**1. Introduction**

Tribology is a wide-spread field with various application fields including the automobile industry. For instance, in passenger cars, friction occurs in the engine, transmission, tires and brakes. According to Holmberg et al. [1], the direct frictional losses, with braking friction excluded, represent 28% of the fuel energy. In total, 21.5% of the fuel energy is used to move the car. Therefore, there is an enormous potential to reduce the global energy consumption by enhancing the tribological performance of engine parts. There are also other fields of application, e.g., bionic engineering where, for example, the sandfish's skin, which is structured in a way that friction is reduced, serves as a model [2].

Laser surface texturing (LST) has shown to be a versatile tool to enhance surface functions. Especially in the field of tribology, it has been used to reduce the coefficient of friction (COF) as well as to extend the oil film lifetime (e.g., [3–8]). Several research studies in the field of LST and tribology deal with the fabrication of microholes or dimples forming closed surface structures. Important to mention is the pioneer work of Etsion [9], who has already given an overview in this field by outlining the variety in shape, size and textured density of dimples and the influence of these parameters on different tribological effects. In particular, three different working principles for dimples have been

distinguished, denoting that they can serve as (i) micro-reservoirs for lubricant in cases of starved lubrication conditions; (ii) micro-traps for wear debris in either lubricated or dry sliding conditions and (iii) micro-hydrodynamic bearings in cases of full or mixed lubrication. For the last point, cavitation is an important factor because it is necessary to achieve a positive pressure build-up which permits to reduce friction. Therefore, an overall low pressure level is required to be able to reach cavitation pressure. Further information about the cavitation phenomenon has been described by Braun [10].

An important parameter that significantly affects the tribological properties of surfaces is related to the fact if the textures have an open or closed characteristic. For example, if a lubricant can be displaced through open surface textures, starvation in the tribocontact might occur which results in a higher COF. On the other hand, lubricant can also be transported actively in open surface textures which might act as a supply of lubricant. The following paragraph therefore outlines the tribological effect of open surface textures.

For example, Wahl [5] compared the behavior of dimples (close) and channel-like (open) structures with different orientation with regard to the sliding direction. For the channels, the width was varied from 60 to 300 μm and the depths were 10 and 50 μm. It was found that for high sliding velocities (>0.5 m/s), dimples (close morphology) could reduce the coefficient of friction whereas the cross-like channels (open morphology) show a significant increase of the coefficient of friction, both compared to a polished reference sample. Nonetheless, Stark et al. [11] found that channel-like structures in a cross-like pattern can also reduce the COF, when using smaller feature sizes. In that work, two different laser texturing methods were applied, namely Direct Laser Writing (DLW) and Direct Laser Interference Patterning (DLIP). For the DLW method, the widths of the channels were 12 and 30 μm and the structure depth was varied between 3 and 24 μm. It was found, that the overall reduction for the mixed lubrication regime can be up to 33%. Additionally, it was found that similar results can be obtained applying DLIP method when fabricating channels with a cross-like pattern geometry. With this method the period of the textures were 4 and 8 μm and the depths around 1 μm. DLIP allows higher throughput compared to DLW [12]. A drawback, however, is that, due to their limited depths, DLIP textures are erased quickly inside the tribocontact where wear occurs. During the testcycle for the tribological evaluation it could be shown that the DLIP textures were erased quickly after the start of the cycle but afterwards the COF stayed at a low level. This was the finding of a temporal investigation of the relation between wear picture and reduction of COF (not shown). Since the DLIP textures inside the tribocontact are erased they cannot be responsible for keeping the COF low. However, in the direct surrounding of the contact point the textures are still present. They form a seamless transition into the tribocontact. The motivation of this work is to investigate whether lubricant can be transported actively in the DLIP textures and how this transport can avoid starvation. Starvation may lead to increased friction, wear and could result in the total malfunction of a machine. Starvation in the tribocontact occurs if oil is displaced from the tribocontact due to various effects such as inertial or thermal phenomena. A solution is to guarantee oil presence in the tribocontact.

