1 Introduction

The most desirable lubrication conditions for a bearing contact are a high load carrying capacity together with low bearing friction. The latter can be best attained with full film lubrication such as hydrodynamic or elastohydrodynamic regime, which is governed by the viscous effect of the lubricant due to the absence of asperity contacts. Using a thin lubricant results in low viscous friction, but it also leads to the loss of bearing load carrying capacity, i.e. the reduction in film thickness. The selection of a lubricant with an appropriate viscosity to ensure a sufficiently high load carrying capacity and possibly the least friction relies on the practice of professional tribologists or lubrication engineers. However, practical situations exist in which reducing the viscous friction or increasing the load carrying capacity might not be achieved by varying the choice of lubricants of different viscosities. For example, an overly viscous lubricant may lead to the complete halt of micro-machineries, whereas a very thin lubricant may not be robust enough to facilitate the complete separation of two running surfaces.

In the last couple of decades, an interfacial phenomenon called boundary slip has attracted considerable interest due to its potential positive effects on bearing lubrication. Boundary slip refers to the scenario in which the adjacent liquid molecular layer slides on the solid surface. Early studies on slip in lubrication focussed on the change in lubrication characteristics due to localised slip regions in the bearing contact. The works were conducted based on the idea that the onset of boundary slip would occur once the shear stress attained certain critical values (referred to as critical shear stress). Jacobson et al. [1, 2] and Zhang and Wen [3] described the phenomenon using the viscoplastic model and the limiting shear stress (or critical shear stress) of the oil film, respectively. The occurrence of boundary slip in bearing contacts depends on the running conditions of the bearing, i.e. the induced shear stress at the liquid/solid boundary. Following the advancement in oleophobic surface treatment technology [4], studies have evolved to pre-treat the surfaces and directly investigate the effect of slip surfaces in bearings [5].

Spikes [6, 7] proposed a new plane pad slider bearing composed of a stationary slider with a homogeneous slip surface and a non-slip moving surface. The lubricant adheres to the non-slip moving surface to facilitate entrainment while slides on the slip slider surface resulting in a very low Couette friction. Choo et al. [8, 9] verified experimentally the idea of Spikes [6, 7] using a low-load tribometer, in which the slip surface was provided by a smooth lyophobized sapphire surface [8] and the treatment with friction modifier additives [9]. Furthermore, Kalin and Polajnar [10] inferred that boundary slip occurred resulting in friction reduction in the test with DLC/DLC contacts at the elastohydrodynamic lubrication regime. They reported that the reduction could be more than 30% when compared with steel/steel contacts at the same testing conditions. Even the affirmative effect of boundary slip on bearing friction reduction has been substantiated, the boundary slip also reduces the hydrodynamic load capacity. Spikes [6] analysed the effect of boundary slip with a special case of zero critical shear stress, which allows boundary slip to occur on the entire stationary slider surface. The resultant hydrodynamic load capacity reduced by half of that of the non-slip boundary condition.

Recent studies [11,12,13] on the critical shear stress effect on the load carrying capacity of plane slider bearing show that the load support is reduced notably with moderate values of critical shear stresses and approaches the level of the Reynolds load support with very large critical shear stresses equivalent to no-slip conditions. The great loss of load support or reduction in film thickness is possibly due to the acquired partial slip (heterogeneous) on the slider surface, which differs from the full-slip (homogeneous) or no-slip surface. Fortier and Salant [14], Guo and Wong [15, 16], Wu et al. [11] and Tauviqirrahman et al. [13] analysed the effect of heterogeneous-slip on hydrodynamic lubricated contacts and indicated that the lubrication behaviour of the bearing is governed by where the slip domain on the static sliding surface is. On the basis of their calculations, a bearing with low friction and high load capacity can be obtained through a careful design of the slip area. More information about numerical studies on partial-slip sliders is available from a review paper by Senatore and Rao [17]. Many studies about the partial-slip surface effect have been conducted; however, they are theoretical, and no relevant experimental results have been reported thus far, especially for heterogeneous-slip bearings with increasing load capacity.

The lubrication performance of a bearing is determined by low friction and high load carrying capacity, which can be theoretically facilitated by partial-slip or heterogeneous-slip surface design. Accordingly, in this study, we numerically analyse the proliferation of slip region on a slider surface with the change in critical shear stress, which is the criterion for the occurrence of boundary slip, for specific running conditions. Insights are extracted from the theoretical study to derive effective slip/no-slip patterns on a slider surface to improve the hydrodynamic performance even beyond what the classical no-slip bearing can provide. The proposed slip/no-slip patterns are validated experimentally through slider bearing tests.

