Visualization of Lubrication Conditions Using the Electrical Impedance Method Considering Surface Roughness

1. Introduction

Rolling bearings are widely utilized as key components to effectively support rotating parts in various mechanical devices. When lubricants such as oil or grease are supplied, extremely thin oil films form between the surfaces. Under these conditions, elastohydrodynamic (EHD) lubrication occurs, which reduces friction and wear by preventing direct metallic contact [1]. This mechanism enables bearings to achieve smooth and stable rotational operation even under high load conditions. The Hamrock-Dowson equation has been established as the standard prediction model for EHD film thickness and is widely used for evaluating oil film thickness in rolling bearings under fully flooded lubrication [2].

For stable low-torque operation of rolling bearings, monitoring technology is essential for accurately capturing operating conditions and for maintaining optimal oil film thickness. While condition monitoring methods are diverse, vibration analysis is the most widely used technique, and its effectiveness has been demonstrated in numerous experiments. When damage occurs on metal surfaces and track surface smoothness is compromised, vibration magnitude increases rapidly. Furthermore, since inner rings, outer rings, and rolling elements each have characteristic passing frequencies, it has been reported that damage locations can be identified through frequency analysis [3,4,5,6]. Research has also been conducted on acoustic emission, which is elastic waves released during material damage [7]. However, these monitoring methods can only detect damage after it has progressed enough for significant signals to be generated, limiting their capability for failure prediction and immediate optimization of operating conditions.

Meanwhile, approaches for measuring oil film thickness in the EHD contact area have attracted attention, with fundamental tests using optical interferometry being conducted [8,9,10,11]. However, these require transparent materials and are difficult to apply to metallic rolling bearings. Therefore, electrical methods have been proposed. The contact-resistance method applies a small DC voltage across the two surfaces; when the oil film collapses and metallic contact forms, the circuit closes and a sharp decrease in resistance is detected. This change in DC resistance serves as an indicator of oil film breakdown and has been shown to correlate strongly with film thickness and other lubrication parameters [12,13]. Additionally, capacitance methods, which treat the EHD contact area as capacitors, have demonstrated the ability to quantify oil film thickness in both fundamental testing and actual bearing testing [14,15,16,17]. More recently, Maruyama and Nakano proposed the electrical impedance method (EIM) [18]. The EIM applies an AC voltage to the contact area and simultaneously estimates thickness and breakdown ratio of oil films from the resulting complex impedance. Therefore, it is superior to conventional capacitance-based and DC contact-resistance methods. Oil film thickness obtained by EIM for single EHD contact agrees well with optical measurement results and the Hamrock-Dowson equation, and applicability to deep groove ball bearings and to needle roller bearings with line contact has also been confirmed [19,20]. Furthermore, Iwase et al. demonstrated that oxidative degradation of lubricants can be detected from dielectric properties obtained by sweeping the applied frequency [21], and Maruyama et al. showed the possibility of in-situ wear measurement from abnormal increases in dielectric constant in steel–steel contacts [22]. Additionally, monitoring of lubrication conditions through EIM application to gears, which similarly have sliding components, has been actively conducted [23,24].

However, current EIM models approximate the contact area separated by an oil film as a parallel-plate capacitor and compute mean film thickness from the capacitance. Consequently, they fail to capture the real film thickness distribution and the influence of surface roughness. Given the well-established effect of surface roughness on capacitance, various analytical approaches explicitly incorporating this factor have been widely employed in electronics and materials science [25,26,27]. However, because these analyses assume an electrode separation distance is much larger than the characteristic surface roughness, they have not yet been applied at the scale of the EHD contact area. In EHD contacts, the oil film thickness is typically on the order of tens to hundreds of nanometers, which is comparable to the surface roughness amplitude. Therefore, the assumption of large electrode separation relative to roughness does not hold in this regime. Surface roughness in the tribology field has been evaluated from multiple perspectives, including statistical contact analysis [28], prediction of peak density and curvature using Power Spectral Density [29], and consideration of surface roughness in Reynolds equations [30], as they affect friction and wear characteristics as well as fluid flow. Electrical approaches have also been explored in tribology: Morris et al. analyzed electric field strength in rough EHD conjunctions [31], and Sunahara et al. developed a device for observing grease film breakdown through electrical measurement [32]. Furthermore, electrochemical impedance spectroscopy has been applied to evaluate surface roughness and morphology [33]. However, none of these studies directly addressed how surface roughness alters capacitance in the EHD contact area.

