The Shear Dependence of Viscosity at Elevated Pressure

CHAPTER EIGHT

The Shear Dependence of Viscosity at Elevated Pressure 8.1 INTRODUCTION In Section 7.1, we saw examples of organic liquids of low molecular weight, M , 400 kg/kmol, that shear thin in the same manner as a liquid polymer with M 5 28,000. The shear stress required to obtain the nonlinear response in Fig. 7.1, however, is greater for the low molecular weight material by the same ratio that the molecular weight is smaller. The classical Newtonian film thickness formulas such as Eq. (1.18), if they were validated, must have been validated with oils with similarly low molecular weight. For the automotive applications, in particular, the use of neat low viscosity mineral oil to arrive at the desired viscosity of a lubricant is becoming uncommon. Mineral oils are generally blended with polymer thickener to reduce the temperature dependence of the viscosity and frequently now the base stock is of higher molecular weight than before. Both of these practices may result in shear-thinning within the inlet zone of the lubricated contact where the film thickness for the contact is established. Examples of measurements of film thicknesses for which the Newtonian assumption overestimates the thicknesses were shown in Fig. 1.6. These measurements were made in a large glass versus steel contact. The shear-thinning response of the same liquids can be seen in Fig. 8.1, where the data were obtained with a pressurized Couette viscometer described in the previous chapter. The interval of shear stress of 0.1
2 MPa is important for the generation of pressure within the inlet zone for the large glass versus steel contact used in most film thickness experiments. The pressure profile in this region establishes the film thickness for the Hertz zone and, therefore, will influence friction. The low molecular weight (M 5 423 kg/kmol) branched alkane, squalane, can be seen in Fig. 8.1 to be reasonably constant in viscosity to the highest shear stress investigated, τ 5 0:3 MPa, and in other work [1], the Newtonian High Pressure Rheology for Quantitative Elastohydrodynamics. DOI: https://doi.org/10.1016/B978-0-444-64156-4.00008-8

© 2019 Elsevier B.V. All rights reserved.

223

224

High Pressure Rheology for Quantitative Elastohydrodynamics

Figure 8.1 Flow curves at elevated pressure for the liquids used in the film thickness measurements shown in Fig. 1.6. The interval of shear stress from 102 to 2 3 103 kPa is apparently very important to the film-forming ability of a liquid.

limit has been found to be reached at about 6 MPa. The Newtonian assumption embodied in the Hamrock and Dowson film thickness Eq. (1.18) provides an accurate prediction of the film thickness in Fig. 1.6 only for the low molecular weight squalane. This film thickness calculation is accurate for the Newtonian case, even when the pressure dependence of the viscosity is very different from that assumed by Hamrock and Dowson [2] for Eq. (1.18), because the particular pressure
viscosity coefficient, α defined by Eq. (1.3), is a reasonably accurate means to quantify the piezoviscous response of a Newtonian liquid for the purpose of film formation in the inlet zone. The PAO650 oil, a high viscosity polyalphaolefin, generates a substantially thinner film than the Newtonian prediction and the discrepancy increases with increasing velocity in Fig. 1.6. The viscosity in Fig. 8.1 for this liquid can be seen to decrease with shear stress in the important interval of stress above τ 5 0.1 MPa. Another type of shear-thinning is seen in the traction fluid, S50, in Fig. 8.1. The viscosity decreases from its lowshear value, μ, in the range of 0:003 , τ , 0:02 MPa to a second Newtonian value, μ2 . Measurements at still higher shear stress [3] have shown that the second Newtonian plateau of S50 extents to shear stress

The Shear Dependence of Viscosity at Elevated Pressure

225

greater than 10 MPa. The film thickness of S50 in Fig. 1.6 is less than the Newtonian prediction but the velocity dependence is the same as the Newtonian prediction, suggesting that the film is governed by the Newtonian response of the second Newtonian plateau. Then the thickness of an elastohydrodynamic lubrication (EHL) film depends upon some average of the viscosity in the interval of stress important to film-forming and the velocity dependence of the thickness of an EHL film is governed by the shear dependence of the viscosity in the important stress interval. Compared with film thickness, the effect of shear-thinning on traction is somewhat more complicated to describe, while at the same time more simple to calculate and in some circumstances more subtle in its effect. At low contact pressures, for highly shear-thinning liquids, the departure from Newtonian traction behavior can be dominated by the shear dependence of viscosity as is illustrated by Fig. 8.2 for the PAO650. In this figure, the experimental data were generated at INSA, Lyon [4] in a point contact between a steel ball and glass plate at Hertz pressure of pH 5 0.528 GPa. The shear-thinning traction calculation utilizes the single Newtonian version of the Carreau [5] equation,

Figure 8.2 The concentrated contact traction response of the PAO650 in Fig. 8.1. Calculations for both the Newtonian assumption and the Carreau equation (8.1) fitted to the data of Fig. 8.1 were based on uniform shear rate within the contact using film thicknesses from the Newtonian calculation of Eq. (1.18), broken curves, and from interferometric film measurements, solid curves.

226

High Pressure Rheology for Quantitative Elastohydrodynamics

"




μγ_ η 5 μ 11 G

2 #ðn21Þ=2 (8.1)

fitted to the data of Fig. 8.1 to give G 5 3:1 3 104 Pa and n 5 0:74. The shear rate was assumed to be uniform throughout the Hertz region and related to the slide-to-roll ratio, Σ, by γ_ 5 uΣ=hc . The dependence of the low-shear viscosity on pressure, μðpÞ, was described by the isothermal Tait
Doolittle free volume model, Eqs. (4.6) and (6.43), with parameters given in Ref. [4]. In Fig. 8.2, the value of film thickness used in the calculation was obtained by both interferometric measurement (solid curves) and the Newtonian formula (1.18) of Hamrock and Dowson (broken curves). The traction coefficient was found by dividing the average shear stress, τ, by the average pressure, p 5 2pH =3. The average shear stress was found by integration of the local shear stress over the circular contact area assuming the hemispherical Hertz pressure distribution. Clearly the Newtonian assumption is not helpful for any easily measurable value of traction coefficient in Fig. 8.2. The slide-to-roll ratio must be reduced to Σ , 3 3 1024 , where the traction coefficient based on a Hertz zone calculation is about τ=p 5 5 3 1025 , to achieve a traction coefficient within 10% of the Newtonian calculation. For these conditions, it is likely that the pure rolling friction generated by drawing liquid into the inlet would result in a traction force comparable to that generated within the Hertz zone. As of this writing, the author is not aware of any traction measurements that have been sufficiently sensitive to extract an accurate value of the averaged low-shear viscosity from contact-based measurements, even for very low molecular weight liquids. It can also be seen in Fig. 8.2, that when the shear-thinning substantially affects the film thickness, as it does in this case, the Newtonian-based film thickness calculation will result in reduced accuracy of the traction prediction. Thus, the first rheological assumption of classical EHL, the Newtonian inlet, has actually delayed progress in understanding the real effect of shear dependence on friction. Soon after publication of the first edition of this book, a complete isothermal EHL numerical simulation [6] of this EHL contact resulted in strikingly accurate reproduction of the film thickness and friction as will be shown in Chapter 13, Application to Elastohydrodynamic Friction. In the complete isothermal numerical simulation, the assumption of Hertz pressure distribution was replaced by the claculted pressure. The assumptions of Hertz pressure and constant film thickness are helpful for a

