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

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.

Despite a large volume of literature on texture-induced friction reduction, most applications rely on a case by case approach in texture optimization and there is no universal principle for LST design. Theoretical analysis solely based on the hydrodynamic effect have often proven cumbersome, system specific and could differ significantly from the experimental results when a single testing parameter (load, velocity, etc.) was changed [55]. Secondary lubrication mechanism provided by the residual lubricants within the texture are widely acknowledged, rarely quantified and as important as, if not more than, the hydrodynamic effect to the lubrication [49,58,65,69,70].
It is well documented that metallic surface textures have some degree of friction or wear reduction under almost all lubrication conditions (starved, boundary, mixed, hydrodynamic, etc.) suggesting some common mechanisms at work. Based on this hypothesis and a survey of literature, we present here a simple analysis of LST induced friction reduction based on the secondary lubrication hypothesis and the surface roughness ratio of the textured surface. A simple model is proposed to corelate the surface roughness ratio with normalized friction coefficient and validated using data collected from the literature.

A Friction Reduction Model Based on Surface Roughness Ratio
A common finding in almost every LST study in literature is a possible relation between friction reduction and a secondary lubrication mechanism provided by residual lubricants within the textured pits (often referred as microrepository for lubricants, lubricant reservoir, etc.) [41,49,50,55,58,[62][63][64][65]. Although LST induced secondary lubrication can significantly impact friction reduction, it is not itself a quantifiable property; qualitative and subjective assessments remain as the 'gold standard' for describing this effect. One possible way in which LST could affect tribological properties is to change the solid-lubricant interfacial area during sliding. Orthogonal array of micro-pits is the most Fig. 1 A simple illustration of the roughness ratio (r) which is defined as the quotient of the actual solid surface area (A′) and the projected surface area (A) of a textured surface. Surface texture generally introduces extra area of the lubricant layer that is in contact with the solid, which may increase the secondary lubrication effect during sliding common and extensively studied texture in literature and is often defined using three parameters: the pit diameter (D), depth (d) and area fraction (f) as shown in Fig. 1. Extra surface area per a single repetitive texture unit as illustrated in Fig. 1 could be calculated as Dd∕L 2 (see Appendix A for a full discussion on the effect of pit geometry). Here, L is the center-to-center distance of neighboring pits. The actual surface area for the whole surface, A′, is the sum of the extra side area of the pits and the projected surface area, A: The surface roughness ratio (r) is defined as the ratio between the actual and projected solid surface area (r = 1 for an ideal smooth surface and r > 1 for a rough one): As the area fraction of pits, f, could be written as Equation 2 could be rewritten as In its definition, roughness ratio reflects the relative increase of counterface's real fluid-solid interface area from a flat surface when wetted with lubricants. There are quantities of reports in tribology literature that the increased fluid-solid interface area in textured surfaces often lead to increased wettability towards the lubricants [64,71,72]. Based on these observations, the fluid-solid interface area may similarly affect the secondary lubrication mechanism as the increased interfacial area may retain more lubricants during sliding from increased surface wettability or decreased spreading parameter [73][74][75]. And it is reasonable to expect a possible relation between the roughness ratio and the friction reduction on textured surfaces.
To test the above hypothesis, we collected a sampling of independent measurements on micro-pits texture induced friction lubrication from 24 independent studies [6, 11, 33, 40-47, 49-51, 54-56, 58-60, 62-65]. Together, they covered a wide range of material, load, speed and lubrication conditions. Figure 2 shows the normalized friction coefficient plotted against the roughness ratio. The normalized friction coefficient is defined as the ratio between friction coefficient on textured counterface and that on untextured counterface under identical sliding conditions within each study (load, speed, lubricants, etc.). Generally, the normalized friction coefficient decreases with increased roughness ratio which supports our hypothesis. The best performance data with the highest friction reduction within each independent study were tabulated in Table 1 in the Appendix B.
Because there is no theory to predict any particular relationship a priori, we fit the complete dataset of 100 data points in Fig. 2 with a power-law function: Because Eq. 5 has to satisfy (r, u/u 0 ) = (1, 1) as the ideal smooth surface is set as the reference, the value of a is ensured to be unity. Quadrature regression analysis was conducted, and Fig. 2 shows the best fit in the dashed line which has a k value of 3.19. The grey region represents the best-fit ± mean deviation of the data from the model.

