*2.1. Materials*

For the texturing, the fluid transport evaluation and the tribological experiments, martensitic stainless steel X90CrMoV18 (1.4112) was used. The sheets were cut in rectangular shaped parts of 16 mm × 16 mm and 16 mm × 6 mm for the fluid transport and the tribological evaluation, respectively. All samples were polished by lapping in order to obtain a roughness of *Rz* = 0.20 μm. Before the experiments, the polished samples were treated with the laser except for the reference sample. All samples were cleaned using isopropanol in an ultrasonic bath for 10 min. Afterwards, the samples were dried and then rinsed with petrolether (boiling temperature 40 to 60 ◦C). This procedure was repeated before the laser process, before the fluid transport experiments and before the tribological evaluation.

As a lubricating oil, a Polyalphaolefin (PAO) with a viscosity of 39.5 mm2/s at 25 ◦C (operating condition) was used. It is the base oil of the commonly used grease Isoflex Topas L32. This oil has no additives in order to avoid side effects which are not induced by the surface structures. The properties of this oil were also used for the numerical model. There, it was assumed that the oil has the behavior of a Newtonian fluid.

As a tribological counterbody, a 100Cr6 ball with a diameter of 12.7 mm was used.

## *2.2. Numerical Methods*

This section introduces the mathematical formulation of the diffuse interface model. In our method the Cahn-Hilliard phase-field equation and the Navier-Stokes equations are coupled.

#### 2.2.1. Phase-Field Approach for Interface Evolution

In the phase-field method, the phase distribution of two phases, liquid (L) and gas (G), is described by an order parameter C. This parameter takes respective values *C*<sup>L</sup> = 1 and *C*<sup>G</sup> = −1 for the corresponding bulk phases and varies rapidly but smoothly in a transition layer, which is called the diffuse interface. The interface dynamics is governed by an evolution equation for C, the convective Cahn-Hilliard equation. It is important to note that it is the diffusion process of C that governs the motion of the contact line. The Cahn-Hilliard equation reads

$$\frac{\partial \mathcal{C}}{\partial t} + \nabla \cdot (\mathbf{u} \mathcal{C}) = \kappa \Delta \Phi,\tag{4}$$

where **u** is the velocity field, *t* is the time, *κ* the mobility and Φ the chemical potential, which is defined as

$$
\Phi = \frac{\lambda}{\varepsilon^2} \mathbb{C}(\mathbb{C}^2 - 1) - \lambda \Delta \mathbb{C}. \tag{5}
$$

Here, *ε* is an interfacial thickness parameter and *λ* is the energy mixing parameter,

$$
\lambda = \frac{3\sigma\epsilon}{2\sqrt{2}}.\tag{6}
$$

As can be seen, it depends on the surface tension *σ* and *ε*. The Cahn number Cn = *ε*/*L*ref relates the interfacial width to the macroscopic length scale.