Therefore, this study introduces surface roughness concepts into EIM and theoretically analyzes the effects of oil film thickness distribution on complex impedance. Specifically, by constructing a modified capacitance model incorporating statistical roughness parameters and incorporating it as a correction term into conventional models, an improvement in oil film thickness estimation accuracy was proposed. This paper reports on the theoretical framework and verification results through fundamental experiments.

2. Measurement Principle

In the EIM, the EHD contact area S is represented by an RC parallel circuit. The resistive branch stands for metallic contact, and the capacitive branch stands for the lubricant film. Fitting the measured complex impedance Z to this circuit gives the breakdown ratio of oil films α and the mean film thickness h₁. In the conventional approximation, the capacitive branch is treated as a parallel-plate capacitor. The capacitance C₁ is then estimated by regarding whole contacts as two flat electrodes separated by a uniform gap h_1,flat,

$$C_1={\epsilon }_0{\epsilon }_r{\displaystyle \frac{(1-\alpha )S}{h_{1,\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}}}=\epsilon {\displaystyle \frac{(1-\alpha )S}{h_{1,\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}}}$$

(1)

where ε is the permittivity of the lubricant.

The conventional circuit also places the capacitance C₂ of the region that surrounds the contact in parallel with C₁. Using the measured impedance magnitude ∣Z∣ and phase angle θ at each angular frequency ω, the model calculates the mean film thickness h_1,flat. Given that r_b is the ball radius and c is the Hertzian contact radius, the film thickness predicted by the electrical impedance method, h_1,flat is then

$$h_{1,\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}={\displaystyle \frac{\left(1-\alpha \right)c^2}{2r_{\mathrm{b}}}}/\mathfrak{W}\left({\displaystyle \frac{\left(1-\alpha \right)c^2}{2{r_{\mathrm{b}}}^2}}\mathrm{e}\mathrm{x}\mathrm{p}\left(1-{\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{2\pi \epsilon \omega r_{\mathrm{b}}\left|Z\right|}}\right)\right)$$

(2)

where $\mathfrak{W}$ denotes the Lambert W function. Let ${\overline{h}}_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$ be the mean oil film thickness over the entire contact area S; then the following relation holds.

$${\overline{h}}_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}=(1-\alpha )h_{1, \mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$$

(3)

In practice, surface roughness causes the local film thickness to vary spatially, reflecting the underlying surface height distribution. The lubricant film at any point on the contact (x, y) can therefore be written as h(x, y). This non-uniform film induces a three-dimensional electric field distribution E(x, y, z); field crowding near asperity summits, for example, can raise the apparent capacitance. To treat this rigorously, the local electrostatic energy density u is introduced,

$$u\left(x,y,z\right)={\displaystyle \frac{1}{2}}\epsilon {E(x,y,z)}^2$$

(4)

and evaluate the total electrostatic energy $U$.

$$U={\iiint }_{\mathit{V}}u(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z$$

(5)

The capacitance C and the stored energy are related by the following equation:

$$C={\displaystyle \frac{2U}{V^2}}$$

(6)

where V is the applied voltage. Thus, when the opposing surfaces are rough, the capacitance C must be obtained by integrating the spatially varying energy density u over every volume element in the gap. As a practical way to treat the electric-field distribution, the rough surface capacitor is approximated as an array of discrete parallel-plate capacitors connected in parallel. Under this approximation, once the coordinates x and y are fixed, the electrostatic-energy density is constant with respect to z. The following relation is obtained:

$$u\left(x,y\right)={\displaystyle \frac{1}{2}}\epsilon {E(x,y)}^2={\displaystyle \frac{\epsilon V^2}{2{h(x,y)}^2}}$$

