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The role of fragility in EHL entrapment
ARTICLE IN PRESS Tribology International 43 (2010) 277–282
Contents lists available at ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
The role of fragility in EHL entrapment Ashlie Martini a,, Scott Bair b a b
Purdue University, School of Mechanical Engineering, 585 Purdue Mall, West Lafayette, IN 47907, USA Georgia Institute of Technology, Center for High-Pressure Rheology, George W. Woodruff School of Mechanical Engineering, Atlanta, GA, USA
a r t i c l e in fo
abstract
Article history: Received 13 April 2009 Received in revised form 2 June 2009 Accepted 3 June 2009 Available online 10 June 2009
Experimental measurements of time dependent film thickness in entrapped liquids, measurements of viscosity under pressure, and simulations using realistic pressure–viscosity models contribute to improved understanding of the mechanisms of entrapment formation and persistence. The ambient viscosity and pressure–viscosity coefficient affect entrapment only as much as they are predictors of behavior at much higher pressure. Fragile liquids, such as lubricating oils, experience rapid increase in sensitivity of dynamic properties to temperature and pressure as the glass transition is approached. The fragility property of lubricants appears to be of overwhelming importance to entrapment which experimental evidence indicates will reduce starting friction. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Elastohydrodynamic lubrication Lubricant rheology
1. Introduction There has been lately much interest in transient effects in elastohydrodynamic lubrication (EHL) and one of the most curious artifacts is the formation of a central axisymmetric dimple at the center of a circular contact, a liquid-filled entrapment, which may persist for hours after the motion of the bulk of the rollers has vanished. Entrapment may occur by a sudden halt to rolling/sliding motion [1,2] or by impact [3,4]. Entrapment may be more than merely a curiosity. The captured liquid is compressed to very high pressure and may support a significant portion of the contact load. The friction at the start-up of sliding should therefore be reduced when an entrapment is present if the force to shear the pressurized liquid is less than the force to shear the circular solid conjunction which it replaces. To demonstrate the feasibility of entrapment eased start-up, in a preliminary experiment, friction was measured at the initiation of sliding with and without entrapment as illustrated in Fig. 1. Using the experimental arrangement described below, slow sliding was initiated after a gradual stop to prevent entrapment and also after a sudden stop which formed an entrapment. Both cases tend toward the same steady friction after 2 s, as expected. However, the entrapment apparently results in a lower residual shear force as shown by the data for negative time in Fig. 1 where the contact is stationary. The sliding friction is lower while the entrapment exists as shown by the plateau at 0.5–1.0 s where the friction coefficient is reduced from about 0.066–0.036. This effect can be repeated. After some sliding the entrapment is displaced to the edge whereupon the entrapped liquid drains away and the friction then returns to the entrapment free level after 1.5 s. The
Corresponding author. Tel.: +1765 494 6726.
E-mail address: [email protected] (A. Martini). 0301-679X/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2009.06.006
relatively slow time response of this friction measurement results from the large elastic compliance of the instrument which is designed for steady rolling traction measurements. These results show that entrapment is not only an interesting phenomenon from a scientific standpoint, but may be a means of improving start-up efficiency. However, we cannot take full advantage of this utility until we understand which lubricant properties are active in entrapment formation and persistence. Towards this end, a transient EHL model-based study and qualitative experimental validation are presented with the goal of elucidating the lubricant properties that are important to entrapment. This particular analysis is unique in the use of a realistic description of the pressure-dependence of viscosity. Viscosity has not been adjusted to yield agreement with film thickness measurement. Instead, the viscosities of the experimental liquids have been measured in viscometers and fitted to pressure–viscosity models which are capable of describing pressure– fragility, a property of glass-forming liquids long ignored in EHL. Fragility is found to be a property of overwhelming importance to EHL entrapment.
