The Temperature Dependence of Divergence Pressure

Abstract

The so-called controversy in elastohydrodynamic lubrication (EHL) regarding the nature of the shear dependence of viscosity, Eyring versus Carreau, is truly a controversy regarding the pressure and temperature dependence of low-shear viscosity. Roelands removed data that contradicted his claims of accuracy for his correlation. The Roelands hoax became acceptable in EHL because ignoring the universal previtreous piezoviscous response made the traction calculated with the Eyring assumption appear to be reasonable. Traction and minimum film thickness calculations sometimes require the description of viscosity at pressures up to the glass transition pressure. There have been few measurements of viscosity at pressures up to glass pressure. Therefore, a need exists for a piezoviscous model that extrapolates accurately, and the Hybrid model fills that need. Here, an improved relation for the temperature dependence of divergence pressure is offered and extrapolation is demonstrated for a polyalphaolefin and propylene carbonate. A linear dependence of divergence pressure with temperature is more useful than previous versions. An improvement in the capability of high-pressure viscometry is suggested based upon the fractional Stokes Einstein Debye relation and the relatively simple measurements of DC conductivity.

1. Introduction

The description of the pressure and temperature dependence of viscosity has been an unresolved dilemma for classical elastohydrodynamic lubrication (EHL) since its founding. Because of the widespread use of the Roelands hoax [1], classical EHL may be seen to be the study of the lubrication of concentrated contacts by a hypothetical liquid that may never be discovered. Quantitative EHL employs viscosity measured in viscometers and fitted to correlations which are, at least, capable of describing the universal previtreous response [2] of glass-forming liquids, a group of liquids that includes lubricating oils. The classical approach has treated the effect of pressure on viscosity as a parameter to be adjusted to validate the favored hypothesis; for example, note the method of extracting pressure–viscosity coefficients from film thickness measurements [3]. These coefficients are unrelated to the piezoviscous strength [4] of the liquid.

The universal piezoviscous response is faster than exponential [5] as glass pressure is approached from lower pressure. Usually, the super-Arrhenius behavior occupies the greatest range of pressure from ambient to glass pressure and cannot be ignored. Correlations that are accurate for this regime are written in terms of a pressure for which the viscosity would be unbounded, i.e., the divergence pressure or idealized glass transition pressure, $p_{\infty }$. One of the first such expressions was proposed by Johari and Whalley [6] in 1972. The existence of glass transition pressure is rarely acknowledged in classical EHL. Therefore, the previtreous response is unanticipated and the divergence pressure is not considered. Furthermore, ignoring the previtreous piezoviscous response makes the traction calculated with the Eyring assumption appear to be reasonable [7]. The difficulty in measuring very large viscosities means that there have been few measurements of viscosity at pressures up to glass pressure. Therefore, a need exists for a piezoviscous model that extrapolates accurately, and the Hybrid model fills that need [8]. Here, an improved relation for the temperature dependence of divergence pressure is offered.

2. The Hybrid Model

Correlations that are appropriate for modeling the pressure and temperature dependence of the viscosity of glass formers are based upon free-volume theory or thermodynamic scaling. However, these are inaccurate for pressure extrapolation, and the former is not as precise as the viscometer. A phenomenological model for the same purpose is the Hybrid [9], which is the only model known to extrapolate well to pressures greater than those for which there are data [8], possibly because it is a precise description of the phenomenon.

The isothermal form of the Hybrid model [9] is

$$\mu ={\mu }_0{\left(1+{\displaystyle \frac{\alpha }{q}}p\right)}^q\exp \left({\displaystyle \frac{C_{F}p}{p_{\infty }-p}}\right)$$

(1)

The Hybrid model is useful for pressure extrapolation [8] and can be made temperature-dependent in the following manner:

$${\mu }_0={\mu }_{\infty }\exp \left({\displaystyle \frac{D_F{T}_{\infty }}{T-T_{\infty }}}\right),\alpha =a_2{\left({\displaystyle \frac{1}{T}}\right)}^2-a_1\left({\displaystyle \frac{1}{T}}\right)+a_0,q=b_1\left({\displaystyle \frac{1}{T}}\right)+b_0$$

