**Appendix B**

#### *Determination of Roughness Parameters*

In this study, calculations were performed using the roughness measured by images extracted from the laser microscope; these images are obtained from roughness height measurements. In this appendix, one of these images is shown as an example, and the equations for calculating the roughness parameters are presented.

#### Roughness Measurement

In order to calculate the boundary lubrication component, the roughness measurement is essential. As mentioned in the article, in the case of the deterministic approach, the real measured height of asperities is needed so as to calculate the separation between the opposing surfaces. In order to achieve the height of the asperities, the surface topography for textured surfaces is measured by using microscopic images. These analyses have been performed using a Keyence Color 3D Scanning Microscope (Keyence, Osaka, Japan), which uses a violet Laser λ = 388 nm. The result from a roughness measurement in the case of a chevron textured sample is presented in Figure A2, (standard lens 50× is employed).

**Figure A2.** Surface image by Laser Scanning Microscope.

The roughness data measured by confocal microscope is illustrated in Figure A3.

**Figure A3.** Surface roughness profile based on the Laser Scanning Microscope measurements (*Ra* = 0.11 μm).

Figure A4 shows the real measured probability density of asperities against the dimensionless asperity height, in order to compare the measured roughness with the Gaussian roughness distribution. The red line represents the Gaussian probability density distribution of one surface.

**Figure A4.** Real measured and Gaussian distribution of surface heights as a function of the dimensionless asperity height (*s*/σ).

The Summit is defined as a point that is higher than its eight neighbour points (see Figure A5).

**Figure A5.** Definition of a Summit.

To determine the radius of an asperity the 3-point definition is employed. In Figure A5, Δ*x*, Δ*y* are the steps or pixel size, and the asperity radii in both the *x* and *y* directions can be calculated as:

$$
\beta\_\chi^{-1} = \frac{z\_{\chi = \Delta \chi, y} - 2z\_{\chi, y} + z\_{\chi + \Delta \chi, y}}{\Delta x^2} \tag{A1}
$$

$$
\beta\_y^{-1} = \frac{z\_{\text{\$x \rightarrow \Delta x, y \,}} - 2z\_{\text{\$x, y \,}} + z\_{\text{\$x \rightarrow \Delta x, y \,}}}{\Delta y^2} \tag{A2}
$$

In Equation (A1), β*x* is the asperity radii in x direction respectively and in Equation (A2), β*y* is the asperity radii in *y* direction and *zx*,*<sup>y</sup>* is the local surface height at location (*x*, *y*). The combined summit radius β*<sup>i</sup>* of the radii in the two perpendicular directions β*x* and β*y* is obtained by:

$$
\beta\_l = \sqrt{\beta\_{xl} \cdot \beta\_{yl}} \tag{A3}
$$

To calculate the average summit radius (β) we have:

$$\overline{\beta} = \frac{1}{n} \sum\_{l=1}^{n} \beta\_l \tag{A4}$$

Therefore, the calculated average radius of asperity (β), is equal to 4.8 <sup>×</sup> <sup>10</sup>−<sup>8</sup> m.
