In Situ Measurement of Grease Capacitive Film Thickness in Bearings: A Review

Abstract

The majority of bearings in the world are lubricated by grease, and nearly 80% of premature bearing damage is attributed to lubrication issues. Accurate measurement and prediction of film thickness are crucial aspects of understanding the lubrication mechanism in grease-lubricated bearings. This work focuses on grease film thickness measurement using the capacitance method in real bearings. It comprehensively reviews the current status, identifies key challenges, and proposes solutions. Mechanisms of mainstream electronic components in capacitance measurement were reviewed for the first time. It enables more accurate capacitance measurement. A new capacitive model and electric network to measure film thickness in fully flooded, starved, and mixed regimes are developed. It is more comprehensive compared to current models. Classic dielectric models are reviewed, and suitable ones for lubricants are proposed. It facilitates a more precise film thickness measurement. Finally, a new grease film thickness model (bearing raceway) is proposed based on the 113 literature capacitive film thickness data points from five different authors. The satisfied R-squared value indicates a strong correlation.

1. Introduction

The film thickness of lubricants plays a vital role in predicting the performance and life of rolling element bearings. For oil-circulated bearings, where a fully flooded condition is typically assumed, the two surfaces are completely separated, and the film thickness is not affected by a change in oil flow at the inlet. This allows for relatively accurate predictions using the classic Hamrock and Dowson film thickness equation [1]. Grease, however, presents a great challenge. Unlike oil, grease-lubricated bearings are usually considered to operate under starved lubrication or “starvation” conditions.

While the rolling surfaces remain separated, the film thickness is highly dependent on the lubricant supply at the inlet. The physics and chemistry of lubricating grease in a rolling bearing are not well understood [2]. Poll et al. reviewed the generation mechanism and criteria of starvation in rolling contacts [3]. Although significant efforts have been conducted, a reliable model to predict the grease film thickness in rolling element bearings has yet to be developed.

Accurate measurement of lubricant film thickness is crucial for model development and verification. Albahrani et al. comprehensively reviewed in situ film thickness measurement methods in the elastohydrodynamic lubrication (EHL) regime [4]. These methods fall into three categories reflective of their fundamental principles: electrical methods (voltage discharge, resistance, capacitance); optical methods (X-ray, interferometry, infrared, Raman, fluorescence, particle image velocimetry); and acoustic methods. Liu et al. compared the EHL experimental techniques with numerical methods and found good agreement in most cases [5].

For grease film thickness measurement specifically, optical interferometry [6,7,8,9], X-ray [10], ultrasonic [11], resistance [12,13,14,15], and capacitance methods [16,17,18,19,20,21,22] have been explored. Optical interferometry provides high resolution for film thickness measurement. However, it cannot mimic the starved lubrication condition in a rolling element bearing. Most optical measurements of grease were conducted in fully flooded conditions using a scoop. X-ray, ultrasonic, and resistance methods are limited in resolution, typically only reaching the micrometer level, whereas grease film thickness falls into the nanometer range in starved conditions. Consequently, the capacitance method has emerged as the most suitable tool for in situ film thickness measurement in grease-lubricated bearings. It exhibits comparable resolution with optical interferometry [23] and can be applied to real bearings without complicated modifications. The characteristics of each method are summarized in Table 1.

The capacitance method was invented by Crook in 1958 [24]. The evolution of this method has been reviewed by Cen et al. [25] and Glovnea et al. [26] and will not be elaborated here. Beyond film thickness measurement, capacitance has been applied widely, such as in lubricant degradation monitoring [27,28,29,30,31,32,33,34,35,36,37,38] and metallic surface contact detection [39,40,41,42,43,44,45,46,47].

The capacitance method applied to film thickness measurement has been reviewed in terms of lubricant type, rig configuration, central film thickness range, and lubrication regimes (summarized in Table 2). The majority of the research to date has focused on oil lubrication in the fully flooded conditions. However, several critical knowledge gaps remain:

• Capacitance measurement methods: Currently, no commercial bearing testing rig is available. Several homemade rigs were recently reported to measure bearing capacitance [17,23,48,49,50,51]. A deeper understanding of the physics behind capacitance measurement methods is important for future capacitance testing rig design and optimization;
• Capacitive film thickness models and electric networks: Current models primarily focus on the film thickness in fully flooded conditions. A more comprehensive model encompassing all three lubrication regimes (fully flooded, starved, and mixed) is necessary to reflect bearing operation in the real world;
• Dielectric constant of lubricants: Accurate film thickness determination relies heavily on the lubricant’s dielectric constant, which is a function of temperature and pressure (among other parameters). A suitable model for grease is needed;
• Grease starvation: It is necessary to develop a grease starvation model based on current capacitive film thickness data.

This review aims to bridge these knowledge gaps in the field of grease-lubricated bearings and pave the way for future advancements.

Table 1. Characteristics of different grease film thickness measurement methods.

Elementary Particle/Fundamental Carriers	Method	High-Spatial Resolution (~nm)	Real-Bearing Test	In Situ Measurement
Electron	Capacitance	√	√	√
	Resistance		√	√
Photon	Optical	√		√
	X-ray		√
Phonon	Ultrasonic		√	√

Table 1. Characteristics of different grease film thickness measurement methods.

Elementary Particle/Fundamental Carriers	Method	High-Spatial Resolution (~nm)	Real-Bearing Test	In Situ Measurement
Electron	Capacitance	√	√	√
	Resistance		√	√
Photon	Optical	√		√
	X-ray		√
Phonon	Ultrasonic		√	√

Table 2. Summary of lubricant film thickness measurement using capacitance methods.

