Classical EHL Versus Quantitative EHL: A Perspective Part II—Super-Arrhenius Piezoviscosity, an Essential Component of Elastohydrodynamic Friction Missing from Classical EHL

The pressure dependence of viscosity was necessary to explain the presence of a film sufficiently thick to separate the roughness features of engineering surfaces in elastohydrodynamic lubrication (EHL) [1]. Therefore, piezoviscous response is at the foundation of the field. Faster-than-exponential, or super-Arrhenius pressure dependence of viscosity (in reference to the dependence of a material property which relies on an Arrhenius law), has been a feature of accurate measurements of viscosity at high pressure for 90 years [2]. This behavior has surprisingly been absent from the dialog of classical EHL since its beginning, and this absence is at least partly responsible for the spectacular failure of the field to provide an understanding of EHD friction using the thermophysical properties of the liquid. In this second installment [3] of a series differentiating classical from quantitative EHL, it will be shown that, although missing from the classical approach, super-Arrhenius piezoviscous response is the natural behavior of glass-forming liquids and that it is indispensable for understanding the behavior of liquids at high pressure and high shear stress.

Faster-than-exponential pressure-viscosity behavior has been observed in nearly every type of viscometer. The guided falling cylinder viscometer used by Nobel Laureate, Bridgman [2] in 1926 was perhaps the first to achieve sufficiently high pressure (1.2 GPa) to see the transition, inflection in log(viscosity) versus pressure, from slower to faster than exponential in simple, low-viscosity liquids. However, by 1959, Lowitz et al. [4] using a rolling ball viscometer had obtained the inflection in diphenylethane at a pressure of only 75 MPa. In 1973, Hutton and Phillips [5] employed a Couette viscometer to demonstrate faster-than-exponential response to refute the incorrect slower-than-exponential behavior that had been derived from an EHL film-thickness analysis based on Newtonian viscosity. Jones et al. [6] in 1975 found faster-than-exponential response at low pressures for two lubricating oils with a capillary viscometer. Piermarini et al. [7] in 1978 and later Cook et al. [8] used diamond anvil cells as dropping ball and rolling ball viscometers, respectively, to observe the inflection in simple liquids such as methanol. In 2012 faster-than-exponential pressure response was reported for a mineral oil in an oscillating quartz viscometer at the Technical University of Clausthal [9].

Moreover, the primary dielectric relaxation time may be proportional to viscosity according to Harrison [10] and may be used to extend viscosity measurements on polar liquids to very high pressures and very large viscosities. Dielectric spectroscopy is relatively easy at high pressure compared to viscometry, requiring only an electrical connection to a sample filled capacitor in the pressure vessel. The technique has been shown to be useful for prediction of the pressure dependence of viscosity of lubricants over the years [11‐13]. The pressure dependence of the dielectric relaxation time is always seen to be super-Arrhenius [14] since short relaxation times (low viscosities) are not accessible to the technique when applied within a pressure vessel. However, excellent agreement in the derivative analysis has been demonstrated for propylene carbonate [15] for viscosity compared with relaxation time. Viscosity and dielectric relaxation time for dibutyl phthalate can be described by the same super-Arrhenius function for pressure to 1.4 GPa [16]. In fact, the Paluch [17] equation for the pressure-viscosity effect at high pressure, below, is the analog of the Johari and Whaley equation for the pressure dependence of relaxation time.

The dielectric relaxation time for di-isobutyl phthalate (DiBP) from [18] is plotted in Fig. 1. New viscosity measurements from Georgia Tech are also plotted. The relaxation times have been multiplied by a constant, k = 0.4 GPa, to compare with the viscosity. The measurements extend across eleven orders-of-magnitude. The overlapping curves show that the pressure dependence of the shear viscosity is the same as the pressure dependence of the relaxation time and that it should be possible to calculate the viscosity up to 10⁹ Pa s from a simple measure of relaxation time. This high viscosity is not accessible to viscometers. The equation fitted to the data in Fig. 1 is the hybrid model [19], Eq. (2), combining the McEwen equation for slower-than-exponential with the Paluch Eq. (1) for faster-than-exponential response.

The parameters are given in Table 1.

To summarize, faster-than-exponential pressure-viscosity behavior has been observed in

Therefore, the absence of this effect in the classical EHL description of the pressure dependence of viscosity cannot be justified.

