## 1. Introduction

The lotus effect, which describes the low wettability of a surface, is an important example for the wetting of super-hydrophobic surfaces. This is due to the surface microstructuring and the hydrophobic properties of epicuticular waxes on the leaf surface [

1,

2]. Water rolls off in drops and takes all dirt particles on the surface of the lotus leaf with it. There are various technical applications of hydrophobic, dirt-repellent surfaces such as self-cleaning roofing tiles, paintings, profiles as well as icephobic coatings for the prevention of ice accumulation [

3,

4,

5].

The wettability of a surface can be tailored by the chemical composition of the surface and the degree of surface roughness [

6]. The surface wettability is typically characterized by the contact angle, which represents the shape of the testing liquid on the solid. Contact angle measurements are the most surface sensitive of any common analysis technique having an analysis depth of ca. 0.5–1 nm [

7]. The roughness induced wetting is widely discussed in the literature.

In previous investigations, it was shown that if the diameter of the drop is three orders of magnitude larger than the scale of mean roughness value (

R_{a}) of the investigated surface the roughness does not affect the contact angle [

8,

9].

Contact angles of real surfaces, in contrast to ideal surfaces according to Young [

10], are described by the roughness of the surface. In general, two types of wetting states are observed besides wetting on a flat substrate (see

Figure 1a) on rough surfaces: The Wenzel and the Cassie state. As far as the Wenzel state is concerned, the surface grooves are filled by the water drop (see

Figure 1b). This leads to the pinning of the drop to the surface. The wetting liquid penetrates completely into the depressions of the rough surface, which is called homogeneous wetting. In the case of heterogeneous wetting of a rough and chemically homogeneous surface, the so-called Cassie state (see

Figure 1c), the drop does not penetrate the rough surface due to the entrapment of air. The resulting contact angle is larger than in the case of the Wenzel model because the interface between the two substances is smaller. Solid surfaces are divided into four categories. If the contact angle is less than 10 degrees, then the surfaces are superhydrophilic. Hydrophilic surfaces have contact angle values between 10 and 90 degrees. Contact angle values between 90 and 150 degrees are known as hydrophobic surfaces. Superhydrophobic surfaces such as the lotus leaf with its self-cleaning properties have contact angle values above 150 degrees or have a low tilting angle of 10 degrees [

11,

12]. Superhydrophobicity of surfaces can be adjusted by choosing an appropriate morphology or surface texture. In this way, a superhydrophobic surface is obtained instead of a hydrophobic one. The surface morphology can have micro-and/or nanoscale textures [

13]. The lotus leaf with its hierarchical structure consisting of nanoscale wax protrusions on microscale roughness exhibits superhydrophobic properties having a stable Cassie state. Air is trapped in the cavities and as a result the Cassie-Baxter state is stabilized, which produces superhydrophobicity [

14].

Different techniques to produce artificial superhydrophobic surfaces based on hierarchical structures have been studied. Especially, the fabrication of the rough structures with polymers as substrates are described in the literature [

15,

16,

17].

In contrast to ideal surfaces, real surfaces (see

Figure 1a) cannot be characterized by a single stable macroscopic contact angle, which is called the apparent contact angle. Consequently, there are different macroscopic contact angles [

18]. These angles, which are described by metastable states, are due to the locally different inclination of the topography and thus correspond to several local minima of the free enthalpy of a liquid drop on a solid surface. Energy barriers exist between these minima. In an energetic equilibrium, where the Gibbs energy has the lowest value, the system is in its most stable state. The corresponding most stable macroscopic contact angle is called θ

_{eq} [

19,

20]. It is calculated from the mean of the advancing and receding contact angles [

18,

21]. As a prerequisite for measuring the contact angle according to Marmur, a ratio between the drop diameter and the lateral extension of the roughness structures of at least three orders of magnitude is required [

18].

Contact angle hysteresis, which is the difference between an advancing and receding contact angle has been investigated and discussed in the literature for a long time. However, the underlying mechanisms are still controversial. Possible causes are the surface roughness [

22,

23], the chemical heterogeneity of the surface [

24,

25,

26], and time-dependent interactions of a solid with a liquid interface, resulting in swelling, liquid penetration into the surface area, and reorientation of the surface of functional groups [

27,

28]. Extrand and Kumagai stated that the range of the contact angle hysteresis was mostly a property of the system liquid-polymer [

29,

30].

