## 3 Thermodynamic Modelling of Cement Hydration
### Discrete Solid Phases (DSP) Derivation for M-S-H and Si-Hydroagmet  
The M-S-H series of DSP was defined using the pure end-members and their thermodynamic properties in terms of Gibbs free energy (G, J/mol), enthalpy (H, J/mol), entropy (S, J/K/mol), heat capacity (Cp, J/K/mol), and molar volume (V, cm\({}^{3}\)/mol) (see Table 3) [21] and used to calculate the solubility constant, log K, and its variation with temperature for dissolution reactions of each end-member (Table 4) using Equation (1). The a, c, and d co-efficients are calculated as functions of enthalpy (\(\Delta\)rH, J/mol), entropy (\(\Delta\)rS, J/K/mol), and heat capacity (\(\Delta\)rCp, J/K/mol) of the reaction [13]. The b, e, and f co-efficients are equal to zero.  
\[\log\mathrm{K}=\mathrm{a}+\mathrm{b}\cdot\mathrm{T}+\mathrm{c}/\mathrm{T}+ \mathrm{d}\cdot\log_{10}(\mathrm{T})+\mathrm{e}\cdot\mathrm{T}^{-2}+\mathrm{f} \cdot\mathrm{  
A similar methodology was applied to derive the DSP for the siliceous hydrogarnet, as described in [25] using the thermodynamic data in Tables 6 and 7 resulting in the phases shown in Table 8.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Component** & **G**\({}^{\text{i}}\) & **H**\({}^{\text{i}}\) & **S**\({}^{\text{ii}}\) & **Cp**\({}^{\text{ii}}\) & **V**\({}^{\text{iii}}\) \\ \hline M075 SH & \(-\)3,218,427 & \(-\)3,505,729 & 270.26099 & 318 & 95.00 \\ \hline M15 SH & \(-\)2,355,660 & \(-\)2,594,220 & 216 & 250 & 74.00 \\ \hline Mg\({}^{+2}\) & \(-\)453,990 & \(-\)465,930 & \(-\)138.07 & \(-\)21.66 & \\ \hline SiO\({}_{2}\) & \(-\)833,411 & \(-\)887,856.2 & 41.338 & 44.4654 & \\ \hline H\({}_{2}\)O & \(-\)237,183 & \(-\)285,881 & 69.923 & 75.3605 & \\ \hline OH\({}^{-}\) & \(-\)157,270 & \(-\)230,010 & \(-\)10.71 & \(-\)136.34 & \\ \hline \hline \end{tabular} \({}^{\text{i}}\)/mol. \({}^{\text{ii}}\)1/mol/K. \({}^{\text{iii}}\)cm\({}^{3}\)/mol.  
\end{table}
Table 3: Thermodynamic properties of M-S-H as taken from [25].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**M075 SH** \\ \hline (MgO)\({}_{1.5}\) (SiO)\({}_{2}\) (H\({}_{2}\)O)\({}_{2.5}\) \(\rightarrow\) 1.5 Mg\({}^{+2}\) + 2 SiO\({}_{2}\) + 3 OH\({}^{-}\) + 1 H\({}_{2}\)O \\ log K \(-\)28.80 & \\ \hline analytical\_expression parameters \({}^{\text{i}}\) \\ \hline \hline \(a\) = 189.588043, \(c\) = \(-\)12,153.394331, \(d\) = \(-\)71.58795 \\ \hline
**M15 SH** \\ \hline (MgO)\({}_{1.5}\) SiO\({}_{2}\) (H\({}_{2}\)O)\({}_{2.5}\) \(\rightarrow\) 1.5 Mg\({}^{+2}\) + SiO\({}_{2}\) + 3 OH\({}^{-}\) + 1 H\({}_{2}\)O \\ log K \(-\)23.57 & \\ \hline analytical\_expression parameters \({}^{\text{i}}\) \\ \hline \hline \(a\) = 182.029762, \(c\) = \(-\)10,551.415516, \(d\) = \(-\)68.75740 \\ \hline \hline \({}^{\text{i}}\)\(a\) = (\(\Delta_{\text{r}}S-\Delta_{\text{r}}\)C\({}_{\text{p}}\)-(1 + LN(298.15)))/(R-LN(101)), \(b\) = \(c\) = (298.15\(\cdot\Delta_{\text{r}}\)C\({}_{\text{p}}\) – \(\Delta_{\text{r}}\)H)/(R-LN(101)), \(d\) = \(\Delta_{\text{r}}\)Cp/R, \(e\) = 0, and \(f\) = 0, where R is the gas constant (8.31451 /mol/K).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Phase**} & \multicolumn{2}{c}{**Mode Fractions**} & \multicolumn{4}{c}{**Analytical Expression Parameters**} & \multirow{2}{*}{**Vol. cm\({}^{3}\)/mol**} \\ \cline{2-2} \cline{4-7}  & \(\text{i}\) & \(\text{j}\) & & & & \(\boldsymbol{a}\) & \(\boldsymbol{c}\) & \(\boldsymbol{d}\) \\ \hline M075 SH & 1 & 0 & \(\begin{pmatrix}\text{MgO}\text{O}\text{O}_{1.5}\text{(SO)}_{2}\text{(H)}_{2} \text{(H)}_{2}\text{(O)}_{25}\text{ = 1.5