The Shear Dependence of Viscosity at Elevated Pressure

227

simple friction prediction but cannot compete with the full simulation for accuracy. This example is typical only for low pressure contacts. As the contact pressure is increased, the effect that shear-thinning has on traction is reduced. At the highest contact pressures, a linear regime develops, for small Σ, where the traction varies linearly with Σ as a result of the elastic response of the solid rollers. The traction coefficient at large Σ for high pressures is dominated by a liquid failure that has come to be known as limiting shear stress where the film cannot support a shear stress greater than some characteristic fraction of the average pressure. Mechanical shear localization and the resulting limit to the shear stress will be the subject of Chapter 10, Shear Localization, Slip and the Limiting Stress. For intermediate pressures, the sinh-law traction behavior of Eq. (1.19) presents itself as a result of the interplay of the pressure
viscosity response and the limiting stress. The parameters of Eq. (8.1) are listed in Table 8.1 for measurements made in the author’s laboratory on some low molecular weight liquids using a pressurized Couette viscometer. The remainder of this chapter is devoted to providing a suitable description of the shear-thinning which has been observed in liquid lubricants so that analyses of lubricated contact behavior may incorporate realistic shear response of the liquid. An example of the necessity of having this type of shear dependence in an EHL analysis is shown in Fig. 8.3 where shear-thinning measurements are presented for the gear oil of Fig. 5.1. In Fig. 5.1, the measured pressure dependence of the low-shear viscosity of the gear oil is shown to be much greater than the pressure dependence derived from a film thickness measurement [7] based upon the Newtonian inlet assumption, the first assumption of classical EHL. Fig. 8.3 indicates strong shear dependence of the viscosity obtained in the viscometer (data points) in the range of shear stress, 0.1
2 MPa, important to film-forming in a large glass/steel contact employed for the film measurements. The shear dependence of the viscosity of this oil was characterized as the dashed curves in Ref. [7] from the second assumption of classical EHL, the equivalence of a flow curve to a friction curve. The resulting viscosity function is the sinh-law often associated with Eyring.
 _ μτ τE 21 μγ   η5 sinh (8.2) or η 5 _ τ γ τ E sinh τ=τ E E

228

High Pressure Rheology for Quantitative Elastohydrodynamics

Figure 8.3 The shear-dependent viscosity of the gear oil that that was represented in the pressure
viscosity plots of Fig. 5.1. The film-derived viscosity in Fig. 5.1 is incorrect because this oil is shear-dependent in a range of stress important to filmforming.

See Eq. (1.19). Sharif et al. [7], by fitting Eq. (8.2) to friction measurements, obtained for the “Eyring stress,” τ E 5 3.0 MPa. This relation is plotted for comparison in Fig. 8.3 as the dashed curves. The second assumption of classical EHL, Eq. (8.2), supports the first assumption of a Newtonian inlet since the assumed viscosity function is Newtonian to shear stress of greater than 2 MPa. However, both assumptions are clearly incorrect for PG460. It will be shown in Chapter 12, Application to Elastohydrodynamic Film Thickness, that the film thickness is correctly predicted by the Carreau description of shear dependence (solid curves in Fig. 8.3).

8.2 NORMAL STRESS DIFFERENCES AT ELEVATED PRESSURES In lubrication studies, non-Newtonian behavior is almost exclusively described by the generalized Newtonian model, Eq. (2.7). In a sim_ and the other velocity gradients are ple shear experiment, @u=@z 5 γ, zero. For a generalized Newtonian liquid, σxz 5 σzx 5 τ 5 ηðγ_ Þγ_ or

The Shear Dependence of Viscosity at Elevated Pressure

229

_ The other shear stresses are zero and the first and second norτ 5 ηðτ Þγ. mal stress differences are for the generalized Newtonian model, N1 5 σxx 2 σzz 5 0 and N2 5 σzz 2 σyy 5 0. See Fig. 2.1 for a definition of the stresses. This is, however, an incomplete description of the stress field. When the shear stress becomes sufficiently large that the viscosity becomes shear dependent, the differences in the normal stresses become easily measurable. The magnitude of the primary normal stress difference becomes comparable to the magnitude of the shear stress. This can be illustrated by the measurements shown in Fig. 8.4 for a commercial motor oil. The subject of these measurements is a 10W-40 multigrade oil, which has been investigated for shear-thinning using a pressurized Couette viscometer and for normal stress difference using a parallel plate rheogoniometer (Figs. 7.13 and 7.14). In Fig. 8.4, N1 reaches to nearly 1 MPa when the shear stress is less than 0.1 MPa. The viscosity in Fig. 8.4 has two transitions. The first transition from Newtonian to what appears to be the development of a second Newtonian plateau is the response of the polymer thickener and the

Figure 8.4 The reduced viscosity, η=μ, and the normal stress difference, N1 , for a multigrade engine oil at elevated pressure generated in the parallel disc rheometer of Figs. 7.13 and 7.14. The curve is a Double Modified Carreau equation fitted to the data generated in a pressurized Couette viscometer for the same oil. Polymerblended oil can generate large normal stress effects. The shear-thinning of the base oil at stress greater than 3 MPa is evident.

230

High Pressure Rheology for Quantitative Elastohydrodynamics

second transition to stronger shear thinning is the response of the base oil. It has been challenging to find an appropriate model to describe this complex shear dependence. The relation plotted as the curve in Fig. 8.4 is a viscosity function for multicomponent systems to be presented in a later section in this chapter. The very existence of differences between the normal stresses in the principle shear rate directions during simple shear may seem foreign to lubrication engineers and it is appropriate to discuss the implications of these stress differences here. Consider the case of a bearing surface that is infinitely wide in the y-direction. The one-dimensional Reynolds equation for a Newtonian liquid is,
 d ρh3 dp d ðρhÞ 5 12u (1.11) dx μ dx dx Integrating and applying the condition that h0 is the film thickness at any position for which the pressure gradient vanishes, dp=dx 5 0, gives for the incompressible case, h3 dp 5 12uðh 2 h0 Þ μ dx

(8.3)

Then for locations along the flow for which h . h0 , the pressure is increasing in the flow direction and for h , h0 , the pressure is decreasing. When the same pressure boundary conditions are applied at both ends of the bearing surface, x 5 0 and x 5 L, the position for h 5 h0 is within the bearing surface and Eq. (8.3) can be integrated to provide a pressure distribution. If the film thickness is always converging in the flow direction, dh=dx , 0, then the average pressure, p, is greater than the pressure at the boundaries and the pressure distribution can be integrated along the flow direction to find the bearing load that is supported by the pressure profile, ðL ðL W5 ð 2σzz Þdx 5 pdx (8.4) 0

0

The load support from the converging gap of a Newtonian hydrodynamic bearing comes about from the viscous shear stress that generates a pressure gradient which gradually boosts the pressure as the pressure gradient acts along the bearing surface, the wedge effect. Tanner [8] analyzed the specific case of a plane slider bearing lubricated by the theoretical second-order liquid. For a second-order liquid, the viscosity is independent of shear, the second normal stress difference is

The Shear Dependence of Viscosity at Elevated Pressure

231

N2 5 0 and the first normal stress difference is N1 5 Ψ 1 γ_ 2 . The load equation must now be written in a different way [8], ð ðL ðL 1 L ð 2σzz Þdx 5 pdx 1 N1 dx (8.5) W5 4 0 0 0 where p is the pressure generated for the Newtonian case by the wedge effect. The first normal stress difference, N1, acts to directly add to the load in a hydrodynamic bearing. Then the relative additional load support provided by the first normal stress difference is of the order of N1 =p. The largest measured value of N1 5 13 MPa appears to have been obtained at elevated pressure in a polymer thickened mineral oil [9]. Therefore for a hydrodynamic bearing operating with average pressure, p, no greater than about 100 MPa, lubricated by polymer-blended oil, it is possible to derive a small but useful portion of the load from normal stress effects. On the other hand, the load supported by the liquid film in an EHL contact can be assumed to be entirely due to the pressure distribution owing to the much larger value of p for these contacts. The most interesting aspect of the load supported by a nonNewtonian lubricated slider bearing may be that a nonzero load can be generated in a parallel film. For a parallel film, the wedge effect vanishes, p 5 0, yet Eq. (8.5) predicts load support if N1 . 0. The very large value of N1 that can be generated compared to the shear stress for polymer-blended oil has great significance for the onset of shear cavitation voids in high shear viscosity measurements at ambient pressure. Eq. (7.9) gives the principal normal ffistress cavitation criterion in terms of the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cavitation stress as τ cav 5 p2 2 pN1 . Fig. 8.5 shows a master flow curve for a reference liquid, squalane plus 15% weight of 4 3 104 kg/kmol molecular weight polyisoprene. Both elevated and ambient pressures were used. The ambient pressure data were obtained with a commercial “high shear” viscometer and, for this instrument, cavitation has altered the apparent viscosity for shear stress greater than about 40 kPa. It seems that N1 must be about 84 kPa when the shear stress is 40 kPa, a quite reasonable value compared with rheogoniometer measurements [10] for this liquid. In Fig. 8.4, for the motor oil, the value of N1 is 300 kPa when the shear stress is 40 kPa. The high-temperature high-shear (HTHS) viscosity is indicated in Fig. 8.5 by the circle. This is a specified rheological property for the SAE J300 classification for engine oils. As shown in Fig. 8.5, the HTHS condition, T 5 150 C and γ_ 5 1 3 106 s21 , does not yield sufficient shear stress to access the shear dependence of the liquid under severe conditions

