24th International Colloquium Tribology - January 2024 263 On the Relation between Friction and Surface Topography - Models and Challenges Charlotte Spies 1,2* , Arshia Fatemi 1 1 Robert Bosch GmbH, Renningen, Germany 2 University of Freiburg/ Department of Microsystems Engineering, Freiburg, Germany * Corresponding author: charlotte.spies@de.bosch.com 1. Introduction Sliding contact of different material combinations can be found in applications of various types. Naturally, the surface roughness is considered to influence friction and their relation is of interest for many researchers. With this knowledge, friction could be controlled in such a way to a.o. increase the efficiency or load carrying capacity of engineering parts. In the following, an introduction to the models and challenges of relating friction and surface topography for nominally flat surfaces is given. Therefore, in Section 2 and 3 models and parameters for surface roughness are given. Different types of friction and their relation to roughness are introduced in Section 4. Finally, a discussion on open questions and challenges, and a conclusion are given in Sections 5 and 6 respectfully. 2. Modeling roughness Surface topography and thus roughness can be considered with different approaches. For multi asperity models the roughness is assumed to consist of a multitude of asperities of the same order of magnitude. Examples of these multi asperity models are the theories by Greenwood and Williamson [1] and Bush, Gibson and Thomas-[2]. The surface roughness is assumed to consist of many asperities with a known height distribution. The asperities are spherical [1] or paraboloidal [2] and have the same radii or principal curvatures. With both theories, the real area of contact can be computed and is found to depend on different roughness parameters respectfully. Here, for the contact of rough surfaces the real area of contact is the area where asperities are in contact and differs from the larger nominal area of contact. In contrast, multi scale models take a different approach. Here, it is assumed that roughness occurs on different scales. When magnifying, roughness of a smaller scale would be found and will occur up to the atomic scale. The first to approach roughness with this multi scale theory was Archard [3]. In current research, Persson’s theory [4] and its adaptations and extensions use multi scale roughness. There, the real area of contact can also be related to the surface roughness. 3. Roughness parameters When defining roughness parameters, the multi scale nature of roughness is not necessarily considered. The amplitude and hybrid parameters given in norms and guidelines [5] can be computed for both types of models for roughness. The occurring challenges will be discussed in Section 5. Here, two roughness parameters, the root mean square (RMS) roughness Rq and the RMS slope Rdq, are highlighted. They are defined as (1) (2) where z(x) is the surface height and stands for averaging [5]. These parameters can be used to describe a rough surface and are also used with the approaches of Greenwood and Williamson [1] or Bush, Gibson and Thomas-[2] to model roughness. However, these roughness parameters are not sufficient to accurately describe multi scale roughness. For this, two different characteristics are needed [6], see Fig.-1. The probability density function gives the contribution of each surface height to the total roughness and is often assumed to be Gaussian [6,7]. The power spectral density-(PSD) describes the contribution of the respective wavelength to the total roughness [6]. Since many engineering surfaces show a self-affine fractal behavior from a specific scale on [7], the PSD shows a horizontal roll-off region and for higher wave vectors a linear decrease when plotting it on a log-log scale [7]. With these two characteristics, multi scale roughness can be described and related to a.o. real area of contact or friction. Fig. 1: Power spectral density and probability density function to describe a multi scale surface roughness. On the Relation between Friction and Surface Topography - Models and Challenges 264 24th International Colloquium Tribology - January 2024 4. Friction Friction can be distinguished based on its source. While other friction sources can occur, a.o. adhesive and deformative friction can be observed. In addition, viscoelastic friction can occur when at least one of the contacting partners is of viscoelastic material. There are different models to compute each type of friction, which also give their respective relation between friction and roughness. Bowden and Tabor use the real area of contact for the calculation of the adhesive friction [8], which itself can be determined e.g. with the Greenwood and Williamson [1] or Bush, Gibson and Thomas [2] theory. There, an increase in RMS roughness Rq or a decrease in RMS slope Rdq will result in an increase of adhesive friction. In contrast, a higher RMS slope will introduce a higher deformative friction [9]. Using Persson’s theory for viscoelastic friction, a relation between the obtained coefficient of friction and the PSD of the multi scale roughness can be shown [4]. In the same manner, the real area of contact when considering adhesion can be computed using Persson’s theory [10]. For the Persson theory, self-affine fractal surfaces with a Gaussian height distribution are assumed. 5. Discussion When relating friction and roughness a variety of challenges are arising. Especially the scale dependency of the roughness parameters must be taken into account. The different roughness parameters, e.g. the RMS roughness-Rq or RMS slope Rdq, are dominated by specific scales of roughness. While Rq is determined by long wavelengths, the short wavelengths determine Rdq [7]. This proposes multiple open questions for roughness measurements: Which cut-off wave lengths must be chosen? How can provided roughness measurements be interpreted and compared? In which way should requirements of roughness be defined? In addition, the multi scale nature of roughness is not always considered for the existing friction models. How can the models be adapted to account for multi scale roughness? Which scales are dominating for which type of friction? Can this knowledge be used for guidelines on roughness measurements of engineering parts? 6. Conclusion While there are many models and theories both for surface roughness and friction, their relation is an ongoing topic of research. Multi asperity models and roughness parameters can be used to relate adhesive or deformative friction and roughness. However, they do not consider the multi scale nature of roughness and the scale dependency of the roughness parameters. Various open questions due to these challenges were presented. References [1] Greenwood, J. A. and Williamson, J. B. P., Proceedings of the Royal Society of London Series A, vol. 295, no. 1442, pp. 300-319, 1966. [2] A. W. Bush, R. D. Gibson, and T. R. Thomas, Wear, vol. 35, no. 1, pp. 87-111, Nov. 1975. [3] Archard, J. F., Proceedings of the Royal Society of London Series A, vol. 243, no. 1233, pp. 190-205, 1957. [4] B. N. J. Persson, The Journal of Chemical Physics, vol. 115, no. 8, pp. 3840-3861, Aug. 2001. [5] “DIN EN ISO 21920-2: 2022-12”. Beuth Verlag GmbH. [6] B. Sista and K. Vemaganti, Wear, vol. 316, no. 1-2, pp. 6-18, Aug. 2014. [7] B. N. J. Persson, O. Albohr, U. Tartaglino, A. I. Volokitin, and E. Tosatti, J. Phys.: Condens. Matter, vol. 17, no. 1, pp. R1-R62, Jan. 2005. [8] F. P. Bowden, D. Tabor, Clarendon Press, 2001. [9] I. Hutchings and P. Shipway, in Tribology, Elsevier, 2017, pp. 37-77. [10] B. N. J. Persson, Eur. Phys. J. E, vol. 8, no. 4, pp. 385- 401, Jul. 2002.