**1. Introduction**

The corrosion of steel in reinforced concrete (RC) structures in coastal zones is mainly induced by chloride ingress, which represents a permanent risk of degradation. Chlorides penetrate into the cover concrete under a concentration gradient between seawater and the concrete pore solution and/or liquid pressure gradient in partially saturated concrete when heat and moisture transfers occur (tidal areas or marine fogs). When chlorides reach the rebars with a threshold concentration, they generate rebar depassivation and corrosion [1]. Given the high cost of maintenance/repair, nowadays the durability of RC structures in their environments is one of the main challenges mentioned in the specifications of construction.

In the last decades, several experimental and numerical studies were developed to propose methods and tools for predicting chloride transport in cementitious materials. The standard migration test is used to determine the chloride diffusion coefficient in a steady state or non-study state [2–8]. Moreover, single-species modelling was used to predict chloride transport in cementitious materials [2]. After that, many multispecies approaches were developed, considering several ions in the pore solution and more chemical and physical interactions during the transport [9–20]. Xia and Li [21] proposed numerical modelling of ion transport in saturated cementitious materials based on Poisson–Nernst– Planck (PNP) equations, considering the chemical interaction between the monovalent ions in the pore solution in order to monitor the impact of the interactions on the chloride ingress. Fenaux et al. [22] proposed a chloride transport modelling in saturated concrete taking into account monovalent and divalent ions of the pore solution: Cl−, Na+, K+, OH<sup>−</sup> and Ca2+. The diffusion, migration and chemical activity were considered. The chemical activity was calculated using the Pitzer model. The numerical results highlighted the influence of the composition of the pore solution and the chemical activity on the chloride penetration.

Furthermore, recent research discussed the impact of the thermodynamic equilibria on the chloride reactive transport in cementitious materials [23–27]. Yu and Zhang [28]

**Citation:** Kribes, Z.-E.; Cherif, R.; Aït-Mokhtar, A. Physico-Chemical Modelling of Chloride Migration in Cement-Based Materials Considering Electrode Processes. *Mater. Proc.* **2023**, *13*, 37. https://doi.org/10.3390/ materproc2023013037

Academic Editors: Katarzyna Mróz, Tomasz Tracz, Tomasz Zdeb and Izabela Hager

Published: 20 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

LaSIE, UMR, CNRS 7356, La Rochelle University, 17000 La Rochelle, France

proposed a model to predict the leaching of cement paste in an ammonium nitrate solution taking into account ion transport, chemical kinetics and thermodynamic equilibria. The ion transport was calculated by the PNP equation, while, the chemical activity was calculated using the Davies model. Tran et al. [29,30] developed a coupling between chloride transport and thermodynamic equilibrium, considering kinetic control to predict chloride fixation in concrete. Jensen et al. [31,32] developed a multispecies approach for reactive mass transport in a saturated mortar exposed to chlorides for 180 days, including the chemical equilibria. Cherif et al. [33] proposed a coupling between multispecies transport and thermodynamic equilibria in cementitious materials containing mineral additions. The modelling considered all the ions of the pore solution, the portlandite dissolution and Friedel's and Kugel salt precipitation during chloride transport. Ion fluxes were calculated using Nernst–Planck equation, while the thermodynamic coupling was based on the low mass action and the rate of dissolution/precipitation of the solid phases.

The models mentioned above are generally based on the PNP equation with a limited number of ions in the pore solution, whose concentrations are considered significant. The other models consider all the multispecies interactions in the material, but they are based on Fick's law, which misdescribes the chloride ion transport. Researchers dealing with chloride migration, considering electrode processes remain very little in the literature.