232

High Pressure Rheology for Quantitative Elastohydrodynamics

Figure 8.5 Flow curves at elevated pressure and at ambient pressure for a reference liquid containing polymer. Cavitation occurs in the ambient pressure viscometer at stress greater than about 40 kPa.

of stress. The HTHS viscosity is η 5 5.1 mPa s and the limiting low-shear viscosity is μ 5 5.7 mPa s. The first normal stress difference may add to the load support in a hydrodynamic bearing; however, the first normal stress difference can have a substantial effect on the apparent viscosity measured in an ambient-pressure high-shear viscometer, leading to an overstatement of the shear dependence. An interesting implication of the first normal stress difference is that the principal shear stress axes are not coincident with the principal shear rate axes. Normal stress effects have been measured in low molecular weight liquids such as mineral oil [9]. This will be important later in the discussion of the liquid failure which results from slip within the film at a critical shear stress. Slip will not occur in the flow direction.

8.3 THE ORIGIN OF NON-NEWTONIAN BEHAVIOR IN LOW MOLECULAR WEIGHT LIQUIDS 8.3.1 The viscoelastic hypothesis In Chapter 2, An Introduction to the Rheology of Polymeric Liquids, the corotational Maxwell model was introduced to illustrate an example of a theoretical model which was capable of at least qualitatively describing

The Shear Dependence of Viscosity at Elevated Pressure

233

shear-thinning, normal stress differences and the analogy between steady and period shear. Viscoelastic liquids can both store energy elastically and dissipate energy viscously. These models can be used to explain the relationship between viscosities measured in continuous and oscillatory shear. Viscoelastic models were widely explored as candidates for the description of the constitutive behavior of liquids in EHL contacts during the 1970s and early 1980s. Elastic compliance of the liquid lubricant film was invoked to explain the linear small slide/roll ratio traction behavior at high contact pressures and the large-strain behavior of the viscoelastic model was used to explain nonlinear traction response. Other explanations for these phenomena have since been found and validated. The monograph by Harrison [11] provides an excellent review of the rheological experimental techniques of the period that involve the propagation of a plane shear wave for a short distance into the liquid. Measurements were reported at pressures to 1.4 GPa. These acoustic methods were performed at ultrasound frequencies of up to ω 5 109 s21 and provide a means to investigate the small-strain oscillatory shear response which was modeled using the single mode Maxwell model in Section 2.4.1. For a time varying shear strain of γ 5 γA sin ωt, the Maxwell model (2.13) gives a time dependent shear stress of   μ2 ωγA =G G τ5 cos ωt 1 ω sin ωt (2.25) 1 1 ðμω=GÞ2 μ The component of stress in phase with the strain is τv 5

ðμ2 ω2 =GÞγA sin ωt  2 1 1 μω=G

(8.6)

The component of the dynamic shear modulus in phase with the shear strain is GvðωÞ 5

τv μ2 ω2 =G 5  2 γ 1 1 μω=G

(8.7)

and the limiting high frequency value of Gv is then GN 5 lim GvðωÞ 5 G ω-N

(8.8)

Therefore, for a single mode Maxwell liquid, a small-strain oscillatory shear experiment can be used to measure the shear modulus property if

234

High Pressure Rheology for Quantitative Elastohydrodynamics

sufficiently high frequency is available to observe a limiting high frequency value, GN , of the component of the dynamic shear modulus in phase with the shear strain, Gv. Generally Gv approaches the limiting value more slowly [11, p. 145] than expected from the single mode Maxwell model, Eq. (8.7). Hereafter in this book, GN will denote the elastic shear modulus property and G will be reserved for the shear stress at the limit of Newtonian response or the material modulus associated with λ, the characteristic time employed in a shear-thinning model, by G 5 μ=λ. The reason for making this distinction will be apparent from the following example where experimental measurements of GN and G have been made for the same material, di(2-ethylhexyl)phthalate (DOP), over a similar range of state variables. Measurements of GN for DOP were obtained by Hutton and Phillips [12] for 230 # T # 30 C and to pressures to 0.815 GPa using high frequency transverse sound. They fitted their results to GN 5

1:29 3 1013 Pa 1:74 3 105 p 2 1 ðT 116:7KÞ2 ðT 292:8KÞ

(8.9)

which is in fair agreement with Barlow et al. [13] who obtained at 30 C, GN 5 0:5 GPa 1 2:45p for pressures to 1.4 GPa. Where comparison can be made, the ultrasonic measurements of Hutton [14] compare very well with direct mechanical measurements from the author’s laboratory of the shear modulus for the model mineral oil, LVI 260, under high pressure near to the glass transition. Then we can accept Eq. (8.9) as an accurate representation of the shear modulus of a well-defined liquid, DOP. The shear-dependent viscosity of DOP has also been measured by the author [1] for temperatures and pressures similar to those for the shear modulus measurements presented above. These data are shown as the points in Fig. 8.6. When these results were originally presented [1], the author fitted the Carreau equation (8.1) to obtain n 5 0:41 and G 5 6 MPa. Slightly different values of n and G, to be introduced in the next section, were used to draw the Carreau curves in Fig. 8.6 for a purpose to be explained in the next section. Experimentally, then, the limit of Newtonian behavior is approximately at τ  G 5 6 MPa. For the temperatures and pressures of Fig. 8.6, Eq. (8.9) gives 1480 , GN , 1770 MPa. Clearly the magnitudes of G and GN are unrelated for DOP and the viscoelastic model using directly measured

The Shear Dependence of Viscosity at Elevated Pressure

235

Figure 8.6 Flow curves for di(2-ethylhexyl)phthalate for pressures and temperatures similar to those for which the elastic shear modulus, GN , has been measured. The curves are the Carreau equation with the characteristic time, λ 5 μ=G, set equal to the Einstein
Debye rotational relaxation time, λEB .

values of GN cannot result in a quantitative description of shear-thinning. For the large-strain viscoelastic model, Eq. (2.21), the limit of Newtonian behavior is roughly at τ  GN , at least two orders-of-magnitude too large.

8.3.2 The kinetic theory of molecules The Einstein
Debye relation [15] for the rotational relaxation time of a molecule is λEB 5

μVm μM 5 ρRg T kB T

(8.10)

where Vm is a molecular volume and kB is the Boltzmann constant. Debye [16] developed a theory, from Einstein’s explanation [17] of Brownian motion, for the liquid molecules that are electrically polar, having an electrical moment. In liquid lubricants, polarity usually arises from the chemical addition of oxygen to a hydrocarbon structure. Thermal (Brownian) motions constantly reorient molecules that may have had, at some instant, some specific orientation. The reorientation is retarded by the viscosity. Debye [16] defined the rotational relaxation time “as the time required for the moments of the molecules to revert to practically a random distribution after the removal of the impressed field.”