In the following, examples are given of the necessity of super-Arrhenius piezoviscosity in explaining the response of liquids at high pressure and high shear stress.

An example can be made of the EHD friction of a polyolester (POE) and a polyalphaolefin (PAO) shown in Fig. 2. The measurements were performed in a skewed roller tribometer [20] which can resolve the friction at very low slide/roll ratio in a point contact. Here the rolling velocity was 2 m/s. For Hertz contact pressure of 1.4 GPa at 57 °C and slide-to-roll ratio of 10⁻³–10⁻¹, this POE generates a substantially greater friction coefficient than does the PAO. The reason can clearly be seen in the pressure-viscosity response in Fig. 3. In spite of the POE having a lower initial pressure-viscosity coefficient, d ln (μ) / dp | _p = 0 = 17.6 GPa⁻¹ compared to 21.4 GPa⁻¹ for POE, the viscosity at high pressure (>600 MPa) is much greater for the POE. The equation fitted to the data is the hybrid model (2), combining the slower-than-exponential with the faster-than-exponential response. The parameters are given in Table 1.

Models which describe the inflection, such as Eq. (2) or free-volume [8], do not exist in classical EHL. Rather, the viscosity is described by a fictional story about a Barus equation [21] or by the most popular, Roelands equation [22].

Here the pole pressure and the pole viscosity have universal values of \(p_{p} = - 0.196{\text{ GPa}}\) and \(\eta_{p} = 6.31 \times 10^{ - 5} {\text{ Pa}}\,{\text{s}}\), respectively [22]. In Fig. 4, this equation has been fitted to the viscosities as instructed by Roelands. That is, only data at pressures less than the inflection pressure are fitted, up to 500 MPa for POE and up to 700 MPa for PAO. The parameters are given in Table 2. The results reported in Fig. 4 show that the viscosities at the high pressures of the friction generating regions of an EHD contact cannot be distinguished using the descriptions employed by classical EHL.

There is an interesting and common problem illustrated in Fig. 4. The Roelands model cannot accurately match the curvature of the PAO data at low pressures so that the value of μ ₀ as regressed, 24 mPa s, is much greater than the measured value, 20 mPa s. To relieve this problem, a smaller pressure interval may be selected for the data fitting, say to 350 MPa. When this is done in Fig. 5, the viscosity of the PAO at pressure above 300 MPa is predicted to be greater than the POE, which is obviously incorrect. In defense of Roelands, it must be mentioned that he did not recommend his correlation for elastohydrodynamic pressures [22], only for hydrodynamic pressure.

Clearly, the friction behavior of these two oils, a PAO and a POE, cannot be explained using the pressure-viscosity models most often employed in classical EHL, and the initial pressure-viscosity coefficients give no indication of the viscosity at high pressure where friction is generated. When the inflection cannot be modeled and two liquids with similar low pressure behavior have different inflection pressures, then the difference in viscosity in the Hertz zone cannot be accounted for. Different liquids may appear to have the same properties at low pressure while having quite different viscosity in the Hertz contact region. Inaccurate descriptions of the pressure dependence have influenced even the way that the shear dependence of viscosity has been represented as shown in the next section.

A particular aspect of the sliding speed dependence of EHD friction, the shape of the friction versus sliding velocity curve, once received much attention and was the motivation for the use of a thixotropy model to describe shear-thinning in classical EHL. For constant rolling speed, contact pressure and temperature, when the friction coefficient or average shear stress, \(\bar{\tau }\), is plotted against the sliding speed, average shear rate, \(\bar{\dot{\gamma }}\), or slide-to-roll ratio, a substantial portion of the friction gradient is logarithmic. That is, friction plotted versus the logarithm of sliding speed displays an interval of data which lie on a straight line in a region of low slide-to-roll ratio often held to be isothermal. Many laboratories observed this response, [23, 24] for examples. In Fig. 2, the friction coefficient for the POE varies with slide/roll ratio in logarithmic fashion.

The simplest explanation of this behavior has been to assume that the logarithmic response of the film over the range of pressures in the contact was exactly the shear response of the liquid under constant pressure and temperature [23, 24]. This would be the sinh-law for thixotropy which was given a theoretical foundation by Eyring [25].