Lubricants	Rig Configuration	Capacitance Measurement	Central Film Thickness Range (µm)	Lubrication Regime	Research Area	Reference
Oil and grease	Twin-disk rig and ball bearing test rig	Measure the capacitor voltage change during charging	0.2~1.2	Fully flooded and starved	Bearing currents and electrical erosive wear study	[16]
Grease	Deep-groove ball bearing and optical rig (WAM5)	Lubcheck + oscilloscope	0~0.2	Fully flooded and starved	Difference in the lubrication mechanism between ultra-low speed and medium speeds	[18]
Oil and grease	Deep-groove ball bearing	Capacitive voltage divider (Lubcheck)	0.04~0.2	Mix to fully flooded	Film thickness and condition monitoring (metallic contact time fraction)	[49]
Grease	Deep-groove ball bearing	Lubcheck Mk3	0.5~3	Starved	Grease starvation quantification	[50]
Grease	Angular contact ball bearing	Constant–current charge	0~0.7	Starved	Influence of lubricating grease composition on grease service life and tribological performance characteristics in high-speed rolling bearings	[48]
Oil	FEA simulation on a ball bearing		10~1000	Fully flooded	Analyze bearing current discharges and their effect on bearing damage by using simulation	[52]
Gear oil	Pair of gears	RLC bridge and oscilloscope	5~140	Fully flooded	Studied the effect of change in speed, oil viscosity, and helix angle on the load-carrying capacity of the oil film	[53]
SAE 30, 60	Engine crankshaft journal and bearing shell	Transformer ratio arm bridge	0.65~8	Fully flooded	Measured the relative effects of oil rheology and engine operating condition	[54]
Grease	Deep-groove ball bearing	Lubcheck Mk3		Starved	Grease replenishment study	[55]
Mineral oil	Twin-disk machine	Capacitance bridge	0.5	Fully flooded	In general terms, it appears the elasto-hydrodynamic theory may have slightly underestimated the film thicknesses	[56]
Oil	Engine connecting–rod bearing	Capacitance bridge	0~10	Fully flooded	Thermal effects on oil film thickness of an engine connecting–rod bearing	[57]
Sunflower oil	Ball-on-disk tribometer	LCR meter	0.001~0.01	Boundary	Measured the thickness of boundary films under a pure sliding contact between metallic surfaces	[58]
Air/oil	Online transportation tube	Electrical capacitance tomography sensor	60~140	Air/oil transportation	Monitoring of the in-tube air/oil flow	[59]
Oil	Single-cylinder diesel engine	Capacitance probe	0.2~8	Fully flooded	Oil film thickness between engine cylinder liner and piston ring	[60]
Oil	Motored engine	Capacitance probe	0~4	Fully flooded	Oil film thickness between engine cylinder liner and piston ring	[61]
Oil	Internal combustion engine	Capacitance transducers	0~80	Fully flooded and starved	Measured capacitance between the sensor and piston ring	[62]
Grease	Angular contact ball bearings		Relative film thickness	Starved	Combined grease life testing with film thickness measurement	[63]
Grease	Radial ball bearings	High-frequency oscilloscope		Starved	Bearing film thickness measurement	[64,65]
Oil	Single-cylinder diesel engine	Capacitance transducer	6~18	Cavitation	Film between piston ring and liner	[66]
Oil	Diesel engine	Capacitance transducer	2~8	Fully flooded	Film between piston ring and cylinder	[67]
Oil	Modified MTM test rig	Digital storage oscilloscope with large memory	0.05~0.9	Fully flooded	Electric discharge behavior and current damage in EV motor bearings	[68]
Oil	Four-disk machine	Capacitance gauge	4~26	Fully flooded	Measured film shape, pressure, and temperature profiles	[69]
Oil	Diesel engine	Capacitance gauge	0.4~2.5	Fully flooded	Piston rings and the cylinder liner	[70]
Oil	Diesel engine	Capacitance gauge	1~11	Fully flooded	Piston rings and the cylinder liner	[71]
Oil	Two-spherical ball contact	LCR meter	0~2.5	Fully flooded	Measured the film thickness from pure rolling to pure sliding	[72]
Oil	Thrust pad bearing	Capacitance probes	2.5~25	Fully flooded	Compared the deflection of a circular pad with theory	[73]
Oil	Twin disk	LC oscillator	0.2~1.8	Fully flooded	Prediction of lubrication film thickness under conditions of different surface velocity directions	[74]
Oil	Bearing in a diesel engine	Transducer	10~60	Fully flooded	Measured the hydrodynamic oil film thickness present in slide bearings	[75]
Oil	EHD rig	Impedance/gain-phase analyzer	0.015~0.2	Fully flooded	Quantitative measurements of film thickness and consideration of cavitation	[76]
Oil with different polarity	EHD rig	Impedance analyzer	0.01	Fully flooded	Influence of a lubricant’s polarity on capacitance measurements	[77]

Table 2. Summary of lubricant film thickness measurement using capacitance methods.

Lubricants	Rig Configuration	Capacitance Measurement	Central Film Thickness Range (µm)	Lubrication Regime	Research Area	Reference
Oil and grease	Twin-disk rig and ball bearing test rig	Measure the capacitor voltage change during charging	0.2~1.2	Fully flooded and starved	Bearing currents and electrical erosive wear study	[16]
Grease	Deep-groove ball bearing and optical rig (WAM5)	Lubcheck + oscilloscope	0~0.2	Fully flooded and starved	Difference in the lubrication mechanism between ultra-low speed and medium speeds	[18]
Oil and grease	Deep-groove ball bearing	Capacitive voltage divider (Lubcheck)	0.04~0.2	Mix to fully flooded	Film thickness and condition monitoring (metallic contact time fraction)	[49]
Grease	Deep-groove ball bearing	Lubcheck Mk3	0.5~3	Starved	Grease starvation quantification	[50]
Grease	Angular contact ball bearing	Constant–current charge	0~0.7	Starved	Influence of lubricating grease composition on grease service life and tribological performance characteristics in high-speed rolling bearings	[48]
Oil	FEA simulation on a ball bearing		10~1000	Fully flooded	Analyze bearing current discharges and their effect on bearing damage by using simulation	[52]
Gear oil	Pair of gears	RLC bridge and oscilloscope	5~140	Fully flooded	Studied the effect of change in speed, oil viscosity, and helix angle on the load-carrying capacity of the oil film	[53]
SAE 30, 60	Engine crankshaft journal and bearing shell	Transformer ratio arm bridge	0.65~8	Fully flooded	Measured the relative effects of oil rheology and engine operating condition	[54]
Grease	Deep-groove ball bearing	Lubcheck Mk3		Starved	Grease replenishment study	[55]
Mineral oil	Twin-disk machine	Capacitance bridge	0.5	Fully flooded	In general terms, it appears the elasto-hydrodynamic theory may have slightly underestimated the film thicknesses	[56]
Oil	Engine connecting–rod bearing	Capacitance bridge	0~10	Fully flooded	Thermal effects on oil film thickness of an engine connecting–rod bearing	[57]
Sunflower oil	Ball-on-disk tribometer	LCR meter	0.001~0.01	Boundary	Measured the thickness of boundary films under a pure sliding contact between metallic surfaces	[58]
Air/oil	Online transportation tube	Electrical capacitance tomography sensor	60~140	Air/oil transportation	Monitoring of the in-tube air/oil flow	[59]
Oil	Single-cylinder diesel engine	Capacitance probe	0.2~8	Fully flooded	Oil film thickness between engine cylinder liner and piston ring	[60]
Oil	Motored engine	Capacitance probe	0~4	Fully flooded	Oil film thickness between engine cylinder liner and piston ring	[61]
Oil	Internal combustion engine	Capacitance transducers	0~80	Fully flooded and starved	Measured capacitance between the sensor and piston ring	[62]
Grease	Angular contact ball bearings		Relative film thickness	Starved	Combined grease life testing with film thickness measurement	[63]
Grease	Radial ball bearings	High-frequency oscilloscope		Starved	Bearing film thickness measurement	[64,65]
Oil	Single-cylinder diesel engine	Capacitance transducer	6~18	Cavitation	Film between piston ring and liner	[66]
Oil	Diesel engine	Capacitance transducer	2~8	Fully flooded	Film between piston ring and cylinder	[67]
Oil	Modified MTM test rig	Digital storage oscilloscope with large memory	0.05~0.9	Fully flooded	Electric discharge behavior and current damage in EV motor bearings	[68]
Oil	Four-disk machine	Capacitance gauge	4~26	Fully flooded	Measured film shape, pressure, and temperature profiles	[69]
Oil	Diesel engine	Capacitance gauge	0.4~2.5	Fully flooded	Piston rings and the cylinder liner	[70]
Oil	Diesel engine	Capacitance gauge	1~11	Fully flooded	Piston rings and the cylinder liner	[71]
Oil	Two-spherical ball contact	LCR meter	0~2.5	Fully flooded	Measured the film thickness from pure rolling to pure sliding	[72]
Oil	Thrust pad bearing	Capacitance probes	2.5~25	Fully flooded	Compared the deflection of a circular pad with theory	[73]
Oil	Twin disk	LC oscillator	0.2~1.8	Fully flooded	Prediction of lubrication film thickness under conditions of different surface velocity directions	[74]
Oil	Bearing in a diesel engine	Transducer	10~60	Fully flooded	Measured the hydrodynamic oil film thickness present in slide bearings	[75]
Oil	EHD rig	Impedance/gain-phase analyzer	0.015~0.2	Fully flooded	Quantitative measurements of film thickness and consideration of cavitation	[76]
Oil with different polarity	EHD rig	Impedance analyzer	0.01	Fully flooded	Influence of a lubricant’s polarity on capacitance measurements	[77]

2. Capacitance Measurement Methods

This section focuses on the mechanisms of various capacitance measurement methods, including oscilloscope, AC bridge, LCR meter, impedance analyzer, voltage divider, and capacitance probe/transducer.

