## 4 Effects of Surface Force  
### Analyses of Liquid Bridge Force, Van der Waals force, and Electrostatic Force  
In general, when the particle size is less than several tens of micrometers, the effects of the surface force generated from the van der Waals force, electrostatic force, and liquid bridge force are strengthened and this surface force becomes greater than the force of gravity [11]. Figure 5 shows a schematic diagram of the surface force between the stylus tip and the measurement surface. Because the fiber stylus has a 5-um-diameter sphere on its tip, the measurement surface draws the stylus tip closer when the stylus tip approaches it, and the distance between the stylus tip and the measured surface is less than the regular displacement. When the stylus tip comes into contact with the measured surface, it adheres to the surface and cannot be separated from it. 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, measurement accuracy reduced because of the bending of the stylus shaft due to the adhesion. The observed adhesion, which is influenced by environmental factors (e.g., humidity) and the roughness of the measured surface, is not reproducible. These surface forces are analyzed with the aim of investigating the causes of adhesion. First, we examine the liquid bridge force. This force has a large effect on adhesion in a humid environment; it is generated by the capillary action of condensed water near the contact zone of the stylus tip and measured surface. This force increases with increasing relative humidity. The liquid bridge force \(F_{1}\) experienced in the case of a stylus tip of diameter \(d\) and under a relative humidity of \(\%RH\) can be calculated using the empirical formula in Equation (3) [11]. This is the empirical formula of the liquid bridge force between a particle and a plate with a hard and clean surface. As indicated by this formula, the liquid bridge force increases with increasing relative humidity. \[F_{1}\,=\,0.15d\{0.5\,+\,0.0045\times(\%RH)\} \tag{3}\]  
Next, we examine the van der Waals force. This force acts between molecules, and it is a general force of interaction that acts between objects at all times. The van der Waals force \(F_{\mathrm{V}}\) can be calculated using the Hamaker constant \(A\) and the distance \(z\) between the stylus tip and the measured surface as given in Equation (4) [11]. The Hamaker constant \(A\) is a constant specific to materials, and it is expressed as in Equation (5), where \(n\) and \(\Lambda\) are the number of atoms per unit volume and the constant of proportionality of London forces, respectively. When the molecules of the stylus tip and  
Figure 5: Schematic diagram of surface force between stylus tip and measured surface. the measured surface come close to each other by the van der Waals force, intermolecular repulsive force is generated between the molecules of the stylus tip and the measured surface. The molecules are stabilized at the most stable position, which corresponds to the position with the smallest potential energy. At this instant, the separation distance \(z\) is about 0.4 nm. \[F_{\mathrm{V}} = Ad/(12z^{2}) \tag{4}\] \[A = n^{2}\,\pi^{2}\,\Lambda \tag{5}\]  
Finally, the electrostatic force generated by an interaction between a charged particle and an uncharged surface is examined. When a charged particle comes into contact with an uncharged surface, the generated electrostatic force \(F_{\mathrm{e}}\) can be calculated using Equation (6) in the case that the distance between the stylus tip and the measured surface is negligible because it is remarkably small compared with the diameter of the stylus tip [11].