Model Validation and Discussion
A sensible model should not only reflect the physics of secondary lubrication, but also be able to predict the effects of key design parameters on friction reduction. In this section, we use the present model to analyze the effects of two parameters frequently adopted in literature on micro-pits

Fig. 2
Normalized friction coefficient plotted against counterface roughness ratio for a collection of literature results using orthogonal arrays of micro-pits texture under fluid lubrication. Normalized friction coefficient is defined as the ratio between friction coefficient on textured counterface and that on untextured counterface under identical sliding conditions (load, speed, etc.). Counterface roughness ratio is calculated using Eq. 4 and texture parameters provided in original studies. A power law function is used to fit the data and the best fit is shown in the dashed line. Grey area represents the mean deviation of the data from the model texture design: pit's depth-to-diameter ratio and area fraction.
A large body of literature suggest micro-pits could act as small dynamic plain bearings under good lubrication and the pit depth-to-diameter ratio directly determines the hydrodynamic lubrication and the surface's load bearing ability [6,7,33,41,49,50,55,56,60,61,67]. This is further supported by the fact that friction reduction correlates more strongly with the pit depth-to-diameter ratio than with depth or diameter alone [1,23,39,54,60,66]. Pit depth-to-diameter ratio with maximum friction reduction predicted using the hydrodynamic lubrication theory often lies between 0.01 and 0.2 [43-45, 55, 66, 68]. However, such optimums differ significantly between studies and not always coincide with the experimental results [55,56].
In the proposed model in Eq. 5, normalized friction coefficient is a strong function of the pit depth-to-diameter ratio and should decrease with increased d/D. To test the predictability of the model, a lower bound of normalized friction coefficient could be written as using the maximum area fraction value (f = 1). 1 In theory, Eq. 6 represents the best possible case of friction reduction based on the roughness ratio hypothesis. Figure 3 plots the normalized friction coefficient against the d/D for the complete dataset in Fig. 2 and the lower bound was shown with the dashed line. The mean deviation of the data from the lower bound within the -high depth/diameter ratio domain (d/D > 0.15) is 1.9 times of that within the low depth/diameter ratio domain (d/D < 0.15), and the two deviations were shown with two different shades of grey in Fig. 3. In summary, 91% of all data points were above the predicted lower bound and the date generally fits the trend predicted by Eq. 6, especially below 0.15 pit depth-to-diameter value. Figure 6 in the Appendix B plots the normalized friction coefficient against the d/D for the best performance data within each independent study. Figure 3 suggests the proposed model in this study reasonably reflects the general data trend in the collected literature: the normalized friction coefficient decreases rapidly when d/D increases from 0 to 0.2, and further increase in d/D only marginally affects the friction coefficient. Interestingly, this critical domain of d/D coincides with the optimum Fig. 3 Normalized friction coefficient plotted against the ratio between pit depth (d) and diameter (D) for the complete dataset in Fig. 2. 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. 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. 91% of data points were above the predicted lower bound which supports our model of friction reduction hypothesis d/D domain (0.01, 0.2) predicted in literature based on the hydrodynamic lubrication theory.
In Fig. 3, it is easier to understand the curve's tailing off at higher d/D using the classical hydrodynamic or secondary lubrication theory as higher d/D reduces the hydrodynamic effect and makes lubricant exchange across the pit edge more difficult. It is more difficult to consider the impact of higher d/D on the lubricant film thickness from the surface wettability point of view. In physics, wetting of geometrically structured surfaces has been a focus of interest for decades. Studies on nanostructured surfaces have revealed that the initial fluid filling of a single pit does not depends on whether it stands alone or is part of an array [76]; whereas when the pits are close to saturation, the amount of fluids adsorbed depends strongly on the array as a whole and the detailed relations remain uncertain [77]. The key to solving such problem is to deepen our understanding of fluid adsorption and wetting transitions near individual wedges and cones [78] which is beyond the scope of this work.
Another way to check the effectiveness of the model is to plot the normalized friction coefficient against the area fraction f for the complete dataset in Fig. 2. The result is illustrated in Fig. 4 in which a lower bound was shown with the dashed line and could be written as (7) 0 min = (1 + 4f ) −3.19 using the maximum d/D value (d/D max ~ 1) in the collected data. There are three interesting results here: first, 91% of all data points were above the predicted lower bound; second, mean deviation of the data from the lower bound is insensitive to area fraction (σ f>0.2 = 1.12σ f<0.2 ); third, no optimum area fraction was noticed in the overall dataset. In Fig. 4, the diminished correlation between the model and the data supports the literature hypothesis that textured pit's area fraction plays a much weaker role than the depth/diameter ratio in friction reduction [1,60].
At last, it is also desirable to compare the sampled data with the proposed model within different lubrication regimes as the effect of friction reduction may vary considerably with lubrication conditions. A simple analysis was conducted in Appendix C in which we categorized the sampled data into two lubrication regimes based on the bearing characteristic number. The results were similar to Figs. 3 and 4 and suggested the proposed model were reasonably valid in both regimes. This, in other words, may imply that the secondary lubrication is insensitive to the lubrication conditions.