Capacitance, C is defined as the ability of a system to store an electric charge. It is the ratio of the change in charge, Q and the corresponding change in the electric potential V.

$$Q=C·V$$

(1)

$$I={\displaystyle \frac{dQ}{dt}}={\displaystyle \frac{d(C·V)}{dt}}=C·{\displaystyle \frac{dV}{dt}}+V·{\displaystyle \frac{dC}{dt}}$$

(2)

Grease film thickness in the starved condition is relatively stable (assuming no degradation, evaporation, or leakage), and the capacitance does not change with time, so (2) can be reduced to

$$I=C·{\displaystyle \frac{dV}{dt}}$$

(3)

2.1. Oscilloscope

Numerous instruments can measure voltage and current. An oscilloscope is the most versatile one. It allows the user to record the circuits’ voltage over time [78]. The capacitance can then be computed for a known current as follows:

$$C=I·{\left({\displaystyle \frac{dV}{dt}}\right)}^{-1}$$

(4)

When subject to a constant current, capacitance is inversely proportional to the slope of the voltage–time curve. This is also called a “constant current charging-up method”, which was applied by Franke and Poll [48].

$$V={\displaystyle \frac{I}{C}}·t$$

(5)

2.2. AC Bridge

Wilson used a capacitance bridge to measure the film thickness in a grease-lubricated bearing [20]. The alternating current bridge is one of the most widely used methods of measuring circuit constants at audio frequencies, as shown in Figure 1. When such a bridge is balanced, and no current flows through the null detector located in the center, then

$${\displaystyle \frac{Z_a}{Z_b}}={\displaystyle \frac{Z_c}{Z_d}}$$

(6)

where Z is the impedance of each electric component.

Figure 1. An alternating current bridge and a simple capacitance bridge.

For a simple capacitance bridge, $C_S$ is a standard capacitor, and $C_X$ is an unknown capacitor. $R_a$ and $R_c$ are standard resistors. $R_c$ is adjustable. When the bridge is balanced,

$${\displaystyle \frac{R_a}{\frac{1}{\omega C_S}}}={\displaystyle \frac{R_c}{\frac{1}{\omega C_X}}}$$

(7)

$$R_a{C}_s=R_c{C}_X$$

(8)

$$C_X={\displaystyle \frac{R_a}{R_c}}{C}_s$$

(9)

$$\omega =2\pi f$$

(10)

In AC circuits, impedance Z is a combination of resistance R and reactance X.

$$Z=R+jX$$

(11)

Reactance X represents the opposition due to the reactive components in the circuit. Inductive reactance $X_L$ opposes changes in current due to the magnetic field generated by the inductors, which is always positive. Capacitive reactance $X_C$ opposes changes in voltage due to the electric field stored by the capacitors, which is always negative. When a capacitor and an inductor are connected in series,

$$Z=Z_L+Z_C$$

(12)

$$Z_L={jX}_L=j\omega L$$

(13)

$${Z_C=-jX}_C=-{\displaystyle \frac{j}{\omega C}}$$

(14)

where $f$ is the AC signal frequency; $\omega$ is the angular frequency; $L$ is the inductance, and $C$ is the capacitance.

2.3. LCR Meter and Impedance Analyzer

For an LCR meter, $C$ stands for capacitor, and $R$ stands for resistor. $L$ stands for inductor. It is closely related to the capacitor. The rate of current change in an inductor is proportional to the voltage applied across it:

$$V=L{\displaystyle \frac{dI}{dt}}$$

(15)

An LCR meter measures the capacitance by determining the capacitive reactance $X_C$ at a specific frequency. When an AC signal is applied to the capacitor, the LCR meter measures the impedance $Z$, which is composed of resistance $R$ (real part) and capacitive reactance $X_C$ (imaginary part). The imaginary unit, or $j$ factor, accounts for the 90° leading phase shift of current versus voltage [78].

$$Z=R-j{X}_C$$

(16)

$$X_C=\left|Z\right|\mathit{sin}\theta$$

(17)

$$R=\left|Z\right|\mathit{cos}\theta$$

(18)

$$\theta ={\mathit{tan}}^{-1}{\displaystyle \frac{X_C}{R}}$$

(19)

$$e^{j\theta }=cos\theta +jsin\theta$$

(20)

$$Z=\left|Z\right|e^{j\theta }$$

(21)

where $\theta$ is the measured phase angle between the voltage and current.

The capacitive reactance $X_C$ is given by

$$X_C={\displaystyle \frac{1}{\omega C}}$$

(22)

For a parallel plate capacitor, the capacitance is calculated as

$$C={\displaystyle \frac{1}{\omega \left|Z\right|\mathit{sin}\theta }}$$

(23)

The contact resistance is calculated as

$$R={\displaystyle \frac{1}{\omega C\mathit{cot}\theta }}$$

(24)

Some types of LCR meters can only measure the basic parameters of passive components: inductance (L); capacitance (C); and resistance (R). However, an impedance analyzer is able to measure the complete impedance, including both magnitude and phase angle. It provides a comprehensive picture of the electrical behavior across a wide frequency range. It is used for broad-spectrum characterization. An impedance analyzer has been reported for bearing capacitance measurement [17]. A low-voltage (50 mV) and high-frequency (100 kHz) current source was set to avoid grease film breakdown during starvation.

2.4. Capacitive Voltage Divider

Figure 2 shows a simple capacitive voltage divider. $V_{in}$ is the voltage input, and $V_{out}$ is the voltage measured on the unknown capacitor $C_x$. $C_s$ is a standard capacitor. Since the two capacitors are connected in series,

$$I={\displaystyle \frac{V_{in}}{Z_x+Z_S}}={\displaystyle \frac{V_{out}}{Z_x}}$$

(25)

$${\displaystyle \frac{V_{in}}{\frac{1}{j\omega C_x}+\frac{1}{j\omega C_s}}}={\displaystyle \frac{V_{out}}{\frac{1}{j\omega C_x}}}$$

(26)

$${\displaystyle \frac{\frac{1}{j\omega C_x}+\frac{1}{j\omega C_s}}{\frac{1}{j\omega C_x}}}={\displaystyle \frac{V_{in}}{V_{out}}}$$

(27)

$$1+{\displaystyle \frac{C_x}{C_s}}={\displaystyle \frac{V_{in}}{V_{out}}}$$

(28)

$$C_x={\displaystyle \frac{V_{in}-V_{out}}{V_{out}}}{C}_s$$

(29)

The SKF Lubcheck method is based on a capacitive voltage divider [51]. The following relationship was reported, which is the same as (29).

$$C_{total,meas}={\displaystyle \frac{10-V_{cap}}{V_{cap}}}{C}_{ref}$$

(30)

where $C_{total,meas}$ is bearing electrical capacitance; $V_{cap}$ is output voltage, and $C_{ref}$ is reference capacitance.

Figure 2. A simple capacitive voltage divider.

2.5. Capacitance Probe/Transducer

The capacitance probe is a well-established technique for measuring oil film thickness in internal combustion (IC) engines [54,60,61,70,79]. It can be used to measure the film between the piston ring and cylinder liner, as well as the film between the crankshaft journal and the bearing shell [79]. In the ring and liner scenario, the capacitor forms between the electrode inside the probe and the piston ring. However, probes are not particularly suitable for making direct measurements of oil film thickness when it is less than 5 µm [70].

2.6. Summary

The oscilloscope, AC bridge, LCR meter, impedance analyzer and voltage divider are able to measure thin film with high resolution. From the measurement range perspective, the LCR meter and impedance analyzer seem more suitable for analyzing complex electrical behavior.