(2)

This α is not a pressure–viscosity coefficient because the Johari and Whalley term influences the pressure dependence at low pressures. The divergence pressure (ideal glass transition pressure) has been given [8] until now as

$$p_{\infty }=c_1\left({\displaystyle \frac{1}{T}}\right)+c_0$$

(3)

Equation (3) combined with (1) and (2) has been shown to provide reasonable results over an interval of temperature of about 60 degrees when applied to EHL reference liquids [10]. The pressure fragility parameter, C_F, has been assumed to be equal to the temperature fragility parameter, D_F, because evaluating them separately does not improve the accuracy. The curve fitting was accomplished by minimizing the relative standard deviation with the Solver add-in for Microsoft Excel. Here, graphs of viscosity versus pressure are plotted with viscosity on a logarithmic scale because the data cover many orders of magnitude.

3. Temperature Dependence of Divergence Pressure

The framework above is deficient when applied to the viscosity of a silicone oil reported in the 1953 pressure–viscosity report of the ASME Research Committee on Lubrication [11]. This problem is shown in Figure 1, where the Hybrid model with Equation (3) successfully describes the viscosity at 0 °C and 99 °C; however, at the intermediate temperature of 38 °C, it is shown that the divergence pressure specified (by Equation (3)) is overstated and must be reduced to match the data. The predicted viscosity at 615 MPa and 38 °C is only 78% of the measured value. The parameters regressed from the ASME data are given in

Table 1. Parameters of the Hybrid model for a silicone oil presented in the 1953 ASME pressure–viscosity report.

$p_{\infty }(T)$	Equation (3)	Equation (4)
${\mu }_{\infty }/$mPa·s	2.082	1.558
$T_{\infty }/$K	141.2	129.5
$a_0/$GPa−1	31.8	40.0
$a_1/$K/GPa	266	303
$a_2/$K2/GPa	−1.543 × 106	−2.126 × 106
$b_0$	−4.33	−0.8887
$b_1/$K	2101	709
$C_F=D_F$	4.23	5.31
$c_0/$GPa	3.92	−1.712
$c_1/$K·GPa	−862	-
$c_{-1}/$GPa/K	-	0.00923
AARD	5.1%	3.3%

. Dimethyl silicones are extremely compressible and, therefore, the sigmoidal shape of a log viscosity versus pressure curve can be observed at relatively small ranges of pressure, as in Figure 1. The free volume fraction is significantly reduced with unusually small increases in pressure because the total volume is significantly reduced with no change in occupied volume.

This silicone oil was oil 2 in Roelands’ Figure IV-4 [12] with pressure viscosity index, Z = 0.50, and slope index, S = 0.48 (see Figure 2). However, Roelands showed only 12 of the data points, those which could be successfully fitted to his correlation. For three temperatures, the data not discarded by Roelands and his correlation are given in Figure 2, where it is shown that the correlation cannot describe the pressure dependence at some low pressures and at all high pressures [1].

The physics literature regarding the previtreous response [13] suggests a linear relation between the divergence pressure and temperature.

$$p_{\infty }=c_0+c_{-1}T$$

(4)

There is no theoretical foundation for the linear dependence, only an empirical observation. To investigate the use of a linear form for $p_{\infty }\left(T\right)$, the regression was repeated, substituting Equation (4) for (3). This model is shown in Figure 1 and the parameters are listed in Table 1. The average absolute relative deviation (AARD) has improved from 6% to 3% and the agreement at the intermediate temperature is now excellent.

It is useful to apply the Hybrid model with linear $p_{\infty }\left(T\right)$ to the EHL reference liquids of Reference [10]. The resulting parameters are listed in

Table 2. Parameters of the Hybrid model with Equation (4) for reference liquids.