(7)

$$U=\iint u\left(x,y\right)h(x,y)dxdy$$

(8)

$$C_1=\epsilon \iint {\displaystyle \frac{dxdy}{h\left(x,y\right)}}$$

(9)

Let H be a random variable representing the film thickness, with probability density function f(h), so that

$$C_1={\int }_{h_{\mathrm{m}\mathrm{i}\mathrm{n}}}^{h_{\mathrm{m}\mathrm{a}\mathrm{x}}}\epsilon {\displaystyle \frac{(1-\alpha )S}{h}}f\left(h\right) \mathrm{d}h=\epsilon \left(1-\alpha \right)S\mathbb{E}\left[{\displaystyle \frac{1}{H}}\right]$$

(10)

where 𝔼[･] is the expectation operator. Given that the capacitance C is obtained experimentally, Equations (1) and (10) lead to the following relation.

$${\displaystyle \frac{1}{h_{1,\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}}}=\mathbb{E}\left[{\displaystyle \frac{1}{H}}\right]$$

(11)

Because the function

$$g\left(x\right)={\displaystyle \frac{1}{x}}$$

(12)

is convex on the positive real line, Jensen’s inequality applies,

$$g\left(\mathbb{E}\left[H\right]\right)\le \mathbb{E}\left[g\left(H\right)\right]$$

(13)

These results indicate that the predicted capacitance considering surface roughness is consistently greater than that predicted by the ideal parallel-plate model [34].

Under fully flooded lubrication, the minimum film thickness h_min remains positive and poses no difficulty. However, once metallic contact occurs, certain points on the real surface profile have zero gap, causing the capacitance integral to diverge. To circumvent this problem, metallic contact areas are excluded and only regions that retain a lubricant film thick enough to act as a capacitor are considered. In addition, when the local electric field exceeds the dielectric breakdown strength E_b of the oil, the lubricant no longer behaves as a dielectric. Regions in which the field surpasses E_b are therefore likewise omitted. With these constraints, h_min can be treated as positive.

To evaluate $\mathbb{E}$[1/H], the film thickness distribution represented by the random variable H is described by an appropriate probability density function. Although surface roughness on engineering components is often well approximated by a normal distribution, this choice is inconvenient here because its domain includes zero, which makes $\mathbb{E}[1/H]$ undefined. A truncated normal distribution with domain (> 0) avoids this singularity yet yields no explicit elementary expression for $\mathbb{E}$[1/H]. A more practical choice is the gamma distribution, whose shape can approximate a normal distribution for appropriate parameters and whose support is strictly positive. As Greenwood and Williamson examined the exponential distribution as an alternative to the normal [28], it can be considered reasonable to employ the gamma distribution, which generalizes the exponential distribution, for the similar purpose. When the film thickness H is assumed to follow a gamma distribution with shape parameter a and rate parameter b, the probability density function is

$$H~\mathrm{G}\mathrm{a}(a,b), f\left(h | a,b\right)={\displaystyle \frac{b^a}{\Gamma \left(a\right)}}{h}^{a-1}\mathrm{e}\mathrm{x}\mathrm{p}(-bh) ,h>0$$

(14)

where Γ(⋅) denotes the gamma function. The gamma parameters a and b can be obtained from the mean film thickness considering roughness $h_{1,\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}$ and the standard deviation of the film thickness distribution σ by the method of moments:

$$a= {\left({\displaystyle \frac{h_{1,\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}}{\sigma }}\right)}^2,b= {\displaystyle \frac{h_{1,\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}}{{\sigma }^2}}$$

(15)

Unlike the normal or truncated-normal distributions, the gamma distribution allows a simple closed-form expression for the E [1/H]. For a gamma-distributed film thickness, $\mathbb{E}$[1/H] is

$$\mathbb{E}\left[{\displaystyle \frac{1}{H}}\right]={\int }_0^{\infty }{\displaystyle \frac{1}{h}}{\displaystyle \frac{b^a}{\Gamma \left(a\right)}}{h}^{a-1}\mathrm{e}\mathrm{x}\mathrm{p}(-bh)dh={\displaystyle \frac{b^a}{\Gamma \left(a\right)}}{\int }_0^{\infty }{h}^{a-2}\mathrm{e}\mathrm{x}\mathrm{p}(-bh)dh$$