236

High Pressure Rheology for Quantitative Elastohydrodynamics

Debye, in his development of the relaxation time, clearly meant that the field should be electrical; however, there is nothing in the development of Eq. (8.10) that should restrict its application to only electrical effects. Suppose instead that a viscous shear stress provides the “impressed field.” Also suppose that the impressed field has associated with it a periodic character arising from the interaction of individual molecules and that the frequency of these interactions may reasonably be approximated _ It should be expected that the rotational alignment of by the shear rate, γ. a molecule by the impressed field would serve to ease the flow, to reduce the viscosity. Therefore, if there is insufficient time for the orientation of a molecule arising from an interaction with another molecule to be removed by thermal motion, the alignment will accumulate and the viscosity will be reduced. As a result, a transition to shear-thinning would be observed when the product λEB γ_ is of order one. This hypothesis has been tested in Fig. 8.6 where the curves are the single Newtonian Carreau equation,

ðn21Þ=2 η 5 μ 11 ðλEB γ_ Þ2 (8.11) with the characteristic time λ 5 λEB . The power-law exponent has been adjusted to n 5 0:38 to provide the good fit to the data in Fig. 8.6. It was assumed that the molecular volume, Vm , in Eq. (8.10) is given by the total liquid volume per molecule, Vm 5

M kB ρ Rg

(8.12)

where the ratio, Rg =kB , is Avogadro’s number and the formula molecular weight of DOP is M 5 390:6 kg=kmol. An equation of state for density, ρ, was obtained by fitting the temperature modified Tait framework, Eqs. (4.6), (4.12), and (4.14), to the density data for DOP reported in the ASME  Pressure
Viscosity Report [18] with K 00 5 10:647, aV 5 8:33 3 1024 K21 , K00 5 11:005 GPa, β K 5 6:327 3 1023 K21 , TR 5 25 C, and 3 ρR 5 1=VR 5 981 kg=m . In the case of DOP, Fig. 8.6 shows that the Einstein
Debye rotational relaxation time is an excellent predictor of the onset of shear-thinning when used with the Carreau model and μM λ 5 λEB 5 (8.13) ρRg T Dynamic viscosity measurements have also been made for DOP at ambient pressure by Yamaguchi and coworkers [19] with a shear

The Shear Dependence of Viscosity at Elevated Pressure

237

Figure 8.7 Experimental dynamic viscosity for di(2-ethylhexyl)phthalate showing frequency dependence. The broken curve is the single Newtonian Carreau equation written for dynamic viscosity and the solid curve represents the same equation with the parameters from steady shear measurements at elevated pressures.

impedance spectrometer described in [20] at frequencies of 5
200 MHz. These viscosities are shown in Fig. 8.7 as η0 =μ plotted versus μω. See Section 2.4.1 and Eq. (2.26). The broken curve is the single Newtonian Carreau equation written for dynamic viscosity as   ðn21Þ=2 μω 2 0 η 5 μ 11 (8.14) G with n 5 0:82 and G 5 10 MPa. The solid curve represents the same equation with the parameters from steady shear measurements at elevated pressures. Notice that the departure from Newtonian response occurs at similar values of μω; however, the rate dependence is considerably less for the dynamic viscosity. Again, the value of G is about two orders of magnitude less than GN and the single-mode viscoelastic explanation fails. Yamaguchi [21] found somewhat the same relationship between steady and oscillatory shear-thinning using nonequilibrium molecular dynamics (EMD) simulation of a Leonard
Jones liquid. The fact that there is a frequency dependence and that the characteristic times have approximately the same dependence on viscosity as for steady shear are evidence that the steady and oscillatory shear-thinning have the same origin and that it is alignment of molecules. The periodic impressed field described by Debye is clearly the periodic shear stress. If

238

High Pressure Rheology for Quantitative Elastohydrodynamics

there is insufficient time for the orientation of a molecule arising from shear stress in one direction to be removed by thermal motion, the alignment will remain and the viscosity will be reduced for the next cycle of shearing. Alignment of molecules in shear of simple low molecular weight liquids is evidenced as well by flow birefringence [22] and by the first normal stress difference [9] both of which result from anisotropy.

8.3.3 Nonequilibrium molecular dynamic simulations Contributed by Clare McCabe, Vanderbilt University The same information regarding viscosity and normal stress differences as is found through experimental measurements can be obtained from molecular simulations. Molecular simulation methods for calculating viscosity can be broadly classified into EMD and NEMD methods. In an EMD simulation, Newton’s equations (or a variant thereof) describing the motion of the atoms in the model system are solved as a function of time. The viscosity, η, is then given as an integral of the stress
stress autocorrelation functions determined during the simulation, viz ðλ V η5 dt , σxy ðtÞσxy ð0Þ . (8.15) lim kB T τ-N 0 where V is the volume of the system, kB is Boltzmann’s constant, T is temperature, and t is time. The quantity σxy ðt Þ is the value of the xy component of the traceless symmetric stress tensor at time, t, and so σxy ðt Þσxy ð0Þ is the stress
stress autocorrelation function and , σxy ðtÞσxy ð0Þ . is its ensemble average measured during the course of the simulation. The time over which the stress
stress autocorrelation function needs to be computed in order to obtain high-accuracy results from Eq. (8.15) is at least several hundred relaxation times [23]. EMD methods based on the application of Eq. (8.15) (and similar approaches [24]) have been applied routinely and successfully to low molecular weight fluids [25
27]. While EMD has been used extensively for calculating transport properties, in recent years the NEMD method has become increasingly popular since the key algorithm, SLLOD [28], is a direct implementation of the experimental method for measuring viscosity, and because it can also probe the non-Newtonian regime (Fig. 8.8). The NEMD SLLOD algorithm for viscosity involves applying a planar Couette flow field at strain

The Shear Dependence of Viscosity at Elevated Pressure

239

Figure 8.8 Schematic of an NEMD simulation. The fluid is sheared between two surfaces—the upper surface moves at velocity V relative to a fixed lower surface. At steady-state, this sets up a velocity profile across the gap of width h, which is charac_ The force/unit area required to maintain the steady state terized by the strain rate γ. velocity of the moving plate has the units of pressure and is the xy component of the stress tensor. The viscosity is simply the constant of proportionality between σxy _ and γ.

rate γ_ 5 @u=@y, which characterizes the constant change in streaming velocity in the x direction with vertical position y. The viscosity at strain rate γ_ is then computed from D E σ0;s xy η52 (8.16) γ_ where σ0;s xy is the traceless symmetric stress tensor computed during the course of the simulation. The result is a strain-rate-dependent shearthinning viscosity. The critical strain rate transition above which we see shear-thinning and below which we see a Newtonian plateau (η indepen_ typically occurs at γ_  λ21 , where λ is the longest relaxation dent of γ) time at equilibrium (i.e., in the absence of shear). This is usually the rotational relaxation time. In systems with long relaxations times (i.e., O (0.1 ns)) for which EMD appears to be impractical, NEMD is the only practical route to compute the Newtonian viscosity. However, when applied to systems of high molecular weight at extreme states (λ $ 10 ns), the computational burden associated with NEMD increases dramatically [29,30]. While the viscosity converges rapidly at high strain rate, the length of simulation needed to obtain reliable averages for , σxy . increases nonlinearly as the strain rate is reduced; in particular, for γ_ , λ21 , , σxy . must be averaged through at least several, and preferably many, rotational relaxation times. Snapshots from a molecular dynamics simulation of a C100 melt are given in Fig. 8.9 and clearly demonstrate the alignment of the molecules at high shear rates, the molecular origin of shear thinning.

240

High Pressure Rheology for Quantitative Elastohydrodynamics

Figure 8.9 Snapshot of a configuration from a molecular dynamics simulation of a C100H202 melt at equilibrium (top) and undergoing shear at a high shear rate (bottom). The molecules are colored randomly for ease of identification. J.D Moore, Private Communication.

The NEMD method has been employed successfully to study the viscosity and transport properties of a wide range of fluids. In particular, Kioupis and Maginn [31] have examined the rheological properties of three isomers of C18 to high pressures by EMD and NEMD and McCabe and coworkers have studied the studied the high-pressure rheology of 9-octylyheptadecane and squalane (2,6,10,15,19,23-hexamethyltetracosane) [29,32,33]. This work provided the first experimental verification of NEMD simulation in the non-Newtonian regime through a comparison

The Shear Dependence of Viscosity at Elevated Pressure

241

of NEMD simulation results and experiment using the standard rheological analysis technique of time
temperature
pressure superposition.

