## 2 Derivation of Discrete Solid Phases (DSPs) from Cemdata07 Solid Solution End-Members  
The first step in deriving a series of DSPs was defining 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 1) [7]. These thermodynamic properties could then be used to calculate the solubility constant, log K and its variation with temperature for the dissolution reactions of each end-member (Table 2). As shown, the end-members were listed in order of increasing solubility and the analytical expressions described the variation of log K as a function of temperature (in Kelvin) where a, c and d were calculated using Equation (1) as functions of enthalpy (\(\Delta\)rH, J/mol), entropy (\(\Delta\)rS, J/K/mol) and heat capacity (\(\Delta\)rCp, J/K/mol) of reaction [7], b, e and f = 0, and T denotes temperature. Each of the end-members shown in Table 2 and their respective analytical expressions were written directly into PHREEQC to suit its data input syntax. \[\log\text{K}=\text{a}+\text{b}\cdot\text{T}+\text{c}/\text{T}+\text{d}\cdot \log\text{10}(\text{T})+\text{e}\cdot\text{T}^{-2}+\text{f}\cdot\text{T}^{2} \tag{1}\]  
Next, appropriate mole fractions were chosen to discretise the solid solutions and create the series of DSPs. In the current paper, mole fraction increments of 0.2 were used to describe the two-end-member CSH model in terms of \(\text{X}_{\text{i}}\) and \(\text{X}_{\text{j}}\), where the sum of the mole fractions in each solid solution had to equal one. Using these mole fractions, the solid phase composition (CaO, \(\text{SiO}_{2}\), \(\text{H}_{2}\text{O}\)), aqueous reaction components (H\({}^{+}\), Ca\({}^{+2}\), \(\text{H}_{2}\text{O}\) and \(\text{SiO}_{2}\text{(aq)}\)), mass (g/mol) and volume (cm\({}^{3}\)/mol) were determined for each DSP based on the original end-members. The solubility constant was determined for each DSP in the CSH model using Equation (2), where \(\text{K}_{\text{i}}\) and \(\text{K}_{\text{j}}\) are the \(\log\text{K}\) values of the end-members. \[K=(\text{K}_{\text{i}}\text{-X}_{\text{i}})^{\text{Xi}}\text{-}(\text{K}_{ \text{j}}\text{-X}_{\text{j}})^{\text{Xi}} \tag{2}\]  
The modelling of OPC hydration required a credible account of the kinetic dissolution of the clinker phases as a function of time [24], the oversaturation of the precipitating hydrate phases during the first 12 h of hydration [25] and the release and uptake of alkali \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Component** & **G \({}^{\text{i}}\)** & **H \({}^{\text{i}}\)** & **S \({}^{\text{ii}}\)** & **Cp \({}^{\text{ii}}\)** \\ \hline Tobermorite-II & \(-\)1,744,356 & \(-\)1,915,813.3 & 80 & 132.372 \\ \hline Jennite & \(-\)2,480,808 & \(-\)2,723,484.3 & 140 & 210.805 \\ \hline Ca\({}^{+2}\) & \(-\)552,790 & \(-\)543,069 & \(-\)56.483997 & \(-\)30.922515 \\ \hline SiO\({}_{2}\)(aq) & \(-\)833,411 & \(-\)887,856.17 & 41.338001 & 44.465416 \\ \hline H\({}_{2}\)O & \(-\)237,183 & \(-\)285,881 & 69.923 & 75.3605 \\ \hline H\({}^{+}\) & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular} \({}^{\text{i}}\) \({}^{\text{j}}\)/mol. \({}^{\text{i}}\) \({}^{\text{j}}\)/mol/K. Molar volumes for tobermorite-II and jennite taken as 59 cm\({}^{3}\)/mol and 78 cm\({}^{3}\)/mol, respectively.  
\end{table} 
Table 1: Thermodynamic properties of tobermorite-II and jennite as taken from the cendata07 database [7]. Letters assigned to each end-member are shown in parentheses.