3. Capacitive Film Thickness Models and Electric Networks

Several test rig configurations have been employed for film thickness measurement using the capacitance method. These studies have described the geometry and electrical network models of the rigs, from benchmark testing rigs (twin disks [16,19,24,56,63,69,80,81] and pin-on disk [41,76]) to full-bearing rigs (ball-bearing [22,23,43,51,82,83,84,85,86] and roller-bearing [20,87]). While the majority of research focuses on fully flooded lubrication conditions, some studies have explored mixed mode [23,40,41], starved lubrication [51], and parched EHL [46,47,88]. Details are summarized in Table 3.

Table 3. Summary of geometric models and electric networks to convert capacitance to film thickness.

Testing Rig	Lubrication Regime	Capacitance Calculation and Some Assumptions	Reference
Twin disk and ball bearing	Fully flooded	For single contact, the total capacitance of a contact can be calculated by the following equation, if $C_{Entry}$ and $C_{Exit}$ are known or if one assumed or calculated a value for $k_C$: $C_{total}=C_{Entry}+C_{Hertz}+C_{Exit}=k_C·C_{Hertz}$ For multipoint contact, all contacts at one ring are in parallel with each other and then in series with the other rings’ capacitances. The full film EHL is considered as a capacitor connected in parallel with a resistor. Bartz approximated the kC factor for ball bearings as 3~4. Gemeinder used 3.5 for calculation. Schneider extended the factor to a wider operating range.	[16,19,63,85,89,90]
Ball bearing under radial load	Fully flooded	The total capacitance of a loaded inner ring contact is divided into five zones: the inlet; outlet; Hertzian; and two side zones. The inlet, outlet, and side capacitances were determined using line contact curve-fitted approximations.	[22]
Ball bearing	Mixed	It is theoretically shown that the oil film thickness and breakdown ratio can be simultaneously measured from the complex impedance. Contact resistor: $R_1=kn\left|Z\right|/l\cos \theta$ Oil film capacitor: $C_1+C_2=-l\sin \theta kn\omega \left|Z\right|$	[23]
Twin disk	Fully flooded	Assume inlet is full of oil, and outlet has two equally thick oil layers adhering to it.	[24]
Steel–oil–mercury system	Mixed	The total impedance can be calculated as $Z=R_0+{\displaystyle \frac{1}{\frac{1}{R}+j\omega C_2+\frac{1}{R_2}}}$	[40]
Pin-on-disk	Mixed	h2 is the maximum oil film thickness in the surrounding area (defined as the vertical position of the center of the ball). $h_2=h_1+\sqrt{b^2-a^2}$ Studied the relationship between air entrainment ratio and film thickness.	[41]
Unloaded ball bearing	Fully flooded	Using the impedance method: $Z_{rc}={\displaystyle \frac{R_K\frac{1}{j\omega C_K}}{R_K+\frac{1}{j\omega C_K}}}$	[43,84]
Ball bearing under axial load	Starved and fully flooded	he starved inlet distance is based on Dowson and Hamrock. $H_c=H_{ff}{\left({\displaystyle \frac{m-1}{m^{\ast }-1}}\right)}^{0.29}$ $m^{\ast }=1+3.06{\left[{\left({\displaystyle \frac{R_x}{b}}\right)}^2{H}_{ff}\right]}^{0.58}$	[51]
Twin-disk machine	Fully flooded	For the outlet zone, it is assumed the relative composite dielectric constant of the two-phase fluid can be described as ${\epsilon }_{r_{comp}}={\epsilon }_{r_{oil}}\times f_{oil}+{\epsilon }_{r_{air}}\times f_{air}$ where $f_{oil}$ and $f_{air}$ are the volume percentages of oil and air, respectively.	[56]
Four-disk machine	Fully flooded	The outlet region is assumed to be made up of two oil layers, and the remainder of the gap is made up of air. Each oil layer is assumed to be half of the central film thickness.	[69]
Pin-on-disk	Fully flooded	Considered the inlet and contact as the flooded region, $C_{flooded}={\int }_{A_{flooded}}{\displaystyle \frac{{\epsilon }_0{\epsilon }_{oil}}{{h_c+h}_{gap}\left(x,y\right)}}dxdy$ where ${\epsilon }_{air}$ is 1; hc is the central film thickness, and hgap is the gap between the solid bodies, given by the Hertzian deformation for dry contacts. In the cavitation region, $C_{cav}={\int }_{A_{cav}}{\displaystyle \frac{{\epsilon }_0}{\frac{h_c}{{\epsilon }_{oil}}+\frac{h_{gap}\left(x,y\right)}{{\epsilon }_{air}}}}dxdy$ The cavitation region was estimated to be about 25% of the total area surrounding the contact. $C_{total}=C_{cav}+C_{flooded}$	[76]
Twin disks	Fully flooded	For the Hertzian region, the dielectric constant is normally estimated at mean surface temperature and mean contact pressure.	[80,81]
Ball bearing	Fully flooded	The total measured capacitance includes the capacitance of the Hertzian, inlet, and outlet regions.	[82,83]
Ball bearing under combined load	Fully flooded	Inner ring and outer ring connected in series. $C_{total}={\sum }_{j=1}^z{\displaystyle \frac{C_{inner,j}·C_{outer,j}}{C_{inner,j}+C_{outer,j}}}$	[85]
Ball bearing	Fully flooded	Accounted for the geometry change in surfaces outside the Hertzian contact zone due to the elastohydrodynamic pressure. The relationship between the Hertzian contact area capacitance and total capacitance was determined using an empirical formula based on numerical simulation.	[86]
Roller bearing and ball bearing	Fully flooded	For the roller bearing, the Hertzian contact area is considered a flat rectangular surface, $C_{hertz}=2{\epsilon }_0{\epsilon }_{r}L{\displaystyle \frac{a}{h_c}}$ where L is the length of the roller elements. For the cavitation region, assuming a long flat over for the raceway, it can be calculated as $C_{cav}=2{\epsilon }_0\underset{a}{\overset{r^{\prime }}{\int }}{\displaystyle \frac{L}{h_c+\frac{x^2}{2r}}}dx$	[87]
Ball bearing	Fully flooded	The cavitation domain can be modeled numerically as $C_{ij}={({\displaystyle \frac{h_{ij}\left(1-{\theta }_{ij}\right)}{{\epsilon }_0{\epsilon }_{r,oil}dxdy}}+{\displaystyle \frac{h_{ij}{\theta }_{ij}}{{\epsilon }_0{\epsilon }_{r,air}dxdy}})}^{-1}$	[91]

A new film thickness measurement model that covers all three lubrication regimes (fully flooded, starvation, and mixed) is proposed here. The model is based on the following assumptions (not limited to):

• The capacitance of the film thickness can be modeled as a parallel plate capacitor;
• Film in the contact region is composed of oil without any thickener;
• A deep-groove ball bearing under axial load is used so the load on each ball is considered equally distributed;
• A polymer cage is used to simplify the electronic network;
• The effect of surface asperities on resistance measurement is neglected;
• The temperature gradient through the inlet, contact, and outlet regions is neglected.

3.1. Fully Flooded

The total measured capacitance $C_{total}$ is the sum of bearing capacitance $C_{bearing}$, and background capacitance $C_{background}$ is calculated as follows:

$$C_{total}=C_{bearing}+C_{background}$$

(31)

Background capacitance can be measured with a ceramic ball bearing with the same geometry. In engineering practices, any two conductors in close proximity may act as a capacitor, such as the interference between adjacent circuits. In addition, some unwanted electrical signals may be induced by background capacitance due to a high-frequency alternating source. Assuming inner and outer-race capacitance is connected in series, and the capacitances of the different balls are connected in parallel:

$$C_{bearing}={\displaystyle \frac{z}{\frac{1}{C_{inner}}+\frac{1}{C_{outer}}}}$$

(32)

where $z$ is the number of balls.