2. Methodology 2.1. Experiment Viscosities were measured in falling body viscometers which apply a shear stress sufficiently low (o100 Pa) so that the viscosities can be considered to be the limiting low shear values. Two types of film thickness experiments were performed, sudden halting and impact. The quiescent loaded contact conditions were the same for either case: A sapphire disc is loaded against a steel ball of 7.6 mm radius with a force of 11.3 N by a weight suspended from the disc. The contact surface of the disc
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A. Martini, S. Bair / Tribology International 43 (2010) 277–282
Nomenclature h hc K0 K 00 K00 p pg pN pN p+, p
r T t
film thickness, m central film thickness, m isothermal bulk modulus at p ¼ 0, Pa pressure rate of change of isothermal bulk modulus at p¼0 K0 at zero absolute temperature, Pa pressure, Pa glass transition pressure, Pa pressure at which viscosity diverges, Pa tension at which the associated viscosity term vanishes, Pa parameters in the Bair and Kottke pressure–viscosity model, Pa
radius, m temperature, K time, s local pressure–viscosity coefficient, Pa1 conventional pressure–viscosity coefficient, Pa1 reciprocal asymptotic isoviscous pressure coefficient, Pa1 pressure–viscosity coefficient of the liquid at the glass transition, Pa1 parameters of the Irving and Barlow model, Pa1 temperature coefficient of K 0 , K1 limiting low-shear and Newtonian viscosity, Pa s viscosity at the glass transition, Pa s mass density, kg/m3 mass density, at p ¼ 0, kg/m3
a a0 a* ag an, ap, bk
m mg r r0
has a semi-transparent metal coating to enhance optical interference so that the details of the film may be observed. The contact is illuminated with light filtered to have a narrow distribution of wavelength centered at 600 nm. The micrographs were calibrated with a stage micrometer. For film thickness measurement, the refractive index was assumed to be equal to 1.5 and the phase change on reflection was adjusted to yield a power-law relation between rolling velocity and film thickness. Sliding motion results in the formation of a film about 300 nm thick. Slowly reducing the sliding speed to rest results in uniform contact. However, using a mechanical stop to rapidly bring the sliding to rest can result in a central entrapment. This method was applied to entrap a Mineral Oil and a Heavy polyalphaolefin (PAO). Entrapments can also be formed by normal motion of the surfaces. The disc with attached weight were lifted 1.8 mm above the ball and released in efforts to entrap a Light PAO and a polyol ester (POE).
coordinates can be written as [5] ! 3 @ rh @p @ðrhÞ r ¼ 12r m @r @r @t
(1)
where h is film thickness, r is radial position, p is pressure, m is viscosity, r is density, and t is time. The model includes elastic deformation of the solids as described by the Boussinesq approximation, load balance, the Tait pressure–density relationship, and multiple pressure–viscosity models described in the next section. The general solution approach is similar to that given in [6]. Some of the numerical methods described in [7] and references therein were employed for accurate and efficient solution of these equations. The computational domain is 0.22 mm square subdivided into 128 discrete units. The transient iteration scheme is solved using time steps of 0.716, 1.791, and 3.581 ms which were found to be small enough to capture even nanometer scale effects of lubricant properties on entrapment behavior. The larger step sizes were used only after (and if) the dynamics of the system slowed sufficiently to allow accurate numerical solution. The Tait pressure–density model [8] was employed because it is known to be accurate both at moderate pressures and when extrapolated to very high pressures such as those required for
2.2. Simulation Entrapment formation was modeled using a full numerical transient solution for EHL point contact using only the Poiseuille and squeeze terms of the Reynolds equation which in spherical
0.15
0.1
Friction Coefficient
With Entrapment Without Entrapment
0.05
0 -1.5 -0.05
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Time / seconds
-0.1
-0.15 Fig. 1. Friction coefficient at start up with and without entrapment illustrating the potential utility of entrapment as a means of improving start up efficiency.
