## 4 Results and Discussions  
### Microscopic Characteristics of NS-Modified Silty Clay  
As a basic soil property, pore size distribution (PoSD) is used to analyse soil-water characteristic curves [30]. The pore size density distributions of NS-modified Qinghai-Tibet silty clay with various NS additions are demonstrated in Figure 3. The results showed that the pore size density curves of NS-modified silty clay varied in the left direction with the increase of NS content. For the corresponding pore diameter of the peak pore size density, compared to the plain silty clay, the NS additions of 0.25%, 1%, and 3% decreased the peak point by 19.18%, 34.49%, and 66.04%, respectively. This indicates that the pore diameters in the soil-water system become smaller. This is because nanoparticles fill the pore space between soil particles, making the soil pores become smaller, and the soil becomes more compacted. In Figure 3, the modified silty clays with 1% and 3% additions show a bimodal distribution, with a secondary peak for pore diameters greater than 1000 nm. The most probable causes are: 1 the NS addition of less than 1% was insufficient to fill the pore directly; 2 NS had strong water absorption, the clay aggregates took NS as the core in the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Probability Distribution** & **Parameters** & **Distribution Function** \\ \hline Normal distribution & \(\mu>0,\ \sigma>0\) & \(f(x)=\frac{1}{\sqrt{2\pi}}e^{-(x-\mu)^{2}/2\sigma^{2}}\) \\ Lognormal distribution & \(\mu>0,\ \sigma>0\) & \(f(x)=\begin{cases}\frac{1}{\sqrt{2\pi}}e^{-\frac{(\frac{1}{2\pi}-\mu)^{2}}{2 \sigma^{2}}},&x>0\\ 0,&x\leq 0\end{cases}\) \\ Gamma distribution & a \(>0,\ \text{b}>0\) & \(f(x)=\begin{cases}\frac{x^{4}-1}{e^{x/b}},&x>0\\ 0,&x\leq 0\end{cases}\) \\ Weibull distribution & a \(>0,\ \text{b}>0\) & \(f(x)=\begin{cases}\frac{k}{\sigma}\left(\frac{k}{\Lambda}\right)^{k-1}e^{-( \frac{1}{2})^{k}},&x>0\\ 0,&x\leq 0\end{cases}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The density functions of probability distributions. modified soil; and 3 the original pores connected with the closed pores and formed a local bigger pore. The formation of these large pores was random and non-uniform. Figure 4 shows the accumulated pore radius distribution curves of NS-modified Qinghai-Tibet silty clay. According to the average pore diameter classification standard from the International Union of Pure and Applied Chemistry (IUPAC) [31], the boundary diameter of meso- and macro-pores is 50 nm. Meanwhile, the size of the NS particles used in this study was about 15 nm. Therefore, 50 nm and 150 nm were selected as the boundary diameters of small pores, medium pores, and larger pores. From Figure 5, the volume content of larger pores was 52.53%, and the medium and smaller pores of the plain silty clay were 29.93% and 17.54%, respectively. The volumetric content of the medium pores increased to 31.59%, 35.83%, and 46.59% for the NS additions of 0.25%, 1%, and 3%, respectively. The volume of the macro-pores, on the other hand, decreased from 52.53% to 33.10% when the NS particle content increased from 0 to 3%, and the volumetric content of the smaller pores had little change with the addition of NS. In general, a small quantity of addition was not enough to fill the void of soil-water systems directly. However, due to the strong water absorption of NS, the voids may be filled by small clay clusters formed with NS as the core. Figure 4: The accumulated pore diameter distribution curves of NS-modified silty clay samples. Figure 3: Pore radius density curves of NS-modified silty clay samples. Measured Median Pore Diameters and Cumulative Volume of Pore Diameters Smaller Than 0.1 \(\mu\)m \(V_{D<0.1\ \mu\)m  
As a porous medium, the soil-water system consists of soil particles, a pore solution, and air, making a four-component medium with the incorporation of NS.