The relationship between the inner and outer-race films was calculated based on the contact geometry and temperature difference [51]:

$${\displaystyle \frac{h_{c,o}}{h_{c,i}}}={\left[{\displaystyle \frac{R_o\left(R_i+R_b\right)}{R_i\left(R_o-R_b\right)}}\right]}^{0.476}{\left({\displaystyle \frac{{\eta }_{T_o}}{{\eta }_{T_i}}}\right)}^{0.67}$$

(33)

It is also assumed that this relationship works for both fully flooded and starved conditions [51].

The following discussion focuses on the inner raceway contact only. There are five regions: inlet; Hertzian contact; outlet; and two side-leakage areas [22]. Due to the small area of side leakage, those regions are neglected by the majority of the literature [16,19,22,23,40,41,43,51,63,69,76,81,82,83,84,85,86,87]. This simplification has also been adopted in the present research.

The electric network is shown in Figure 3. The inlet, contact (Hertzian), and outlet (cavitation) regions are connected in parallel:

$$C_{inner}=C_{inlet}+C_{Hertz}+C_{outlet}$$

(34)

$$C_{inner}={\epsilon }_0\left({\displaystyle \frac{{{\epsilon }_{oil}A}_{inlet}}{h_{inlet}}}+{\displaystyle \frac{{{\epsilon }_{Hertz}A}_{Hertz}}{h_{Hertz}}}+{\displaystyle \frac{{{\epsilon }_{outlet}A}_{outlet}}{h_{outlet}}}\right)$$

(35)

$$C_{inner}={\epsilon }_0\left({\epsilon }_{oil}{\iint }_{A_{inlet}}{\displaystyle \frac{dxdy}{h_{inlet}\left(x,y\right)}}+{\displaystyle \frac{1}{h_c}}{\iint }_{A_{Hertz}}{\epsilon }_{Hertz}dxdy+{\iint }_{A_{outlet}}{\displaystyle \frac{{\epsilon }_{outlet}dxdy}{h_{outlet}\left(x,y\right)}}\right)$$

(36)

where $h_c$ is the central film thickness in the contact region.

Figure 3. Electronic network of the bearing (a) and a single-raceway contact (b,c).

For the inlet region as shown in Figure 4, ${\epsilon }_{oil}$ is assumed to be equal to the dielectric constant of oil under ambient pressure. The inlet section is assumed to be full of oil under a fully flooded condition, the same as Crook [24,92]. The geometry boundary is set to $h_{inlet}=51h_c:$

$$h_{inlet}=h_c-{\displaystyle \frac{b^2}{2R}}+{\displaystyle \frac{x^2}{2R}}$$

(37)

$$x=\sqrt{100h_{c}R+b^2}\approx 10\sqrt{h_{c}R}$$

(38)

$$C_{inlet}={\epsilon }_0{\epsilon }_{oil}{\int }_{-a}^a{\int }_{-10\sqrt{h_{c}R}}^{-b}{\displaystyle \frac{1}{h_c-\frac{b^2}{2R}+\frac{x^2}{2R}}}dxdy$$

(39)

Figure 4. Front view (a) and top view (b) of the fully flooded contact region.

For the Hertzian contact area, the film thickness is assumed to be equal to the central film thickness since it dominates. And the dielectric constant in this region is determined by the average surface temperature and mean contact pressure [80,81]. The total Hertzian contact area is $\pi ab$.

$$C_{hertz}={\displaystyle \frac{{\epsilon }_0{\epsilon }_{Hertz}\pi ab}{h_c}}$$

(40)

For the outlet/cavitation regime, Crook assumes that the outlet section has two oil layers of equal thickness adhering to each disk [24]. The remainder of the outlet section is composed of air. Chittenden assumed the composite dielectric constant of the two-phase fluids was a function of the oil and air percentages [56]. In this research, it is assumed that there are three layers at the outlet, two layers of oil, and one layer of air in between (Figure 5). Each oil layer is assumed to be half of the central film thickness [24,69,80,81]. The pressure in the cavitated region of the lubricant film is approximately constant and near the atmospheric or ambient pressure [93].

$${\displaystyle \frac{1}{C_{outlet}}}={\displaystyle \frac{1}{C_{oil}}}+{\displaystyle \frac{1}{C_{air}}}+{\displaystyle \frac{1}{C_{oil}}}$$

(41)

$$C_{outlet}={\iint }_{A_{outlet}}{\displaystyle \frac{{\epsilon }_0}{\frac{h_c}{{\epsilon }_{oil}}+\frac{h_{outlet}-h_c}{{\epsilon }_{air}}}}dxdy$$

(42)

$$C_{outlet}={\iint }_{A_{outlet}}{\displaystyle \frac{{\epsilon }_0}{\frac{h_c}{{\epsilon }_{oil}}+\frac{x^2-b^2}{2R{\epsilon }_{air}}}}dxdy$$

(43)

Figure 5. Electronic network of the outlet region.

After applying the same geometry boundary at the outlet,

$$C_{outlet}={\int }_{-a}^a{\int }_b^{10\sqrt{h_{c}R}}{\displaystyle \frac{{\epsilon }_0}{\frac{h_c}{{\epsilon }_{oil}}+\frac{x^2-b^2}{{2R\epsilon }_{air}}}}dxdy$$

(44)

and combining (39), (40) and (44),

$$C_{inner}={\epsilon }_0\left({\epsilon }_{oil}{\int }_{-a}^a{\int }_{-10\sqrt{h_{c}R}}^{-b}{\displaystyle \frac{1}{h_c-\frac{b^2}{2R}+\frac{x^2}{2R}}}dxdy+{\displaystyle \frac{{\epsilon }_{Hertz}\pi ab}{h_c}}+{\int }_{-a}^a{\int }_b^{10\sqrt{h_{c}R}}{\displaystyle \frac{1}{\frac{h_c}{{\epsilon }_{oil}}+\frac{x^2-b^2}{{2R\epsilon }_{air}}}}dxdy\right)$$

(45)

3.2. Starved

For the starved condition, the boundary of the inlet regime is determined by the inlet meniscus (Figure 6). The starvation model applied here is similar to that reported by Shetty [51]. The starvation condition is described with a dimensionless meniscus distance $m$ ($m=x/b$, and $x$ is the meniscus distance). $m^{\ast }$ is the critical location of $m$, which is a function of central film thickness [94]:

$$m^{\ast }=1+3.06{\left[{\left({\displaystyle \frac{R_x}{b}}\right)}^2{H}_{ff}\right]}^{0.58}$$

(46)

where $H_{ff}$ is the dimensionless, fully flooded Hamrock and Dowson film thickness.

Figure 6. Front view (a) and top view (b) of the starved contact region.

If the inlet distance m is less than $m^{\ast }$, starvation occurs; otherwise, it is fully flooded. The starved dimensionless central film thickness can be expressed as

$$H_c=H_{ff}{\left({\displaystyle \frac{m-1}{m^{\ast }-1}}\right)}^{0.29}$$

(47)

The analytical results and the experimental results were also found to be in good agreement for the film thickness reduction parameter $(m-1)/(m^{\ast }-1)$ [95].

Similarly, the inlet capacitance can be expressed as

$$C_{inlet}={\epsilon }_0{\epsilon }_{oil}{\int }_{-a}^a{\int }_{-mb}^{-b}{\displaystyle \frac{1}{h_c-\frac{b^2}{2R}+\frac{x^2}{2R}}}dxdy$$

(48)

The calculation of $C_{hertz}$ and $C_{outlet}$ are the same as in the fully flooded condition.

$$C_{inner}={\epsilon }_0\left({\epsilon }_{oil}{\int }_{-a}^a{\int }_{-mb}^{-b}{\displaystyle \frac{1}{h_c-\frac{b^2}{2R}+\frac{x^2}{2R}}}dxdy+{\displaystyle \frac{{\epsilon }_{hertz}\pi ab}{h_c}}+{\int }_{-a}^a{\int }_b^{10\sqrt{h_{c}R}}{\displaystyle \frac{1}{\frac{h_c}{{\epsilon }_{oil}}+\frac{x^2-b^2}{{2R\epsilon }_{air}}}}dxdy\right)$$

(49)

3.3. Mixed

The mixed model is inspired by Maruyama [23]. There are additional assumptions for the mixed regime:

• It is assumed there will be sufficient lubricant supply in the mixed regime; otherwise, it may cause premature bearing damage;
• It is assumed that there is no tribofilm formation on the surface. Tribofilms significantly increase contact resistance.

