# Chemo-Mechanical Model for the Expansion of Concrete due to Alkali Silica Reaction
## 3 Mass Transport at Macro-Scale  
### Transport of Alkali  
Alkalis in the reactive aggregates are released gradually as the silica dissolves [1; 43]. While the exact mechanism of the alkali release is not completely understood yet, it is believed that the release process mainly depends on the ion exchange reaction between sodium from siliceous aggregate and calcium from the pore solution [1; 30]. The released amount is related to the volume of the reaction layer of the aggregate. Since the ASR reaction is much slower than hydration, it is assumed that all the alkalis in cement was released into the pore solution, for a simplification.  
The total concentration of alkali per volume of concrete includes the free alkali in the pore solution, the alkali in the reactive aggregates, and the alkali bound in the ASR gel, i.e.,  
\[C_{t}=\phi_{w_{e}}C_{f}+C_{gg}+C_{b}, \tag{14}\]  
where \(C_{t}\) is the total alkali content in a unit volume of concrete (mol/m\({}^{3}\) of concrete), \(\phi_{w_{e}}\) is the volume fraction of evaporable pore water (m\({}^{3}\) solution/ m\({}^{3}\) concrete), \(C_{f}\) is the free alkali content in the pore solution (mol/m\({}^{3}\) of pore solution), \(C_{ag}\) is the alkali content in the aggregate which will decrease due to its release (mol/m\({}^{3}\) of concrete), and \(C_{b}\) is the alkali content bound in the ASR gel (mol/m\({}^{3}\) of concrete). The relation between \(\phi_{w_{e}}\) and humidity (\(h\)) could be obtained according to the model developed by [PERSON], [PERSON], and [PERSON] (BSB model) [44], given in Appendix A.  
The alkali content in the aggregate, \(C_{ag}\), will decrease due to release caused by dissolution. The alkali content bound in the ASR gel, \(C_{b}\), will increase due to the gel formation and decrease due to calcium recycling. It depends on the reaction front of ASR and alkali-recycling front as given in Equations (3) and (6). They are expressed as  
\[C_{ag} = N_{a}\frac{4}{3}\pi z_{7}^{3}2\bar{\rho_{a}}, \tag{15}\] \[C_{b} = N_{a}\frac{4}{3}\pi(z_{c}^{3}-z_{7}^{3})\bar{\rho_{s}}\beta_{as}, \tag{16}\]  
where \(N_{a}\) is the number of reactive aggregates per volume of concrete (1/m\({}^{3}\) of concrete), which is calculated by dividing the total mass of the aggregates by the mass of the representative aggregate, \(\bar{\rho_{a}}=\rho_{a}/m_{a}\), \(\bar{\rho_{s}}=\rho_{s}/m_{s}\), and \(\beta_{as}=0.5\) is the alkali-to-silica stoichiometric mass conversion ratio.  
The momentary rate of the total alkali content can be calculated as  
\[\frac{\partial C_{t}}{\partial t}=\frac{\partial\phi_{w_{e}}C_{f}}{\partial t }+\dot{C}_{ag}+\dot{C}_{b}. \tag{17}\]  
The alkali mass balance governing equation is given by  
\[\frac{\partial C_{t}}{\partial t}=\mathbf{\
abla}\cdot(D_{f}\mathbf{\
abla}C_{f}), \tag{18}\]  
where \(D_{f}\) is the diffusion coefficient of free alkali, which is a nonlinear function of temperature and relative humidity, as given in Appendix A.  
By combining Equations (17) and (18), the alkali mass balance governing equation can be obtained and simplified as  
\[\frac{\partial\phi_{w_{e}}C_{f}}{\partial t}-\mathbf{\
abla}\cdot(D_ {f}\mathbf{\
abla}C_{f})=I_{agg}-I_{gel}, \tag{19}\] \[I_{agg}=-\dot{C}_{ag}=-8\pi\bar{\rho_{a}}N_{a}z_{7}^{2}z_{r},\] (20) \[I_{gel}=\dot{C}_{b}=4\pi\beta_{as}\bar{\rho_{s}}N_{a}z_{c}^{2} \dot{z}_{c}-4\pi\beta_{as}\bar{\rho_{s}}N_{a}z_{7}^{2}\dot{z}_{r}, \tag{21}\]  
where \(I_{agg}\) being the term representing the alkali released from aggregate, and \(I_{gel}\) being the sink term reflecting alkali fixed in the gel. The first term of \(I_{gel}\) means the alkali recycling rate by calcium and the second means the binding rate due to formation of the new basic gel.