2 Test rig of Lubrication Study

Lubrication experiments were conducted using an optical slider-on-disc test rig [18], the schematic of which is shown in Fig. 1. The sliding contact was composed of a stationary slider and a rotating glass disc. Interferometry was adopted to measure the lubricant film thickness. A coherent light beam was projected onto the contact, and the interference images can be captured with a high-speed camera. To increase image quality, a thin chromium layer was coated on the top surface of the glass disc (20% reflectance for beam splitting). An additional SiO2 layer (200 nm thick) was coated on top of the chromium layer for protection. Therefore, the lubricant was sandwiched by the slider and the glass disc during the experiment. The inclination of the slider can be fixed and changed by adjusting eight backing bolts. The inclination can be obtained directly with the number of interference fringes shown in the interferogram. When the glass disc begins to rotate, the slider is lifted because of the hydrodynamic effect. The lubricant film thickness can be obtained directly using conventional interference equations and the multibeam interference method [19,20,21] by recording the change in intensity during the start-up (accelerating) or die-down (decelerating) phase.

Fig. 1
figure 1

Schematic of the applied test rig

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3 Specimen Surface and Working Conditions

The oleophobicity treatment of the surface can be facilitated by sputtering thin oleophobic films on the surface, such as DLC [22]. However, the coated films are of finite thickness, typically about 1 μm, which makes it not suitable for the present study as it forms a physical step at the edge of the coated domain. The alternative is to modify the wettability of liquid lubricants on the steel surface by using physio-absorbed or chemisorbed boundary films of different molecular structures [23]. The present study adopted the latter approach by using a fluorocarbon-CFx-based coating (EGC, a proprietary oleophobic thin film coating). The specimen sliders used in the study were made of steel, and the size of the sliding surface was 4 mm × 4 mm. EGC was coated on the slider surface to generate boundary slip at the specific zone. It was applied simply by spinning coating, and ultra-thin physio-adsorbed boundary films were formed on the specimen surface. Two slip patterns were fabricated as shown in Fig. 2. They were evaluated together with a non-slip slider (original steel surface) and an entirely EGC-coated slider for benchmarking. Figure 3 shows a surface profile measurement that covers the EGC-coated surface domain and the uncoated original steel slider surface. The coating thickness is probably in nanoscale because it is hardly identified in Fig. 3. Any step effect on the lubricating film formation can thus be ruled out. The roughness, Ra, of the steel slider and EGC coating is 9 and 39 nm, respectively.

Fig. 2
figure 2

Two specimen slip patterns: a lateral central stripe and b inverse triangle

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Fig. 3
figure 3

Surface profile measured across the EGC-coated domain (left) and uncoated original surface (right)

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The contact angle (CA) and contact angle hysteresis (CAH) of EGC and steel surfaces were measured with PAO 40 and tabulated in Table 1. The CA and CAH of EGC are more than double and around 30% less than those of steel, respectively. The effects of wettability on hydrodynamic lubrication have been studied by various research groups. Guo et al. [18] experimentally investigated the surface effects on hydrodynamic lubrication by measuring the film thickness by using different oleophilic/oleophobic surfaces and proved that bearing surfaces with a small CA would result in high lubricating film thickness. Guo et al. [24, 25] evaluated the correlations of various interfacial parameters with the hydrodynamic lubricating film thickness and concluded that CAH is the key parameter to represent the strength of the interface between the liquid lubricant and the solid bearing surface. Surfaces with a small CAH are prone to generating boundary slip at the solid/liquid interface. EGC has a large CA and small CAH (Table 1), which is why it is adopted in this study to form two different heterogeneous slip/no-slip surface patterns, as shown in Fig. 2. PAO 40 was applied as the lubricant in this study, and its viscosity was 0.83 Pas at room temperature (23 °C).