8.4 TIME
TEMPERATURE
PRESSURE SUPERPOSITION Flow curves of real shear-thinning behavior, when plotted as log of viscosity versus log of shear rate or shear stress or when plotted as log of shear stress versus log of shear rate have a useful property. Any curve can be shifted vertically and horizontally without rotation to superimpose over another curve, often within experimental accuracy. The only exception appears to be the portion of the curve influenced by a second Newtonian plateau. The ratio of the first Newtonian viscosity to the second Newtonian viscosity, μ=μ2 , often is found to vary with temperature and pressure [34]. This graphical shifting of flow curves to superimpose has been called time
temperature
pressure superposition, or alternatively the method of reduced variables, and is dramatically illustrated by the reduction of the disparate data for squalane in Fig. 8.10A to the master curve of Fig. 8.10B using reduced variables. These data were obtained from the NEMD technique of the previous section and experimental measurement [33]. See Section 2.5 as well. Consider a plot of logη versus logγ_ as in Fig. 8.6. If the various curves are to superimpose, the terminal regimes must superimpose and the terminal behavior is determined only by the low-shear viscosity. The reduced variable for viscosity is therefore, η^ 5 η=μ. The shear rate at the transition to shear-thinning is, for all states, 1=λ or G=μ, and therefore the choice _ 5 γμ=G. _ for the reduced shear rate is γ^_ 5 γλ This is the definition, assuming that λ 5 λEB , of the reduced rate that was successfully applied to the data of Fig. 8.10A to arrive at the master curve of Fig. 8.10B. The reduced shear stress must then be τ^ 5 η^ γ^_ 5 τ=G 5 τλ=μ. The principle that has been laid out above is extremely important for EHL calculations. Flow curves (even a single flow curve) that have been obtained at temperatures and pressures of experimental convenience may be used to describe the shear dependence of viscosity at the often more severe conditions of the lubricated contact. It is only necessary to find μðT ; pÞ and G ðT ; pÞ to provide the shifting rules, and the various models for μðT ; pÞ have been presented in detail in Chapter 6, Correlations for the Temperature and Pressure and Composition Dependence of Low-Shear Viscosity. When a

242

High Pressure Rheology for Quantitative Elastohydrodynamics

_ for squaFigure 8.10 (A) Flow curves plotted as shear stress, τ, versus shear rate, γ, lane obtained from experiment (E) with a pressurized Couette viscometer (solid points) and from simulation (S) using nonequilibrium molecular dynamics (open points). (B) The same data replotted as shear stress versus reduced shear rate, Wi 5 γλ _ EB , to obtain a master flow curve.

second Newtonian is present, an expression for μ2 ðT ; pÞ will also be required, although the assumption of constant μ2 =μ will often suffice for a tribology calculation. For the examples of Figs. 8.6 and 8.10B, the Einstein
Debye relation was used to set G ðT ; pÞ 5 ρRg T =M. More often, Eq. (8.13) is not taken as an exact relationship for shifting but utilized for the temperature and pressure dependence only to give another shifting rule, ρT G 5 GR (8.17) ρ R TR

The Shear Dependence of Viscosity at Elevated Pressure

243

the standard or Ferry rule [35]. Here the subscript “R” indicates a value obtained for a reference state, TR and pR . A very simple rule that is often sufficiently accurate is a constant G. G 5 GR

(8.18)

the Vinogradov
Malkin rule [35]. Measurements made over a wide range of temperature and pressure such as the capillary measurements of Fig. 7.7 often show that G decreases with temperature, which is in contradiction to the increase predicted by the Einstein
Debye relation (8.10) if molecular volume increases with temperature at a rate less than in proportion to temperature. In these cases
m μ G 5 GR (8.19) μR with 0 # m # 0:25 may be helpful [36,37]. This is the shifting rule applied to the data in Figs. 7.16, 8.4, and 8.5 with m 5 0.1 and in Fig. 7.7 with m 5 0.2 [34]. The pressure
viscosity and the temperature
viscosity coefficients for the low-shear viscosity were defined in Eqs. (5.1) and (5.2), respectively. αðT ; pÞ 5

1 @μ μ @p

β ðT ; pÞ 5 2

1 @μ μ @T

(5.1) (5.2)

It should be possible to define similar coefficients for the generalized viscosity, η. For the terminal regime, the definitions must be identical to those above.     1 @η 1 @η 5 α and 2 5β (8.20) η @p T ;γ-0 η @T p;γ-0 _ _ As was the case for thermal softening in Section 7.3.3, it will be necessary to specify the nature of the process to determine the temperature and pressure sensitivity. The single Newtonian generalized Newtonian models of Table 2.1 reduce, in the limit of high shear rate or stress, to
12n
ð1=nÞ21 G n G η-μ 5μ (8.21) γ_ τ

244

High Pressure Rheology for Quantitative Elastohydrodynamics

If the shifting rule of Eq. (8.19) is invoked, in the limit of high shear rate or stress,

 GR 12n G ð1=nÞ21 ðn1m2mnÞ ð11ðm=nÞ2mÞ η-μ 5μ (8.22) μR m τ μR m γ_ and the pressure
viscosity coefficient for η is in the limit of high shear    1 @η m (8.23) αη;τ 5 - 11 2m α η @p T ;τ n for stress control and αη;γ_ 5



1 @η η @p

 T ;γ_

-ðn 1 m 2 mnÞα

(8.24)

for rate control. Similar relationships can be derived for the temperature
viscosity coefficients by replacing α by β. For the case of Vinogradov
Malkin shifting, m 5 0 and αη;τ 5 α and αη;γ_ 5 nα. This is in agreement with the result, β η;γ_ 5 nβ, obtained experimentally by Winter [38].

8.5 THE COMPETITION BETWEEN THERMAL SOFTENING AND SHEAR-THINNING Two sections of the previous chapter dealt with the thermal softening of the liquid that results from viscous heating in viscometers. The effect is clearly of major concern for the measurement of shear-dependent viscosity of low molecular weight liquids as it is usually the first limitation that is encountered as a measurement is extended toward and into the shear-thinning regime. Thermal softening is so much more severe in the reduction of viscosity with shear than is the constitutive behavior that it is not even possible to correct with accuracy for the viscous heating effect. See Fig. 7.16 for an example. In Couette shear, thermal softening will impose a limit to the steady stress, corresponding to Na 5 3:5138, the limit to real solutions of Eq. (7.24), which may be less than the stress at which shear-thinning begins. This competition between thermal softening and shear-thinning has implications beyond just the measurement of viscosity. In steady shear, if the Nahme
Griffith number, Na, becomes

The Shear Dependence of Viscosity at Elevated Pressure

245

larger than unity before the Weissenberg number, Wi, approaches one, then regardless of the circumstances, it will be unlikely that shear-thinning will play a significant role. This is true of liquid films in lubricated contacts as well as in viscometers. To provide guidance in deciding whether or not shear-thinning will be operative in any film undergoing simple shear a new dimensionless group [39] is helpful. B5

Na β 5 h2 G 2 2 Wi μk

(8.25)

Dividing Na by the square of Wi cancels the shear rate (or shear stress) so that B is independent of the kinematics. Only the film thickness and liquid properties are required. For B . 1, thermal softening will overwhelm the shear dependence of viscosity and a Newtonian description will be sufficient; however, if B , 1, it still cannot be said that thermal softening will not be important at any magnitude of shear. If typical values, for EHL pressures, of the temperature viscosity coefficient, β, and the liquid thermal conductivity, k, are assumed and if G 5 3 MPa is typical for a mineral oil, then thermal softening will preclude shear-thinning when the film is thicker than h 5 0:3 μm where μ 5 1PaUs and when the film is thicker than h 5 30 μm where μ 5 104 PaUs.

8.6 MULTICOMPONENT SYSTEMS The single Newtonian Carreau model (8.1) has proven to be an accurate form for monodisperse liquids. With shear rate as the independent variable, this is useful for calculations of the friction generated within the Hertz zone. A similar viscosity function for a single-component lubricant is the single Newtonian version of the modified Carreau equation [34].   ð12ð1=nÞÞ=2 τ 2 (8.26) η 5 μ 11 G With shear stress as the independent variable, this function is useful for film thickness calculations within the inlet zone. The modified Carreau has a broader transition than the Carreau form; however, for n . 0.5, they are surprisingly similar.