Surface asperity contact frequently occurs in the mixed mode regime, as illustrated in Figure 7a. The contact area is modeled as a resistor, and the separated area is modeled as a capacitor, as shown in Figure 7b. Resistors and capacitors are connected in parallel. The breakdown ratio [23] $\alpha$ is defined as

$$\alpha ={\displaystyle \frac{breakdown area}{Hertzian contact area}}={\displaystyle \frac{contact resistance measured under static load}{contact resistance measured under dynamic load}}$$

(50)

Figure 7. Illustration (a) and electronic network (b) of the mixed mode.

A simplified geometry model is shown in Figure 8. The inlet and outlet are the same in the fully flooded conditions. Only the Hertzian region needs to be modified.

$$C_{inner}={\epsilon }_0\left({\epsilon }_{oil}{\int }_{-a}^a{\int }_{-10\sqrt{h_{c}R}}^{-b}{\displaystyle \frac{1}{h_c-\frac{b^2}{2R}+\frac{x^2}{2R}}}dxdy+{\displaystyle \frac{{\epsilon }_{Hertz}\pi ab(1-\alpha )}{h_c}}+{\int }_{-a}^a{\int }_b^{10\sqrt{h_{c}R}}{\displaystyle \frac{1}{\frac{h_c}{{\epsilon }_{oil}}+\frac{x^2-b^2}{{2R\epsilon }_{air}}}}dxdy\right)$$

(51)

In this regime, the admittance $Y$ (inverse of impedance $Z$) needs to be measured since the capacitors connect in parallel with the resistor.

$$C_{inner}=C_{inlet}+C_{Hertz}+C_{outlet}$$

(52)

$$Y={\displaystyle \frac{1}{Z}}={\displaystyle \frac{1}{Z_R}}+{\displaystyle \frac{1}{Z_{C_{inner}}}}={\displaystyle \frac{1}{R_c}}+j\omega C_{inner}$$

(53)

$${\displaystyle \frac{1}{R_c}}=\left|Y\right|\mathit{cos}\theta$$

(54)

$$\omega C_{inner}=\left|Y\right|\mathit{sin}\theta$$

(55)

$$R_c={\displaystyle \frac{1}{\left|Y\right|\mathit{cos}\theta }}$$

(56)

$$C_{inner}={\displaystyle \frac{\left|Y\right|\mathit{sin}\theta }{\omega }}$$

(57)

The contact resistance $R_c$ and capacitance are calculated as a function of admittance and phase angle.

The model for fully flooded, starved, and mixed lubrication conditions is a proposal in the current research. Its accuracy relies on the precise dielectric constant of lubricants under high pressure and elevated temperature. The model evaluation and comparison will be included in the future work.

Figure 8. Front view (a) and top view (b) of the mixed mode.

3.4. Program Flow Chart of Film Thickness Measurement

The procedures of film thickness measurement are illustrated in Figure 9.

(1)

Measure the background impedance and admittance using a hybrid bearing under the same contact pressure;

(2)

Measure both the static and dynamic impedance and admittance using the testing bearing;

(3)

If the electrical contact resistance is finite, the bearing is in the mixed mode. The $C_{inner}$ and $C_{outer}$ are calculated with admittance. Then, the breakdown ratio $\alpha$ and the central film thickness are calculated. The program ends;

(4)

If the electrical contact resistance approaches infinity, the bearing is in either fully flooded or starved mode. $m$ and $m^{\ast }$ are calculated to determine the starvation degree. If $m$ < $m^{\ast }$, calculate the film thickness with the starved model. Otherwise, use the fully flooded model.

Figure 9. Program flow chart of film thickness measurement.

4. Dielectric Constant of Lubricants

Models and measurements of dielectric constants of lubricants are summarized in Table 4. For non-polar base oils, the classic Clausius–Mossotti (CM) relationship was suggested by many authors [56,58,69,72,76,91,96,97]. Paraffinic-based mineral oil and PAO can be considered a non-polar group. The CM equation describes the dielectric constant as a function of density. A widely applied relationship between density and pressure was described by Dowson and Higginson [98]. However, its validity under high pressure is doubtful. Bair [99] claimed that the Tait equation [100] resulted in higher accuracy than Dowson and Higginson’s equation for EHL calculations in the absence of compressibility data.

Table 4. Summary of dielectric constant models of lubricants.

Lubricants	Dielectric Constant Models and Measurements	Reference
Grease and oil	The dielectric constant of the grease (measured at laboratory temperature, atmospheric pressure, and a frequency of 100 kHz) was 3.07 in the unused condition and 2.60 after being sheared between the rollers, while that of the base oil was 2.32. The dielectric constant of the grease decreased on shearing to approach but did not equal that of the base oil. A further investigation, not reported in this paper, showed that the remaining difference was caused by the polar constituents of the grease and by the presence of fragments of disrupted soap fibers.	[20]
Shell Turbo 68 oil	The dielectric constant was reported as 2.65, 3, and 1 in the inlet, Hertzian zone, and outlet, respectively.	[22]
Grease	The dielectric properties of the greases were determined experimentally in a separate setup consisting of a plate capacitor with exactly known geometry.	[48]
Lithium grease	The measured capacitance and Hamrock–Dowson film thickness equation were used to back-calculate the dielectric constant.	[51]
SAE 20, SAE 60	The dielectric constants of both fresh and used oils were measured using a cylindrical brass capacitor and an excitation signal of 100 kHz. The dielectric constant of the oil was determined as the ratio of the capacitance of the cell filled with oil and air. Measurements were made at 100 °C, 125 °C, and 150 °C.	[54]
Shell HVI 160 medium viscosity mineral oil	The relationship between the dielectric constant and the pressure was described by the Clausius–Mossotti (CM) relationship. ${\displaystyle \frac{\epsilon -1}{\left(\epsilon +2\right)\rho }}=constant$	[56]
Sunflower oil	The dielectric constant of the bulk lubricant at the test temperature was then calculated as follows: ${\epsilon }_b={\displaystyle \frac{C_b}{C_{air}}}{\epsilon }_{air}$ Dyson and Galvin found the CM equation to be relatively accurate for non-polar fluids (such as mineral oil). The equation overestimates the dielectric constant at high pressures. Chua mentioned that the discrepancy might be because the polarizability did not change linearly with density. The Onsager formula was considered more suitable.	[58]
Naphthenic oil	Non-polar oil should obey the Clausius–Mossotti equation. Some direct measurements on similar mineral oils by Galvin, Naylor, and Wilson (1963) suggested a better relationship: $\epsilon ={\epsilon }_0(1+0.376P-0.628P^2)$	[69,101]
NYE 182	The dielectric constant of the lubricant at pressure can be approximated by making use of the Clausius–Mossotti equation.	[72]
Paraffinic oil	The resulting variations in dielectric constant with temperature and pressure are substantially in agreement with those found by Galvin, Naylor, and Wilson in 1963.	[73,101]
PAO4, PAO40	For non-polar lubricants, the dielectric constant can be calculated using the Clausius–Mossotti equation.	[76]
Glycerol, PEG, and PAO	For polar fluids, the CM equation is not valid. More complex Onsager or Kirkwood relationships must be applied.	[77]
Paraffinic oil	From 0 to 350 MPa, the dielectric constant decreases with an increase in temperature and increases with an increase in pressure.	[80]
Oil	A modified CM equation can be used based on the work of Bondi and Schrader.	[91,97]
PAO4	The dielectric constant of a non-polar lubricant at the contact pressure is estimated using the Clausius–Mossotti equation.	[96]
Lithium grease	The dielectric constant of lithium grease depends primarily on the polarity of the dispersion medium (e.g., additives.) Organic additives (LZ-318, USI, DPA, Khloref 40) have a stronger influence on the dielectric constant than inorganic additives (mica, graphite, and MoS2). When lithium greases are heated to close to their dropping point (180–200 °C), a sharp increase in the dielectric constant is observed.	[102]
Mineral oil, synthetic oil, and additives	Hydrocarbon lubrication oils have a dielectric constant from 2.1 to 2.8, which depends on the viscosity of the oil, the paraffinic/naphthenic content, and the additive package. The dielectric constant of hydrocarbon fuels (which contain smaller carbon chains than lubrication oils) correlates with fuel density (kg/m3), within an accuracy of 2% $\epsilon =0.001667\rho +0.785$	[103]

