# Predicting Dissolution Kinetics of Tricalcium Silicate Using Deep Learning and Analytical Models
## 5 Analytical Model Development  
The abovementioned results demonstrate that the DF model can produce predictions of the dissolution rate of C\({}_{3}\)S in a high-fidelity manner. However, the use of machine learning (ML) techniques can have some limitations, such as the \"black-box\" issue, where the underlying input-output correlations learned by the model are difficult to interpret. Additionally, ML models may not be accessible to end users who do not have a programming background. To address these issues, this section introduces an original, closed-form analytical model that has been distilled from the DF model. This model can be used to predict the dissolution rate of C\({}_{3}\)S and provide a better understanding of the input-output correlations involved. The development of a reliable analytical model involves a wise selection of input variables. The inclusion of influential variables is vital to enhance the performance of the analytical model. Simultaneously, the exclusion of inconsequential variables reduces the complexity of the model. The new analytical model is developed based on [PERSON]'s model [PERSON] (1983), and some new input variables are added to it. We selected [PERSON]'s model as the baseline model because it is the most used model to predict C\({}_{3}\)S dissolution kinetics. This model accounts for the SSA of C\({}_{3}\)S, solvent pH, temperature, and ions in solvents. The feature importance, shown in Figure 2, also confirms that those parameters dominated the dissolution rate of the C\({}_{3}\)S. It is worth pointing out that only data from [PERSON] et al. [PERSON] et al. (2017); [PERSON] et al. (2017) was employed to develop the analytical model. This is because the SSA of C\({}_{3}\)S is not applicable in [PERSON] and Gallucci [PERSON] and [PERSON] (2018). In the baseline model, the Gibbs free energy of the overall reaction is one of the major influential variables. To properly quantify this variable, it is important to understand the dissolution mechanism of C\({}_{3}\)S. The dissolution process of C\({}_{3}\)S can be considered as an inverse nucleation process [PERSON] (1973), which is controlled by two major factors: interfacial properties and the driving force. The interfacial properties include chemical composition, chemical bond, surface defects, and impurities in crystals. Generally, the dissolution process can be divided into three steps: (1) horizontal movement at the atomic scale to form a 2D vacancy; (2) etch pit formation at dislocation; and (3) step retreat at pre-existing roughness [PERSON] (1973); [PERSON] (1973). The driving force of the C\({}_{3}\)S dissolution reaction is defined as the energy to overcome the activation energy barriers for the first two steps of the dissolution process. The equation to calculate the driving force is shown in Equation (1) [PERSON] (1973); [PERSON] (1973):  
\[\sigma=\frac{\Delta\mu}{kT}=\frac{\Delta G^{*}}{RT}=\ln\biggl{(}\frac{IAP}{K_{ SP}}\biggr{)} \tag{1}\]  
Here, \(\sigma\) is the undersaturation coefficient; \(\Delta\mu\) is the difference in chemical potential; \(k\) is the Boltzmann constant; \(T\) is the temperature; \(\Delta G^{*}\) is the free energy difference between the undersaturated solution and the solution in equilibrium; \(R\) is the gas constant; \(IAP\) is the ion activity product to reactant species; and \(K_{SP}\) is the mineral solubility products. The dissolution reaction of C\({}_{3}\)S is expressed in Equation (2) [(8)], and the \(IAP\) is defined in Equation (3). a\({}_{\rm i}\) is the ion activity of species \(i\). The chemical equilibrium constant (\(K_{SP}\)) for C\({}_{3}\)S dissolution has been estimated as \(10^{-17.65}\)[PERSON] (1973); [PERSON] (1973). \[(CaO)_{3}SiO_{2}+3H_{2}O\to 3 Ca^{2+}+H_{2}SiO_{4}^{2-}+4 OH^{-} \tag{2}\]  
\[IAP=a_{{\rm C}\alpha^{2+}}^{3}.a_{OH^{-}}^{4}.a_{H_{2}SO_{4}^{2-}} \tag{3}\]  
Equation (3) suggests that the value of \(IAP\) is solely determined by calcium and hydroxide ion activity.