# Experiment and Analytical Model for Pore Structure of Early-Age Composite Cement Pastes by LF-NMR
## 3 Results and Discussion
### Analytical Pore Structure Model
#### 3.3.2. [PERSON] and [PERSON] ([PERSON] and B.) Pore Structure Model  
The pore size distribution according to [PERSON] and [PERSON]'s [41] lognormal mixture model is expressed as follows:  
\[P(d)=\sum f_{i}p(d,\mu_{i},\sigma_{i}),\sum f_{i}=1. \tag{3}\]  
where \(f_{i}\) is the weight coefficient of the ith logarithmic normal distribution function; \(\mu i\) and \(\sigma i\) are the location parameter and shape parameter of the logarithmic normal distribution function, respectively; and \(p(d,\mu_{i},\sigma_{i})\) is expressed as Equation (4):  
\[p(d,\mu_{i},\sigma_{i})=\frac{1}{\sqrt{2\pi\sigma_{i}^{2}x}}\exp\left[-\frac{ 1}{2}\left(\frac{\log(d)-\mu_{i}}{2}\right)^{2}\right] \tag{4}\]  
As shown in Figure 10, a comparison is made between the experimental data of WC40, FA30, SL30, and SI10 and the corresponding predicted results, with the model performance metrics listed in Table 5. From the figure, the predictions from the S. and B. model align closely with the experimental results, outperforming the V.B. model. Both the predicted and experimental cumulative volume probabilities decrease as the pore size increases, showing a consistent trend. Additionally, the gap between the S. and B. model's predictions and the experimental values is relatively small. The curves of both the simulated values and the experimental data display an inverse sigmoid pattern. It should be noted that the S. and B. model slightly underestimates the experimental data for pore diameters smaller than 1 nm and those between 10 nm and 30 nm. Conversely, for pore diameters ranging from 1 nm to 10 nm, as well as those exceeding 30 nm, the model tends to overestimate the experimental values. However, the differences between the predicted values and experimental results are minimal. According to Table 5, the correlation coefficients of the model are above 0.96, and the standard error is low, around 0.10. Nevertheless, the S. and B. model is more complex and involves numerous parameters. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Sample** & **Age (Day)** & \(a\) & \(n\) & **Correlation** & **Standard Error** \\ \hline \multirow{4}{*}{WC40} & 3 & 0.610 & \(-\)0.457 & 0.7011 & 0.2896 \\  & 7 & 0.604 & \(-\)0.460 & 0.7022 & 0.2884 \\  & 14 & 0.603 & \(-\)0.461 & 0.7025 & 0.2880 \\  & 28 & 0.596 & \(-\)0.465 & 0.7029 & 0.2873 \\ \hline \multirow{4}{*}{FA30} & 3 & 0.657 & \(-\)0.437 & 0.6474 & 0.3458 \\  & 7 & 0.630 & \(-\)0.451 & 0.6556 & 0.3362 \\  & 14 & 0.625 & \(-\)0.454 & 0.6601 & 0.3315 \\  & 28 & 0.543 & \(-\)0.497 & 0.6997 & 0.2844 \\ \hline \multirow{4}{*}{SL30} & 3 & 0.635 & \(-\)0.448 & 0.6630 & 0.3299 \\  & 7 & 0.574 & \(-\)0.481 & 0.6840 & 0.3039 \\  & 14 & 0.570 & \(-\)0.483 & 0.6818 & 0.3046 \\  & 28 & 0.530 & \(-\)0.504 & 0.6950 & 0.2860 \\ \hline \multirow{4}{*}{SI10} & 3 & 0.558 & \(-\)0.489 & 0.6906 & 0.2958 \\  & 7 & 0.526 & \(-\)0.508 & 0.6960 & 0.2857 \\ \cline{1-1}  & 14 & 0.526 & \(-\)0.507 & 0.7045 & 0.2780 \\ \cline{1-1}  & 28 & 0.526 & \(-\)0.507 & 0.6993 & 0.2824 \\ \hline \end{tabular}
\end{table} 
Table 4: Parameters of the V.B. model of neat cement pastes.