Table 1 CA and CAH formed between PAO 40 and specimen surfaces
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4 Results and Discussion

4.1 Theoretical

Spikes [6] proposed a simple critical shear stress slip model that illustrates the onset of boundary slip on the basis of the attainment of the critical shear stress. The critical shear stress model can be expressed as

$$\left\{ {\begin{array}{*{20}c} {~\tau _{{{\text{theoretical}}}} < \tau _{{{\text{c,}}}} ~~~~~\tau = \eta \dot{\gamma }~{\text{ for no - slip}}} \\ {\tau _{{{\text{theoretical}}}} \ge \tau _{{{\text{c,}}}} ~~~~~\tau = \tau _{{\text{c}}} {\text{ for slip}}} \\ \end{array} } \right.$$
(1)

where τ is the shear stress, \(\tau_{{\text{c}}}\) is the critical shear stress, \(\eta\) is the viscosity and \(\dot{\gamma }\) is the shear rate. A modified Reynolds equation for the hydrodynamic lubrication of sliding contact of a non-slip moving surface and a slip stationary surface, that is governed by a critical shear stress [Eq. (1)], can be derived [6] as

$$\frac{\partial }{\partial x}\left( {\frac{{h^{3} }}{\eta }\frac{\partial p}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {\frac{{h^{3} }}{\eta }\frac{\partial p}{{\partial y}}} \right) = 3\left( {u_{1} \frac{\partial h}{{\partial x}}} \right) - \frac{3}{2\eta }\left( {\frac{{\partial \left( {{\text{sgn}} \left( {u_{s} } \right)\left| {\tau_{hx} } \right|h^{2} } \right)}}{\partial x} + \frac{{\partial \left( {{\text{sgn}} \left( {v_{s} } \right)\left| {\tau_{hy} } \right|h^{2} } \right)}}{\partial y}} \right)$$
(2)

where \(p\) is the pressure;\(\tau_{{{\text{hx}}}}\) and\(\tau_{{{\text{hy}}}}\) are the shear stress at the static surface (subjected to slip and governed by [Eq. (1)] in the x and y directions, respectively;\(u_{1}\) is the velocity of the no-slip moving surface; and sgn\(\left( {u_{{\text{s}}} } \right)\) and sgn\(\left( {\nu_{{\text{s}}} } \right)\) are the sign of the slip velocities in the x and y directions, respectively. Taking the assumption that \(\tau_{{\text{c}}}\) is a system (a particular pair of a surface and an oil) constant, one can use Eq. (2) to determine the effect of \(\tau_{{\text{c}}}\) on the bearing load support for a given running condition. Figure 4 depicts the variation of film thickness (i.e. load carrying capacity) with critical shear stress of the slider bearing (4 mm × 4 mm) used in this study for load = 4 N, \({u}_{1}\) = 25 mm/s, and \(\eta\) = 0.83 Pas (PAO 40). The insets depict the acquired slip/no-slip domains on the slider surface at different critical shear stresses. The two extremities in critical shear stress in Fig. 4 represent the perfect-slip and no-slip conditions, respectively, as illustrated by insets (1) and (5). At the limit \(\tau_{{\text{c}}} = 0\) (perfect slip), Eq. (2) becomes,

$$\frac{\partial }{\partial x}\left( {\frac{{h^{3} }}{\eta }\frac{\partial p}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {\frac{{h^{3} }}{\eta }\frac{\partial p}{{\partial y}}} \right) = 3\left( {u_{1} \frac{\partial h}{{\partial x}}} \right)$$
(3)
Fig. 4
figure 4

Change in film thickness with critical shear stress (Slider area: 4 mm × 4 mm, load: 4 N, speed: 25 mm/s, viscosity: 0.83 Pas)

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The classical Reynolds equation for sliding pad bearing [26] assuming non-slip boundary conditions is

$$\frac{\partial }{\partial x}\left( {\frac{{h^{3} }}{\eta }\frac{\partial p}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {\frac{{h^{3} }}{\eta }\frac{\partial p}{{\partial y}}} \right) = 6\left( {u_{1} \frac{\partial h}{{\partial x}}} \right)$$
(4)