(16)

The condition a > 1 is necessary because the integral in Eq. (16) diverges when a ≤ 1, corresponding to Γ(a−1) being undefined. By employing the gamma function relation in Eq. (17) gives a compact expression for $\mathbb{E}$[1/H] in terms of the shape and rate parameters a and b:

$${\int }_0^{\infty }{h}^{a-2}\mathrm{e}\mathrm{x}\mathrm{p}(-bh)dh={\displaystyle \frac{\Gamma (a-1)}{b^{a-1}}}$$

(17)

$$\mathbb{E}\left[{\displaystyle \frac{1}{H}}\right]={\displaystyle \frac{b^a\Gamma (a-1)}{b^{a-1}\Gamma \left(a\right)}}={\displaystyle \frac{b}{a-1}} (a>1)$$

(18)

Thus, using equation (15), $\mathbb{E}$[1/H] can be expressed solely in terms of the $h_{1,\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}$ and the σ.

$$\mathbb{E}\left[{\displaystyle \frac{1}{H}}\right]={\displaystyle \frac{h_{1,\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}}{{h_{1,\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}}^2-{\sigma }^2}}$$

(19)

Substituting this relation into Eq. (11), yields a quadratic equation in h_1,rough, retaining the positive root expresses h_1,rough explicitly as a function of h_1,flat.

$${h_{1,\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}}^2-h_{1,\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}{h}_{1,\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}-{\sigma }^2=0$$

(20)

$$h_{1,\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}={\displaystyle \frac{h_{1,\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}}{2}}\left(1+\sqrt{1+{\left({\displaystyle \frac{2\sigma }{h_{1,\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}}}\right)}^2}\right)$$

(21)

When the metallic contact region is included, the overall mean film thickness ${\overline{h}}_{\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}$ can be written in the same form as Eq. (3):

$${\overline{h}}_{\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}=(1-\alpha )h_{1,\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}$$

(22)

By applying this equation, the conventional value ${\overline{h}}_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$ can be corrected using the σ to obtain the roughness adjusted mean film thickness ${\overline{h}}_{\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}$.

3. Experimental Details

3.1. Measurement Apparatus

Figure 1 shows the ball-on-disc type apparatus used in this study. A steel ball with a radius of 12.7 mm rolls against a BK7 glass disc (diameter 100 mm, thickness 10 mm). The disc was coated with a 5 nm semi-reflective chromium film, followed by an electrically conductive 1 µm ITO spacer layer. The ball is fixed to a horizontal shaft driven by a servo motor, and the disc is mounted on a vertical shaft driven by another motor. The two spindles are synchronized so that their circumferential speed v is the same, 0.05 m/s to 0.50 m/s, which sets the slip ratio to zero. The lower hemisphere of the ball dips into the oil bath and entrains lubricant into the contact. White light from a microscope illuminates the contact (see Figure 2). Part of the beam is reflected at the chromium layer; the remainder passes through the ITO spacer and oil film, then reflects at the ball surface. A beam-splitter divides the interfered light. One branch is recorded directly by a high-speed camera to visualize the lubricant condition, while the other is directed through a slit within the spectrometer to a second camera that provides quantitative film thickness measurements. Since rough ball surfaces scatter light diffusely, optical interferometry was performed only on the smooth ball. An LCR meter applies a sinusoidal voltage of 0.2 V rms at 100 kHz across the contact. The resulting current is measured to obtain the complex impedance Z. Slip rings on each shaft maintain electrical connection, and rubber couplings insulate the disc from the rest of the shaft. Post-processing of Z yields the oil film thickness without considering surface roughness ${\overline{h}}_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$ and the breakdown ratio α. A torque sensor on the vertical spindle records the torque required to rotate the disc under a load of F_N = 22.4 N. The friction coefficient μ is expressed in terms of the measured torque M, the track radius R_t, and the applied load F_N as follows

$$\mu ={\displaystyle \frac{M}{R_{\mathrm{t}}{F}_{\mathrm{N}}}}$$

(23)