However, the dielectric constant of polar lubricants does not change linearly with density. Polarity may be introduced by the aromatic/naphthenic content and additives, which leads to an increase in the dielectric constant [103]. In this case, the CM equation is no longer valid, while Onsager or Kirkwood relationships should be applied [58,77]. The dielectric constant of a grease differs from its base oil. Wilson found that the dielectric constant difference between grease and its base oil is around 10% [20]. It may be attributed to the polar constituents and the presence of soap fiber fragments.

There are some efforts to correlate both the rheological and dielectric behaviors of lubricants. For dielectric loss, the surrounding liquid medium offers a resistance to these motions that is roughly proportional to its viscosity [97]. Khanmamedov et al. found that there was a linear relationship between the oil viscosity and the temperature, corresponding to the maximum dielectric loss tangent with a fixed frequency of measurement [104]. Once the constant of proportionality has been determined, measurements of viscosity can be obtained from a measurement of the dynamic permittivity under pressure [99].

Dielectric relaxation time can be correlated with fluid viscosity. The average motion of these neighboring molecules might be described by replacing them with a continuous medium with the properties of a macroscopic viscous fluid. This possibility leads to the model developed by Debye [105], in which a dipolar molecule is considered to be a sphere moving in a continuous, viscous fluid and obeying the macroscopic equation of flow.

4.1. Revisiting Classic Dielectric Models

When deciding the approximate model for a given material, one has to judge (a) whether the basic model represents the actual material and (b) whether the mathematization holds for the given range of concern [106]. Here, different dielectric models are revisited. Classical theories calculate the relative permittivity with a continuum approach: the molecule is placed in a cavity surrounded by the material treated as a continuum. For a spherical-shaped sample, the general equation of relative permittivity is

$${\displaystyle \frac{\epsilon -1}{\epsilon +2}}={\displaystyle \frac{4\pi }{3}}{\alpha }_{P}N+y_0$$

(58)

where ${\alpha }_P$ is the polarizability of the molecule; $N$ is the number density, and $y_0$ is the dimensionless dipole strength function.

When $y_0=0$, the equation reduces to the Clausius–Mossotti (CM) equation. It is valid for non-polar molecules.

When $y_0={\displaystyle \frac{4\pi N{\mu }_0^2}{9k_{B}T}}$, it equates to the Debye equation [105], where ${\mu }_0$ is the permanent dipole moment; $k_B$ is the Boltzmann constant, and $T$ is the temperature. It holds approximately for gases and dilute solutions of molecules carrying a permanent dipole. But, it is not applicable for polar liquids because of neglecting the inner field.

When $y_0={\displaystyle \frac{\left(\epsilon -{\epsilon }_{\infty }\right)\left(2\epsilon +{\epsilon }_{\infty }\right)}{\epsilon {\left({\epsilon }_{\infty }+2\right)}^2}}$, the resulting equation is the Onsager equation [107]. It accounts for the orientation polarization of a molecule influenced by the surrounding dielectric due to long-range interactions at high densities. The high-frequency relative permittivity ${\epsilon }_{\infty }$ is commonly calculated from the Maxwell relationship ${\epsilon }_{\infty }\approx n^2$, where $n$ is the inner refractive index. It can also be calculated based on the CM equation. The Onsager equation works quite well for liquids when the dipole moment is not too high.

The Kirkwood–Fröhlich equation [108] accounts for short-range correlation by introducing the effect of angular correlations between dipolar molecules. The original model is too complicated to be applied to a real material. Valisko et al. [109] developed an empirical extension based on the Kirkwood–Fröhlich equation to calculate the relative permittivity of dipolar liquids and mixtures in terms of the molecular dipole moment, the refractive index, density, and temperature.

4.2. Evaluation of Dielectric Models

The density and permittivity of 57 non-polar and dipolar molecular liquids at different temperatures (143 sets, up to 100 °C) and pressures (555 sets, up to 500 MPa) were collected and analyzed [110]. However, no equation was found that could accurately predict the liquid permittivity in the range of the pressures and temperatures tested. Molecular liquids exhibit significantly larger pressure and temperature dependence on permittivity compared to density. This indicates the strong influence of these external factors on the molecular structure. While density remains relatively unaffected, permittivity can be dramatically altered by several factors, including volume change, molecular deformation, shifts in dipole–dipole interactions, conformational equilibrium changes, and even modifications to the structure of OH complexes.

4.3. Summary

The results of the evaluation and analysis herein suggest modifying the CM and Onsager models with measured data. An engineering model with better accuracy can then be developed. Direct measurements of the dielectric constant under elevated temperature and high contact pressure are recommended for model validation.

5. Grease Starvation Factor

Table 5 summarizes different starvation models. These models collectively account for the factors perceived to influence the starvation of oil and grease-lubricated contacts. Starvation models have been proposed by many authors, based on either benchmark or full-bearing test results (both experimental and simulation results). Wedeven [111], Hamrock, and Dowson [94] proposed starvation models as a function of inlet meniscus distance. Some models have been discussed in the previous section. Cann [8] conducted the experiments on an oil-lubricated optical EHL rig and developed a starvation criterion as a function of lubricant volume, contact width, surface tension, critical speed, and velocity. Cen [50] measured the grease film thickness in a deep-groove ball bearing using capacitance methods. It was found that the ratio between grease film and fully flooded film depended on speed, load, temperature, and grease properties.

Table 5. Summary of different starvation models of both oil and grease.

