1 Introduction

Natural joint lubrication has been subject to study since the 1960s [1,2,3]. To understand the lubrication mechanisms at work in total joint replacements, numerical and experimental approaches have been undertaken to relate the behavior of the friction coefficient (COF) and wear with lubricant film-thickness [1, 4,5,6,7]. The COF between two rubbing surfaces in dry contact is defined as the ratio of lateral friction force to normal load. When a lubricant is introduced, the dragging forces associated with the shearing stresses within the fluid moving between the counterparts must be considered. Most mathematical models for lubrication rely on the definition of suitable, simplified viscosity functions to mimic the non-Newtonian nature of synovial fluids, without fully grasping their complex rheological behavior [8].

A number of theoretical and experimental studies of lubrication regimes have focused on the COF between different tribopairs, using either synthetic or biological fluids [9,10,11,12,13,14]. For instance, under some experimental conditions, mixed lubrication was reported for CoCrMo on ultra-high molecular weight polyethylene (UHMWPE), lubricated with fetal bovine serum (FBS) [15, 16]. On the other hand, Murakami and Otsuki [17] reported using silicon oil to study the behavior of the lubricant film between the surfaces of a specific model of total knee replacement (TKR). Their results showed a clear dependence between the thickness of the lubricant film and the phase of a non-standardized walking cycle.

Other researchers have applied the theory of elastohydrodynamic lubrication (EHL), both numerically and experimentally, to predict the behavior of the lubrication mechanisms in joint-related tribopairs [18,19,20,21]. To this respect, Hook and Huang [22] demonstrated that, for a line contact, the viscoelastic properties of the soft material in the tribopair might play an important role in the lubrication process. From the experimental side, lubrication regimes in hip joint replacements have been amply investigated using optical interferometry [23]. The main conclusion of this line of research is that protein rheology, complicated by protein-protein and protein-surface interactions, renders the EHL theory useless to explain the thickness of the observed films. The tribopairs were not the ones used in real-life devices and, due to the restrictions of the experimental technique, the loading and kinematics at the contact point reproduced only a few clinically significant values of the corresponding walking cycle.

Experimental intervals for the independent variables were established for the present work from previously observed behavior of the COF during the walking cycle, reported by Barceinas et al. [24] for a specific commercial model of knee prosthesis were the loading pressure (Pm), entrainment speed (Vm), and sliding-to-rolling ratio (SRR) were obtained from the loading cycle and arthrokinematics prescribed by the ISO 14243-3:2014 standard [25]. Due to the constant variations of the parameters along the walking cycle, which is generally divided into stance and swing phases [26], the lubrication regime of the tribosystem gets masked. During the swing phase, we found that SRR is larger than 2 for the most part, peaking at about 20 [24]. Under this condition, both surfaces of the tribopair move in opposite directions. For clarity and simplicity purposes, we designate the kinematics for which SRR> 3 as super-sliding.

To improve our understanding of protein assisted lubrication (PAL) during the rubbing of AISI 316L on UHMWPE under super-sliding conditions when lubricated with FBS solutions at 37 ∘C, we performed a systematic study of the behavior of the COF within the chosen experimental intervals. Presumably, under these experimental conditions, protein interactions should have less of an effect on lubrication, leading to EHL as the expected lubrication regime for the tribosystem.

2 Materials and methods

2.1 Experimental setup

Figure 1 depicts a schematic representation of the experimental apparatus, described by Barceinas et al. [24]. In this experimental setup, an AISI 316L stainless steel ball (diameter: 19.05 mm, maximum surface roughness of 0.127 μ m), is loaded against the face of an UHMWPE disc, backed with an AISI 316L stainless steel disc. The instrument only takes integer values for load L, in newton. The lubricant cup possesses a temperature control graded in Celsius.

Fig. 1
figure 1

Schematic representation of ball-on-disc tribometer. (1) AISI 316L ball, (2) Lubricant, (3) UHMWPE disc, and (4) AISI 316L support

Full size image

The ball-on-disc tribometer measures the COF as a function of \(SRR=\left | V_{r}/V_{m}\right |\), where the relative and entrainment speeds at the contact point are defined as \(\left |V_{r}\right |=\left |V_{1}-V_{2}\right |\) and \(\left |V_{m}\right |=\left |(V_{1}+V_{2})/2\right |\), respectively. Both the ball and the disc rotate independently from one another, thus creating a variety of kinematic conditions: rolling without sliding (SRR = 0), rolling while sliding (0 < SRR < 2), sliding (SRR = 2), and super-sliding (SRR > 2).

The input variables in the MTM2 tribometer from PCS Instruments (London, UK) include the sliding-to-rolling ratio (SRR), the entrainment speed (Vm), the load (L), and the temperature (T) of the lubricant fluid.

