## 2 Materials and Methods
### Evaluation of the Effect of Lubricants on the Lubrication Regime, Friction, and Wear  
#### 2.3.1 Lubrication Regime Estimation  
A general approach to estimate the friction and wear behavior of a lubricating system is to determine the lubrication regime in which the system will operate: boundary, mixed, or hydrodynamic/elastohydrodynamic lubrication. These regimes are strongly dependent on the properties of lubricants. Each regime predicts different behavior, which can produce high or low friction--even mild or severe wear. In boundary lubrication, high friction and wear usually occur because the lubricant film is so thin that it allows larger contact between the surfaces. Contrarily, in hydrodynamic/elastohydrodynamic lubrication, the lubricant film is thick enough to eliminate the direct contact, thus reducing wear and friction. Mixed lubrication is a combination of the two previous modes, where the lubricant film is thicker to separate the surfaces, but there is still slight contact that could produce friction and wear. To deepen the characteristics of each lubrication regime, different studies [3; 4; 8] can be consulted.  
Knowing the minimum film thickness (\(h_{min}\)), a specific type of lubrication can be determined by calculating the lambda ratio (\(\lambda\)) or the film parameter. The boundary lubrication condition can be expected when \(\lambda<1\), mixed when \(1<\lambda<3\), and hydrodynamic/elastohydrodynamic lubrication can be expected when \(\lambda>3\). The film parameter can be calculated by Equation (1), where \(\sigma_{1}\) and \(\sigma_{2}\) are the surface roughnesses of Elements 1 and 2, respectively:  
\[\lambda=\frac{h_{min}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}}. \tag{1}\]  
The minimum film thickness can be obtained by the equations of [PERSON] and [PERSON] [18] for point contacts, as applied in [6]. For that purpose, the values of the pressure-viscosity coefficient (\(\alpha_{p}\)) of the lubricants are required. In this work, the pressure-viscosity coefficient was calculated following the procedure reported earlier in appendix A in [6]. In the previous work, the estimation of the \(\alpha_{p}\) was obtained from the properties of the lubricants by the methodology of [PERSON] [19]. This procedure first involves determination of the density and viscosity properties of a liquid lubricant at ambient pressure at two or more different temperatures. Then, the coefficients of compressibility (\(\beta\)) and thermal expansivity (\(\alpha_{T}\)), as well as the slope of the graph of the logarithm of viscosity vs. the reciprocal of temperature (\(S_{V/T}\)) are determined from these properties to obtain the \(\alpha_{p}\) by the relationship expressed in Equation (2):  
\[\alpha_{p}=-\frac{\beta(S_{V/T})}{2\alpha_{T}T^{2}}. \tag{2}\]  
Nevertheless, as an exception to this method, in the present study, the coefficient of compressibility (\(\beta\)) was calculated from the reciprocal of the isothermal secant bulk modulus (\(\beta_{T,S}\)) expressed in Equation (3) for any specific fluid and constant temperature conditions [20]. In Equation (3), \(v\) represents the kinematic viscosity at atmospheric pressure (cSt), \(T\) the temperature (\({}^{\circ}\)C), and \(P\) the pressure (bar):  
\[\beta_{T,S}=[1.30+0.15\log(v)]\Big{[}10^{0.0023(20-T)}\Big{]}\times 10^{4}+5.6P. \tag{3}\]