# Reasons for the Formation and Properties of Soliton-Like Charge Waves in Membrane Systems When Using Overlimiting Current Modes

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Department of Applied Mathematics, Kuban State University, Krasnodar 350040, Russia

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Department of Applied Mathematics, Kuban State Technological University, Krasnodar 350042, Russia

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Author to whom correspondence should be addressed.

Received: 22 July 2020 / Revised: 11 August 2020 / Accepted: 13 August 2020 / Published: 16 August 2020

(This article belongs to the Special Issue Electromembrane Processes: Experiments and Modelling)

The study of ion transport in membrane systems in overlimiting current modes is an important problem of physical chemistry and has an important application value. The influence of the space charge on the transport of salt ions under overlimiting current modes was first studied in the work of Rubinstein and Shtilman and later in the works of many authors. The purpose of this research is to study, using the method of mathematical modeling, the reasons of formation and properties of the local maximum (minimum) space charge in membrane systems under overlimiting current conditions. It is shown that, in the diffusion layer of the cation-exchange membrane (CEM), the local maximum of the space charge appears due to the limited capacity (exchange capacity) of the membrane at a given potential jump, i.e., the local maximum of space charge appears due to the presence of a local minimum of space charge at the surface of the CEM. The local maximum of the space charge moves as a single soliton-like wave into the depth of the solution. Unlike real solitons, this charged wave changes its size and shape, albeit quite slowly. In the section of the desalination channel, the situation is completely different. First, the space charge of the anion-exchange membrane (AEM) has a negative value, so we should be talking about the local minimum (or the maximum of the absolute value of the charge). However, this is an insignificant clarification. Secondly, the space charge waves of different signs begin to interact, which leads to a new effect, namely the effect of the breakdown of the space charge. The dependence of the local maximum on the input parameters—the cation diffusion coefficient, the growth rate of the potential jump, and the initial and boundary concentrations—is studied.