A comparison between Eqs. (3) and (4) shows that a perfect-slip bearing has only half of the load carrying capability of the non-slip counterpart. As the load support of a bearing is directly proportional to the square of the film thickness, the perfect-slip film thickness is about 70% of the non-slip, as shown in Fig. 4. When the critical shear stress is very small, corresponding to inset (1), full-slip occurs on the static bearing surface, where boundary slip occurs on the entire slider surface. Its film thickness is about 70% of the non-slip at inset (5), which can be obtained with the classical Reynolds equation [Eq. (4)]. With the increase in the critical shear stress, the static bearing surface may still be at full slip, as shown in inset (2). When the magnitude of the critical shear is large enough and the last two terms on the right-hand side of Eq. (2) become significant, the film thickness reduces largely. This film thickness reduction ceases at the minimum value [inset (3)] once the full-slip condition stops. A tiny region at the inlet of the static bearing surface turns into non-slip [inset (3)]. Figure 4 shows that the largest film thickness is the Reynolds’, i.e. no boundary slip exists [inset (5)]. With the decrease in the critical shear stress, boundary slip would firstly occur at the outlet of the contact. As the outlet film thickness is always the smallest, its shear stress is thus large and readily attains the critical value. Inset (4) depicts that boundary slip exists in a large trailing region of the bearing contact, which prompts the oil to move out of the bearing and leads to a severe drop in film thickness.

The results in Fig. 4 elucidate that no matter how serious the slip on the bearing surface is, the bearing load carrying capacity can never be better than that under the condition with Reynolds load support (of the non-slip surface). However, the film thickness would be largely affected if the bearing surface is composed of slip and non-slip domains (partial-slip surface). The theoretical data shown in Fig. 4 illustrate a strong downward trend in the film thickness curve upon the partial-slip stimulus. Following the same line of thinking, the downward trend can be reversed and the film thickness can be possibly augmented through the appropriate design of the partial-slip pattern on the bearing surface.

We devised and tested two candidate partial-slip patterns, namely, lateral central stripe and inverse triangle, as shown in Fig. 2. The former design is based on the concept that the non-slip domains help retain the lubricant in the mid-region of the bearing contact to increase the load support. The latter design is largely the opposite of the partial-slip pattern shown in inset (4) in Fig. 4, such that the lubricant entrainment is augmented at the inlet slip domain and the outflow through the outlet and the two side edges is relatively restricted by the non-slip boundary conditions. Boundary slip is allowed to exist in the slip domain only, whereas the other area is non-slip (of infinite critical shear stress). Their characteristic curves of film thickness against critical shear stress were obtained using the critical shear stress model [Eq. (1)] and are plotted in Fig. 5, showing completely different characteristics from that of the entire bearing surface subjected to slip, as shown in Fig. 4. As \(\tau_{{\text{c}}}\) approaches zero, in which boundary slip exists in the whole slip domain, significant increases from the Reynolds film thickness (large \(\tau_{{\text{c}}}\) and no-slip) are realised, as shown in Fig. 5. Both partial-slip surface patterns show a positive effect on enhancing the load carrying capacity of the bearing, as indicated by the increase in film thickness. The increases in film thickness achieved by the central stripe pattern and the inverse triangular pattern are 46% and 85%, respectively.

Fig. 5
figure 5

Change in film thickness with critical shear stress for slip-patterned bearing surfaces (Symbols indicate the measured film thickness. Load: 2 N, speed: 25 mm/s, viscosity: 0.83 Pas)

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4.2 Experimental

To verify the accuracy of the applied test rig, an original steel slider (non-slip) was initially applied. Two loads at 2 and 4 N were adopted, and the inclination was fixed to 1/1797. Figure 6 shows the change in the measured film thickness with speed. To increase the measurement accuracy, each experiment was repeated six times, and the average value was adopted. The result showed that the test has good repeatability; therefore, no error bar was plotted here. The measured film thickness increases with speed and decreases with load, which is consistent with experiences in lubrication. Furthermore, the theoretical values obtained from the classical Reynolds equation [Eq. (4)] were provided and compared with experimental data. Apparently, the measured lubricant film thicknesses are almost identical to the classical lubrication theory solutions. This finding indicates the accuracy of the test rig and the non-slip assumption of the conventional lubrication theory. Then, the steel slider that was entirely coated with EGC was adopted to repeat the experiments. The measured film thickness with the EGC slider under 4 N load is also illustrated in Fig. 6. The film thickness in the whole speed range is much lower than the non-slip theoretical values. As mentioned previously, a 30% reduction in the non-slip film thickness is acquired if the critical shear stress approaches zero. Nevertheless, the reduction in EGC film thickness is greater than that value.