Motivated by this need, multispecies modelling of chloride migration is proposed in this study, considering the electrode processes. The modelling is applied on Portland cement submitted to standard chloride migration test. The electrode processes reflect the generation of OH<sup>−</sup> and H+ in the cathode and anode, respectively. The electrode processes ensure the electroneutrality in the migration cell (sample and compartments). The concentrations of OH<sup>−</sup> and H<sup>+</sup> are calculated from the current density measured during the test, using Faraday's law. The charge passed is deduced from the current density measured. Ion fluxes are calculated by the Nernst–Planck equation, which describes the diffusion and migration of the species. The Langmuir model is used to simulate the chloride chemical fixation by the material (chloride isotherms). The chemical activity is neglected according to [3]. The considered ions are Cl−, Na+, K+, OH−, H+, Ca2+ and SO4 <sup>2</sup>−. The migration cell used is composed of two compartments: (1) upstream containing 25 mM NaOH and 83 KOH and 500 mM NaCl; (2) downstream containing only 25 mM NaOH and 83 KOH (boundary conditions). The composition of the pore solution of the material tested was considered the initial condition. An electrical field of 300 V·m−<sup>1</sup> was applied at the sample boundaries and monitored using two calomel reference electrodes. The latter were placed at each side of the sample tested in order to maintain the electrical field constant modelling outputs are as follows:


#### **2. Methodology**

#### *2.1. Modelling Principle*

The time evolution of the ion concentration (Ci) during the migration test is calculated by using the mass balance equation (Equation (1)), which takes into account the porosity of the material tested (ϕ), the chloride concentration bonded to the cement matrix (Ci,b) calculated by the Langmuir's model and the ion flux (Ji) calculated by the NP equation (see Equation (2)). The internal electrical potential between ions is neglected in front of the applied electrical field of 300 V·m−1. The mass exchange term (qi), added to the mass balance equation, describes the ion gain/loss in the pore solution due to the dissolution/precipitation of the solid phases considered (C-S-H, portlandite, monosulfoaluminates and trisulfoaluminates). Further details about the calculations of the term (qi) and the

thermodynamic equilibrium constants used are shown in [8]. The ions considered in this study are: Cl−, Na+, K+, OH−, H+ Ca2+ and SO4 <sup>2</sup>−. Note that the proposed modelling concerns ion transport in saturated materials that do not require coupling with convection and moisture transfer.

$$\frac{\partial \mathbf{C}\_{\mathrm{i}}}{\partial \mathbf{t}} + \frac{((1-\varrho)\mathbf{C}\_{\mathrm{i},\mathrm{b}})}{\partial \mathbf{t}} = -\mathrm{div}(\mathbf{J}\_{\mathrm{i}}) - \frac{\partial \mathbf{q}\_{\mathrm{i}}}{\partial \mathbf{t}}\tag{1}$$

$$J\_i = -D\_{E,i} \left( \nabla \mathbf{C}\_i + \frac{z\_i \mathbf{C}\_i FE}{RT} + \mathbf{C}\_i \nabla In \gamma\_i \right) \tag{2}$$

where *DE,i* [m2.s−1] is the effective diffusion coefficient of the ion *i*, *zi* is the valence of the ion *<sup>i</sup>*, *<sup>F</sup>* [C·mol−1] is the Faraday constant, *<sup>E</sup>* [V·m−1] is applied electric field, *<sup>R</sup>* [J·K−1·mol−1] is the ideal gas constant and *T* [K] is the temperature *γ<sup>i</sup>* is the ion activity coefficient.

The electrode processes responsible for the generation of OH− in the catholyte (upstream) and H<sup>+</sup> in the anolyte (downstream) are given in the following. Note that noncorrodible Platine electrodes were used.

$$2\text{ Cathode:}\ 2\ \text{H}\_2\text{O} + 2\ \text{e}^- \rightarrow \text{H}\_2 + 2\ \text{OH}^-\tag{3}$$

$$\text{Anode: }\mathrm{H\_2O} \to 0.5\,\mathrm{O\_2} + 2\,\mathrm{H^+} + 2\,\mathrm{e^-}\tag{4}$$