#### *1.1. Solution Approach*

Klima [13] followed the approach of active lubricant transport towards the tribocontact with the objective to avoid starvation. The driving force for the transport in the channel-like textures is the capillary effect. For this effect, Lucas [14] and Washburn [15] derived the so-called Lucas–Washburn equation which describes a square-root dependence of the fluid column length *Lu*(*t*) from the time:

$$L\_u(t) = \sqrt{\frac{b\gamma \cos(\theta)}{4\eta}} t = \sqrt{W}\sqrt{t},\tag{1}$$

where *W* is a factor combining multiple parameters which essentially governs the speed of fluid advancement [13]. Equation (1) was verified experimentally for circular tubes [15]. However, there have also been investigations for microchannels with different cross-sections (e.g., [16,17]). They found a Washburn-like behavior with the same time law *Lu*(*t*) <sup>∝</sup> <sup>√</sup>*t*. Gruetzmacher et al. [18] investigated the flow behavior of a typical lubricant in DLIP textures and found an anisotropic spreading behavior with a higher spreading velocity parallel to the surface patterns compared to perpendicular to them. In the study of Klima [13], the textures ended at the transition region into the tribocontact in order to avoid fluid displacement. The difference in this study is that the textures are created throughout the entire sample, except for the last part, were selective structuring is applied. Since the DLIP textures have a low depth and therefore are erased quickly inside the tribocontact, the lubricant transport in the surrounding channel-like textures might be important. Since the DLIP textures are in the lower micrometer-range, microfluidics play an important role. Therefore, an important non dimensional number is introduced. Non dimensional numbers are of the utmost importance to compare the relative importance of different physical phenomena [19]. The Bond number compares the relative importance of gravity and surface tension:

$$\text{Bo} = \frac{\rho g l^2}{\gamma},\tag{2}$$

where *l* is a characteristical length, in our case the channel width. When the length *l* = *lc*, Bo= 1. *lc* is the capillary length which is defined as

$$d\_c = \sqrt{\frac{\gamma}{\rho g}},\tag{3}$$

where *γ* is the surface tension between the liquid and an adjacent fluid, typically air, and *g* is the gravitational acceleration. For this study, *lc* is around 2 mm, which is much larger than our characteristic length for the DLIP textures which leads to a small Bond number. A small Bond number indicates that surface forces dominate gravity. This is typical for microfluidics and often allows the neglect of gravitational influence.

In order to better understand the microfluidic transport processes, a numerical model is applied. For this model, the open-source platform OpenFOAM-<sup>R</sup> is applied. The two-phase wetting phenomena are described by the phase-field method. In this method, the Cahn-Hilliard equation describing the multiphase system is coupled with the Navier Stokes equations governing the flow phenomena. Due to the diffusive character of the Cahn-Hilliard equation, this method allows a motion of the contact line in combination with a no-slip boundary condition at the solid wall [20]. The first implementation of this method in OpenFOAM-<sup>R</sup> was done by Cai et al. [21] who validated the numerical model for different test cases and investigated the wetting phenomena of a droplet on a flat substrate. Fink et al. [22] investigated the hydrophobicity of micro-textured surfaces and showed good agreement between the numerical model and experiments, at least for the advancing phase.

#### *1.2. Outline of Article*

This study aims at elucidating the underlying phenomena of oil transport in DLIP textures and oil transition into the tribocontact. To this end, first, a numerical model is introduced and applied. It allows the investigation of the spreading (advancing phase) of a fluid in channels of different sizes which are typical for the case of laser textured surfaces (widths down to 10 μm). The results of the numerical investigation convey the idea that the capillary effect inside the channels is the driving force for the oil transport in the channels towards the tribocontact (similar to the work from Klima [13]). In a second step, experiments for understanding the fluid transport in DLIP channels are conducted. These are twofold. On the one hand, the fluid transport inside the channel-like textures is examined and on the other hand one possible transition process of the oil out of the channels into the tribocontact is investigated.