The system polytetrafluoroethylene (PTFE)/water has been investigated by the dynamic sessile drop and tilting angle methods in several studies. Schulze et al. have received the hysteresis-free contact angle value, which is considered as a thermodynamic equilibrium contact angle from sessile drop measurements on different rough PTFE surfaces [

31]. Other authors such as Extrand and In Moon investigated the contact angle on flattened PTFE surfaces and PTFE spheres [

32]. Ruiz-Cabello et al. studied smooth PTFE surfaces (

R_{a} < 0.1 µm) and found a general disagreement between the sessile drop and the captive bubble methods [

33]. Pericet-Camara et al. investigated PTFE surfaces by the tilting plate technique and the sessile drop method. As far as the tilting angle drop method is concerned, the sessile drop platform is inclined in steps of a 0.5° tilting angle with respect to the horizontal plane. Gravity moves the drop downwards in the inclined plane at the upper side. The drop is brought into an asymmetrical shape and only moves when the drop has reached a certain size. The advancing contact angle is the angle at the bottom, the angle at the top is the receding contact angle. They obtained contact angle hysteresis values with high values between 40–60° [

34].

Contact angle goniometry is a powerful technique to observe the contact angle between the tangent to the liquid-gas and liquid-solid interfaces at the three-phase contact line. This method is widely used as a screening experiment for smooth surfaces [

35]. To estimate the contact angle from the drop profile different methods are used: Spherical cap approximation [

36], polynomial fitting [

37], tangent line or Young-Laplace equation [

38]. It was shown that different algorithms give different values of contact angles [

39,

40].

The contact angle goniometry is not very accurate for rough and hydrophobic surfaces, because the contact point between the axial location of the base line and the projected droplet boundary can appear distorted [

41]. There are substantial inaccuracies as far as image processing is concerned, especially for surfaces which are superhydrophobic. Optical errors lead to systematic errors with respect to the determination of the droplet shape and tangent line [

42]. It is difficult to determine the location of the baseline. The deviations of the measured contact angles can be large [

43]. Contact angle measurements using the sessile drop technique depend on the experience and skills of the user. There are large deviations even if an experienced user performs the measurements [

44]. Vuckovac et al. have shown that errors increase for superhydrophobic surfaces. The increase of the image resolution can be reduced slightly [

42]. However, Heib and Schmitt have developed the so-called high-precision drop shape analysis (HPDSA), which involves a transformation of images from sessile drop experiments to calculate physically meaningful contact angles and to improve the disadvantages of the sessile drop goniometer method [

45].

In comparison with the contact angle goniometry, the Wilhelmy balance technique has many advantages. The method is fully automated, and the influence of the experimenter is significantly reduced. Furthermore, it has a precise definition of the kinetic stages of advancing and receding and is efficient in the measurement of advancing and receding states (e.g., immersion rate) [

46]. Recently, it was shown that the modified Wilhelmy balance technique can also be used for irregular shaped specimens instead of regular shaped samples having a constant perimeter [

47]. In previous works, the contact angle hysteresis of elastomers was correlated with roughness factors which were obtained from white light interferometry measurements [

19,

48]. Little work has been done to correlate contact angles and contact angle hysteresis values of PTFE with roughness parameters, such as fractal dimension and height profile data [

49,

50,

51]. To address this issue, we used various smooth and rough PTFE surfaces to investigate the influence of roughness parameters similar to the mean value

R_{a} and surface descriptors such as the fractal dimension

D_{f} on the contact angle and contact angle hysteresis. The surface descriptors such as the fractal dimension was calculated from the white light interferometry data. For this purpose, the roughness length (height difference correlation (HDC) function) and the cube counting methods were used to calculate the fractal dimension

D_{f} and the surface descriptors.

