# Effect of Hydrophobic Silica Nanochannel Structure on the Running Speed of a Colloidal Damper
## 2 Structure and Fractal Description of the Silica Gel Nanochannel Structure
### Fractal Description of Nanochannels  
Based on the above analysis, it was assumed that the silica gel nanochannels were composed of a series of independent capillaries, for each of which one end was located on the surface of the particles and the other end was closed inside the particles [17]. The inner surface area of a capillary is equal to the inner surface area of particles, and all flowable spaces are equal to the pore volume of the particles. In practice, a capillary is not a uniform straight circular pipe. It was assumed that each capillary in the porous silica gel had different cross-sectional area and had a certain tortuosity degree \(\tau\). In order to characterize the channel structure in the porous silica gel, two fractal characterization parameters were introduced: the pore area fractal dimension \(D_{f}\) and the pore tortuosity fractal dimension \(D_{t}\).  
The cross-sectional area and tortuosity degree of capillaries obey the distribution of fractal power law, which was expressed as [12]  
\[N(L_{0}\geq\lambda_{min})=\left(\frac{\lambda_{max}}{\lambda}\right)^{D_{f}} \tag{1}\]  
where \(N\) is the number of capillaries, \(L_{0}\) is the characteristic length of the fractal capillary set, \(\lambda_{min}\) is the minimum diameter of a capillary, and \(\lambda_{max}\) is the maximum diameter of a capillary. The values of \(\lambda_{min}\) and \(\lambda_{max}\) are shown in Table 1.  
The expression of the pore area fractal dimension \(D_{f}\) is as follows [10]:  
\[D_{f}=D_{E}-\frac{\ln\phi}{\ln(\lambda_{min}/\lambda_{max})} \tag{2}\]  
where \(D_{E}\) is the Euclidean dimension. \(D_{E}\) is equal to 2 because of the two-dimensional space selected in this paper. \(\phi\) is the porous medium's porosity. The prerequisite for the application of fractal theory is that the size distribution of capillaries in the fractal porous medium satisfies the following formula: \((\lambda_{min}\lambda_{max})^{D_{f}}\cong 0\). The expression of the pore tortuosity fractal dimension \(D_{t}\) is as follows:  
\[D_{t}=1+\frac{\ln\tau_{av}}{\ln(L_{0}/\lambda_{av})} \tag{3}\]  
where \(\lambda_{av}\) is the average pore size. In this paper, \(\lambda_{av}\) equals \(2\tau_{a}\). \(\tau_{av}\) is the average tortuosity of the capillaries. Choosing the appropriate feature length \(L_{0}\) is the key to obtaining \(D_{t}\). Assuming that the fractal capillaries are concentrated on a square with side length \(L_{0}\), \(L_{0}\) can be calculated according to the following formula:  
\[L_{0}=\sqrt{\frac{(1-\phi)\pi\lambda_{max}^{2}D_{f}}{4\phi\left(2-D_{f}\right)}} \tag{4}\]  
Figure 2: Pore structure of hydrophobic silica nanochannels.  
The average tortuosity in Formula (3) can be calculated by the following formula:  
\[\tau_{av}=\frac{1}{2}\left(\frac{\sqrt{1-\phi}}{1-\sqrt{1-\phi}}\sqrt{\left(\frac {1}{\sqrt{1-\phi}}-1\right)^{2}+\frac{1}{4}}+\frac{1}{2\sqrt{1-\phi}}+1\right) \tag{5}\]  
The length of the curved fractal capillary, \(L_{t}\), is as follows:  
\[L_{t}(\lambda)=\lambda_{av}^{(1-D_{t})}L_{0}^{D_{t}} \tag{6}\]  
The average tortuosity values and fractal dimensions of silica gels with different pore structures can be calculated by Formulas (2)-(5). The results are shown in Table 2. It can be seen that the value of \(D_{f}\) ranged from 1.58 to 1.83 and increased with an increase in the number of grafted molecules, whereas it decreased with increasing pore size. The value of \(D_{t}\) ranged from 1.07 to 1.18, showing the opposite trend to \(D_{f}\) decreasing with an increase in the number of grafted molecules and increasing with increasing pore size.