## 2 Basic Transport Formulation of Unsaturated Concrete
### Material Models  
#### 2.2.1 Capacity and Diffusivity Coefficients of Multi-Species in Non-Saturated Concrete  
The moisture capacity \((\partial w/\partial H)\) of concrete can be calculated as the average of the moisture capacities of the cement paste and aggregate [37]. The chloride binding capacity \((\partial C_{l-cl}/\partial C_{f-cl})\) is defined as the ratio of the change in the free chloride content to the total chloride content [38]. The other species (Na\({}^{+}\), Ca\({}^{2+}\), K\({}^{+}\), OH\({}^{-}\)) and temperature capacity are employed as one. The heat capacity \((\partial Q/\partial T)\) of concrete is assumed to have a specific heat and density. The change in moisture due to temperature \((\partial w/\partial T)\) is negligible under normal temperature conditions and can be ignored.  
\[\left[\begin{array}{cc}\frac{\partial Q}{\partial T}&\frac{\partial Q}{ \partial H}&\frac{\partial Q}{\partial C_{f-i}}\\ \frac{\partial w}{\partial T}&\frac{\partial w}{\partial H}&\frac{\partial w}{ \partial C_{f-i}}\\ \frac{\partial C_{l-i}}{\partial T}&\frac{\partial C_{l-i}}{\partial H}&\frac{ \partial C_{l-i}}{\partial C_{f-i}}\end{array}\right]=\left[\begin{array}{ cccccc}\frac{\partial Q}{\partial T}&\frac{\partial Q}{\partial H}&0&0&0&0&0\\ \frac{\partial w}{\partial T}&\frac{\partial w}{\partial H}&0&0&0&0&0\\ 0&0&\frac{\partial C_{l-cl}}{\partial C_{f-cl}}&0&0&0&0\\ 0&0&0&\frac{\partial C_{l-Na}}{\partial C_{f-Na}}&0&0&0\\ 0&0&0&0&\frac{\partial C_{l-K}}{\partial C_{f-K}}&0&0\\ 0&0&0&0&0&\frac{\partial C_{l-OH}}{\partial C_{f-OH}}&0\\ 0&0&0&0&0&0&\frac{\partial C_{l-Ca}}{\partial C_{f-Ca}}\end{array}\right] \tag{13}\]  
The temperature diffusion coefficient \((D_{T-T})\) refers to the thermal diffusivity of concrete, which is approximately on the order of \(10^{-6}\) m\({}^{2}\)/s. This can vary depending on factors such as moisture content, porosity, and the type of aggregate used.  
The moisture diffusion coefficient \((D_{H-H})\) of concrete is derived using composite theory and can be computed through the relationship with the diffusion coefficient of aggregate and cement paste [39]. The ion diffusion coefficient \((D_{i-i})\) is adopted in references [14,34,38]. The diffusion coefficient of chloride ions \((D_{cl-cl})\) in concrete can be estimated using the multifactor method [38]. This coefficient depends on factors such as water-cement ratio, the composite action of aggregates, relative humidity, temperature, and free chloride concentration. The diffusion coefficient of other species (Na\({}^{+}\), Ca\({}^{2+}\), K\({}^{+}\), OH\({}^{-}\)) can be obtained from various references [14,34]. The remaining coupling parameters\[\left(D_{T-H},D_{H-T},\,D_{T-i},\,D_{i-T},D_{H-i},D_{i-H}\right)\,\,\,\text{are discussed in the following section.}\]  
\[\left[\begin{array}{llll}D_{T-T}&D_{T-H}&D_{T-i}\\ D_{H-T}&D_{H-H}&D_{H-i}\\ D_{i-T}&D_{i-H}&D_{i-i}\end{array}\right]=\]  
\[\left[\begin{array}{llll}D_{T-T}&D_{T-H}&D_{T-cl}&D_{T-Na}&D_{T-K}&D_{T-OH}&D _{T-Ca}\\ D_{H-T}&D_{H-H}&D_{H-cl}&D_{H-Na}&D_{H-K}&D_{H-OH}&D_{H-Ca}\\ D_{cl-T}&D_{cl-H}&D_{cl-d}&D_{cl-Na}&D_{d-K}&D_{cl-OH}&D_{cl-Ca}\\ D_{Na-T}&D_{Na-H}&D_{Na-cl}&D_{Na-Na}&D_{Na-K}&D_{Na-OH}&D_{Na-Ca}\\ D_{K-T}&D_{K-H}&D_{K-cl}&D_{K-Na}&D_{K-K}&D_{K-OH}&D_{K-Ca}\\ D_{OH-T}&D_{OH-H}&D_{OH-cl}&D_{OH-Na}&D_{OH-K}&D_{OH-OH}&D_{OH-Ca}\\ D_{Ca-T}&D_{Ca-H}&D_{Ca-cl}&D_{Ca-Na}&D_{Ca-K}&D_{Ca-OH}&D_{Ca-Ca}\end{array}\right] \tag{14}\]