## 2 Basic Transport Formulation of Unsaturated Concrete  
### Governing Equation  
Concrete is made up of three physical phases: aggregate, cement paste, and porosity. Due to the dense and hard nature of the aggregate, the diffusivity of ionic species is minimal within it. However, the pore system of cement paste provides a pathway for the migration of ionic species. The presence of multi-species ions (Cl\({}^{-}\), Na\({}^{+}\), Ca\({}^{2+}\), K\({}^{+}\), OH\({}^{-}\)) in pore solutions depends on the type of cement and mineral admixtures used. The total chloride content includes both bound and free chlorides. Once the water-soluble free chlorides reach a certain threshold, the passive film on the reinforced rebar surface can be damaged. To predict the service life of concrete structures, the content of chloride ions is computed with a mathematical model. While this implementation is simple and convenient, it is not able to demonstrate the deterioration of concrete including ionic interactions inside the pore system of concrete under micro-climate conditions. Therefore, a mathematical model that accurately reflects the diffusion of ionic species in the system may be necessary to better understand real-world phenomena. For the flux of each species in the pore solution of saturated concrete, the Nernst-Planck equation is used, as follows:  
\[J_{i}=-\bigg{[}D_{i}\
abla C_{f-i}+Z_{i}D_{i}\bigg{(}\frac{F}{RT}\
abla\phi \bigg{)}C_{f-i}\bigg{]} \tag{1}\]  
where \(J_{i}\) denotes the flux of ions, \(D_{i}\) represent the diffusion coefficient, \(C_{f-i}\) specifies the concentration of multi-species (Cl\({}^{-}\), Na\({}^{+}\), Ca\({}^{2+}\), K\({}^{+}\), OH\({}^{-}\)), \(Z_{i}\) is the charge number, \(F\) is [PERSON]'s constant, \(R\) is the gas constant, \(T\) refers to the temperature, \(\phi\) indicates the electrostatic potential, and index \(i\) corresponds to the \(i\)-th species [36]. When subjected to an internally induced electric field, all ionic species in the pore solution influence the electro-neutrality condition. There are two physical requirements of the electro-neutrality condition: one is that the total charge should be zero at every point of the domain, and the other is that no electric current should pass through the domain [36]. To account for the electrical potential (\(\Phi\)) inherently caused by the movement of all species, two conditions are required, as follows:  
Eletroneutrality \[\sum Z_{i}C=0\] (2) Null current \[\sum Z_{i}J_{i}=0\] (3)To solve Equation (1), the null current assumption from Equation (3) is employed. The expression for the electro potential can be obtained as in Equation (4). \[\
abla\phi=-\frac{RT}{F}\frac{\sum\limits_{i=1}^{N}Z_{i}D_{i}\
abla C_{i}}{\sum \limits_{i=1}^{N}Z_{i}^{2}D_{i}C_{i}} \tag{4}\]  
The formulation of chloride ions (Cl\({}^{-}\)) and other species (Na\({}^{+}\), Ca\({}^{2+}\), K\({}^{+}\), OH\({}^{-}\)) in non-saturated concrete is based on the Nernst-Planck equation. The transport of all ionic species is associated with the coupled effect of temperature and moisture in concrete. The heat flow is described by [PERSON]'s law of heat conduction. The governing equation of multi-species diffusion in partially saturated concrete affected by the coupling of moisture gradient and temperature gradient (Soret effect) can be written as follows:  
\[J_{i}=-\bigg{[}D_{i}\
abla C_{f-i}+Z_{i}D_{i}\bigg{(}\frac{F}{RT}\
abla\phi \bigg{)}C_{f-i}+D_{i-H}\
abla H+D_{i-T}\
abla T\bigg{]} \tag{5}\]  
where \(J_{i}\) is the flux of ions, \(D_{i}\) denotes the diffusion coefficient, \(C_{f-i}\) represents the concentration of multi-species (Cl\({}^{-}\), Na\({}^{+}\), Ca\({}^{2+}\), K\({}^{+}\), OH\({}^{-}\)), \(Z_{i}\) is the charge number, \(F\) is [PERSON]'s constant, \(R\) is the gas constant, \(T\) is the temperature, \(\Phi\) is the electrostatic potential, and index \(i\) represents the \(i\)-\(th\) species. \(D_{i-H}\) and \(D_{i-T}\) are the coupling parameters of the effect of moisture and temperature on ion diffusion, respectively.