material balance and electroneutrality conditions in these layers are taken into account . The obtained corrected values of concentrations are then used to recalculate the concentrations of each component in all other elementary layers of the membrane and in the aqueous diffusion layer (from the first layer to the last one) taking into account diffusion processes.…”