*2.1. Deterministic Asperity Model*

The Greenwood and Williamson contact model [40] assumes that asperities are spheres with a similar radius, and that asperity heights can vary randomly with a Gaussian probability distribution. However, in reality, this is rarely the case, as all of the asperities have the same radius; also, representing them as spheres or ellipsoids is more accurate. Moreover, the Gaussian height distribution is not an accurate approximation for most of the surfaces. As a recent advancement in optical measurement tools, the interference microscope can provide more accurate digital data of the surface topography which could be applicable for different applications, such as the calculation of the load carried by the deformed asperities of a rough surface when it is in contact with a flat surface (see Figure 1).

**Figure 1.** The schematic illustration of a flat surface and rough surface contact [34].

Figure 1 shows a flat on rough surface contact. For a given separation of two surfaces (*d*), the total force carried by the contact, the real contact area and the number of asperities *s*, are deterministically measurable by summing up the local contributions of the above-mentioned parameters. The asperities are assumed as ellipsoids with different radii (β*xi* are the asperity radii in the sliding direction, and β*yi* is in perpendicular direction), as well as the fact that they deform independently from each other. From Figure 2. the deformation of an asperity can be defined as Equation (1):

$$w\_i = z\_i - d$$

where *zi* is the individual summit height, and *wi* is the indentation of each deformed asperity.

For a given value of *wi*, by adding the individual components of each asperity contact, the asperities' normal load (*FC*) and the real contact area can be determined.
