# Establishment of a Standard Method for Boundary Slip Measurement on Smooth Surfaces Based on AFM
## 3 Standard Process of Measuring Boundary Slip
### Data Processing
As a colloidal probe in deionized water at a certain speed approaching the hydrophilic surface in non-slip condition, the deflection of cantilever can be expressed as [29]:  
Figure 3.4  
Figure 3.5 The extend curve transformed such that the abscissa is the separation distance. where \(V\) is the approximation rate between the sphere and sample surface, and \(k\) is spring constant of the colloid probe. In fact, many studies present methods to calibrate the spring constant of the probe [33; 34; 35]. However, according to Equation (4), the slip length \(b\) is the linear regression line-intercept. As a scale factor, the spring constant does not affect \(b\). Thus, it was not necessary to use the most accuracy way to obtain the spring constant. Thus, the spring constant of the colloid probe combined with the calibration of the cantilever deflection was obtained by the fitting method. The AFM colloidal probe approached a smooth flat hydrophilic surface in non-slip condition with a certain velocity, and then the deflection of the probe was measured. Finally, the deflection value of \(k/6\pi\mu R^{2}\) was obtained using Equation (8). Then, the spring constant was calculated with the known values of \(\mu\) and \(R\) (the sphere radius \(R\) could be measured by the light microscope). For example, the probe was driven approaching a cleaned silicon wafer, which was assumed to be in non-slip condition with a velocity of \(77~{}\upmu\)m/s in deionized water. With such velocity, the electrostatic force was far less than the hydrodynamic force on the sphere. Then, the curve in the range of \(1000\)-\(5000\) nm was fitted to further reduce the effect of electrostatic force and van der Waals force. During the experiment, the radius of sphere was measured to be \(29.3~{}\upmu\)m, and the viscosity of deionized water in room temperature was \(9.8\times 10^{-4}\)kg/m \(\cdot\) s [36]. The value of \(k/6\pi\mu R^{2}\) fitted as \(19~{}\upmu\)m \(\cdot\) s, and the value of \(k\) was calculated as \(0.3~{}\)N/m. After obtaining the spring constant, the deflection was transformed into force on the probe. Then, the relationship between total force of the probe and separation was obtained, as shown in Figure 3.6  
(5) The treatment of electrostatic force, van der Waals force and Stokes force  
When the probe approached the solid surface in the liquid, the force acting on the probe included electrostatic force \(F_{\text{E}}\), van der Waals force \(F_{\text{um}}\), Stokes \(F_{S}\) and hydrodynamic force \(F_{H}\): \(F_{total}\) = \(F_{E}\) + \(F_{un}\) + \(F_{S}\) + \(F_{H}\). Thus, to obtain the hydrodynamic force acting on the AFM colloidal probe, it was necessary to exclude the influence of van der Waals force, electrostatic force and Stokes force. There are two methods for measuring electrostatic force: double-speed and low-speed methods. Hydrodynamic force acting on the colloidal probe (\(F_{H}\)) can be expressed as Equation (1), where hydrodynamic force is proportional to the velocity of the probe while electrostatic force does not influence the velocity of the probe. The double-speed method uses two different speeds for the probe approaching the surface. The low-speed method is based on the principle that, when the speed is low enough, hydrodynamic force is much smaller than electrostatic force, thus the former can be ignored. In this study, the low-speed method was used to measure the electrostatic force, and the driving speed of colloidal probe was chosen as \(0.22~{}\upmu\)m/s. Figure 3.7 Force–distance curve of probe. The van der Waals force between the sphere and flat can be expressed as [37; 38]:  
Figure 3.8  
where A is Hamaker constant, \(R\) is radius of sphere and \(D\) is the separation distance. When ignoring hydrodynamic force, the experimental data measured by AFM minus the van der Waals force is the electrostatic force.