# Material Removal Model for Lapping Process Based on Spiral Groove Density
## 3 Experimental Results
### Oil Film Thickness and Frictional Force  
The oil film thickness and frictional force were monitored during the lapping process to identify changes in the contact interface condition depending on the groove density. The displacement of the lapping head was measured to observe changes in the oil film thickness. Figure 9a shows the displacement of the lapping head over time when the groove density is 0.1. In the initial dry condition, the displacement was approximately 250 \(\upmu\)m on average and exhibited the form of a sine wave due to the run-out caused by the rotating platen and the discrepancy in the shape of the wafer attached to the head. At 7 s of starting supply of the slurry, the head displacement gradually increased to an average displacement of approximately 260 \(\upmu\)m. This indicates the development of an oil film due to the slurry, and the difference in the average head displacement can be considered as the oil film thickness. Figure 9b shows the changes in the frictional force when the slurry is supplied with a groove density of 0.2. A high frictional force is maintained in the initial dry condition without any slurry between the wafer and platen, but the frictional force rapidly decreases as soon as the slurry is supplied. This reduction in frictional force is caused by the formation of an oil film between the platen and wafer; as the oil film develops and the contact interface stabilizes, the frictional force gradually decreases and converges. As shown in Figure 10, the film thickness as a function of the groove density is obtained by calculating the difference in the average displacement of the lapping head between the dry and wet conditions. The oil film thickness decreases rapidly as the groove density increases, converging to approximately 1 \(\upmu\)m at a groove density of 0.3. That is, when the groove density is less than 0.3, the oil film develops at the contact interface between the wafer and platen, which decreases the frictional force upon the supply of slurry in the low groove density range. Figure 9: (**a**) Changes in displacement of the head upon supplying slurry (\(d_{g}=0.1\)); (**b**) changes in frictional force upon supplying slurry (\(d_{g}=0.2\)). Figure 11a shows the oil film thickness and frictional force as a function of the inverse number of groove density, and Figure 11b shows a Stribeck curve that represents the lubrication condition of the typical contact interface. The Stribeck curve shows the friction coefficient and oil film thickness at the hydraulic bearing based on the Sommerfeld number, which is calculated by multiplying the fluid viscosity by the relative velocity and dividing it by the load. Depending on the Sommerfeld number, the lubrication conditions are divided into boundary lubrication, mixed lubrication, and hydrodynamic lubrication conditions [27]. The boundary lubrication condition occurs when the relative velocity and fluid viscosity are low or the load is high, during which the asperities on both surfaces are in direct contact or the two surfaces are separated by an oil film that is monolayered or multilayered. In the mixed lubrication condition, both direct contact of surface asperities and dynamic pressure of the oil film are present, and the load carried by the hydrodynamic pressure increases as the oil film grows in thickness. In the hydrodynamic lubrication condition, both surfaces are completely separated by the fluid, and the entire load is carried by the hydrodynamic pressure [28]. As shown in Figure 11a, the oil film thickness increases and the frictional force decreases in proportion to the inverse of the groove density, and a similar tendency can be observed when compared to the Stribeck curve. When the two graphs are compared, the Sommerfeld number does not vary because the viscosity, velocity, and load remain consistent over the entire range of groove densities.