For this explanation, the value of the Eyring stress is found from the slope of the logarithmic part of the friction curve as \(\tau_{0} = {{{\text{d}}\ln \bar{\tau }} \mathord{\left/ {\vphantom {{{\text{d}}\ln \bar{\tau }} {{\text{d}}\dot{\gamma }}}} \right. \kern-0pt} {{\text{d}}\dot{\gamma }}}\). The logarithmic function plotted for the POE in Fig. 2 yields τ ₀ = 8 MPa while the curve through the PAO data is a power-law. Such a hypothesis requires another assumption, that the viscosity not be strongly dependent upon pressure. If the viscosity, μ, is strongly dependent on pressure and Eq. (4) is correct, the pressure variation across the contact would alter the logarithmic friction response of the film. In classical EHL, which does not employ real viscosity as measured in viscometers, this assumption of weak pressure dependence could not, of course, be tested.

More than 20 years ago, it was found that the slope of the logarithmic friction gradient for two liquids could be quantitatively explained by the assumption of a limiting stress which was proportional to pressure [26] if real super-Arrhenius pressure dependent viscosity, as measured in viscometers, was employed. The constitutive equation isWith this shear response and the Hertz pressure distribution for circular contact, 0 ≤ r ≤ 1,

The shape of the friction curve results from the growth of the stress-limited circular region of the contact [26] and the logarithmic slope was found to bewhere \(\alpha = \left. {{{{\text{d}}\ln \left( \mu \right)} \mathord{\left/ {\vphantom {{{\text{d}}\ln \left( \mu \right)} {{\text{d}}p}}} \right. \kern-0pt} {{\text{d}}p}}} \right|_{{p = p_{H} }}\). Thus, the use of Eq. (4) with \(\tau_{0} = {{{\text{d}}\bar{\tau }} \mathord{\left/ {\vphantom {{{\text{d}}\bar{\tau }} {\ln \left( {\dot{\gamma }} \right)}}} \right. \kern-0pt} {\ln \left( {\dot{\gamma }} \right)}}\) for a constitutive law was not justified. The liquids were a polyphenyl ether, 5P4E, and a mineral oil, LVI 260, which were shown to give logarithmic friction behavior [23]. The limit to shear stress can affect friction at low sliding velocity because of the large value of viscosity at the contact center which, of course, results from super-Arrhenius response.

In the example of the logarithmic function plotted for the POE in Fig. 2, the hybrid model (2) gives α = 19.6 GPa⁻¹ for p _H  = 1.42 GPa. Therefore, Eq. (7) yields Λ = 0.078, a reasonable estimate of the limiting stress coefficient.

In the previous example, the appearance of logarithmic friction response was explained using super-Arrhenius piezoviscosity. In the next example, another type of non-intuitive response will be explained using natural pressure-viscosity behavior.

Höglund and Jacobson [31] employed the Luleå high-pressure chamber [32] to measure the shear stress supported by liquid lubricants sheared at low shear rate. At a specific test temperature, the pressure was raised in small increments after which the liquid was sheared at ostensibly constant shear rate. After a nearly exponential increase in stress with pressure, there was a range of pressure for which the stress variation was linear in pressure and the linear portion began at shear stress equal to about 2 MPa and reached to about 12 MPa. See Fig. 9 which is Fig. 6 of Höglund and Jacobson [31] for a mineral oil and which is also Fig. 9.6 of Jacobson [32]. The same response was seen at 40, 70 and 100 °C. Since this linear behavior is unexpected for a Newtonian liquid, the authors labeled the pressure at the onset of linear response as the “solidification pressure” and labeled the shear stress above that pressure as “limiting shear stress”. This definition differs from that used by the authors [Eq. (5)] where the limiting stress is defined by the appearance of rate-independent response.

This linear pressure-shear stress response can be seen to naturally arise from the pressure and shear dependent viscosity typical of a mineral oil. One of the most thoroughly characterized mineral oils is Shell T9 which has been the subject of numerous EHD friction studies [33‐36] since the thermophysical properties are known to high pressures. The viscosity of this mineral oil is shown in Fig. 10. Note that the measured T9 viscosity dependence with pressure follows a super-Arrhenius response which occurs at the three temperatures investigated (from 40 to 120 °C) beginning at pressures less than 1 GPa. The curves fitted to the data in Fig. 10 represent the improved Yasutomi model [37] below. An equation of state is not necessary to apply this correlation.