246

High Pressure Rheology for Quantitative Elastohydrodynamics

Practical lubricants, however, are seldom comprised of a single monodisperse liquid. The significance of the value of the shear stress that sets the limit of Newtonian behavior, G, for an individual liquid material often tends to be retained when that liquid becomes a part of a two component system. This is dramatically illustrated in Fig. 8.11 where dibenzylethylbenzene (DBEB) has been blended with 10% of a nearly monodisperse polystyrene (M  700 kg=kmol) with polydipersity of 1.06. (A monodisperse liquid has a polydipersity of one.) The flow curve for the neat DBEB shows a transition to shear thinning at a shear stress of 6.2 MPa and this transition is also apparent in the flow curve for the mixture. The mixture shows an additional transition at τ 5 1.5 MPa which must be attributed to the presence of the polystyrene additive [40]. The curves composed of connected straight lines in Fig. 8.11 represent the Spriggs truncated power-law model introduced in Table 2.1. Most lubricant base stocks cannot be described as monodisperse. Their molecular weights are broadly distributed. Fig. 8.11 illustrates how mixing of liquids of different molecular weights broadens the transition from

Figure 8.11 Flow curves for DBEB and for DBEB with 10% by weight of a nearly monodisperse polystyrene. The critical stress (6.2 MPa) for transition to shear thinning of DBEB is retained in the mixture. Reproduced from Bair S. The high pressure rheology of mixtures. J Tribol 2004;126(4):697
702, with permission of the American Society of Mechanical Engineers.

The Shear Dependence of Viscosity at Elevated Pressure

247

Newtonian to power-law response. It can be imagined that the Carreau form with a broader transition would span the gap between the two transitions in Fig. 8.11. A viscosity function that accommodates broadening of the transition is the Carreau
Yasuda [41] model,  a ðn21Þ=a

μγ_ η 5 μ 11



(8.27) G See Table 2.1. For a 5 2, this is the Carreau model and as a decreases, the transition broadens. This broadening is useful for mixtures of base oils of similar molecular weight and for base oils which are naturally polydisperse. If stress should be the independent variable, is possible to write a modified Carreau
Yasuda viscosity function. h τ a ið12ð1=nÞÞ=a

η 5 μ 11 (8.28) G The shear response of blends of polydisperse liquids may become complicated by additional transitions as illustrated by Fig. 8.12. The PAO100 is a 100 centiStoke grade polyalphaolefin. It differs from a previously

Figure 8.12 Flow curves at 20 C for three liquids; a 100 cS grade PAO, a 650 cS grade PAO and a blend of the two. The curves represent Eq. (8.29). From Bair S, Vergne P, Querry M. A unified shear-thinning treatment of both film thickness and traction in EHD. Tribol Lett 2005;18(2):145
152.

248

High Pressure Rheology for Quantitative Elastohydrodynamics

Table 8.1 Carreau parameters for low molecular weight single-component liquids Designation Type μ0 ð40o CÞ ðPa sÞ GðPaÞ n

SHF 8001 PAO-650 SHF 1001 SHF 403 PG460 143AD Z25 BS 98 LVI260 Turbo T9 ISO100 POE Heptamethylnonane

PAO 800 PAO 650 PAO 100 PAO 40 Polyglycol Branched PFPE Linear PFPE Bright stock Mineral oil Mineral oil Polyol ester High purity hydrocarbon Squalane High purity hydrocarbon Dibenzylethylbenzene High purity hydrocarbon Dicyclohexyl High purity methylpentane hydrocarbon

6.6 6.2 1.1 0.38 0.44 0.92 0.23 0.29 0.26 0.008 0.78 0.0024

4:5 3 103 3:1 3 104 1:5 3 106 6:0 3 106 2:7 3 105 5 3 105 7 3 104 3:5 3 106 5:6 3 106 7:0 3 106 4:0 3 106 5:3 3 106

0.67 0.74 0.625 0.40 0.63 0.67 0.82 0.65 0.34 0.35 0.56 0.44

0.016

6:6 3 106

0.46

0.014

6:5 3 106

0.41

0.017

1:3 3 107

0.41

studied [42] PAO100 (see Table 8.1) in having a high molecular weight “tail” in the distribution which apparently causes the first departure from constant viscosity at τ  105 Pa in the second from top flow curve in Fig. 8.12. The PAO650 was introduced in Fig. 8.1. When the PAO650 is blended with the PAO100, the transition at τ  3 3 104 Pa, which is clearly present in the flow curve for the PAO100, appears in the curve for the blend. The author has suggested [4] a multicomponent modified Carreau viscosity function for mixtures that is written "
 #ð12ð1=ni ÞÞ=2 N X τ 2 η5μ fi 11 (8.29) Gi i51 where fi is analogous to the areal fraction Pin the Ree
Eyring model [43] and also, as in the Ree
Eyring model, N i51 fi 5 1. Unlike Ree
Eyring, there is no restriction to N . 1. It is remarkable that the values for Gi that are to be used in Eq. (8.29) for the blend can be obtained from the flow curves for the individual components as shown in Table 8.2. The values of the parameters for a motor oil and a gear oil are also listed. For N 5 2, f1 5 1 2 μ2 =μ, f2 5 μ2 =μ, and n2 5 1, Eq. (8.29) becomes the more usual

249

The Shear Dependence of Viscosity at Elevated Pressure

Table 8.2 Parameters of Eq. (8.29) for the liquids shown in Fig. 8.12 and for a crankcase oil and for a gear oil Designation N i fi ni Gi ðPaÞ

PAO-100

2

PAO-650 20% PAO-650 in PAO-100

1 3

10W-40

2

75W-90

2

1 2 1 1 2 3 1 2 1 2

0.5 0.5 1 0.4 0.3 0.3 0.41 0.59 0.38 0.62

0.8 0.5 0.74 0.28 0.75 0.4 0.62 0.63 0.83 0.40

1 3 105 4 3 106 3:1 3 104 3:1 3 104 1 3 105 4 3 106 4:8 3 103 6:5 3 106 2:0 3 104 2:5 3 106

double Newtonian modified Carreau model [34] that is used for calculating film thickness of a polymer oil solution.   ð12ð1=nÞÞ=2   τ 2 η 5 μ2 1 μ 2 μ2 11 (8.30) G This equation is plotted in Fig. 7.16 with for a motor oil with n 5 0.60, G 5 1300 Pa, and μ2 =μ 5 0.57. There is a practical problem in the application of the series in modified Carreau terms in Eq. (8.29). At large shear rate (large shear stress), the term with the largest value of ni will dominate and, if this term is not associated with the largest value of Gi , there will be an unintended _ A solution to upturn in the slope of logðηÞ versus logðγ_ Þ for large γ. this issue is to write the product of a double Newtonian modified Carreau
Yasuda function and a single Newtonian modified Carreau
Yasuda function. (
 a1 ð12ð1=n1 ÞÞ=a1 ) a2 ð12ð1=n2 ÞÞ=a2

τ

τ η μ2 μ2 11



5 1 12 11



μ G1 G2 μ μ (8.31) This is the Double Modified Carreau
Yasuda model [44] plotted in Fig. 8.4 for a 10W-40 motor oil and in Fig. 8.13 for a 15W-40 Diesel engine oil. The parameters are listed in Table 8.3 along with those for an automatic transmission fluid, ATF. There is no second Newtonian inflection for the ATF or for the Diesel engine oil as can be seen in Fig. 8.13.

250

High Pressure Rheology for Quantitative Elastohydrodynamics

Figure 8.13 The Double Modified Carreau
Yasuda model fitted to flow curves for a Diesel engine oil.

Table 8.3 Parameters of Eq. (8.28) for the 10W-40 crankcase oil of Fig. 8.4 and for an automatic transmission fluid, ATF, and for a 15W-40 Diesel engine oil Designation i ai ni Gi ðPaÞ μ2 =μ

10W-40 ATF 15W-40

1 2 1 2 1 2

2 2.5 2 4 2 5

0.66 0.62 0.915 0.47 0.90 0.55

3:44 3 103 3:2 3 106 1:5 3 102 2:0 3 106 6:0 3 103 2:2 3 106

0.58 0 0

It is very unusual today to find a true second Newtonian plateau with constant viscosity as in Fig. 7.7. It is not unusual in multigrade crankcase oil to find an inflection, a local minimum in d lnðηÞ=d lnðτ Þ, which suggests the development of a plateau at greater stress; however, the plateau usually cannot appear as the base oil will contribute to the shear dependence above a shear stress of a few megaPascal as shown in Figs. 8.4 and 8.13. Gear oils seldom show even an inflection. In the study of lubrication with polymer thickened oil, there has been an expectation of a second plateau with the same viscosity as the base oil and this expectation has resulted in the application of inappropriate viscosity functions [44].

