# Establishment of a Standard Method for Boundary Slip Measurement on Smooth Surfaces Based on AFM
## 2 Measurement Model and Application Scope  
When the boundary slip is measured using the CP-AFM method, according to the measurement model shown in Figure 1, a colloidal probe will be driven towards the flat solid surface immersed in fluid, and then the deflection of the probe induced by the interaction between the probe and the surface in the fluid will be recorded as laser feedback signal offset on the PSD. The interaction between the probe and the surface is mainly hydrodynamic force, which can be affected by the boundary slip. Note here, if there are colloidal particles close to the substrate, the interaction between the probe and the surface will be significantly affected [27; 28]. Here, we only focused on the situation for pure liquid, which means there should be no colloidal particles close to the surfaces.  
The measuring principle is shown in Figure 1. When the surface is an ideally smooth plane while the sphere is also smooth, the relationship between the hydrodynamic force on the sphere and the separation distance is shown as [15; 29]:  
\[F_{H}=f^{*}\frac{6\pi\mu R^{2}}{D}V \tag{1}\]  
where \(R\) is the radius of the sphere, \(V\) is the velocity, \(F_{H}\) is hydrodynamic force, \(\mu\) is liquid viscosity and \(D\) is separation distance between the colloidal probe and the solid surface.  
\(f^{*}\) is the correction coefficient associated with the existence of boundary slip. The sphere and the ideal plane fulfill no-slip boundary condition when \(f^{*}=1\); and, when the ideal plane is subject to a slip length of \(b\) while the sphere has no slip, the correction factor \(f^{*}\) is expressed as [15; 29]:  
\[f^{*}=\frac{1}{4}\bigg{\{}1+\frac{6D}{4b}\bigg{[}(1+\frac{D}{4b})\ln(1+\frac{4 b}{D})-1\bigg{]}\bigg{\}} \tag{2}\]  
Figure 1: AFM (Atomic force microscope) contact mode measurement model and principle.  
Additionally, when slip length of both the sphere and ideal plane is \(b\), the correction factor is defined as [15; 29]  
\[f^{*}=\frac{D}{3b}\bigg{[}(1+\frac{D}{6b})\ln(1+\frac{D}{6b})-1\bigg{]} \tag{3}\]  
The measurement should be based on this theoretical model, thus three important demands for the measurement method are proposed as follows:  
1. Smooth borosilicate glass sphere should be used for the colloidal probe, which is widely considered fulfilling no-slip boundary condition [13; 16; 17; 18].  
2. The measuring surface of the sample is required to be flat and smooth, as obvious errors will occur on the surface with large roughness and curvature. Thus, surface roughness should meet: \(b>>\) R\({}_{x}\).  
3. The measurement is localized since it based on point-contact model. In other words, the surface of sample is required to be homogenic, and several points should be taken on the measured surface to obtain the average value.  
When separation distance is far greater than the slip length (\(D>>b\)), the relationship between the hydrodynamic force and condition of solid-liquid interface boundary slip can be simplified as:  
\[\frac{V}{F_{H}}=\frac{1}{6\pi\mu R^{2}}(D+b) \tag{4}\]  
It can be defined from Equation (4) that the relationship between \(V/F_{H}\) and the separation distance is linear. Thus, the experimental results of \(V/F_{H}\) can be fitted and the linear regression line-intercept on the separation distance \(D\) is boundary slip length \(b\). The advantage of this method is that the calculation of boundary slip \(b\) is no longer dependent on the parameters including radius of the sphere \(R\) and fluid viscosity \(\mu\). Eventually, the linearity of the curve needs to be checked, to see whether different data fitting segments would result in similar boundary slip length with little deviation.