Factors affecting the measurement of roughness factor of surfaces and its implications for wetting studies

Abstract

Roughness factor is widely used for topography characterization of surfaces. The measurement of meaningful values of roughness factor depends on the instrument settings, e.g. spatial resolution and scan-size, the instrument characteristics (voxel dimensions), the post-treatment of discrete data array (tessellation algorithm), and finally the nature of the surface texture (e.g. fractal). To analyze the influence of all these parameters on the value of roughness factor and evaluate the influence of each parameter, different synthetic (mathematically defined) surfaces and acid-etched/passivated titanium surfaces were used. The titanium surface topographies were studied using two different microscopes: white light confocal microscopy (WLCM) and atomic force microscopy (AFM). In decreasing order of influence, roughness factor values are sensitive to the specific surface nature (fractal or non-fractal), the spatial resolution, the scan-size and the tessellation algorithm, whereas the instrumentation does not seem to be an important parameter in this study. The effect of the variability in the roughness factor depending on the above parameters in interpretation of the Young's contact angle and solid–vapor interfacial energy was studied based on the apparent contact angle observed and the Wenzel's equation. It was found that depending on how roughness factor is measured variations up to 20% in the Young's contact angle or solid–vapor interfacial energy may be found.

Introduction

Surface metrology is the science which studies the set of procedures to measure local or global features of surfaces at any scale. Topography data are superposition of surface features found at different scales. These features can be grouped according to their characteristic horizontal distance (i.e. the spatial wavelength). In decreasing order of wavelength, these features are the shape, the waviness, and the roughness. The roughness is related to the surface fluctuations (periodic or non-periodic) of short wavelengths features, whereas waviness describes surface periodic and non-periodic features at intermediate wavelengths. In general, in many surface phenomena (e.g. wetting and friction), shape is less relevant to the characteristic scale of the phenomenon. For example, for a given shape, roughness determines the behavior of a surface in problems related to wetting and tribology, reflecting the surface-nature of these phenomena [1], [2], [3], [4].

Any wavy rough surface z(x,y), which is single-valued, can mathematically be defined as a superposition of two terms, one associated with the waviness f(x,y) and another to the roughness n(x,y). The superposition of waviness and roughness will be different depending on the process which governs the surface evolution. These processes can be grouped as ballistic-wise process [5], where the roughness is added to the waviness in the vertical direction, and growth/erosion process [6], where that addition occurs in the normal direction.

However, practical surfaces are not univocally well defined because a topography acquired by finite-resolution techniques is actually an array of equispatial points (i.e. discrete surface). This limitation explains the dependence of any topographic descriptor on the surface reconstruction (tessellation or meshing), the resolution (length of array) and the sample size (scan-size or scan-length). Hence, depending on the measuring parameters (lateral resolution, scan-size, meshing) [7], [8], [9], [10], [11] and the instrument characteristics (vertical range, pixel size, voxel shape), different value of topography descriptors for the same surface can be found. Furthermore, since many natural surfaces are self-affine fractals [10], [12], [13], an intrinsic scale dependence on geometrical features is observed regardless of the mentioned discretization and instrument effects.

Roughness descriptors based on the amplitude fluctuations of the roughness of a surface are widely used in surface engineering. However, statistical parameters [14] provide no information about the horizontal distances between the surface features and therefore, the possible surface anisotropy.

In order to describe surface roughness and texture properly, hybrid parameters (based on a combination of height and spatial wavelength) [14], [15], such as the surface slope, are employed together with the amplitude parameters. In fact, surface area (three-dimensional interpretation of surface slope) is especially useful in phenomena such as wetting where the liquid–solid interactions depend on their contact area [16], [17].

A special case of dimensionless hybrid parameter relevant to wetting is the surface area ratio or roughness factor, r. This parameter is defined as the ratio between the area (A) of the topography determined by any measurement technique, and the nominal or geometrically projected area (A_g) of the topography:

$$r=\frac{A}{A_{\text{g}}}$$

Since A_g is obtained from the projection of the microscopic area, A, onto a reference surface, it is also known as apparent area. This reference surface is generally a horizontal plane obtained after a vertical filtering process [18]. As such, A_g will be equal to the scan area (product of the transversal and longitudinal scan-lengths).

The scan-size and sampling interval dependence of statistical, spatial and hybrid parameters have been studied by several authors [4], [17], [18], [19], [20]. Scan-size and spatial resolution was studied using different techniques such as white light confocal microscopy (WLCM) [18], profilometry [20] and atomic force microscopy (AFM) [4], [17], [20]. These studies have shown that roughness descriptors are affected by the resolution and the scan-size. The effect of the scan-size over the roughness factor from scanning force microscopy (SFM) topographies have also been studied [19]. However, comparative importance of roughness descriptors and their dependence on measurement parameters is very difficult to understand from above studies as each have looked at a particular system using a particular instrument. As such, a general understanding is lacking currently.

Furthermore, before applying the definition of roughness factor, r, in any measurement technique the waviness of a surface should be filtered (by instrument) in order to isolate the information corresponding to the roughness. The waviness filtering algorithms remove the waviness from topographical data by fitting functions or using different filters [21], [22], [23]. The practical filters used, are generally based on the minimization of vertical offsets [24]. Therefore, such algorithms can be only applied to surfaces governed by ballistic-wise processes. For surfaces governed by growth/erosion processes, a least squares orthogonal distance fitting technique should be used [25], [26]. The effect of the waviness filtering is not discussed in this study as instruments available does not allow manipulation of surface data using various filtering techniques (note that the majority of instruments use the vertical offset procedure as orthogonal distance filtering is generally underdeveloped and computationally expensive to implement).

In wetting phenomena, Wenzel [27] related the effect of topography of a rough, but chemically homogeneous surface to contact angle of that of an ideally smooth surface through Eq. (2):

$$\cos {\theta }_{\text{app}}=r\cos {\theta }_{\text{Y}}$$

where θ_app is the apparent contact angle (experimentally accessible angle) and θ_Y is the Young's contact angle (the angle related to the solid surface energy observed on a smooth surface). Consequently, the estimation of meaningful values of surface energy for rough surfaces will depend partially on a correct evaluation of roughness factor. Recently, several authors have studied the variation of apparent contact angle with roughness factor [17], [28] and the scan-size effect on the determination of roughness factor [19]. The surface roughness also plays an important role in study of superhydrophobic surfaces and its thermodynamic description [29], [30], [31].

The principal aim of this study is to comprehensibly analyze the parameters related to the measurement of r, i.e. scan-size, lateral resolution and meshing, and that of instrument characteristics, by using two different instruments. Therefore, one of the aims of this study is to evaluate, compare and determine the importance of the effects of each of these parameters on r. This is done by experimenting with synthetic (mathematically constructed) surfaces as well as roughened titanium surfaces. Furthermore, the results will be discussed in the context of wetting and contact angle interpretation, as an example. As such, this will be the first study that has examined the evaluation of r using a multitude of parameters and two instruments in a single study, and has analyzed the finding in the context of wetting.