The Shear Dependence of Viscosity at Elevated Pressure

251

8.7 VISCOSITY FUNCTIONS, THE POWER-LAW EXPONENT AND THE SECOND NEWTONIAN Table 2.1 lists many of the viscosity functions that have been employed to correlate shear-dependent viscosity measurements using consistent definitions of the parameters to allow for comparisons. Viscosity is η 5 τ=γ_ and the limiting low-shear viscosity,μ, is equal to η for τ 5 0 or γ_ 5 0. The power-law exponent is n 5 d ln τ=d ln γ_ at infinite shear rate assuming no second Newtonian, μ2 5 0. The characteristic time is the reciprocal of the shear rate at the intersection of the extrapolated powerlaw and terminal regimes. And G 5 μ=λ is the shear stress at the intersection of the extrapolated power-law and terminal regimes. Some of these functions have been applied to the data presented in this work. In Figs. 8.14 and 8.15, the reduced viscosity, η=μ, is plotted versus the _ for n 5 0.4 and 0.8, respectively. These are not Weissenberg number, λγ, unusual values of n for lubricants. The functions plotted are the single Newtonian (μ2 5 0) versions of the Cross equation, η5

μ

12n

1 1 λγ_

(8.32)

Figure 8.14 Comparing the viscosity functions; Cross, Carreau, Carreau
Yasuda, Ellis, and Modified Carreau for n 5 0.4.

252

High Pressure Rheology for Quantitative Elastohydrodynamics

Figure 8.15 Comparing the viscosity functions; Cross, Carreau, Carreau
Yasuda, Ellis, and Modified Carreau for n 5 0.8.

the Ellis equation, η5

μ

ðð1=nÞ21Þ 1 1 τ=G

(8.33)

the Carreau equation (8.1), the Modified Carreau equation (8.26) and the Carreau
Yasuda equation (8.27) with a 5 1 and a 5 4. For n 5 0:4, Fig. 8.14, the Cross equation differs from the others in having a broader transition occupying three or four decades of shear rate. For n 5 0:8, Fig. 8.15, the Cross equation and the Ellis equation have transitions occupying more than seven decades of shear rate. Lubricants do not show these broad transitions. In using the Cross equation, the transition may be narrowed to fit the viscosities measured in an ambient-pressure high-shear viscometer by setting n to a very low value and correcting the shear dependence by invoking a second Newtonian [45] even when one may not exist. Recall that ambient pressure viscometers must cavitate for shear stress less than 100 kPa when the oil is shear thinning and some means of extending the viscosity to greater stress is required if limited to ambient pressure. Then many reports of a second Newtonian viscosity in the tribology literature may be inaccurate [44].

The Shear Dependence of Viscosity at Elevated Pressure

253

We have seen in Section 8.3.2 that the value of the effective modulus, G, for single-component, low molecular weight liquids can be estimated from the ratio of the low-shear viscosity to the Einstein
Debye rotational relaxation time and in Section 8.6 that the description of shear-thinning of a multicomponent system can be estimated from the moduli, Gi , of the individual components. For an EHL analysis, it will be necessary to have a numerical value for the power-law exponent, n, as well and if the lubricant is a polymer solution, a value for the second Newtonian viscosity, μ2 , may also be required. The means to estimate these parameters without a measurement have not yet been found; however, a few trends have been established. Some preliminary associations have been made between the powerlaw exponent and branching of the molecule. Wood-Adams [46] made steady and dynamic shear viscosity measurements on samples of polyethylene, all having molecular weights of about 9 3 104 kg/kmol. She found that as the number of long chain branches increased the value of n decreased. The NEMD simulations of Jabbarzadeh et al. [47] appear to contradict this trend; however, the flow curves displayed in the reference do not appear to have fully developed into the power-law regime. The values of n in Table 8.1 seem to confirm that branching decreases the power-law exponent. For the branched perfluorinated polyether, n 5 0:67, whereas for the linear PFPE, n 5 0:82. In addition, this author [48] has plotted the measured values of n against the measured values of G for lubricant base oils. Low values of n only appeared for large values of G. This may be more closely related to the experience of the formulation of synthetic lubricants where a base oil with small values of both n and G would have poor film-forming ability. The dimethyl silicone oils with both small n and small G are an example of this as they do not easily form films of the normal thickness [48,49]. Alternatively, the more highly branched structures may have lower molecular weight at a given viscosity so that G which is inversely proportional to molecular weight will be larger for the highly branched structures with inherently low n. In order to meet multigrade requirements for lubricants, polymeric viscosity index improvers (VII) are added to base oils. The objective is to minimize the viscosity
temperature variation for base oils. The conventional model assumes that the molecular configuration of the oil soluble

254

High Pressure Rheology for Quantitative Elastohydrodynamics

VII varies with temperature. At low temperature, the dissolved polymer molecule curls up in roughly spherical form, with minimum contribution to base oil viscosity. As temperature is raised the sphere volume increases owing to the increased solubility and results in a corresponding increase in viscosity of the solution. The multigrade lubricants are polymer solutions which often display a second Newtonian plateau or an inflection which suggests a second Newtonian plateau as in Figs. 7.7, 7.16, 8.1, 8.4, and 8.5, with second Newtonian viscosity, μ2 . Sometimes μ2 can be approximated by the viscosity, μb , of the solvent [8]. For lubricants the solvent is the base oil and this is not usually a useful approximation. Novak and Winer [50] investigated the shear-dependent viscosity of polymer-blended mineral oils at 4% polymer concentration in a high pressure capillary viscometer. For a polyalkylmethacrylate polymer, μ=μ2 5 1:7 and μ=μb 5 2:0 so that μ2 =μb 5 1:2 and here the assumption, μ2  μb , may be useful. For a polyalkylstyrene, μ=μ2 5 1:63 whereas μ=μb 5 4:9 so that μ2 =μb 5 3 and the assumption is not useful here. Rarely is μ2 , μb but sometimes this occurs. The solvent in SRM 2490 of Fig. 7.3 is pristane with viscosity, μb 5 0.0029 Pa s at 50 C and ambient pressure, which is about 100 times less than the second Newtonian viscosity, μ2 5 0:4 Pa s. Van Krevelen [51] has associated the viscosity of the second Newtonian with the viscosity of the unentangled polymer when entanglement contributes to the solution viscosity. This may explain the large ratio μ2 =μb for the SRM 2490 where the molecular weight of the polyisobutylene, 106 kg/kmole, is likely above the entanglement limit. For lubricants the polymer is not likely to be entangled and the appearance of the second Newtonian is more likely a result of a saturation of the molecular alignment process. If the solvent or base oil begins to shear-thin at a level of shear stress for which the polymer continues to contribute to the viscosity then, of course, the second Newtonian cannot appear. This has been reported for a polybutene/mineral oil solution and has also been observed in formulated gear oils [52]. Unfortunately, the only means available to set an accurate value for the second Newtonian viscosity is to perform a measurement which must at least display an inflection in the flow curve, a local minimum in d ln ðηÞ=d ln ðτ Þ. For nearly all lubricants, this requires elevated pressures to avoid thermal softening and shear cavitation.

The Shear Dependence of Viscosity at Elevated Pressure

255

Of course there are non-Newtonian polymer solutions for which there is no measurable shear dependence of viscosity due to the presence of the polymer; the only non-Newtonian effect is the normal stress difference. These are the Boger liquids [53]. The tribological behavior of these liquids in hydrodynamic films must be of interest to researchers hoping to contribute to understanding the effects of constitutive behavior on lubrication.

