## 2 Multi-Scale Moisture Diffusion Coefficient Model of Concrete
### Effective Moisture Diffusion Coefficient of Cement Paste  
According to the reference [17], the moisture migration in cement-based materials is affected by both molecular free diffusion and Knudsen diffusion. Therefore, the moisture diffusion coefficient needs to consider the influence of Knudsen diffusion on the basis of molecular diffusion. When considering the influence of Knudsen diffusion, the moisture diffusion coefficient of cement-based materials can be expressed as follows:  
\[D_{cm}=k_{f}D_{cm0} \tag{2}\]  
where \(D_{cm}\) is the effective moisture diffusion coefficient of cement-based materials considering Knudsen diffusion, m\({}^{2}\)/s; \(D_{cm0}\) is the moisture diffusion coefficient without considering Knudsen diffusion; and \(k_{f}\) is the Knudsen diffusion influence coefficient. It is also pointed out in reference [17] that Knudsen diffusion should be considered only when the humidity changes phase, and the value of \(k_{f}\) is related to the average pore size of cement-based materials. However, in essence, due to the wide range of pore size distribution of concrete, there is an order of magnitude difference between the maximum pore size and the minimum pore size, so the water migration on the micro scale needs to comprehensively consider the influence of pore structure. The migration of water in concrete pores is affected by the ratio \(r\) of the average free path \(\lambda\) of water molecules to the pore diameter \(d_{p}\), and there are two forms of free diffusion and Knudsen diffusion. When \(r\leq 0.01\), it is free diffusion; when \(r\geq 10\), it is Knudsen diffusion, that is, the collision between water molecules and pore walls determines the diffusion of molecules; and when the ratio of the two is between 0.01 and 10, it is mixed diffusion. For a single pore, the diffusion coefficient \(D_{fx}\) can be expressed as follows [26]:  
\[D_{fx}= \begin{cases}\frac{\lambda\overline{r}}{3}&,r\leq 0.01\\ \frac{\overline{r}_{3}}{3}\big{[}1-Exp(\frac{-d_{p}}{\lambda})\big{]},0.01<r<1 0\\ \frac{d_{p}\overline{r}}{3}&,r\geq 10\end{cases}  \tag{3}\]  
where \(\overline{v}\) is the average velocity of water molecules relative to air molecules, m/s. Assuming that the mixed gas in the pores is an ideal gas composed of air and water vapor, the free path \(\lambda\) and the average motion rate \(\lambda\) of the binary mixed gas can be expressed as follows [27]:  
\[\lambda=\frac{k_{b}(T+273)}{\sqrt{2}\pi P_{w}d_{w}^{2}+\pi P_{A}\Big{(}\frac{d_{w }+d_{A}}{2}\Big{)}^{2}\sqrt{1+\frac{m_{w}}{m_{A}}}} \tag{4}\]  
\[\overline{v}=\sqrt{\frac{8R(T+273)(m_{A}+m_{w})}{\pi m_{w}m_{A}}} \tag{5}\]  
where \(k_{b}\) is the Boltzmann constant, 1.380649 \(\times\) 10\({}^{-23}\) J/K; T is the temperature, \({}^{\circ}\)C; \(d_{w}\) is the diameter of water molecule, 4 \(\times\) 10\({}^{-10}\) m; \(d_{A}\) is the effective diameter of air molecules, 3.5 \(\times\) 10\({}^{-10}\) m; \(m_{w}\) is the molecular weight of water, 18 g/mol; \(m_{A}\) is the effective molecular weight of air, 29 g/mol; \(R\) is the perfect gas constant, 8.314 m\({}^{3}\)Pa/(mol\({}^{\circ}\)C); and \(P_{w}\) and \(P_{A}\) are the partial pressures of water vapor and air [28], Pa.  
\[\left\{ \begin{aligned} P_{w}=610 RH\cdot Exp\left(\frac{17,3T}{237.3+T}\right)\\ P_{A}=P_{atom}-P_{w}\end{aligned} \right. \tag{6}\]  
where \(RH\) is the relative humidity; \(P_{atom}\) is the standard atmospheric pressure, 100 kPa. According to Equations (3)-(6), even if there is no phase change, the influence of Knudsen diffusion still exists in the mixed diffusion stage, and the pore size distribution of cement paste also determines the influence of Knudsen diffusion to a certain extent.