## 4 Results and Discussion
### Case 1: \(q_{u\,max}\)
Indeed, as the concentration of MWCNTs increases, the probability of agglomeration increases, resulting in a loss of their beneficial properties. On the other hand, the surfactant concentration (x1) and surfactant hydrodynamic diameter (x2) have a negative impact on q\({}_{\text{u max}}\), but, curiously, when combined (x1.x2) the impact becomes positive. It means that these variables depend on each other. Indeed, an increase in the surfactant concentration or surfactant hydrodynamic diameter promotes larger particles' spacing (especially regarding cement particles), resulting in a less strong and stiff stabilized matrix. However, when both effects are combined, the impact on unconfined compression strength (q\({}_{\text{u max}}\)) becomes positive, meaning that these two variables (surfactant concentration and surfactant hydrodynamic diameter) depend on each other. In fact, if a surfactant has a higher hydrodynamic diameter, less molecules will be necessary to cover the surface of the CNTs, and, thus, a lower concentration of the surfactant will be required to guarantee a good dispersion of the CNTs. Thus, the two variables are correlated, as expected. The PLS residual versus leverage is useful to understand the distribution of the observations. Figure 4 confirms the significance of the data set. The leverage of an observation measures its ability to move the regression model as a whole by simply moving in the y-direction. The leverage always takes values between zero and one. A point with zero leverage has no effect on the regression model. If a point has a leverage equal to one, the line must follow the point perfectly. As all points are inside the defined limits and no outlier is present, the data set is significant. The surface plot shows the behavior of q\({}_{\text{u max}}\) with x1 and x3 (Figure 5) and x2 and x3 (Figure 6). As expected, it is a non-linear relation. q\({}_{\text{u max}}\) increases with x2 (higher surfactant \begin{table}
\begin{tabular}{c c c} \hline
**Component** & **q\({}_{\text{u max}}\) (kPa)** & **q\({}_{\text{u max}}\) Standardized** \\ \hline Intercept & 243 & 0 \\
1\(-\)x1 & \(-\)60 & \(-\)1.2 \\
2\(-\)x2 & \(-\)3 & \(-\)1.0 \\
3\(-\)x3 & 34,203 & 3.3 \\
4\(-\)x1x2 & 2 & 1.2 \\
5\(-\)x3x3 & \(-\)3,131,470 & \(-\)3.2 \\ \hline \end{tabular}
\end{table} 
Table 6: Regression coefficients for the q\({}_{\text{u max}}\) model. Figure 4: PLS residual vs. leverage plot for the q\({}_{\text{u max}}\) model. hydrodynamic diameter leads to higher \(\mathrm{q_{u\,max}}\)) and shows an optimum regarding x1 (surfactant concentration) and x3 (CNT concentration). In order to better understand the structure of the composite material, Figure 7 shows a scanning electron microscope (SEM) image of a stabilized soil sample additivated with MWCNTs dispersed in an aqueous solution of Amber 4001. Three kinds of materials can be identified in the SEM image: the needle-shaped lattice materials are calcium silicate associated with the Portland cement reactions with water, the materials in the form of irregular polyhedrons are soil particles, while the long and narrow particles are the MWCNTs. As can be seen from Figure 7, the MWCNTs are adequately dispersed in the matrix due to the effect of the surfactant, thus ensuring that the beneficial properties associated with the presence of the MWCNTs are not lost. The low crystallinity of the MWCNTs [38] ensures good interaction with surrounding particles, in particular with Portland cement particles. Thus, these results suggest that the MWCNTs seem to give some continuity to the calcium silicates produced by the Portland cement, allowing the establishment of denser  
Figure 5: Surface plot of \(\mathrm{q_{u\,max}}\) vs. x1 and x3. Figure 6: Surface plot of \(\mathrm{q_{u\,max}}\) vs. x2 and x3. cementitious bonds, i.e., the CNTs seem to act as a nanofiller and a nano-reinforcement agent, promoting the construction of a denser and more resistant structure.