# Chemo-Mechanical Model for the Expansion of Concrete due to Alkali Silica Reaction
## 2 Mathematical Model for the Chemical Reaction of ASR
### Formation of the Basic Gel  
In the model proposed in Reference [5], the advance of the reaction front of ASR in a reactive aggregate was driven by water diffusion. A part of the reacted aggregate was modeled as the ASR gel [17; 18]. This would result in the same volume of reacted aggregate regardless of the alkalinity. This means that the reactive aggregate may fully react without producing the ASR gel in an asymptotically extreme case, such as a very low alkaline environment. In this case, even if high alkaline environment is supplied later, no more reactive aggregate is left. In addition, alkali release from aggregates is not considered. However, the dissolution rate of silica in reactive aggregates depends on the alkalinity of the solution [35]. The advance of the reaction front in the reactive aggregate would be better described by means of the alkali content in the pore solution, rather than the water [6; 36]. It is better to consider the transport of alkali explicitly because alkali concentration may be influenced by internal alkali release from reactive aggregates, and external alkali supplies, too. With a given reactive aggregate, the reaction between alkali and silica in the reactive aggregate is governed by the diffusion of alkali from the pore solution. To simplify the calculation of the advance of  
Figure 1: Reaction front of alkali silica reaction (ASR) and alkali-recycling front in a representative reactive aggregate. the reaction front, a steady state diffusion process of the alkali is assumed as in the literature [5; 17; 18]. Then the concentration of free alkali in the corresponding reacted layer, \(C_{rf}\) is given as  
\[C_{rf}=C_{f}\frac{1-z_{r}/x}{1-z_{r}/R}\ \ \ \ (z_{r}\leq x\leq R), \tag{1}\]  
where \(C_{f}\) is the concentration of free alkali in the pore solution (mol/m\({}^{3}\) of pore solution), \(R\) is the radius of the reactive aggregate, and \(z_{r}\) is the reaction front of ASR with initial value \(z_{r0}\). The distribution of free alkali is shown in Figure 2a schematically. By assuming the alkali released at the reaction front will be consumed instantly in situ, the mass balance of alkali at the reaction front will be given as  
\[(\beta_{as}\beta_{s}-2\rho_{a})dz_{r}=-D_{af}\frac{\partial C_{rf}}{\partial x }dt, \tag{2}\]  
where \(\beta_{as}=0.5\) is the alkali-to-silica stoichiometric mass conversion ratio, \(\rho_{s}=\rho_{s}/m_{s}\) with the mass density of reactive SiO\({}_{2}\) inside the aggregate \(\rho_{s}\), and the molar weight of SiO\({}_{2}\)\(m_{s}=60.08\) g/mol. \(\rho_{a}=\rho_{a}/m_{a}\) with the mass of releasable alkali per volume of aggregate \(\rho_{a}\), mainly Na\({}_{2}\)O\({}_{e}\), and the molar weight of Na\({}_{2}\)O \(m_{a}=62\) g/mol, \(D_{af}\) is the diffusivity of alkali into the aggregate which depends on temperature and humidity as given in Appendix A. Note that the reactive silica is only a part of the total silica content and depends on the type of aggregates. The consumption of the reactive silica is shown in Figure 2b at different times schematically. The movement speed of ASR reaction front can be determined by substituting \(C_{rf}\), the free alkali Equation (1) into Equation (2)  
\[\dot{z_{r}}=-\frac{D_{af}C_{f}}{z_{r}(1-z_{r}/R)(\beta_{as}\beta_{s}-2\rho_{a} )}. \tag{3}\]  
The growth of the basic ASR gel is shown in Figure 2c. The thickness of the C-S-H layer formed instantaneously by the reaction of the dissolved silica and calcium was determined as \(\delta_{csh}=1.05\) um following the method proposed by Reference [37]. This layer is not considered to imbibe water. The more the calcium is penetrated into the gel layer from the pore solution, the more the basic gel will be converted into C-S-H [1; 33].