Sample	Squalane	DIDP	DiPEiC9	TOTM	FVA3	PAO-4	PC
Temperature/°C	20 to 65	40 to 100	50 to 90	40 to 100	40 to 120	40 to 120	25 to 105
Max Measured Pressure/GPa	1.35	1.00	0.70	1.10	1.20	1.00	1.00
Max Measured Viscosity/Pa·s	3.5 × 105	2.4 × 104	9 × 103	1.13 × 104	2 × 103	1.22 × 102	2.345
${\mu }_{\infty }/$mPa·s	0.0337	0.0396	0.01580	0.01194	0.0477	0.0779	0.0396
$T_{\infty }/$K	150.8	176.0	156.3	140.6	176.8	171.4	96.4
$a_0/$GPa−1	−23.3	−1.209	7.235	0.486	7.40	21.7	−21.9
$a_1/$K/GPa	−17,200	717	−114.8	−4038	319.2	9218	−18,760
$a_2/$K2/GPa	−1.31 × 106	2.145 × 106	1.339 × 106	0.312 × 106	1.153 × 106	2.26 × 106	−3.34 × 106
$b_0$	−4.246	−2.828	1.194	−1.368	−8.319	−5.231	0.7706
$b_1/$K	2863	3395	1019	1520	4952	2964	−87.60
$C_F=D_F$	6.581	5.35	10.12	10.90	5.794	4.374	8.639
$c_0/$GPa	−2.819	−3.480	−4.501	−4.878	−3.709	−0.632	−6.235
$c_{-1}/$GPa/K	0.0175	0.0164	0.0171	0.0225	0.0164	0.0108	0.0292
AARD	2.5%	3.0%	3.6%	2.2%	3.2%	2.0%	1.8%

. These liquids are as follows:

• Squalane;
• Diisodecyl phthalate (DIDP);
• Dipentaerythritol hexaisononanoate (DiPEiC9);
• Tri(2-ethylhexyl) trimellitate (TOTM);
• Gear Research Center, FZG, reference gear oil (FVA3).

There is only a small improvement in AARD. The idealized glass pressure calculated from the Hybrid model with a linear $p_{\infty }\left(T\right)$ is plotted in Figure 3. For the standard form of the Roelands correlation, the viscosity is unbounded only at −135 °C [12] for all finite pressures, as shown in Figure 3.

For an ambient pressure isobar, if the linear relation (4) for $p_{\infty }$ holds for low temperatures, then $T_{\infty }$ should be equal to $-c_0/c_{-1}$. This is not correct, and $T_{\infty }<-c_0/c_{-1}$ in every case. Therefore, the response of $p_{\infty }$ to temperature may not be linear for sub-ambient temperatures, or the pressure fragility parameter, $C_F$ in Equation (1), may not be assumed to be equal to the temperature fragility parameter, $D_F$ in Equation (2). A regression was made for DiPEiC9 assuming $T_{\infty }=-c_0/c_{-1}$ and varying $C_F$ and $D_F$ independently. There was no change in the AARD averaged over the entire data set. However, the accuracy of the prediction at the highest viscosities was poorer. Because the usefulness of the Hybrid correlation lies in pressure extrapolation, any degradation of the accuracy at high pressure is not welcome.

4. Examples of Special Importance

A polyalphaolefin of 4 centistokes grade at 100 °C, PAO-4, is a common base oil for blending oils used in EHL. The viscosity of a PAO-4 has been measured to 1 GPa at 40 to 120 °C and these data were fitted to the improved Yasutomi correlation in Reference [14], as shown in Figure 4. The Yasutomi parameters are given in [14]. The improved Yasutomi correlation is a free volume model that fits the data very well with fewer parameters than the Hybrid. The ten parameters of the Hybrid model with linear $p_{\infty }\left(T\right)$ were fitted to these data and are listed in Table 2. Additional results are available from a 1.4 GPa falling cylinder viscometer [15] at 40 °C and these have been added to Figure 4. The improved Yasutomi model extrapolated to 1.42 GPa predicted the low shear viscosity to be 44% of the measured value, whereas the Hybrid model was within 2%. Clearly, the Hybrid model extrapolates well, although the 2% was clearly fortuitous, and the Yasutomi model should not be used for this purpose. The hybrid model is a precise description of the pressure–viscosity phenomenon, whereas the Yasutomi approach does not necessarily incorporate the super-Arrhenius previtreous response.

