## 2 Multi-Scale Moisture Diffusion Coefficient Model of Concrete
### Effective Moisture Diffusion Coefficient of Cement Paste
Based on Equation (1), for an arbitrary cross-section with an area of \(\Omega\), when the humidity gradient is \(\
abla RH\), the mass pass through all the pores with a pore size of \(d_{p}\) in unit time can be expressed as follows:  
\[Q\big{(}d_{p}\big{)}=D_{fk}\big{(}d_{p}\big{)}A\big{(}d_{p}\big{)}\
abla RH \tag{7}\]  
where \(A\big{(}d_{p}\big{)}\) is the total pore area with pore size \(d_{p}\). Let the relation between the cumulative porosity of cement paste and pore size be \(\phi_{c}=\phi_{c}\big{(}d_{p}\big{)}\), then the total pore area with pore size of \(d_{p}\) can be expressed as follows:  
\[A\big{(}d_{p}\big{)}=\frac{\partial\phi_{c}(d_{p})}{\partial d_{p}}\Omega \tag{8}\]  
According to the conservation of mass in the transmission process, the total mass passing through the cross section per unit time can be expressed as follows:  
\[Q=\sum_{d_{p-min}}^{d_{p-max}}D_{fk}\big{(}d_{pi}\big{)}\Omega\
abla RH\frac {\partial\phi_{c}\big{(}d_{p}\big{)}}{\partial d_{p}} \tag{9}\]  
where \(d_{p-min}\) and \(d_{p-max}\) are the minimum pore size and the maximum pore size in the cement paste, m.  
Therefore, when considering the influence of Knudsen diffusion, the effective moisture diffusion coefficient \(D_{ce}\) of the cross section can be expressed as follows:  
\[D_{ce}=\frac{Q}{\Omega\
abla RH}=\sum_{d_{p-min}}^{d_{p-max}}D_{fk}\big{(}d_{ pi}\big{)}\frac{\partial\phi_{c}\big{(}d_{p}\big{)}}{\partial d_{p}} \tag{10}\]Similarly, the effective moisture diffusion coefficient of the cross section without considering Knudsen diffusion can be expressed as follows:  
\[D_{c00}=\sum_{d_{p-min}}^{d_{p-max}}\frac{\lambda\overline{v}}{3}.\frac{\partial \phi_{c}\big{(}d_{p}\big{)}}{\partial d_{p}} \tag{11}\]  
Then, the Knudsen diffusion influence coefficient \(k_{f}\) can be obtained as follows:  
\[k_{f}=\frac{D_{ce}}{D_{ce0}} \tag{12}\]  
It should be noticed that Equation (11) is the diffusion coefficient in an ideal state within the section \(\Omega\) with a certain free path \(\lambda\). Though it seems that Equation (11) can consider the effect of RH while the \(\lambda\) has considered the effect of RH via Equations (4) and (6), the \(D_{c00}\) cannot be equal to \(D_{cm0}\) directly. This is because the \(D_{ce0}\) lacks consideration for hindered diffusion (in the real moisture diffusion process) and different shapes of pores (\"bottle neck\" effect and \"open-pore\" effect) [29]. But, Equation (12) is still valid as the \(k_{f}\) in Equation (12) is a relative value with the same ideal state assuming of \(D_{ce}\) and \(D_{ce0}\). For the \(D_{cm0}\), since the moisture diffusion coefficient of porous material varies with RH, the moisture diffusion coefficient of cement paste can be described an S-shaped curve [19]:  
\[D_{cm0}=D_{cm0-max}\Big{[}1+(7.5-7.5 RH)^{4}\Big{]}^{-1} \tag{13}\]  
where the \(D_{cm0-max}\) is the maximum moisture diffusion coefficient of cement paste when RH = 1, and it can be approached via Equation (11). Though Equation (11) is the diffusion coefficient in an ideal state within the section \(\Omega\) with a certain RH, the \(D_{cm0-max}\) can be calibrated by using \(D_{ce0-max}\) (the maximum of \(D_{ce0}\) when RH = 1) due to the quasi-steady state of the local thermodynamic equilibrium [24]. Meanwhile, since the pores in cement paste are tortuous with different shapes, the diffusion path will also be tortuous by the pores. Then, the \(D_{ce0-max}\) for the spatial domain can be modified with the pore tortuosity \(\tau\) (which can be defined as a function of porosity \(\tau=1-\sqrt{3}\ln\phi_{c}\)[30]), and the \(D_{cm0-max}\) can be expressed as follows:  
\[D_{cm0-max}=\frac{D_{ce0-max}}{\tau} \tag{14}\]