*3.1. Numerical Results-Droplet Wetting on Line-Like Textures*

In Figure 3, the numerical results for the case of droplet wetting on line-like textures are shown. The wetting process of a droplet with an initial diameter of 140 μm and an impact velocity of <sup>1</sup> × <sup>10</sup>−<sup>6</sup> m/s is investigated for line-like textures with channel widths of <sup>10</sup> to 30 μm. The depth of the channels is kept constant at 10 μm. The equilibrium contact angle for all cases is 60◦. The mesh resolution is the same independent of the channel width and corresponds to a cell length of 1.67 μm in all directions. As a consequence, the minimum number of cells in a channel is 5. This is neccesary for a correct numerical resolution of the gas–liquid interface. In all simulations, the global Cahn number is Cn = 0.02. In Appendix A.2, a full overview of all numerical parameters and settings is given. Two different views are chosen in order to illustrate the spreading behavior along the line-like textures. The first view (left column in Figure 3) shows a cut through the domain at the geometrical center of the droplet. The second view (right column) shows one half of the spreaded droplet filling the channels from the bottom.

Figure 4 shows the quantitative results of the numerical investigations. For this, the spreading factor *χ* is defined as

$$\chi = \frac{\text{total extension of spreached droplet in channel}}{\text{initial droplet diameter}}.\tag{12}$$

This quantity is plotted over a relative simulation time *t* ∗ which is defined as *t* <sup>∗</sup> = *t*/*t*total, where *t*total = 0.039 s for all simulations.

From Equation (12), it is possible to see that the smaller the channel width the higher the spreading factor. The minimal channel width in the simulation is 10 μm. For smaller channels the cell size has to be reduced even further in order to ensure correct representation of the gas–liquid interface. However, smaller cells result in significant higher computational efforts which were not affordable for this study. However, already at a size of 10 μm, the spreading factor reaches a value greater than 2. Since the driving force is the capillary force inside the channels and since this force increases for smaller widths, the demonstrated effect even increases for smaller channel widths.

This phenomenon illustrates the anisotropic flow behavior of line-like textures which can be applied to guide lubricant towards the tribocontact. Therefore, the numerical results can serve as a theoretical demonstration of the possibility for active lubricant transport in laser textures of small sizes and with a line-like pattern.

**Figure 3.** Simulation of oil in channels of different widths at t = 0.039 s. The left column (**a**,**c**,**e**) illustrates a cut through the domain at the geometrical center of the droplet. The right column (**b**,**d**,**f**) shows one half of the spreaded droplet filling the channels from the bottom.

**Figure 4.** Quantitative evaluation of the numerical results comparing the spreading factor *χ* for the different channels widths as a function of the relative time *t* ∗.

#### *3.2. Experimental Results*

#### 3.2.1. Fabrication of Line-Like Surface Textures Using DLIP

Four different laser textures were created as explained in Section 2.3.1 applying DLIP. For all textures a line-like pattern was chosen. Figure 5 illustrates microscope pictures of all textures which are evaluated in this study Figure 5a–d and the corresponding profiles for these textures Figure 5e–h. Structures S1 and S2 have a period of 8 μm and S3 and S4 a period of 4 μm. Table 2 shows the depths of the textures. The value is an averaged value for the entire surface. Figure 6 shows a side view of the channels for texture S1. This image shows exemplarily the integrity of the produced surface textures. For the fluid dynamical behavior it is of utmost importance that no defects of the channels occur because otherwise the fluid is not transported along the channels.

**Figure 5.** Microscope pictures and profiles of laser textures evaluated in this study. Textures and profiles of S1 (**a**) and (**e**), S2 (**b**) and (**f**), S3 (**c**) and (**g**) and S4 (**d**) and (**h**).

**Table 2.** Depths of the laser textures S1–S4.


**Figure 6.** Side view of channels from texture S1. This microscopic image demonstrates in an exemplary manner the integrity of the laser produced textures. It can be seen that there are no discontinuities which lead to open structures.

#### 3.2.2. Fluid Transport Inside Laser Textured Surfaces

For evaluating the fluid transport inside the DLIP textures specially designed samples were used. As can be seen in Figure 7a, the sample has four reservoirs (1) with a capacity of around 1 μl each. The DLIP textures (2) start inside the reservoir. Furthermore, next to the DLIP textures a millimeter-scale (3) is placed in order to measure the traveled distance of the fluid front. One bar at the scale corresponds to 500 μm. This means that for each texture the experiment can be repeated four times. Figure 7b shows a zoom-in of the traveling fluid front at a distance of around 500 μm. The fluid transport in three different DLIP textures was evaluated (S2–S4).

Figure 8 shows the results of the fluid transport experiments. On the vertical axis the traveled distance in millimeter and on the horizontal axis the time in seconds are plotted. All curves show a square-root behavior similar to the Washburn-behavior mentioned in Section 1. This is due to the increasing friction with increasing length of the wetted part of the textures. The texture with a period of 8 μm allows slower transport than textures with a period of 4 μm (smaller channel width). Comparing the depths of the 4 μm textures, a higher depth (1 μm) leads to faster transport than a lower depth (0.7 μm).

**Figure 7.** Experimental investigation of fluid transport in laser textured surfaces. (**a**) Specially designed sample with different regions. 1: Reservoir, 2: Laser surface textures, 3: mm-scale (500 μm per bar); (**b**) fluid front propagating in laser textures at a distance of 500 μm.

**Figure 8.** Quantitative results of flow behavior in different Direct Laser Interference Patterning (DLIP) textures (S2–S4). The sample introduced in Figure 7 was used to measure the distance of the fluid column in different channel-like textures as a function of time. The curves show Washburn-like behavior with the fastest transport in channels with smaller period and higher depth.