Fig. 6
figure 6

Change in film thickness with speed with steel surface and EGC coating

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The two partial-slip patterns (Fig. 2) were evaluated. Sliding hydrodynamic lubrication tests were conducted with EGC-coated sliders in the shape of central stripe and inverse triangle. The working conditions were identical to those of the experiments on steel and entirely EGC-coated sliders. Figure 7 illustrates the measured film thickness with the central stripe pattern under two loads, namely, 2 and 4 N. Evidently, the film thicknesses with the patterned slider are higher than those of the steel slider in the speed range of 5 mm/s to 25 mm/s. The maximum increment of the film thickness reaches 12.5% and 7.7% for 2 and 4 N, respectively.

Fig. 7
figure 7

Film thickness vs. speed of bearing with central stripe slip pattern for two different loads (Hollow symbols: steel slider without any slip pattern; solid symbols: slider with slip pattern)

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The measurement was then repeated with the inverse triangular patterned slider, and the results are plotted in Fig. 8. Similarly, the inverse triangular pattern can increase the lubricant film thickness, achieving an average increment of approximately 6.2% in the entire testing speed range.

Fig. 8
figure 8

Film thickness vs. speed of bearing with inverse triangular slip pattern for 2 N load. (Hollow symbols: steel slider without any slip pattern; solid symbols: slider with slip pattern)

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All the measured film thicknesses are much higher than the roughness of steel and EGC surfaces. Thus, the roughness effect can be ignored. The improvement of the lubricant film thickness with the two partial-slip patterns is the result of the change in boundary conditions on the slider/lubricant interface because of the application of EGC coating on the designed area, which is prone to generate boundary slip for its smaller CAH. However, compared with the predicted film thickness with \(\tau_{{\text{c}}} = 0\) as shown in Fig. 5, the improvement using EGC is relatively marginal, which means that the critical shear stress of the two partial-slip sliders in the experiments is not small enough. As mentioned before, the critical shear stress is determined by the oleophobic properties of the applied coating. Therefore, the load capacity of the designed patterns can be increased further if highly oleophobic coatings are adopted.

Figure 9 illustrates the change in non-dimensional pressure P on the midsection (Y = 0) in the sliding direction X, (non-dimensional terms: \(X=x/L\), \(Y=y/B\), \(P=p({h}^{2}/\eta {u}_{1}L)\), where L and B are the length and breadth of the slider bearing, respectively). The values of \({\tau }_{c}\) for the central stripe and inverse triangular slip patterns adopted in the calculation are 2440 and 2170 Pa, respectively, as extracted from Fig. 5. The maximum pressures with the central stripe and inverse triangular slip patterns are higher than those in the non-slip case, and their locations are close to the outlet of the slider. The maximum pressure along Y = 0 locates at the border of EGC coating region near the outlet for the central stripe pattern. This phenomenon corresponds to the variation of slip velocity at the border, as shown in Fig. 10a, wherein the slip velocity reaches the maximum value and it drops rapidly to zero in the outlet, leading to a high pressure in that location. Figure 10b illustrates the slip velocity distribution with the inverse triangle slip pattern. The slip velocity increases gradually from the inlet to the outlet and reaches the maximum at the rear boundary of the slider.

Fig. 9
figure 9

Change in non-dimensional pressure along Y = 0 with and without EGC coating (2 N, Highlighted zone: region of central stripe)

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Fig. 10
figure 10

Slip speed distribution in the X direction (Load: 2 N, \({U}^{*}={u}_{s}/{u}_{1}\))

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5 Conclusion

This paper experimentally and numerically investigates the effect of oleophobic bearing surface on bearing load carrying capacity. The study was conducted with a two-dimensional lubricated contact of a stationary slip slider surface and a moving non-slip plane. The results indicate that if slip occurs at the entire stationary surface, then the hydrodynamic load carrying capacity must be less than the Reynolds load support, regardless of the magnitude of the boundary slip. The numerical and experimental results demonstrate that the appropriate slip pattern on the stationary bearing surface can improve the bearing load carrying capacity. Such improvement can be considerable if the slip pattern acquires a zero critical shear stress. Two slip patterns, namely, central stripe and inverse triangle, were evaluated in the study. Both slip patterns have a large no-slip area in the trailing region of the stationary slider surface, and they demonstrate the enhancement of the bearing load carrying capacity beyond the predictions of the classical Reynolds equation. The optimal effect of a heterogeneous slip/no-slip surface on bearing performance can be accomplished through the optimization of the slip pattern on the slider surface and the attainment of zero critical shear stress surface treatment. The present results carry implications for enhancing the load capacity and reducing friction in liquid-lubricated micro-machines.