Lubricants	Testing Rig	Starvation Models	Reference
High-viscosity PAO	Pin-on-disk	Fully flooded and starved transition of oil-lubricated contact: $Starvation Degree={\displaystyle \frac{{\eta }_{0}ua}{{h_{oil\infty }\gamma }_s}}$ where ${\eta }_0$ is base oil dynamic viscosity; $u$ is the entrainment speed; $a$ is the contact width; $h_{oil\infty }$ is oil height (lubricant volume), and ${\gamma }_s$ is oil surface tension.	[8]
Grease	Deep-groove ball bearing	Grease starvation depends on speed, load, temperature, and grease properties: ${\displaystyle \frac{h_g}{h_{ff}}}=a\times {\left({nd}_m\right)}^b$ where $n$ is the rotational speed, and $d_m$ is the bearing pitch diameter.	[50]
Grease	Deep-groove ball bearing	A power–law relationship is found between the normalized film thickness and speed x viscosity x contact width: ${\displaystyle \frac{h_g}{h_{ff}}}={\left({\displaystyle \frac{1.274\times {10}^{-4}}{u\eta b}}\right)}^{0.791}$ where $u$ is the entrainment speed; $\eta$ is the dynamic viscosity of the base oil, and $b$ is the semi-minor.	[55]
Oil	Optical interferometry + numerical simulation	Both the Damiens and Van Zoelen models show the same film thickness decay with time: $h\left(t\right)\propto t^{-1/\gamma }$ where $\gamma$ is the resistance to side flow, which is a function of Moe’s dimensionless numbers (M, L) and contact ellipticity $\kappa$.	[112,113]
Oil	Optical EHD rig	The surface tension gradient is considered the driving force for fluid recovery. The amount of lubricant replenishment can be predicted as follows: $∆\zeta ~{\displaystyle \frac{{\gamma }_s}{{\eta }_0}}t$ where ${\gamma }_s$ is oil surface tension; ${\eta }_0$ is dynamic viscosity, and $t$ is the time interval between successive rolling-element passages.	[114]
Oil	Numerical simulation	Effects of inlet supply starvation on film thickness in EHL point contact: ${\displaystyle \frac{h_c}{h_{cff}}}={\displaystyle \frac{r}{\sqrt[\gamma ]{1+r^{\gamma }}}}$ where $h_c{/h}_{cff}$ indicates dimensionless film thickness reduction; $r$ is the dimensionless film thickness on the track, and $\gamma$ is the resistance to the side flow.	[115]
Grease	Angular contact ball bearings	The starvation number for oscillating grease-lubricated bearings: $Starvation number ~ {\displaystyle \frac{{\eta }_{0}a{f}_c}{{\gamma }_{s}2b{O}_{SR}}}$ where ${\eta }_0$ is base oil dynamic viscosity; a is the semi-major; b is the semi-minor; $f_c$ is overrolling frequency; ${\gamma }_s$ is surface tension, and $O_{SR}$ is the oil-separation rate.	[116,117]

Damiens and Van Zoelen [112,113] proposed that the film thickness decayed with time and revolutions. However, in the author’s opinion, it is the grease degradation or change in the permeability that leads to film thickness decay. Chiu [114] proposed a starvation model that claims that the decay is a function of surface tension. Wandel [116,117] concluded that the starvation and relubrication mechanisms in grease-lubricated oscillating bearings were highly dependent on the operating conditions and lubricant rheology. Masjedi [118] simulated the starvation effect in mixed EHL numerically. The starvation degree was strongly correlated with the lubricant mass flow rate for both point and line contacts. Zhang et al. studied the factors affecting grease film formation on an EHD optical rig [119]. The proposed film thickness model considered the factors of working conditions, grease type, and consistency.

Based on the discussion above, the following five parameters are chosen to develop a new dimensionless grease starvation model:

$${\displaystyle \frac{h_g}{h_{ff}}}~{\displaystyle \frac{{\gamma }_s{k}_p}{u\eta b}}$$

(59)

where ${\gamma }_s$ is oil surface tension; $k_p$ is the permeability; $u$ is entrainment velocity; $\eta$ is the dynamic viscosity of the base oil, and $b$ is the semi-minor. After the unit and dimension analysis of each parameter as shown in Table 6, the following dimensionless model was proposed:

$${\displaystyle \frac{h_g}{h_{ff}}}~{\left({\displaystyle \frac{{\gamma }_s}{u\eta }}\right)}^x{\left({\displaystyle \frac{k_p}{b^2}}\right)}^y$$

(60)

where $x$ and $y$ are fitting parameters.

Table 6. Units and dimensions of parameters.

Parameter	Unit	Dimension (m, L, t)
Lubricant/air surface tension ${\gamma }_s$	N/m or kg/s2	$m{t}^{-2}$
Entrainment velocity $u$	m/s	$L{t}^{-1}$
Dynamic viscosity $\eta$	Pa·s or kg/ms	$m{L}^{-1}{t}^{-1}$
Half contact width $b$	m	$L$
Permeability $k_p$	m2	$L^2$

There are few papers mentioning the capacitive grease film measurements within a bearing. Most works focus on the oil film thickness measurement. The available capacitive film thickness data of grease-lubricated bearings were extracted from the literature [17,20,50,51,120]. These data originated from various types of bearings (ball bearing, SRB, CRB), running conditions, grease formulations, testing rigs, and capacitive film thickness models.

Entrainment velocity and the contact semi-minor are calculated from bearing geometry and working conditions. The Kinematic Walther equation and the 40° and 100 °C spec sheet viscosities were used to calculate the test temperature-dependent base-oil viscosity.

Data were collected at the start of the bleeding phase, at which time, the film thickness was relatively stable. The bearings ran for 20~100 h. It is assumed that no grease degradation occurred at this point.

The permeability of a lubricant varies with working conditions (pressure, temperature, and shear) as well as irrecoverable degradation (breakdown of the thickener structure and concentration gradient). Akchurin et al. [121] have captured these influences in terms of the “adjusted friction energy density” and a master curve. This curve relates permeability to adjusted friction energy density. These data are included in the analysis. Grease permeability captures the porous thickener structure and base-oil duality. It also influences the decay of a grease film and is indicative of mechanical degradation.

The surface tension of lubricants is a function of temperature. The data on grease base oil at room temperature can be found in the literature. The surface tensions of mineral oil, semi-synthetic oil, PAO, and ester are reported as 32.3, 32.3, 31.2, 28.9, and 31.8 mN/m, respectively [122]. Empirical results of Ross [123] show that surface tension varies linearly within the temperature range 20~160 °C. The average slope of 24 groups was found to be −0.081. Here, an engineering model to calculate the surface tension of grease base oil is proposed:

For mineral oil and PAO,

$${\gamma }_s=-0.081·\mathrm{T}+33.6$$

(61)

For ester,

$${\gamma }_s=-0.081·\mathrm{T}+30.6$$

(62)

The mineral oil model is assumed to be sufficient when the lubricant-specific surface tension is unknown.

The developed regression model is shown in Figure 10. A total of 113 data points were extracted from different authors. The R-squared value is 0.71, which is reasonably good. Interestingly, ${\gamma }_s/u\eta$ is also reported by Chen et al. [124] as a dimensionless capillary number through numerical analysis. It describes the competition between viscous and capillary forces, which affects the shape of the inlet meniscus. The model needs to be further polished with more accurate film thickness data, which could be included in future work.

Figure 10. Developed regression model based on capacitive film thickness data [17,20,50,51,121].

6. Conclusions

This review focuses on the measurement of capacitive film thickness in grease-lubricated bearings. The current developments in capacitance measurements and models are evaluated. Key challenges are identified, and potential solutions are proposed. It is expected that this work will bring a deeper understanding of grease-lubrication mechanisms in rolling element bearings.

The main conclusions can be drawn as follows:

(1)

The mechanisms of mainstream electronic components in capacitance measurement were reviewed. For analyzing complex electrical behavior, the LCR meter and impedance analyzer seem to be more suitable. It enables more accurate capacitance measurement;

(2)

Current capacitive models and programs can only measure one or two lubrication regimes. A new capacitive model, electric network, and program flow chart to measure lubricant film thickness in fully flooded, starved, and mixed regimes was developed. It is more comprehensive compared to the literature models;

(3)

Current dielectric constant models were reviewed, and suitable ones for lubricants were proposed. Modifying the CM and Onsager models with measured data to develop an engineering model is suggested. It facilitates a more precise film thickness measurement;

(4)

A new dimensionless grease starvation model was developed based on the 113 literature capacitive film thickness data points from five different authors. It is a function of surface tension, entrainment velocity, contact geometry, dynamic viscosity, and permeability. The R-squared value of 0.71 indicated a strong correlation, considering the variability in bearing types, operating conditions, grease formulations, testing rigs, and capacitive film thickness models.

$${\mathsf{\Phi }}_{starvation}={\displaystyle \frac{h_g}{h_{ff}}}~{\left({\displaystyle \frac{{\gamma }_s}{u\eta }}\right)}^x{\left({\displaystyle \frac{k_p}{b^2}}\right)}^y$$