# Chemo-Mechanical Model for the Expansion of Concrete due to Alkali Silica Reaction
## 2 Mathematical Model for the Chemical Reaction of ASR
### Swelling of the Basic Gel  
Swelling of the basic gel by water imbibition may be given by References [5; 18].  
\[\frac{\partial w_{i}}{\partial t} = \frac{A_{i}}{\tau_{i}}, \tag{9}\] \[A_{i} = k_{i}m_{bg}-w_{i},\] (10) \[\tau_{i} = \frac{\delta_{w}^{2}}{D_{wi}}, \tag{11}\]  
where \(A_{i}\) is the thermodynamic affinity of the water imbibition, which is the imbalance between the imbibed water at a thermodynamic equilibrium and the current imbibed water; \(k_{i}=0.22\) is an empirical constant for water imbibition at the thermodynamic equilibrium; \(\tau_{i}\) is the characteristic time of the water imbibition; \(\delta_{w}=R-z_{r}\) is the distance to reach the gel in the water transport process which increases with the advance of the reaction front; and \(D_{wi}\) is the diffusivity of water for the micro-diffusion near the aggregate, which is a nonlinear function of temperature and the relative humidity that is given in Appendix A.  
Figure 2: Schematic description of the progress of phases distribution; (**a**) free alkali, (**b**) reactive silica, (**c**) basic ASR gel (or ASR reaction front), (**d**) free calcium, (**e**) C-S-H gel by alkali-recycling, and (**f**) distribution of different phases at time \(t\).  
\(k_{i}m_{bg}\) in Equation (10) represents the maximum amount of water imbibition by the basic gel. The maximum water imbibition must depend on the expansibility of the gel. This is strongly influenced by the chemical composition of the gel, specifically, the ratio of Na(K)/Si and Ca/Si, etc., in the gel [23; 24]. As the concentration of Ca increases and that of Na(K) decreases, the gel will behave more like C-S-H gel [38]. In such a condition, the expansibility of the ASR product is low, or negligible. Therefore, the thermodynamic affinity of Equation (10) is corrected by the expansibility of the gel as given by  
\[A_{i}=f_{ac}k_{i}m_{bg}-w_{i}, \tag{12}\]  
where \(f_{ac}\) is the expansibility of the gel depending on its chemical composition.  
Because of the complicated nature of the alkali silica reaction, it is difficult to capture the exact composition of the gel at different positions even with numerical simulations. We adopt [PERSON] and [PERSON]'s [20] proposal to estimate the expansibility of the gel, which is proportional the ratio of sodium to calcium in the pore solution. The expansibility of the gel is given by  
\[f_{ac}=\frac{\langle\rho-\rho_{T}\rangle^{+}}{\rho_{0}-\rho_{T}}, \tag{13}\]  
where \(\rho=[Na^{+}]/[Ca^{2+}]\) is the ratio of sodium to calcium in the pore solution, \(\rho_{T}\) is the threshold value for the expansibility which depends on temperature. By referring to the experiment data [39], \(\rho_{T}\) may be given by \(\rho_{T}=\rho_{T_{ref}}\exp[\frac{U}{R}(\frac{1}{T_{ref}}-\frac{1}{T})]\), with \(\rho_{T_{ref}}=198.62\), \(T_{ref}=293.15\) K, and \(U=-7.8\) kJ/mol. \(<X>^{+}\) is equal to \(X\) if \(X>0\) or equal to \(0\) if \(X\leq 0\). \(\rho_{0}\) is the ratio of sodium to calcium when the silica is most reactive, i.e., when the alkali charge-balanced by hydroxide ions is \(1\) mol/L as reported in Reference [35]. It is calculated based on the calcium concentration, which is temperature dependent, as given in Appendix B.  
We consider sodium as alkali in this model. To calculate the ratio of sodium to calcium, both alkalis charge-balanced by hydroxide ions and charge-balanced by other anions, such as chloride ions, are considered. To simplify the calculation, the relative proportion of the alkali contents in the pore solution charge-balanced by different anions were assumed to keep the same proportion during the diffusion of alkalis because of the similarity of diffusion coefficients of the anions that charge-balanced with the alkalis [40; 41]. Another reason of the simplification is the complexities for describing the multi-species ionic diffusion in the pore solution of concrete correctly [41; 42], which is too complicated within the frame of this work.