## 1 Introduction
Researchers presented a mathematical model of ion diffusion in porous media. This model, based on the Nernst-Planck equation, included the Poisson equation to evaluate the electrostatic coupling between ions [16]. Subsequently, the model was expanded to consider chemical activity effects by including ions in non-ideal solutions. A mathematical model was developed by considering the chemical activity terms in the governing equations [17]. More recently, a model was developed for unsaturated cement-based materials related to external sulfate or calcium and hydroxide leaching, based on a set of Nernst-Planck/Poisson equations, considering diffusion, electrostatic coupling, chemical activity, and convective mechanisms under isothermal conditions [18]. [PERSON] et al. proposed a chloride diffusion coefficient in partially saturated cementitious materials by combining composite theory and [PERSON]'s model instead of the Nernst-Planck equation [19]. Recently, a numerical model was developed to analyze mass transport and electrode polarization in concrete under an externally applied voltage, accounting for multi-species electromigration, time-dependent overpotential, and non-equilibrium binding. The model incorporated the Tafel equation to represent overpotential at electrode interfaces and dynamically updated boundary conditions to accurately simulate the ion distribution. Itsreliability was validated against experimental data. The results showed that electrode polarization reduced the effective potential across the concrete, significantly influencing chloride penetration and the spatial-temporal distribution of ionic species [20]. Furthermore, multi-species reactive transport modeling (RTM) has been advanced to better understand the deterioration mechanisms of cement-based materials under various environmental conditions. [PERSON] et al. developed an RTM for saturated systems, incorporating multi-ionic transport via modified Poisson-Nernst-Planck equations, chemical equilibrium, and evolving pore structure, with validated predictions for chloride and ion distribution in mortar exposed to NaCl and seawater. This approach was extended to unsaturated conditions, coupling multi-ionic and moisture transport with a two-phase moisture model driven by capillary pressure. The numerical results under wetting-drying cycles showed that prolonged wetting increased the chloride penetration and deterioration depth, with Friedel's salt formation limiting deeper chloride ingress. An additional study focused on carbonation processes, modeling how pore solution composition and phase changes affect steel corrosion. It revealed alkali ion depletion and migration toward carbonated zones, increased sulfur and chloride concentrations from solid phase decomposition, and enhanced calcite formation due to calcium migration [21; 22; 23]. The actual boundary conditions of concrete structures are often exposed to climate changes, facing periodic moisture and temperature boundary conditions. Moreover, temperature fluctuations and moisture diffusion affect chloride penetration. This effect has previously been integrated with BET isotherms and the Arrhenius law [24; 25]. To better predict the transport of chloride ions in concrete materials, it is necessary to develop a robust mathematical model that considers temperature effects. In general, concrete structures are subject to non-isothermal conditions, and temperature fluctuations can significantly affect the chemical reactions of ions. In other words, the temperature gradient is one of the most important driving forces [26]. A governing equation should be derived for chloride ion penetration based on temperature effects under partially saturated concrete conditions, which has been studied by many researchers based on experimental and numerical studies. [PERSON] et al. and [PERSON] et al. mentioned the relationship between temperature and moisture movement in saturated and unsaturated concrete using mathematical models, experimental data, and numerical programs. In particular, [PERSON] et al.