REFERENCES [1] Bair S. The high pressure rheology of some simple model hydrocarbons. Proc Institut Mech Eng, Part J: J Eng Tribol 2002;216(3):139
49. [2] Hamrock BJ, Dowson D. Ball bearing lubrication
the elastohydrodynamics of elliptical contacts. New York: Wiley-Interscince; 1981. [3] Bair S. Recent developments in high-pressure rheology of lubricants. In: Dowson D, Taylor CM, Childs THC, Dalmaz G, editors. Lubricants and lubrication. Tribology series, vol. 30. Amsterdam: Elsevier; 1995. p. 169
87. [4] Bair S, Vergne P, Querry M. A unified shear-thinning treatment of both film thickness and traction in EHD. Tribol Lett 2005;18(2):145
52. [5] Carreau PJ. Rheological equations from molecular network theories. Transact Soc Rheol 1972;16(1):99
127. [6] Liu Y, Wang QJ, Bair S, Vergne P. A quantitative solution for the full shearthinning EHL point contact problem including traction. Tribol Lett 2007;28 (2):171
81. [7] Sharif KJ, Kong S, Evans HP, Snidle RW. Contact and elastohydrodynamic analysis of worm gears Part 1: theoretical formulation. Proc Institut Mech Eng, Part C: J Mech Eng Sci 2001;215(7):817
30. [8] Tanner RI. Engineering rheology. 2nd ed. Oxford: Oxford Univ. Press; 2000. p. 19, 292-3. [9] Bair S. Normal stress difference in liquid lubricants sheared under high pressure. Rheol Acta 1996;35(1):13
23. [10] Bair S. Reference liquids for quantitative elastohydrodynamics: selection and rheological characterization. Tribol Lett 2006;22(2):197
206. [11] Harrison G. The dynamic properties of supercooled liquids. London/New York: Academic Press; 1976. [12] Hutton JF, Phillips MC. Shear modulus of liquids at elastohydrodynamic lubrication pressures. Nat Phys Sci 1972;238(87):141
2. [13] Barlow AJ, Harrison G, Irving JB, Kim MG, Lamb J, Pursley WC. The effect of pressure on the viscoelastic properties of liquids. Proc R Soc Lond A 1972;327 (1570):403
12. [14] Hutton JF. Reassessment of rheological properties of LVI 260 oil measured in a disk machine. J Tribol 1984;106(4):536
7. [15] Paluch M, Sekula M, Pawlus S, Rzoska SJ, Ziolo J, Roland CM. Test of the Einstein-Debye relation in supercooled dibutylphthalate at pressures up to 1.4 GPa. Phys Rev Lett 2003;90(17):175702. [16] Debye PJ. Polar molecules. New York: Dover; 1929. p. 77
85. [17] Einstein A. On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat. Annalen der physik. 1905;17:549
56.

256

High Pressure Rheology for Quantitative Elastohydrodynamics

[18] Kleinschmidt RV, Bradbury D, Mark M. Viscosity and density of over forty lubricating fluids of known composition at pressures to 150,000 psi and temperatures to 425 F. New York: ASME; 1953. [19] Yamaguchi T, Akatsuka T, Koda S. Rheological bases for empirical rules on shear viscosity of lubrication oils. J Phys Chem B 2013;117(11):3232
9. [20] Yamaguchi T, Hayakawa M, Matsuoka T, Koda S. Electric and mechanical relaxations of LiClO4 2 propylene carbonate systems in 100 MHz region. J Phys Chem B 2009;113(35):11988
98. [21] Yamaguchi T. Stress-structure coupling and nonlinear rheology of Lennard-Jones liquid. J Chem Phys 2018;148(23):234507. [22] Champion JV, Meeten GH. Flow birefringence in simple liquids. Transact Farad Soc 1968;64:238
47. [23] Mondello M, Grest GS. Viscosity calculations of n-alkanes by equilibrium molecular dynamics. J Chem Phys 1997;106(22):9327
36. [24] Daivis PJ, Evans DJ. Comparison of constant pressure and constant volume nonequilibrium simulations of sheared model decane. J Chem Phys 1994;100(1):541
7. [25] Cui ST, Cummings PT, Cochran HD. The calculation of viscosity of liquid ndecane and n-hexadecane by the Green-Kubo method. Mol Phys 1998;93 (1):117
22. [26] Haile JM. Molecular dynamics simulation: elementary methods. Hoboken: John Wiley & Sons, Inc; 1992. [27] Mondello M, Grest GS. Molecular dynamics of linear and branched alkanes. J Chem Phys 1995;103(16):7156
65. [28] Evans DJ, Morriss GP. Nonlinear-response theory for steady planar Couette flow. Phys Rev A 1984;30(3):1528. [29] McCabe C, Cui S, Cummings PT, Gordon PA, Saeger RB. Examining the rheology of 9-octylheptadecane to giga-pascal pressures. J Chem Phys 2001;114(4):1887
91. [30] McCabe C, Cui S, Cummings PT. Characterizing the viscosity
temperature dependence of lubricants by molecular simulation. Fluid Phase Equilib 2001;183:363
70. [31] Kioupis LI, Maginn EJ. Molecular simulation of poly-α-olefin synthetic lubricants: impact of molecular architecture on performance properties. J Phys Chem B 1999;103(49):10781
90. [32] McCabe C, Manke CW, Cummings PT. Predicting the Newtonian viscosity of complex fluids from high strain rate molecular simulations. J Chem Phys 2002;116 (8):3339
42. [33] Bair S, McCabe C, Cummings PT. Comparison of nonequilibrium molecular dynamics with experimental measurements in the nonlinear shear-thinning regime. Phys Rev Lett 2002;88(5):058302. [34] Bair S, Khonsari M. An EHD inlet zone analysis incorporating the second Newtonian. J Tribol 1996;118(2):341
3. [35] Schurz J, Vollrath-Rödiger M, Krässig H. Rheologische Untersuchungen an Polyacrylnitril-Spinnlösungen: Erweiterung des Viskosimeters HV 6 Für Höhere Temperaturen. Rheol Acta 1981;20(6):569
78. [36] Greenwood JA, Kauzlarich JJ. Elastohydrodynamic film thickness for shear-thinning lubricants. Proc Institut Mech Eng, Part J: J Eng Tribol 1998;212(3):179
91. [37] Bair S, Qureshi F. Time-temperature-pressure superposition in polymer thickened liquid lubricants. J Tribol 2014;136(2):021505. [38] Winter HH. Viscous dissipation in shear flows of molten polymers. Adv Heat Transfer 1977;13:205
67. [39] Bair S, Qureshi F. Ordinary shear thinning behavior and its effect upon EHL film thickness. In: Dowson D, Priest M, Dalmaz G, Lubrecht AA, editors. Tribological

The Shear Dependence of Viscosity at Elevated Pressure

[40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53]

257

research and design for engineering systems. Tribology series, vol. 41. Amsterdam: Elsevier; 2002. p. 693
9. Bair S. The high pressure rheology of mixtures. J Tribol 2004;126(4):697
702. Yasuda KY, Armstrong RC, Cohen RE. Shear flow properties of concentrated solutions of linear and star branched polystyrenes. Rheol Acta 1981;20(2):163
78. Bair S, Khonsari M, Winer WO. High-pressure rheology of lubricants and limitations of the Reynolds equation. Tribol Int 1998;31(10):573
86. Ree F, Ree T, Eyring H. Relaxation theory of transport problems in condensed systems. Indust Eng Chem 1958;50(7):1036
40. Bair S. Generalized Newtonian viscosity functions for hydrodynamic lubrication. Tribol Int 2018;117:15
23. Dobson GR. Analysis of high shear rate viscosity data for engine oils. Tribol Int 1981;14(4):195
8. Wood-Adams PM. The effect of long chain branches on the shear flow behavior of polyethylene. J Rheol 2001;45(1):203
10. Jabbarzadeh A, Atkinson JD, Tanner RI. The effect of branching on slip and rheological properties of lubricants in molecular dynamics simulation of Couette shear flow. Tribol Int 2002;35(1):35
46. Bair S. Shear thinning correction for rolling/sliding elastohydrodynamic film thickness. Proc Inst Mech Eng, Part J: J Eng Tribol 2005;219(1):69
74. Wilson WR, Huang XB. Viscoplastic behavior of a silicone oil in a metalforming inlet zone. J Tribol 1989;111(4):585
90. Novak JD, Winer WO. Some measurements of high pressure lubricant rheology. J Lubricat Technol 1968;90(3):580
90. Van Krevelen DW, Te Nijenhuis K. Properties of polymers: their correlation with chemical structure: their numerical estimation and prediction. 4th ed. Amsterdam: Elsevier; 2009. p. 477. Bair S, Qureshi F. The high pressure rheology of polymer-oil solutions. Tribol Int 2003;36(8):637
45. Boger DV. A highly elastic constant-viscosity fluid. J Non-Newton Fluid Mech 1977;3(1):87
91.