3.2. Materials

The ball specimen is made of AISI 52,100 steel with a Young’s modulus of 208 GPa and a Poisson’s ratio of 0.30. The disc specimen is fabricated from BK7 optical glass with a Young’s modulus of 72 GPa and a Poisson’s ratio of 0.26. Surface measurements were performed using optical interference microscopy. Surface roughness parameters, namely the standard deviation σ, skewness γ₁, and kurtosis γ₂ were calculated from the height distribution after removal of the spherical form but without waviness filtering, and are listed in Table 1 for both specimens. Additionally, corresponding two-dimensional height maps and histograms are provided in Figure 3 and Figure 4. The lubricant is a poly-α-olefin (PAO) base oil. It has a kinematic viscosity of 426.6 mm²/s at 40 °C and a relative permittivity of ε_r = 2.1 at 25 °C and 100 kHz.

3.3. Procedure

Prior to testing, the ball, disc, and oil bath were ultrasonically cleaned in hexane, and dried. The components were then assembled, and the bath was filled with PAO. With the ball pressed against the stationary disc, a reference interferogram was captured to determine the spacer-layer thickness, and the static complex impedance was measured. Afterward, the spindles were accelerated to the prescribed speed and the ball was re-loaded against the disc with the specified normal force. For each circumferential-speed condition, film thickness interferograms, electrical impedance data, and friction torque were recorded simultaneously. All experiments were carried out in a laboratory maintained at 25 ± 0.5 °C.

4. Experimental Results

The results obtained by applying the proposed equation to the oil film thickness measured in the ball-on-disc tests are shown in Figure 5. For balls with rough surfaces, the ${\overline{h}}_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$ values represented by open circles, were lower than the Hamrock-Dowson equation in the v range where the contact ratio α and the friction coefficient μ begin to increase. In contrast, the ${\overline{h}}_{\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}$ values represented by filled circles, showed good agreement with the Hamrock-Dowson equation, the central film thickness measured by optical interferometry, and the oil film thickness of Ball 0, which had an extremely smooth surface. This agreement was observed in the speed range above 0.1 m/s, suggesting that the proposed correction successfully compensates for the increase in capacitance caused by surface roughness. In the lower v range, below 0.1 m/s, ${\overline{h}}_{\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}$ converged to an approximately constant value, which was nearly equal to σ.

5. Discussion

5.1. Validity of Assuming Surface Height Distribution as a Parallel-Plate Model

This study assumes that, once the gap distance between infinitesimal surface elements is specified, the electric field remains uniform; in other words, the spatial field distribution is not considered, and the Laplace equation is not solved. To verify the validity of this simplification, we computed the capacitance and compared the results with those obtained from a more rigorous analysis. Surface-profile data for the ball, acquired with an optical interferometer were inverted and offset by a prescribed $\overline{h}$ to generate a two-dimensional array of the film thickness. Because the roughness of the disc is negligibly small relative to that of the ball, σ for the disc surface was set to zero. Cases in which the maximum asperity height exceeded $\overline{h}$, giving non-positive gap values (i.e., solid contact), were excluded. Capacitance for this gap field was evaluated by two methods. The first method employed electrostatic finite-element analysis using COMSOL^® Multiphysics (version 6.2). The gap domain was discretized, constant electric potentials were applied to the two bounding surfaces, and the Laplace equation was solved to obtain the potential distribution. The electric field was then derived from the gradient of the potential, and the capacitance was calculated by theoretical procedure. Because interactions between neighboring elements are fully considered, this method yields the most rigorous solution. Figure 6 shows a representative electric field distribution for Ball 1 at $\overline{h}$/σ = 3 and under an applied voltage of 1.0 V.

