## 3 Results and Discussion
### Calculation of Diffusion Coefficient in Cement Bodies with \(\gamma\)-C\({}_{2}\)S  
It was difficult to measure the concentration of CO\({}_{2}\) in the paste blocks at the carbonation front using the error function (erf) solution given in Equation (6) derived from [PERSON]'s second law, Equation (7). Because the measurements of the CO\({}_{2}\) concentration at the carbonation front were not possible experimentally, the solution using the error function was not used to calculate the CO\({}_{2}\) diffusion coefficient and verify [PERSON]'s second law. Many researchers have simplified this difficulty by using Equation (5), such that the diffusivity constant, \(k\), was calculated from the depth of carbonation, x\({}^{1}\), in addition to the carbonation coefficient using Equation (5) [14; 15; 16; 17; 18]. \[\mathrm{x^{1}=\ k\sqrt{t}} \tag{5}\]  
\[\mathrm{C_{(Xt)}=\ C_{s}\bigg{[}erf\bigg{(}\frac{x}{2\sqrt{D_{t}}}\bigg{)} \bigg{]}} \tag{6}\]  
\[\frac{\partial c}{\partial t}=\ D\bigg{(}\frac{\partial^{2}c}{\partial x^{2}} \bigg{)} \tag{7}\]  
where  
C = concentration in dimensions (quantity/volume), e.g., (mol. m\({}^{-3}\));  
t = time in seconds;  
x = longitudinal separation distance from uphill to downhill position, e.g., (m);  
D = coefficient of diffusion. \[\mathrm{Ca(OH)_{2}(aq)+H_{2}CO_{3}(aq)\ \to\ CaCO_{3}(s)+2H_{2}O} \tag{8}\]  
[PERSON]'s second law states that concentration in the body is a function of both position and time. In other words, an increase in concentration in a cross-section of a unit area with time is simply the difference between the flux in and out of the volume. [PERSON]'s second law implies that the flux of CO\({}_{2}\) into the paste is equal to the flux of CO\({}_{2}\) out plus accumulation, Equation (9). \[\mathrm{F_{in}=\ F_{out}+\Delta C} \tag{9}\]  
where Flux in = mass of CO\({}_{2}\) flowing in (kg. m\({}^{-2}\) s\({}^{-1}\)); flux out = mass of CO\({}_{2}\) flowing out (kg. m\({}^{-2}\) s\({}^{-1}\)). The accumulation quantity, \(\Delta C\), is the amount of CaCO\({}_{3}\) minus the CaO combined during carbonation. [PERSON]'s second law can be proved considering the flux of CO\({}_{2}\) in and out of the paste specimen are not equal if the direction of flow is considered to be unidirectional. On comparing the chemical contents of the uncarbonated pastes in Figure 3a,b with those of the carbonated pastes in Figure 4a,b, it was clear that the carbonation ingress in the paste caused the deposition of CaCO\({}_{3}\) in the pore structure, as shown in Figures 7 and 8, resulting in densification in the carbonated regions. Therefore, because CaCO\({}_{3}\) was proved to have \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Type**} & \multicolumn{3}{c}{**Average Measured Carbonation Depth x\({}^{1}\), (mm)**} \\ \cline{2-5}  & **12 h** & **24 h** & **72 h** & **7 h** \\ \hline C\({}_{2}\)S0 & 3.952 & 8.916 & 10.903 & 14.324 \\ \hline C\({}_{2}\)S5 & 3.204 & 7.552 & 10.322 & 10.871 \\ \hline C\({}_{2}\)S15 & 2.543 & 5.236 & 6.716 & 9.406 \\ \hline C\({}_{2}\)S25 & 2.602 & 4.554 & 4.854 & 7.225 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average carbonation depths measured from the specimens. been formed owing to the carbonation reactions in the OPC and \(\gamma\)-C\({}_{2}\)S composite pastes, it indicated that the fluxes in and out of the pastes were not equal. If a paste sample under carbonation was considered, as shown in Figure 7, making a material balance and applying the law of conservation of matter to Equation (9) results in [PERSON]'s second law, Equation (10). \[\frac{\mathrm{d}\mathrm{C}}{\mathrm{d}t}=\ D\frac{\mathrm{d}^{2}\mathrm{C}}{ \mathrm{d}x^{2}} \tag{10}\]  
Setting boundary conditions (BC) for the diffusion above,  
\[\mathrm{C}\ :\ \mathrm{t}=0,\ 0\ <\ \mathrm{x}\ <\infty,\ \mathrm{C}_{\mathrm{x}}=0 \tag{11}\]  
\[\mathrm{C}\ :\ \mathrm{x}=0,\ 0\ <\ \mathrm{t}\ <\infty,\ \mathrm{C}_{\mathrm{x}}= \mathrm{C}_{0} \tag{12}\]  
The general solution to such an equation is the error function, Equation (13).