## 3 Results and Discussion
### Effective Boundary Slip  
#### 3.3.1 Superoleophilic Surfaces  
* _Hydrodynamic forces._ Figure 6a shows the measured hydrophobic forces \(F_{hydro}\) applied on the AFM probe approaching the superoleophilic surface immersed in hexadecane with varying _Rsm_ roughness as a function of separation distance \(D\). The plots of experimental data exhibit nonlinear behavior at separation distances of 0 nm to 400 nm and are indicative of a slip boundary condition. In order to obtain the slip length, the \(V/F_{hydro}\) applied on the AFM probe is used for superoleophilic surfaces immersed in hexadecane and ethylene glycol as a function of separation distance \(D\), as shown in Figure 6a. In the case of hexadecane, the plots of \(V/F_{hydro}\) on superoleophilic surfaces exhibit linear behavior at separation distances of 0 nm to 800 nm, and show a trend of the linear parts shifting to the left with increasing _Rsm_ from samples _a1_ to _a5_. This means that the interceptions of \(V/F_{hydro}\) on the horizontal ordinate shift to the left, and lead to an increasing slip length from samples _a1_ to _a5_. For ethylene glycol, the plots of \(V/F_{hydro}\) are similar to those of hexadecane. This means that the increasing _Rsm_ roughness leads to a larger slip length, as shown in Figure 6b. It can be noted that the tendencies are more obvious, because the surface tension of ethylene glycol is greater than that of hexadecane. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Type**} & \multirow{2}{*}{\(R_{z}\) **(nm)**} & \multirow{2}{*}{_Rsm_ **(nm)**} & \multicolumn{2}{c}{**Hexadecane**} & \multicolumn{2}{c}{**Ethylene Glycol**} \\ \cline{3-6}  & & & **CA (\({}^{\circ}\))** & **CAH (\({}^{\circ}\))** & **CA (\({}^{\circ}\))** & **CAH (\({}^{\circ}\))** \\ \hline _a1_ & 1040 \(\pm\) 150 & 620 \(\pm\) 90 & 0 & – & 10 \(\pm\) 1 & – \\ _a2_ &  
* _Slip length_. Measured slip length was obtained from the hydrophobic forces according to Equation (1). Figure 7a shows the measured slip length data as a function of \(Rsm\) roughness. For samples \(a1\) to \(a5\), the slip length of the superoleophilic surface immersed in hexadecane and ethylene glycol increased with increasing \(Rsm\) roughness. The effective slip length was obtained according to Equation (11). Figure 7b shows the effective slip length data as a function of \(Rsm\) roughness. Results show that all of the slip lengths are negative and remain constant from samples \(a1\) to \(a5\) after correction by compensation \(\beta^{*}\). This means that the flow velocity at the reference surface is discontinuous due to the rough structure, and the slippage is independent of spacing roughness, where the slip length of the sphere and the location of the reference surface are taken into account. Figure 6: Hydrodynamic force \(F_{lightro}\) and \(V/F_{hydro}\) of a borosilicate sphere on superoleophilic surfaces immersed in (**a**) hexadecane and (**b**) ethylene glycol with various \(Rsm\) roughness at a sphere velocity of 38.5 \(\upmu\)m/s. Figure 7: **(a)** Measured slip length and (**b**) effective slip length on superoleophilic surfaces with varied \(Rsm\) roughness. * _Discussion._ It should be noted that slip boundary conditions at the reference surface are complex, as there is rough topography, and slip length at the interface of liquid, and rough superoleophilic surface is discontinuous, which should be studied by the average hydrodynamic forces detected using AFM. The effect of spacing roughness on slip length can be described by the following model, shown in Figure 8. The boundary of rough topography were replaced by a flat plane of reference surface, and the pitches were divided into square zones, where the average length of side \(\varepsilon\) was the value of \(Rsm\) roughness.