A Simple Analysis of Texture-Induced Friction Reduction Based on Surface Roughness Ratio

Abstract

The effect of surface texture on friction reduction under fluid lubrication has been broadly acknowledged in the tribology community. However, the lack of understanding of the underlying mechanisms remains a challenge for the advancement of textured enhanced lubrication. Numerous models have been proposed, but they are almost all based on the hydrodynamic effect alone and have proven complex, system limited and unreflective of the beneficial secondary lubrication provided by residual lubricants within the texture. This paper presents a simple analysis of texture induced friction reduction based on the actual liquid–solid interface area and the secondary lubrication hypothesis. A simple model based on the surface roughness ratio (the ratio between the actual and projected solid surface area) of the textured surface was proposed which (1) is quantitative, straightforward, intuitive and sensitive to texture shape and area fraction; (2) directly reflects the proposed secondary lubrication mechanisms proposed in literature; (3) reflects the general data trend in the collected literature data. By focusing on the variations of key texture parameters, the proposed model combined with a sampling of independent studies in literature has demonstrated that (1) the effect of increased pit depth-to-diameter ratio (d/D) on friction reduction is most significant between 0.01 and 0.2; (2) further increase in d/D only marginally affects the friction coefficient; (3) texture’s area fraction plays a much weaker role than the depth/diameter ratio in friction reduction. This model may prove useful in gaining more insights into texture-enhanced lubrication by providing a tool to quantitatively studying the secondary lubrication mechanism often cited in the literature.

Graphical Abstract

Appendices

Appendix A

The Effect of Pit Geometry on Roughness Ratio

In Fig. 1, we assumed a cylindrical shape of the pits to simplify our calculation of the interfacial area and the roughness ratio. This is in accordance with the pit geometries in 75% of the studies in Fig. 2. However, in some cases, the pit geometry is best represented by a spherical shape likely introduced by Gaussian distribution of the laser beam intensity. Such is the case for 25% of the studies [6, 41, 42, 56, 58, 65] in Fig. 2. To consider the roughness ratio of pits with a spherical bottom, we adopt similar definitions of pit depth (d) and diameter (D) as the maximum vertical and lateral dimension of a single pit as illustrated in Fig. 5. The radius of the spherical bottom (R), could be calculated as

$$R=\frac{d}{2}+\frac{{D}^{2}}{8d}$$

(8)

Fig. 5

Roughness ratio curves with fixed texture area fractions (f) and varying depth-to-diameter ratios (d/D). With increased d/D, the roughness ratio increases proportionally for pits with a flat bottom and quadratically for pits with a spherical bottom

The surface area within a single pit (S), can be calculated as

$$S=2\pi Rd=\frac{\pi {D}^{2}}{4}+\pi {d}^{2}$$

(9)

And the extra surface area per a single repetitive texture unit as illustrated in Fig. 1 could be calculated as $\pi {d}^{2}/{L}^{2}$. Using the definition of roughness ratio and Eq. 3, the modified roughness ratio ($r^{\prime}$), could be written as

$$ r^{\prime} = \frac{{A^{\prime}}}{A} = 1 + \frac{{\pi d^{2} }}{{L^{2} }} = 1 + 4f\left( {\frac{d}{D}} \right)^{2} $$

(10)

Figure 5 directly compares Eqs. 4 and 10 by plotting the roughness ratio against the d/D for a variety of textured area fractions. In general, cylindrical pits have higher roughness ratio than spherical ones and the difference increases with the depth-to-diameter ratio and area fraction. Interestingly, a more accurate estimation of the roughness ratio in Fig. 2 using Eqs. 4 and 10 based on the pit geometry only slightly (< 0.6%) affected the exponent k in Eq. 5 (plot not shown). Part of the reason, as we speculate, is that most textures in Fig. 2 have low d/D (< 0.2) and area fraction values (< 0.3), and the difference between $r$ and $r^{\prime}$ is less than 10%.

Appendix B

Model Validation with the Best Performance Data

See Table 1 and Fig. 6.

Table 1 Table of the best performance data with the lowest friction reduction within each independent study in Fig. 2

Fig. 6

Normalized friction coefficient of the best performance data within each independent study in Fig. 2 plotted against the pit depth-to-diameter ratio (d/D). The same legend as in Fig. 2 was used. The lower bound of friction reduction predicted by Eq. 6 was shown as the dashed line

Appendix C

Model Validation Within Different Lubrication Regimes

See Figs. 7, 8 and 9.

Fig. 7

Normalized friction coefficient plotted against the bearing characteristic number for the best performance dataset in Table 1. The same legend as in Fig. 2 was used. The bearing characteristic number is defined as the ratio of the product of sliding speed (V, m/s) and lubricant viscosity (η, Pa$\cdot$s) to the contact pressure (P, N/m²)

Fig. 8

Normalized friction coefficient plotted against athe pit depth-to-diameter ratio and b the pits’ area fraction for studies with low bearing characteristic numbers (< 10^–2 μm) in Fig. 7. The same legend as in Fig. 2 was used. Lower bounds of friction reduction predicted by the proposed roughness ratio model were shown as dashed lines. Grey regions represent the mean deviation of the data from the lower bound within the high (d/D > 0.15) and low (d/D < 0.15) depth/diameter ratio domains

Fig. 9

Normalized friction coefficient plotted against a the pit depth-to-diameter ratio and b the pits’ area fraction for studies with high bearing characteristic numbers (> 10^–2 μm) in Fig. 7. The same legend as in Fig. 2 was used. Lower bounds of friction reduction predicted by the proposed roughness ratio model were shown as dashed lines. Grey regions represent the mean deviation of the data from the lower bound within the high (d/D > 0.15) and low (d/D < 0.15) depth/diameter ratio domains