## 4 Effects of Surface Force
### Effect of Relative Humidity on the Liquid Bridge Force  
As stated in the discussion above, the primary cause of adhesion is the liquid bridge force, which increases with increasing relative humidity. When the measured surface is scanned point by point in the touch-trigger mode, the measurement time increases because the stylus tip is required to be separated from the measured surface on account of the surface force. When the measured surface is scanned continuously in the scanning mode, the measurement accuracy reduces owing to the bending of the stylus shaft due to the adhesion. Therefore, in our previous work, a vibration mechanism was introduced to prevent the adhesion of the stylus tip to the measurement surface caused by the surface force [12]. By this mechanism, the stylus tip is set to vibrate in a circular motion, where it traces a circle of diameter 0.4 \(\upmu\)m in the \(X\)-\(Y\) plane. However, the stylus shaft actually traces an elliptical path. The difference between the profile of the perfect circle and that of the actual elliptical path of the stylus tip leads to measurement errors. Moreover, the surface roughness cannot be measured in the scanning mode when using the vibrating fiber probe. Therefore, the measurement system, placed in a vacuum vessel, is constructed suitably to prevent adhesion of the stylus tip to the measured surface caused by the surface force. Figure 8 shows a photograph of the measurement system in the vacuum vessel. The effects of pressure inside the vacuum vessel on the surface force are evaluated experimentally. In the experiment, the relative humidity is 61% and the temperature is 25.1 \({}^{\circ}\)C.  
First, the measurement method of the surface force is explained as follows. There are many techniques for measuring surface forces, such as those involving the surface forces apparatus (SFA), atomic force microscopy (AFM), micro cantilever (MC), optical trapping (OT), _etc._[13; 14; 15]. The SFA can directly measure the force between two surfaces in controlled vapors or immersed in liquids. The force sensitivity of the SFA is about 10 nN. The SFA contains two curved molecularly smooth surfaces of mica between which the interaction forces are measured using a variety of force-measuring springs. AFM is, in principle, similar to the SFA except that forces are measured not between two macroscopic surfaces but between a fine tip and a surface. The force sensitivity of AFM is 1-10 pN. The measuring method using the optical fiber probe is, in principle, similar to the SFA and AFM. As shown in Figure 9, after the stylus tip comes into contact with the measured surface, the stylus tip is displaced in the backward direction by using the piezoelectric stage, due to which it separates from the measured surface. Because the stylus tip is displaced while maintaining its contact with the measured surface, the stylus shaft is deflected and the force required to separate the stylus tip from the measured surface is generated by the deflection of the stylus shaft. With an increase in the displacement of the stylus tip, this required force also increases. Therefore, the stylus tip can be separated from the measured surface when the displacement \(D\) of the stylus tip exceeds a certain value. Figure 7: Relationship between relative humidity and surface force (stylus tip diameter: 5 \(\upmu\)m). The load acting on the stylus tip, _i.e._, the surface force, is calculated by assuming that the optical fiber probe, 2.5 \(\upmu\)m in diameter and 0.38 mm in length, is equal to the cantilever of the fixed support. In other words, the surface force is calculated by the deflection of the stylus shaft. The surface force \(F\) is calculated using Equation (7). \[F=\frac{3 EI}{L^{3}}\cdot D \tag{7}\]  
where \(E\), \(I\), and \(L\) are the Young's modulus, the geometrical moment of inertia, and the length of the stylus shaft, respectively.