The second approach idealizes each pixel of the gap (pixel indices i, j = 1, …, n) as an independent parallel-plate capacitor that does not interact electrostatically with its neighbors. The capacitance of every pixel is evaluated individually from its local separation, and the overall capacitance C is approximated by connecting these elemental capacitors in parallel and summing their contributions. Denoting the surface-height distributions of the disc and ball by d_ij and b_ij, respectively, the local oil film thickness h_ij and the total capacitance are expressed as below.

$$h_{ij}=d_{ij}-b_{ij}+\overline{h}$$

(24)

$$C=\sum _i^n\sum _j^n\epsilon {\displaystyle \frac{S/n^2}{h_{ij}}}$$

(25)

Although this pixel-based model sacrifices accuracy by neglecting electrostatic coupling between elements, it offers a significant gain in computational efficiency. The capacitances predicted by the electrostatic finite-element analysis and the parallel-plate approximation were compared to validate the simpler model. Figure 7 compares the capacitances obtained with the two numerical schemes. The horizontal axis is the $\overline{h}/\mathsf{\sigma }$, which corresponds to the film parameter Λ; the vertical axis is the capacitance normalized by that of an ideal parallel flat capacitor whose gap equals $\overline{h}$. For every film thickness regime the normalized capacitance exceeds unity, indicating that surface roughness always enhances the capacitance relative to the flat-plate configuration. This enhancement becomes markedly larger as the film thickness decreases. The finite-element results (symbols) consistently exceeded those of the parallel-plate model (solid lines). The difference between the two predictions arises because the parallel-plate model does not solve Laplace’s equation and therefore cannot capture the mutual electric-field interactions that develop when asperities are very close to each other. This limitation becomes increasingly pronounced for roughness components with high wavenumbers. A detailed discussion of this effect is given in Appendix A. Nevertheless, the two curves lie so close together that, for the surface textures and film thickness range investigated here, treating each pixel as an independent parallel-plate capacitor can reasonably be considered acceptable.

5.2. Validity of Assuming Oil Film Thickness Distribution as a Gamma Distribution

The validity of describing the gap-height distribution with a gamma distribution was examined. Figure 8 shows, for each ball, the histogram of surface heights at a mean film thickness of $\overline{h}$ = 1200 nm together with the probability-density curve of a gamma distribution whose shape a and scale b parameters were estimated from $\overline{h}$ and σ by the method of moments. In every case the fitted gamma curve reproduces the measured histogram well. To assess goodness of fit as $\overline{h}$ varies, the gamma and normal models were compared using the Kullback–Leibler divergence D_KL, defined for an observed distribution P = {p_j} and a model Q = {q_j} as

$$D_{\mathrm{K}\mathrm{L}}(P|\left|Q\right)=\sum _j{p}_j\mathrm{l}\mathrm{o}\mathrm{g}{\displaystyle \frac{p_j}{q_j}}$$

(26)

with smaller values indicating better agreement [35]. Figure 9 plots D_KL versus $\overline{h}$: the solid line corresponds to the gamma-distribution model, while the dashed line represents the normal distribution model. Since the rate parameter of a normal distribution is independent of $\overline{h}$, its D_KL remains constant. In contrast, the gamma model provides a markedly better fit. Because the polished ball surfaces used in this study exhibit negative skewness, the corresponding oil film thickness distribution, which is the inverted surface profile, has positive skewness. A gamma distribution inherently possesses positive skewness, quantified as $2/\sqrt{a}$, whereas a normal distribution is symmetric with zero skewness; this asymmetry enables the gamma model to match the experimental data more accurately than the normal model. When $\overline{h}/\mathsf{\sigma }$ approached 1, the D_KL obtained with the gamma fit increased. At this point, as indicated by Eq.15, the shape parameter a also approaches 1. As a result, the gamma probability density function progressively develops a long right tail and, at a = 1, converges to the exponential distribution. This theoretical tendency supports the observation in Figure 5 that ${\overline{h}}_{\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}}$ approaches σ in the low-speed regime. However, the actual measured distribution deviates from the exponential distribution, and the gamma distribution can no longer adequately reproduce it, causing D_KL to become large. A rigorous description of the thickness distribution under contact conditions therefore remains an outstanding topic for future work.

