# Effect of Hydrophobic Silica Nanochannel Structure on the Running Speed of a Colloidal Damper
## 3 Fractal Seepage Model  
### Seepage Theory of Nanochannels  
For nanoscale silica nanochannels, the inertia force and volume force in macro-hydrodynamics can be neglected, and the surface force dominates [13]. In order to clearly describe the seepage in the silica gel nanochannels, the roughness of the channels was neglected, and they were simplified to a smooth capillary structure. The analysis of gas/liquid seepage in hydrophobic silica nanochannels was performed based on the following assumptions: the interface resistance of the channel was ignored, the two phases of flow in the channel were incompatible with each other, the viscosities of the two-phase fluid did not affect each other, and the temperature did not change during the process of water infiltration [18]. The gas flow in the hydrophobic silica nanochannels was considered to be slip flow and diffusion flow. According to the analysis of gas/liquid two-phase seepage velocity by [PERSON] [11], it is understood that the gas seepage velocity in hydrophobic silica nanochannels is higher than that of water because of gas diffusion in microflows, so the operation speed of the CD depends on the water seepage velocity in the nanochannels. The flow pattern of water in nanochannels is an important basis for the study of microfluidic flow. It was assumed that the flow pattern of water in nanochannels was Poiseuille flow and the velocity distribution was parabolic. In Reference [19], the flow in the nanoscale colloidal microporous channel was considered as Poiseuille flow, and then the emergence of the \"stagnant ring\" was explained by the \"Neck-Bottle\" theory and the \"Contact Angle\" theory. [PERSON] et al. [20] simulated the Couette and Poiseuille flows of liquids at the nanoscale and found the existence of wall \"slip\" phenomena and density oscillation distributions; [PERSON] et al. [21] showed that the velocity distribution was close to the parabolic quadripartite curve, the temperature distribution was close to the quadrangle curve, and the continuous medium theory results were consistent. Figure 3 shows the flow pattern of water in silica gel nanochannels under external pressure \(p_{m}\). Terms \(p_{\sigma}\) and \(p_{G}\) denote the capillary pressure and gas pressure in the nanochannels, respectively. \(L\) is the length of the nanochannels, \(x\) is the flow distance of water in the nanochannels, and \(\Upsilon\) is the surface tension of the solid-liquid interface. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Trade Name** & **10-100 C4** & **10-100 C8** & **10-100 C18** & **10-200 C18** & **10-300 C18** \\ \hline \(\phi\) & 0.53 & 0.57 & 0.62 & 0.50 & 0.47 \\ \(D_{f}\) & 1.64 & 1.74 & 1.83 & 1.67 & 1.58 \\ \(D_{t}\) & 1.15 & 1.11 & 1.07 & 1.13 & 1.18 \\ \(\tau_{av}\) & 1.41 & 1.35 & 1.30 & 1.46 & 1.52 \\ \(L_{0}\) (nm) & 36.43 & 42.11 & 50.97 & 71.33 & 94.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The fractal dimensions and related parameters of hydrophobic silica gels. In the CD compression stage, when the external pressure \(p_{m}\) is greater than the capillary pressure \(p_{\sigma}\), the bulk water overcomes the capillary pressure \(p_{\sigma}\), gas pressure \(p_{G}\), and other microforces to squeeze air into the nanochannel. In the CD relaxation stage, water seeps through the nanochannel under capillary pressure and gas pressure. The capillary pressure follows the Yang-Laplace formula [8]:  
\[P_{\sigma}=-\frac{4\sigma\cos\theta}{\lambda_{av}} \tag{7}\]  
where \(\sigma\) is the surface tension of water and \(\theta\) is the surface contact angle of the hydrophobic silica gel. Air occupies the silica nanochannels before water infiltration. In this paper, air was considered to be compressible real gas (atmospheric temperature and pressure), without considering gas dissolution in water.