# Establishment of a Standard Method for Boundary Slip Measurement on Smooth Surfaces Based on AFM
## 3 Standard Process of Measuring Boundary Slip
### Data Processing
distance between the probe and surface increases, which impacts the hydrodynamic resistance signal. [PERSON] and [PERSON] [1] found that there is relative movement between the cantilever and the detector because of the piezoelectric ceramic. Thus, even without force, a certain amount of virtual deflection is generated. Thus, when measuring, the impact of this resistance should be excluded in the process. [PERSON] [15] reported that the pressure produced by the piezoelectric ceramic transducer generates a vertical elastic deformation, resulting in a virtual deflection. Their research indicated that the virtual deflection can be removed from the measuring process by subtracting the same slope's linear part from the curve. The extend curve is denoted as \(Def_{ex}\), while the retract curve as \(Def_{re}\). Then, the average curve of extend and retract curves can be calculated as:  
\[Def_{average}=(Def_{ex}+Def_{re})/2 \tag{5}\]  
The remote linear portion of the average curve can be fitted linearly. Denoting the fitting curve as \(Def_{average}^{\prime}\), the corrected extend curve can be obtained as:  
\[Def_{ex}{}^{corrected}=Def_{ex}-Def_{average}^{\prime} \tag{6}\]  
The corrected extend curve no longer had the extra slope of the curve caused by deflection of the instrument, as shown in Figure 6b. The extend curve without virtual deflection is shown in Figure 7. The ordinate of the curve is the probe deflection measured in volts, and the abscissa is still the driving displacement. (3) Transform of the abscissa to separation distance  
Figure 3.0 (**a**) Virtual deflection due to the relative motion of the detector; and (**b**) the deflection curve after removal of virtual deflection. Figure 3.1 Process curve after removing virtual deflection. After the steps above, the abscissa still did not represent the position of the surface. It was necessary to transform the abscissa (driving displacement) into the separation distance, and remove the instrument sensitivity so that the ordinate measured in volts could be transformed into the probe deflection in nm. There are several methods to calibrate the deflection [30; 31; 32]; here, the defection of the cantilever was obtained by the following process:  
The linear region on the left was fit to obtain the intercept \(m\) and the slope of the linear part \(k_{i}\). As noted above, the linear part on the left was where the hard contact occurred, so the change of the deflection should be equal to the change of the driving displacement. Thus, -\(k_{i}\) is the cantilever sensitivity. The curve was divided by -\(k_{i}\) to transform the ordinate into nanoprobe deflection. The curve is denoted as \(Def_{ex}{}^{nm}\). The origin point of the abscissa should be located at the contact position of probe and surface to make the abscissa associated with surface. It was necessary to shift the original abscissa, that is, subtract m from the original abscissa to obtain the new abscissa defined as _Displacement_, which is the driving position. The relationship among driving position, deflection of the probe and _Distance_ (_Distance_ is separation distance between the probe and surface simplified as \(D\)) is:  
Figure 3.2  
The separation distance (\(D\)) can be obtained based on Equation (7). The relationship between separation distance and deflection of the probe is shown in Figure 3.3 After this procedure, the abscissa is the separation distance and ordinate is the deflection of the probe. To confirm the validation of the deflection calibration, the data shown in Figure 8 were checked. When the sphere contacted the surface, the separation distance should be 0 nm, the data with high deflection value (in Figure 8, for example, the checking range is above 120 nm) should always be close to the deflection axis (\(D\) = 0 nm), and the error should smaller than 5 nm. (4) Calculation of the probe spring constant  
After obtaining the corrected probe deflections, the spring constant of the probe was calculated to obtain the total force on the probe.