A diester, di(2-ethylhexyl) sebacate, DEHS or DOS, was the original MIL-L-7808 jet engine oil [16]. It is also the most widely used pressure-transmitting medium for high pressures because of the relatively weak piezoviscous response. It is one of the oils investigated in the 1953 ASME viscosity report [11]. The isothermal form of the Hybrid model, Equation (1), was fitted to the ASME data for 37.8 °C and the parameters are ${\mu }_0=$ 0.0113 Pa·s, $\alpha =$ 12.88 GPa⁻¹, $q=$ 4.79, $C_F=$ 6.36, and $p_{\infty }=$ 3.44 GPa (see Figure 5). There are published viscosities to 1.4 GPa for this oil at this temperature [15] and these have been added to Figure 5. The extrapolation from the ASME data is reasonable, even more so considering that there is no inflection in the ASME data to guide the characterization of $p_{\infty }$. If the glass transition viscosity is 10⁷ Pa·s, then the glass transition pressure at 38 °C is 2.2 GPa. This large pressure range of the liquid state explains the usefulness of DOS as a pressure medium.

5. An Application to High-Pressure Physics

The fractional Stokes Einstein Debye relation [17] correlates viscosity with the DC conductivity, $\sigma$, of a liquid.

$$\sigma {\mu }^s=\mathrm{const}$$

(5)

Here, the exponent, $s$, is less than or equal to one. If $s=1$, this is the Stokes Einstein Debye equation. This relation can be tested using the conductivity data of Thoms and coworkers [18] for propylene carbonate, PC. They measured the conductivity at 77 °C to 1.3 GPa. Casalini and Bair [19] measured the viscosity of PC at 25, 55, and 105 °C up to 1.0 GPa. These two contributions would allow for a test of the fractional Stokes Einstein Debye relation if the viscosities were measured under the conditions of the conductivity measurements. Such a test at 77 °C to 1.3 GPa will require pressure extrapolation and temperature interpolation of the viscosity data shown in Figure 6.

The Hybrid model fitted to the viscosities of propylene carbonate, PC, is shown in Figure 6 and the parameters are listed in Table 2. The predicted viscosity at 77 °C up to 1.3 GPa is shown as the dashed curve in Figure 6. The reciprocal of the electrical conductivity data from Reference [18] are plotted versus the interpolated and extrapolated viscosity in Figure 7. The slope of the curve is $s=$ 0.824 and the Hybrid model and the fractional Stokes Einstein Debye equation are validated.

This result suggests an improvement in the capability of high-pressure viscometry. If the fractional Stokes Einstein Debye Equation (5) is always valid along a pressure–viscosity isotherm, as it is for PC in Figure 7, then viscometer measurements may be extended to greater pressures with conductivity measurements. A measurement of conductivity does not require moving pieces within the compressed liquid sample, other than the motion of charge carriers, making the measurement easier than viscometry.

6. Conclusions

Currently, most high-pressure viscometers used in EHL are limited in pressure capability to about 1 GPa. There are only a few laboratories with greater capability. And the limitation is often to lower pressure because of the difficulty in measuring very large viscosities. The Hybrid model provides useful extrapolation to pressures of the glass transition [10,20] which will be helpful in predicting minimum film thickness [21] and traction. A linear relation for the temperature dependence of divergence pressure improves the fit of the Hybrid model to viscometer measurements. The usefulness of the fractional Stokes Einstein Debye equation for high-pressure viscometry has been demonstrated. Overlapping measurements of viscosity and DC conductivity at low pressures will yield the value of the fraction, s, which may be used with conductivity measurements at higher pressures to calculate viscosity at greater pressures. Students of tribology must no longer be taught the Roelands equation as a description of the pressure–viscosity effect.