Closing Remarks
The tribology community widely acknowledges the beneficial effect of surface texture on friction reduction under fluid lubrication. However, the absence of a universal design principle has limited the adoption of LST in tribological design. Apparent discrepancies between models and experimental Fig. 4 Normalized friction coefficient plotted against the area fraction of pits (f) for the complete dataset in Fig. 2. The same legend in Fig. 2 was used. The lower bound of friction reduction predicted by Eq. 7 was shown as the dashed line. Grey regions represent the mean deviation of the data from the lower bound within the high (f > 0.2) and low (f < 0.2) pit's area fraction domains. 91% of data points were above the predicted lower bound which supports our model of friction reduction hypothesis results in the literature have unclear origins that are likely related to the secondary lubrication differences. Numerous friction reduction models have been proposed, but they were almost all based on the hydrodynamic effect alone and none have been widely adopted due largely to lack of portability, ease of use and sensitivity to the secondary lubrication. The roughness ratio model proposed in this study was based on the secondary lubrication hypothesis and intuitive, quantitative and sensitive to well recognized LST parameters like pit depth-to-diameter ratio and area fraction. A broad sampling of literature measurements showed the model reflected the general data trend in the collected literature. By focusing on the variations of key texture parameters, the proposed model has demonstrated that (1) the effect of increased 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 reduction, (3) pit's area fraction plays a much weaker role than the depth/diameter ratio in friction reduction. Literature strongly suggested the optimum surface textures with the highest friction reduction observed in relevant studies are due to a coupled effect of hydrodynamic and secondary lubrication. Broader application of the proposed model may prove useful for isolating and studying the effect of secondary lubrication in LST and provide more insights into relevant designs which aim at friction reduction and lubrication.

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 The surface area within a single pit (S), can be calculated as And the extra surface area per a single repetitive texture unit as illustrated in Fig. 1 could be calculated as d 2 ∕L 2 . Using the definition of roughness ratio and Eq. 3, the modified roughness ratio ( r ′ ), could be written as 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 (9) 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 ′ is less than 10%.

Model Validation with the Best Performance Data
See Table 1 and Fig. 6.  . 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 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 ⋅ s) to the contact pressure (P, N/m 2 )

Fig. 8
Normalized friction coefficient plotted against athe pit depthto-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 pre-dicted 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 depthto-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 pre-dicted 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