## 2 Basic Transport Formulation of Unsaturated Concrete
### Material Models
#### 2.2.2 Coupling Parameters  
The coupling parameters in the diffusivity matrix have been studied by many researchers. The Soret effect (thermal diffusion) is the occurrence of a diffusion flux due to a temperature gradient. The Dufour effect is the reciprocal phenomenon, where heat flux occurs due to a chemical potential gradient. The temperature effect on chloride diffusion \((D_{cl-T})\) was reported by [PERSON] [35]. The effect of moisture due to chloride diffusion and the influence of chloride due to the moisture gradient were demonstrated by [PERSON] et al. and [PERSON] et al. [5; 40]. More detailed information is described below. \(D_{T-H}\) and \(D_{H-T}\)  
The humidity diffusion coefficient of concrete depends on the diffusion coefficients of both the aggregate and the cement paste. The diffusion coefficient of the aggregates is considered negligible, as the pores within the aggregates are discontinuous and surrounded by cement paste. The humidity diffusion coefficient of cement paste can be evaluated using the empirical formula proposed by [PERSON] et al. [25]. Although the coupling parameters in the diffusivity matrix of Equation (14) for the effect of temperature on moisture diffusion \((D_{H-T})\) and the effect of moisture on temperature diffusion \((D_{T-H})\) were introduced by [PERSON] et al. [26; 27], the flux driven by the temperature gradient (Soret flux) is relatively small based on the limited experimental data available; that is, approximately \(D_{H-T}\approx 0\). In addition, the coupled heat flux \((D_{T-H})\) may be neglected in this numerical model. \(D_{T-i}\) and \(D_{i-T}\)  
In Equation (14), the coupling parameter of temperature's effect on chloride diffusion \((D_{cl-T})\) was as proposed by [PERSON], following Equation (15), which was evaluated based on the experimental study of the effect of temperature on chloride penetration into saturated concrete [35]. This depends on the effect of aging and temperature. \[D_{cl-T}=f_{1}\cdot C_{f-cl}\cdot f_{2}(t)\cdot f_{3}(T) \tag{15}\]  
where \(f_{1}\) is the coefficient \(5\times 10^{-8}\), which depends on the concrete composition; \(C_{f-cl}\) is the free chloride concentration (%); \(f_{2}\) is a factor that accounts for the effect of the aging of concrete, which is due to hydration reactions in cement; and \(f_{3}\) is the factor of temperature's effect. \[f_{2}(t)=\frac{4}{t} \tag{16}\]  
\[f_{3}(T)=Exp\Bigg{\{}0.1\times\frac{U}{R}\times(\frac{1}{T_{ref}}-\frac{1}{T}) \Bigg{\}} \tag{17}\]  
\[\{\hat{\mathbb{C}}_{ij}\}=\{\hat{\mathbb{C}}_{i1},\hat{\mathbb{C}}_{i2},\hat{ \mathbb{C}}_{i3},\cdots\cdots\cdots\cdots\hat{\mathbb{C}}_{in}\} \tag{26}\]  
The nodal variables of relative humidity, ion concentrations, and temperature are determined by substituting the approximated values from Equations (20)-(22) into the governing Equations (9)-(11). The [PERSON] method is then applied to the weak forms, which leads to the finite element matrix as presented in Equation (27):  
\[\frac{d}{dt}\left(\left[C_{e}\hat{\left(\hat{\phi}\right)}\right]\left\{\hat{ \phi}\right\}\right)=\left[K_{e}\hat{\left(\hat{\phi}\right)}\right]\left\{ \hat{\phi}\right\} \tag{27}\]  
where the element matrices and the variable vector are given by Equations (28)-(30) and the element matrices and the variable vector are given in Appendix A. \[\left[C_{e}\right]= \begin{bmatrix}C_{T}&0&0\\ 0&C_{h}&0\\ 0&0&C_{i}\end{bmatrix}  \tag{28}\]  
\[\left[K_{e}\right]= \begin{bmatrix}K_{TT}&K_{Th}&K_{Ti}\\ K_{hT}&K_{hh}&K_{hi}\\ K_{iT}&K_{ih}&K_{ii}\end{bmatrix}