2.2 Surface preparation

We obtained a number of slices from an as received UHMWPE white bar (diameter: 50.8 mm); these were rectified to a thickness of 4.20 ± 0.03 mm, then lathed down to a diameter of 46 mm. After the machining process, the discs were wet ground in running water until obtaining a center-line average roughness Ra= 0.025 ± 0.016 μ m, as measured with an atomic force microscope EasyScan2 from Nanosurf (Liestal, Switzerland). In Fig. 2, we shows a typical AFM image of the sample surface.

Fig. 2
figure 2

Surface topography of the polished UHMWPE surface measured by AFM

Full size image

2.3 Lubricant fluid

As for the lubricant testing fluid, we diluted as received bovine serum S1650 FBS (Biowest SAS, Nuaillé, France) with deionized and sterilized water until achieving a protein mass concentration of 20 g/L. Although the ISO standard makes allowances for the use of anti-microbial reagents, we dispensed with their use for the present work. To avoid sample contamination, we sterilized the laboratory materials for 15 minutes at 125 ∘C at 0.15 MPa in a LS-B7SL autoclave from Labtronic. The dilution was performed under an AHC-40 laminar flow hood from Esco.

2.4 Experimental design

Table 1 summarizes the experimental design that we used. We obtained the contact pressure values (Pm) from Hertz contact theory considerations [24], corresponding to loads of 3, 6, and 9 N. The SRR values range from 3 to 21 in steps of 2, whereas |Vm| increased from 10 to 100 mm/s in steps of 10 mm/s.

Table 1 Experimental design with three factors
Full size table

We used a disc for each level of load, measuring the COF for all possible combinations of SRR and Vm, within their respective range intervals. Since all three levels of load were randomized, each complete repetition of the experiment required three discs.

3 Results and analysis

We measured the COF five times along the entire Vm range, for each combination of the (Pm, SRR, Vm) triads. We depict typical results of these measurements in Fig. 3 for SRR values of 3, 7, 11, and 21, and for load L = 3 N, equivalent to a contact pressure of 19.1 MPa.

Fig. 3
figure 3

Typical Stribeck curves for the maximum contact pressure Pm = 19.1 MPa

Full size image

3.1 Regression analysis

We developed the prediction model using Design Expert™, adopting as significance level α = 0.05. The simplest best-fit model obtained for the experimental data was the power law model in Eq. 1:

$$ COF = {b_{0}}\cdot P_{m}^{b_{1}} \cdot SRR^{b_{2}} \cdot |V_{m} |^{b_{3}} $$
(1)

with determination coefficient R2 = 0.95. The corresponding ANOVA results showed that Pm, SRR, and |Vm| had significant effects on the COF. The power law model in Eq. 1 explains 95% of the variability in the COF. All the three factors related inversely to it; however, the effect of Pm was almost six-fold larger than either SRR or |Vm|, as can be observed in Table 2. The details on the regression analysis can be consulted in Appendix A.

Table 2 Power law model for the COF
Full size table

In Fig. 3, we can observe that the COF is a monotonic decreasing function of |Vm|, rather than following the more complicated variations of the COF reported by Lopez et al. [27] for water, which is a Newtonian fluid.

We obtained the response surfaces of the regression model in Eq.1 for all three levels of contact pressure, then contrasted them with the experimental data. As we show in Fig. 4, the regression model fits fairly well with the experimental data. The highest values of COF are observed at the lowest end of SRR and |Vm|.

Fig. 4
figure 4

Response surface of the models of powers. aPm = 19.1 MPa, bPm = 24.1 MPa, and cPm = 27.6 MPa

Full size image

For the lowest level of contact pressure (see Fig. 4a) the experimental data are distributed above and below the response surface; however, for the 24.1 MPa contact pressure (see Fig. 4b) the model overestimates the experimental COF in almost the entire surface, except for SRR = 3, where the highest value of COF is observed at the lowest value of |Vm|. For the highest level of contact pressure (see Fig. 4c) the experimental data are fairly evenly distributed above and below the response surface, overestimating at low values of SRR and entrainment. In the three cases, the COF is highest at the low ends of the experimental conditions, i.e. low SRR and |Vm|. Under these conditions, the relative speed between the surfaces is 30 mm/s, which is small enough for the proteins to interact with each other, thus contributing to increase the COF [28].

4 Conclusions

The power law expression developed herein explains 95% of the experimental results obtained for the response variable, and it is consistent with regression models obtained for the more complicated variations of the experimental parameters during a stride [8, 24]. Using different types of lubricant fluids, de Vicente et al. [29] developed regression models for Newtonian fluids which also involved powers of SRR, entrainment, and load. In their research, the dependence between the COF and the SRR is direct, rather than inverse. Garcia et al. [8] reported similar results for water under the kinematics and loading conditions of the walking cycle, while Lopez et al. [27] did it under experimental conditions similar to the ones in the present work. We conclude, therefore, that the inverse relation observed between the COF and SRR is due to the presence of proteins in the lubricant fluid. At high values of SRR, a smooth transition from mixed to full-film lubrication occurs, as |Vm| and SRR increase.