## 2 Multi-Scale Moisture Diffusion Coefficient Model of Concrete
### Effective Moisture Diffusion Coefficient of Mortar and Concrete  
Since concrete is a composite binding material, the equivalent diffusion coefficient is essential for engineering applications, and the equivalent moisture diffusion coefficient of composite porous media can be determined by the Maxwell and self-consistent method [31]. [PERSON] assumes that the particles in the inclusion phase are far enough away from each other and randomly distributed in the continuous phase, which is suitable for the case of the low dispersed phase [31]. The self-consistent method can better estimate the effective properties of the composite material embedded in the matrix, but there will be a large deviation when the volume of the inclusion phase is greater than \(1/3\)[31]. If the mortar is regarded as a two-phase composite material composed of fine aggregate (inclusion phase) and cement paste (continuous phase), according to the self-consistent method, the moisture diffusion coefficient relationship between the composite material and the inclusion phase and the continuous phase can be expressed as follows:  
\[\frac{D_{\alpha}-D_{eq}}{D_{\alpha}+2D_{eq}}=\frac{\text{V}_{\beta}}{1-\text {V}_{\beta}}\frac{D_{eq}-D_{\beta}}{D_{\beta}+2D_{eq}} \tag{15}\]where \(D_{\alpha}\) is the moisture diffusion coefficient of the continuous phase, \(D_{\beta}\) is the moisture diffusion coefficient of the inclusion phase, \(D_{\alpha\beta}\) is the equivalent moisture diffusion coefficient of composite materials, and \(\mathrm{V}_{\beta}\) is the volume fraction of the inclusion phase. Considering that the volume fraction of mortar fine aggregate is greater than \(1/3\) in practical engineering, the \"mortar\" containing \(1/3\) volume fraction of fine aggregate can be regarded as a continuous phase, and the remaining fine aggregate can be regarded as an inclusion phase. The self-consistent equation is iterated again to obtain the moisture diffusion coefficient at a higher volume fraction of fine aggregate. Assume that the equivalent moisture diffusion coefficient in Equation (15) is \(D_{\alpha\beta}=D_{\alpha\alpha}\big{(}\mathrm{V}_{\beta},D_{\alpha},D_{\beta} \big{)}\), then the effective moisture diffusion coefficient \(D_{m}\) of mortar with different volume fractions of fine aggregate can be expressed as follows:  
\[D_{m}=\left\{ \begin{array}{ll}D_{\alpha\beta}\big{(}\mathrm{V}_{\beta},D_{ \alpha},D_{\beta}\big{)},&0<\mathrm{V}_{\beta}\leq 0.33\\ D_{\alpha\beta}\big{(}\mathrm{V}_{\beta 1},D_{\alpha 1},D_{\beta}\big{)},&0.33< \mathrm{V}_{\beta}\leq 0.56\\ D_{\alpha\beta}\big{(}\mathrm{V}_{\beta 2},D_{\alpha 2},D_{\beta}\big{)},&0.56< \mathrm{V}_{\beta}\leq 0.70\\ D_{\alpha\beta}\big{(}\mathrm{V}_{\beta 3},D_{\alpha 3},D_{\beta}\big{)},&0.70< \mathrm{V}_{\beta}\leq 0.80\end{array} \right. \tag{16}\]  
where \(D_{\alpha 1}\), \(D_{\alpha 2}\), and \(D_{\alpha 3}\) are the moisture diffusion coefficients of the continuous phase in the three iterations; \(\mathrm{V}_{\beta 1}\), \(\mathrm{V}_{\beta 2}\), and \(\mathrm{V}_{\beta 3}\) are the residual volume fractions of inclusion phase in three iterations. Concrete is usually regarded as a three-phase composite material composed of mortar, coarse aggregate, and ITZ, but the Maxwell and self-consistent method are equivalent methods of two-phase composite materials. In order to deduce the effective moisture diffusion coefficient of concrete, ITZ and aggregate can be regarded as a \"composite\" structure, and then, concrete can be analyzed as a two-phase composite material composed of mortar and the \"composite\", as shown in Figure 1.  
Since the volume fraction of coarse aggregate in the \"composite\" is much larger than \(1/3\), if the self-consistent method is used, a large deviation will be generated.