# Predicting Dissolution Kinetics of Tricalcium Silicate Using Deep Learning and Analytical Models
## 5 Analytical Model Development
Thus, a high calcium ion activity leads to an equilibrium for the C\({}_{3}\)S dissolution, resulting in a slower dissolution rate compared to a solvent without calcium ions [6; 70]. Similarly, a basic solvent significantly decreases the dissolution rate of C\({}_{3}\)S by containing a large amount of hydroxide ions. In this study, only H\({}_{2}\)SiO\({}_{4}\)\({}^{2-}\) was considered in the _IAP_ calculation because H\({}_{4}\)SiO\({}_{4}\) and H\({}_{3}\)SiO\({}_{4}\)\({}^{-}\) can deprotonate to form H\({}_{2}\)SiO\({}_{4}\)\({}^{-}\)[71; 72]. To clearly observe the influence of _IAP_ on dissolution rate, Figure 3 shows the correlation between the degree of undersaturation (_IAP/K\({}_{sp}\)_) and the dissolution rate of C\({}_{3}\)S. The general trend of the correlation and order of magnitude of changes in the dissolution rate observed herein are in good agreement with previous studies [6; 28]. It is not surprising that the dissolution rate of C\({}_{3}\)S decreases as the degree of undersaturation increases, as a high degree of undersaturation indicates that the solution is approaching an equilibrium, which reduces the driving force for dissolution. As previously discussed in the introduction, [PERSON] [29] has demonstrated that H\({}^{+}\) and OH\({}^{-}\) leach mineral surfaces in different activation energies. Since [PERSON]'s model only accounts for the H\({}^{+}\), the new model includes the ion activities of both H\({}^{+}\) and OH\({}^{-}\) in order to interpret the leaching process. Moreover, especially for C\({}_{3}\)S dissolution, OH\({}^{-}\) is one of the main products of the dissolution reaction, as shown in Equation (2). Previous studies [32; 33; 34; 35; 36] have also shown that the concentration of major cations (excluding H\({}^{+}\)) in the solvent can influence the dissolution rate, and this is supported by the data shown in Figure 2, which highlights the importance of Ca concentration in the analytical model. However, previous studies have not explored the relationship between the activity of Ca\({}^{2+}\) and the dissolution rate of C\({}_{3}\)S. Using data from [PERSON] et al. [37; 38], we show this relationship in Figure 4, which plots the natural logarithm of the dissolution rate of C\({}_{3}\)S against the initial activity of Ca\({}^{2+}\). The correlation is observed as linear (shown as the red line). This means the relationship between C\({}_{3}\)S dissolution rate and Ca\({}^{2+}\) activity is exponential. Some outliers can be seen in the Figure, which may be due to the influence of other parameters, such as temperature and the specific surface area of C\({}_{3}\)S, on the dissolution rate. If all other parameters are kept constant, a more ideal linear relationship should be observed. After embodying OH\({}^{-}\) and Ca\({}^{2+}\), the new analytical model, with seven input variables, is formed as Equation (4). Here, \(C_{i}\) is the coefficient for each attribute; \(T\) is Temperature (\({}^{\circ}\)C); \(A\) is the specific surface area of C\({}_{3}\)S (m\({}^{2}\)/g); \(a_{ij}\) is ion activity of \(i\) species at initial/final state (unitless); \(I\) is ion strength of initial state (mM); \(IAP\) is ion activity product of final state (unitless); \(K_{sp}\) is C\({}_{3}\)S solubility product (\(\approx\)10\({}^{-17.65}\)[8; 69]). \[rate=e^{C_{0}}*e^{\frac{C_{1}}{T}}*A^{C_{2}}*e^{C_{3}p_{C},initial}*O_{OH, initial}*a_{H,initial}^{C_{5}}*I^{C_{6}}*\left(\frac{IAP}{K_{sp}}\right)^{C_{7}} \tag{4}\]  
Figure 3: The correlation between the degree of undersaturation (_IAP/K\({}_{sp}\)_) and the dissolution rate of C\({}_{3}\)S. The x-axis shows in a logarithmic scale due to the small magnitude of the degree of undersaturation. Table 6 shows the optimal coefficients of the analytical model for _alkaline_ solvent scenario. Predicted results of the C\({}_{3}\)S dissolution rate, as produced by the analytical model based on the coefficient in Table 6, are demonstrated in Figure 5b.