Figure 10 shows the $\mathbb{E}$[1/H] × $\mathbb{E}$[H] for Ball 1, Ball 2, and the gamma distribution. This value is theoretically equal to the capacitance ratio C/C_flat shown in Figure 7. The results show closer agreement with Ball 2 than with Ball 1, suggesting that the distribution of the gap formed between the two surfaces of Ball 2 is better represented by a gamma distribution. Nevertheless, in both cases the deviation from the gamma approximation remains small, indicating that the gamma distribution provides a reasonable approximation for the capacitance increase due to surface roughness. Therefore, the analytically accessible value of $\mathbb{E}$[1/H] derived from the gamma approximation can be considered adequate for subsequent analysis of the parallel-plate model. It should be noted that an alternative method for computing $\mathbb{E}$[1/H] applicable to a wider range of non-normal distributions is presented in Appendix B.

5.3. Inverse Determination of Surface Roughness from Film Thickness Measurements

Up to this point the surface roughness σ has been regarded as known, and the focus has been on correcting the impedance-derived film thickness h_1,flat. In contrast, the problem can be inverted. If h_1,flat is forced to agree with a reference film thickness obtained by an independent method, then the roughness amplitude can be determined in situ. Rearranging Eqs. (21) and (22) to solve for σ explicitly, when the target value is h_HD, the solution takes the form,

$$\sigma ={\displaystyle \frac{h_{1,\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}}{2}}\sqrt{{\left({\displaystyle \frac{2h_{\mathrm{H}\mathrm{D}}}{{(1-\alpha )h}_{1,\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}}}-1\right)}^2-1}$$

(27)

Figure 11 plots the roughness estimates extracted from ball-on-disc experiments together with the surface roughness measured directly using an optical interference microscope. For entrainment speeds v ≥ 0.15 m/s the estimates coincide with the measurements, demonstrating high sensitivity of the method. Above v ≈ 0.30 m/s, the scatter increases because the difference between h_1,flat and h_HD becomes small at high speed, even a slight correction to h_1,flat strongly influences the computed σ. The estimated roughness decreases as v is lowered. This speed range corresponds to the regime in which the shape parameter a approaches 1, and the reduction is attributed to the mismatch that arises when the gap-height distribution is assumed to follow a gamma distribution. It should be noted that for Ball 0, whose surface roughness measured by white-light interferometry was negligibly small, the inverse analysis overestimated the roughness amplitude. This overestimation is attributed to the macroscopic film thickness variation inherent in EHD contacts, which the present model interprets as surface roughness. A detailed discussion of this effect is given in Appendix C.

6. Conclusions

In this study, the metallic contact region and the lubricating oil film region were represented by an RC parallel circuit, and the oil film thickness was modeled as a random variable H whose spatial distribution reflects the surface roughness profile. This modeling approach allowed us to construct a theoretical framework for an EIM that incorporates surface roughness. We showed that the capacitance increases in proportion to $\mathbb{E}$[1/H] and, since 1/H is a convex function of H, Jensen’s inequality directly explains why any roughness-based model necessarily predicts a larger capacitance than the ideal parallel-plate model. Because the gamma distribution is strictly positive and its expectation of the reciprocal admits a closed-form expression, we adopted it as the statistical model for film thickness. Using parameters obtained via the method of moments, we derived a correction formula that converts the conventional EIM mean film thickness h_1,flat into a roughness-compensated value. When this formula was applied to ball-on-disc experiments, the film thickness that had been underestimated for rough specimens was brought into better agreement with both the Hamrock-Dowson equation and interferometric measurements, indicating that the proposed method can correct oil film thickness estimates. However, in the thin-film regime where the ratio $\overline{h}/\mathsf{\sigma }$ approaches unity, the gamma distribution degenerates toward an exponential distribution, so it no longer reflects the actual distribution, and the accuracy degrades.