## 2. Materials and Methods

A polytetrafluoroethylene TECAFLON PTFE naturally produced by Ensinger GmbH, Nufringen, Germany was investigated in this study. The PTFE test specimens (length: 3 cm, width: 1 cm, thickness: 2 mm) were covered on both sides with different types of SiC sandpaper using Matador Nassschleifpapier P60-ST7000 (Starcke GmbH & Co. KG, Melle, Germany) and pressed for 3 min at 30 bars between two polished press plates using the vulcanization press WLP63/3.5/3 (Wickert Maschinenbau GmbH, Landau in der Pfalz, Germany) at a temperature of 25 °C. The samples were subsequently cleaned with 2-propanol (p. a., Merck) in the ultrasonic bath Sonorex Super (Bandelin electronic GmbH & Co. KG, Berlin, Germany) for 2 hours at (23 ± 1) °C. Different grits were used (2000, 1000, 400, 240, 60, and 30) which correspond to the coarseness of the abrasive particles or coarseness of the surface. It is a dimensionless number; the larger this number is to be considered, the smaller is the diameter of the grinding grains (grain size). This corresponds to the grain sizes 10.3, 18.3, 35, 58.5, 269, and 642 according to ISO 6344-2 and 3:1998 [

52,

53]. Additionally, unmodified PTFE specimens were used as reference material (denoted as unmodified).

A white light interferometer (“FRT-CWL 300”, lateral resolution: <2 μm; height resolution: 10 nm) from FRT (Fries Research & Technology GmbH, Bergisch Gladbach, Germany) was used to investigate the topography of the seven different PTFE surfaces and to obtain the height profiles of the surfaces. An area of 4 mm^{2} was measured with 1000 × 1000 measuring points per sample. In addition, an area of 64 mm^{2} was examined for the very rough surfaces with the grits 30 and 60.

The measured raw data for the representation of the topography were considered using the software program “Igor Pro” after deduction of the plane. “Igor Pro” was also used to calculate the surface descriptors, the fractal dimension, as well as the mean roughness (

R_{a}) (according to DIN EN ISO 4287) [

54]. Fractal dimensions were calculated also using the cube counting method in the software Gwyddion (GPL, Brno Czech Republic).

For the conventional Wilhelmy method, by which the surface tension of the liquids is determined, a rectangular, DIN-standardized, and roughened platinum plate PT 11 (DataPhysics Instruments GmbH, Filderstadt, Germany) is applied (see

Figure 2a) [

55]. We have used the DCAT 11 Dynamic Contact Angle Tensiometer (DataPhysics Instruments GmbH, Filderstadt, Germany) for our investigations. A precision clamp PSH 11 (DataPhysics Instruments GmbH, Filderstadt, Germany) with defined dimensions, which is available for the DCAT 11 system was used instead of a platinum plate (see

Figure 2b).

Figure 2c reveals the immersion and emersion cycles of the PTFE samples for the determination of advancing and receding contact angles. The specimens were first cleaned for 2 h in an ultrasonic bath using 2-propanol as liquid and after the fabrication process the samples were cleaned with deionized water (DI) and dried before use at room temperature (23 ± 1) °C. Before dipping the sample in a test liquid, the surface tension of DI water was determined by the Wilhelmy Pt-Ir-Plate. The surface tension of DI was 72.7 mN m

^{−1} at (23 ± 1) °C.

First, the sample is attached to the sample holder which is shown in

Figure 2b. The holder with the attached sample is mounted on the force sensor holder of the tensiometer. After balancing the weight force (

$mg=0$), the test specimen is immersed in the water and emerged with a scan rate of 0.1 mm/min. The technical design of the device ensures that the weight force was balanced. The accuracy of the balance is ±100 μg. The forces

F_{adv} and

F_{rec} (s. Equations (1) and (2)) are measured as a function of the immersion depth

h.

$V=h\xb7b\xb7d$ is the volume and

$l=2\xb7\left(b+d\right)$ is the wetted length of the test specimen, and γ

_{lv} designates the surface tension and

${\rho}_{lv}$ the density of the solvent.

By linear regression to the immersion depth zero, the buoyancy force

${F}_{a}=V\xb7\rho \xb7g$ can be eliminated from the recorded force-distance diagrams. If the sum of buoyancy and weight force is equal to zero, the resulting force corresponds to the wetting force. Hence, the corresponding measured forces

F_{adv} and

F_{rec}_{,} from which the contact angles

${\theta}_{adv}$ and

${\theta}_{rec}$ can be calculated are obtained separately for the extrapolation of h to 0 (i.e.,

$V=0$) [

56].