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279
entrapment. The Tait model for the effect of pressure on density, r, is r0 1 p ¼1 ð1 þ K 00 Þ (2) 0 ln 1 þ r K0 1 þ K0 where r0 is the low-pressure density, and K0 and K 00 are the bulk modulus and the pressure rate of change of the bulk modulus, respectively, at ambient pressure. The bulk modulus is a temperature dependant parameter given by K 0 ¼ K 00 exp bK T . The compressibilities of the test liquids have not been experimentally determined, therefore following [8], we use K00 ¼ 9 GPa, bK ¼ 6.5e3 1 K1, and K 00 ¼ 11 as representative of general lubricants. The elastic modulus and Poisson ratio for the steel ball and sapphire flat surface were E0 ¼ 210 and 365 GPa and n ¼ 0.29 and 0.25, respectively. The ball radius was 7.6 mm and the applied load 11.3 N. The temperature (used in the density calculation) was 25 1C for the Mineral Oil and Heavy PAO, and 50 1C for the Light PAO and POE. The assumed initial conditions were Hertz (hemispheric) pressure distribution (pHertz ¼ 0.97 GPa) and a uniform film thickness of 300 nm for all cases. The intention is to qualitatively simulate the condition of the interface either immediately after a sliding motion has been abruptly stopped or at some instant after a dropped ball impacts a lubricant layer. Note that, while there is certainly some surface roughness that contributes to experimental results, the ball and disk are modeled as ideally smooth. 2.3. Rheology Models are necessary to describe the pressure-dependence of viscosity in the Reynolds equation. For glass-forming liquids, the temperature and pressure dependences of the viscosity become greater as the glass transition is approached from conditions of lower viscosity. This effect has been termed ‘‘fragility’’. Fragility is therefore a property of liquid lubricants [9] which must be included in high-pressure calculations. Pressure fragility may be quantified by the local pressure–viscosity coefficient at the glass pressure, j@ ln m=@pjp¼pg . However, the most reasonable metric for pressure fragility is still a matter of debate [10]. An early model which captures fragile behavior in the pressure dependence was introduced by Irving and Barlow in 1971 [11]. a þ ap a0 a0 an exp m ¼ m0 exp n expðap pÞ expðan pÞ
ap
ap
(3) This model is, however, incapable of reproducing the pressure versus log viscosity inflection when the inflection pressure is very high, as is the case for polyalphaolefins. In fact, the inflection that is present in the data for Heavy PAO at about 0.5 GPa in Fig. 2 is absent in the Irving and Barlow equation fitted to the data. Bair and Kottke [8] found a relation with one more parameter which removes that restriction. p pþ pþ p exp (4) m ¼ m0 exp þ p1 p1 p1 p p p1 The entrapment formation capabilities of four lubricants designated Heavy PAO (Mobil SHF403), Mineral Oil (Hygold L750), Light PAO (polyalphaolefin described in [8]), and POE (polyol ester described in [8]) are evaluated here. The viscosity data for the Mineral Oil and Heavy PAO given in Table 1 were fitted to Eq. (3), and the data for the POE and the Light PAO from Ref. [8] were fitted to Eq. (4). In addition, a hypothetical liquid was constructed from Eq. (4) which has the low-pressure character of the Heavy PAO but with a low glass pressure, that is to say, an inflection occurs at a low pressure (0.2 GPa) that is more characteristic of a Mineral Oil. This model lubricant is referred
Fig. 2. Variation of viscosity with pressure both measured (data points) and predicted by empirical models (curves).
Table 1 High pressure viscometer measurements for the Mineral Oil and Heavy PAO. Pressure (MPa)
Viscosity (Pa s) Mineral Oil
0.1 25 50 100 150 200 250 300 400 500 600 700 800
Heavy PAO
0.383 0.886 1.87 9.26 40.7 183 1012 5580
0.752 2.17 5.68 30.1 134 480 1850 6350 2300 9040
Table 2 Pressure–viscosity model parameters for five lubricants considered in this study.
m0 (Pa s) a* (GPa1) a0 (GPa1) ap (GPa1) an (GPa1) pN (GPa) pN (GPa) p+ (GPa) p (GPa) pg (GPa) m0 a* (ns) m0 a0 (ns)
Mineral Oil
Heavy PAO
Low pg PAO
Light PAO
POE
0.383 32.3 37.63 1.8203 16.016 – – – – 0.55 12.4 13.5
0.752 20.0 26.293 0.1 12.417 – – – – Insufficient data 15.0 19.4
0.752 20.1 23.2 – – 1.329 0.2715 14.112 1.1366 0.83 15.1 17.4
0.0201 16.4 20.5 – – 3.38 0.422 37.51 3.011 2.12 0.33 0.41
0.0169 14.4 17.6 – – 3.38 0.381 71.29 1.503 1.71 0.24 0.30
A dash indicates that the lubricant viscosity was not described by the model corresponding to that parameter.