The parameters are given in Table 3. The shear dependent viscosity [33] is given bywith a = 5 and n = 0.35. The shear does not localize for this mineral oil until the stress approaches 0.083 p [33].

The shear stress, \(\tau = \eta \dot{\gamma }\), is plotted for pressure steps of 25 MPa and the indicated temperatures in Fig. 11 where it is assumed that the shear rate is \(\dot{\gamma } = 10\) s⁻¹. The same linear response beginning at τ ≈ 2 MPa as in Fig. 9 is present in Fig. 11. It is the faster-than-exponential piezoviscosity combined with shear-thinning which produces the sharp transition to linear behavior. It is not necessary that the shear rate be estimated accurately. Increasing the shear rate simply shifts the curves to the right in Fig. 11, whereas the slope of the linear part is determined by the local pressure-viscosity coefficient.

A recent article from classical EHL [38] has addressed the thermal non-Newtonian EHL of another Shell turbine oil, T33, also a mineral oil. For the shear dependence, they employed the sinh-law (4) with τ ₀ = 10 MPa. For the pressure and temperature dependence they used the full Roelands equation with parameters listed in Table 2.

Roelands specified a universal value of the divergence temperature, T _∞ = −135 °C. The viscosity predicted by this model is plotted as the dashed curves in Fig. 10. Now the reader should recognize that this is not the pressure or temperature dependence of a mineral oil and that this was not the intended use of Roelands’ equation.

The shear stresses for the Roelands and Eyring assumptions are plotted for pressure steps of 25 MPa and the indicated temperatures in Fig. 12 where it is again assumed that the shear rate is \(\dot{\gamma } = 10\) s⁻¹. The response reported in Fig. 12 is quite different from Figs. 9 and 11. There is no clear break point at stress of 2 MPa going from nearly exponential to linear. It is more difficult to identify any interval of linear behavior in Fig. 12.

Ninety years of measurements of the dynamic properties of supercooled van der Waals liquids to high pressure have established that faster-than-exponential response to pressure is universal and must be accounted for in a quantitative approach to EHL. For ordinary lubricants, the low pressure response is most often slower-than-exponential, resulting in an inflection point in a log viscosity versus pressure plot. The absence of this real behavior from classical EHL has had serious consequences and is at the heart of the failure to understand the mechanism of friction.

For example, when the inflection is not acknowledged, and two liquids with similar low pressure behavior have different inflection pressures, then the difference in viscosity in the Hertz zone cannot be accounted for. Different liquids may appear to be the same at low pressure (similar pressure-viscosity coefficients) while having quite different viscosity in the Hertz contact region. Differences in friction between similar liquids can be explained by differences in inflection pressure. This behavior is critical for the understanding of friction in classical EHL, especially at low sliding speeds. By examples it was shown that the response of liquids to combined high pressure and high shear stress can only be understood when realistic pressure dependence of viscosity is employed.

Contact-based measurements have not provided accurate estimates of the pressure dependence of viscosity (see [3] for instance). Film thickness is sensitive only to the piezoviscous response near ambient pressure. If a perfect film thickness formula existed and a perfect definition of pressure-viscosity coefficient existed, then for a Newtonian liquid with the idealized compressibility, it would be possible to extract a pressure-viscosity coefficient from a perfect film thickness measurement carried out at perfectly stationary conditions. However, having just the coefficient gives no information regarding the functional form of the pressure-viscosity behavior.

Friction in the very low slide/roll regime is affected by roller elastic creep. At slightly greater slide/roll ratio, a true Newtonian regime may appear; however, it has not proven possible to deconvolve the pressure-viscosity function from friction measurement which is necessarily the average of the local shear stress over an area for which the pressure is variable and the surface not well defined. In the high sliding regime, the details of the piezoviscous response are obscured by thermal and non-Newtonian effects [35].

The departure from exponential pressure dependence is related to fragility, a property which quantifies the rapid increase in the sensitivity of the dynamic properties to changes in temperature and pressure as the glass point is approached from the liquid side. While the pressure fragility is related to the temperature fragility [14], there is no known formula for predicting one from the other.

It should be clear that, had classical EHL employed realistic pressure dependence of viscosity from its beginning, the field would have been in a better position today to solve engineering problems which involve the differences among lubricant chemical structures.