# Effect of Hydrophobic Silica Nanochannel Structure on the Running Speed of a Colloidal Damper
## 3 Fractal Seepage Model
### Fractal Seepage Model of Nanochannels  
The pore size of the porous media obeys the fractal power distribution, and the distribution is random. This can be expressed by a probability density function [22].  
\[f(\lambda)=D_{f}\lambda_{max}^{D_{f}}\lambda_{av}^{-(D_{f}+1)} \tag{12}\]  
Assuming that the porous hydrophobic silica gel nanochannels are composed of curved capillary bundles, as in Reference [11], by combining Formulas (6), (11), and (12), the flow \(Q\) through a unit cross section of the capillary can be obtained by integrating a group of fractal capillaries.  
\[\begin{split} Q=-\int\limits_{\lambda_{min}}^{\lambda_{max}}q( \lambda)d_{N(\lambda)}=\frac{\pi\Delta p}{128\mu L_{0}^{D_{f}}}D_{f}\lambda_{ max}^{D_{f}}\times\\ \left[\left(\frac{8l_{max}^{D_{f}-D_{f}+2}}{D_{1}-D_{f}+2}- \frac{\lambda_{max}^{D_{f}-D_{f}+3}}{D_{1}-D_{f}+3}\right)-\left(\frac{8l_{min }^{D_{f}-D_{f}+2}}{D_{1}-D_{f}+2}-\frac{\lambda_{min}^{D_{f}-D_{f}+3}}{D_{1} -D_{f}+3}\right)\right]\end{split} \tag{13}\]  
In fractal porous media, \((\lambda_{min}/\lambda_{max})^{D/\cong 0}\), \(1<D_{f}<2\), and \(1<D_{t}<2\), so \((\lambda_{min}/\lambda_{max})^{Dt-D/t3}\cong 0\). Therefore, Formula (13) can be simplified as follows [12]:  
\[Q_{m}=\frac{\pi}{128\mu}\frac{\Delta pD_{f}}{L_{0}^{D_{t}}}\times\left(\frac{ 8l\lambda_{max}^{D_{t}+2}-\lambda_{max}^{D_{t}+2}\lambda_{min}^{D_{t}-D_{f}+2 }}{D_{t}-D_{f}+2}+\frac{\lambda_{max}^{D_{t}+3}}{D_{t}-D_{f}+3}\right) \tag{14}\]  
The above formula reveals that the flow rate is related to the channel structure and fractal dimension when silica gel nanochannels are regarded as a group of fractal capillaries. The fractal seepage model can characterize the flow rate in regions with characteristic length \(L_{0}\) under different pore structures when the pressure difference is \(\Delta p\). And the total cross-sectional area \(A\) of a fractal capillary unit can be expressed as follows [9]:  
\[A=\frac{\pi}{4}\frac{D_{f}}{2-D_{f}}\frac{1-\phi}{\phi}\lambda_{max}^{2} \tag{15}\]  
According to the total cross-sectional area \(A\), the number of fractal capillary units \(n=MV_{a}/AL_{0}\) in hydrophobic silica gel with mass \(M\) can be calculated. The total flow rate \(Q_{\text{m}}\) in hydrophobic silica nanochannels can be obtained by combining with Formula (14):  
\[Q_{m}=\frac{\pi}{128\mu}\frac{MV_{a}}{A}\frac{\Delta pD_{f}}{L_{0}^{D_{t}+1}} \times\left(\frac{8l\lambda_{max}^{D_{t}+2}-\lambda_{max}^{D_{t}+2}\lambda_{ min}^{D_{t}-D_{f}+2}}{D_{t}-D_{f}+2}+\frac{\lambda_{max}^{D_{t}+3}}{D_{t}-D_{f}+3} \right). \tag{16}\]  
When the bulk water is compressed, all of it infiltrates the silica gel nanochannels. According to the fact that the macroflow rate in the high-pressure cavity of the CD and the microflow rate in the nanochannels are conservative, the running speed of the CD, \(v_{m}\), can be obtained [11]:  
\[v_{m}=\frac{4Q_{m}}{\pi D^{2}}=\frac{1}{32\mu D^{2}}\frac{MV_{a}}{A}\frac{\Delta pD _{f}}{L_{0}^{D_{f}+1}}\times\left(\frac{8l\lambda_{max}^{D_{+}+2}-\lambda_{max}^ {D_{+}+2}\lambda_{min}^{D_{-}+2}}{D_{t}-D_{f}+2}+\frac{\lambda_{max}^{D_{t}+3}} {D_{t}-D_{f}+3}\right) \tag{17}\]  
where \(D\) is the piston diameter of the CD. Formula (17) characterizes the relationship between pore structure parameters of the silica gel and the CD operating speed under pressure difference \(\Delta p\). Given pressure difference \(\Delta p\), the operating speeds of silica gel CDs with different pore structures can be calculated by combining the parameters in Tables 1 and 2. Figure 4 shows the fractal percolation model velocity of 10-100 \(CM\) silica gel (\(M\) = 4, 8, 18) under constant pressure difference \(\Delta p\) (\(\Delta p\) = 1 MPa, 2 MPa). Figure 5 shows the fractal percolation model velocity of 10-2\(r_{a}\) C18 silica gel (\(2r_{a}\) = 100 A, 200 A, 300 A) under constant pressure difference \(\Delta p\) (\(\Delta p\) = 1 MPa, 2 MPa).