to as the Low pg PAO. Fitted parameters are given in Table 2 and both the experimental data and model predictions are illustrated in Fig. 2. Fragility is apparent in Fig. 2 from the upturn in the log viscosity at high pressure. Eq. (4) is unbounded for p ¼ p1 and the glass transition is accommodated by truncating the viscosity at 109 Pa s. Also reported in Table 2 are the glass transition pressure, pg, defined as the pressure for viscosity equal to 109 Pa s, and two pressure–viscosity coefficients. The initial or conventional
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Fig. 3. Images taken from experiment 3 s after impact (top), simulation-predicted film thickness (middle) and pressure (bottom) distributions varying with time for the Light PAO (left) and POE (right).
pressure–viscosity coefficient is dðln mÞ a0 ¼ dp p¼0
(5)
and the reciprocal asymptotic isoviscous pressure coefficient is Z 1 mðp ¼ 0Þdp 1 a ¼ (6) mðpÞ 0
3. Results and analysis It has been observed before that the low-pressure viscosity and pressure–viscosity coefficient affect the entrapment capability of a liquid [1,12]. Here the effect on entrapment of the pressure– viscosity response of liquids is investigated over a large range of pressure with specific focus on the glass transition pressure and fragility. To that end, two groups of lubricants, low viscosity (Light PAO and POE) and high viscosity (Heavy PAO and Mineral Oil), were investigated. Within each group, the fluids vary significantly only in their high-pressure rheology.
Consider first the low viscosity lubricants, Light PAO and POE. Neither is able to sustain an entrapment for more than a minute. This may be expected based on their relatively small low-pressure viscosities (0.0201 and 0.0169 Pa s, respectively) and pressure–viscosity coefficients (16.4 and 14.4 or 20.5 and 17.6 GPa1 depending on definition). Fig. 3 provides a qualitative comparison between simulation results (lower figures) and experimental observations (upper figures) for these lubricants. The experimental images captured 3 s after impact reveal that the Light PAO and POE entrapment behavior is quite different. The film thickness difference between fully developed light and dark fringes is about 100 nm. From these images, it can be observed that the Light PAO may exhibit at most a 50 nm thick entrapment at this time while the POE supports a small central dimple with steep sides about 200 nm thick. This is consistent with the simulation results which are illustrated as 2-dimensional film thickness profiles in Fig. 3. The model clearly predicts that, in a timeframe on the order of a few seconds, the POE will be able to support an entrapment while the Light PAO will not. This behavior is attributable directly to the high-pressure characteristics of the lubricant; the POE
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281
Fig. 4. Experimental images from the rapid stop tests with Mineral Oil and Heavy PAO at times from 14 to 2430 s after the motion is mechanically stopped.
Fig. 5. Entrapment width as a function of time measured experimentally (solid data points indicate maximum and minimum measurements) and predicted by the EHL simulation (hollow points).
viscosity increases more with pressure than the Light PAO at the high pressures as shown in Fig. 2. Or, in another way of looking at the effect, the glass pressure is reached over a greater portion of the contact for the POE. Low-pressure comparisons of these lubricants give just the opposite rheological picture. Both the ambient pressure viscosity and the two pressure–viscosity coefficients are greater for the Light PAO than for the POE, reinforcing the importance of the high-pressure behavior. The calculated pressure distributions are plotted at the bottom of Fig. 3. The shear stress in the vicinity of the pressure spike exceeds 5 MPa. These liquids should therefore shear-thin [8] in these regions, reducing the sharpness of the spike. Next, the behavior of entrapments formed by high viscosity fluids is evaluated. These lubricants, Mineral Oil and Heavy PAO,
are expected to be able to retain entrapments for longer times based on their relatively large low-pressure viscosities (0.383 and 0.752 Pa s, respectively) and pressure–viscosity coefficients (32.3 and 20.0 or 37.6 and 26.3 GPa1 depending on definition). This is indeed what is observed experimentally as illustrated in Fig. 4. The non-circular form of the entrapment is believed to result from small scratches on the ball surface. Although both lubricants retain entrapments for more than a minute, the Mineral Oil-based entrapment is thicker, larger in diameter, and persists for a much greater time than the Heavy PAO. This is in contradiction to predictions based upon the lubricant parameter [12], defined as the product of the ambient viscosity, m0, and the pressure viscosity coefficient. The lubricant parameter is listed in Table 2 using both definitions, Eqs. (5 and 6), of the pressure–viscosity
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coefficient. In both cases, the lubricant parameter is greater for the Heavy PAO than for the Mineral Oil. The EHL simulation was used to evaluate the effect of rheology with high viscosity fluids. The ability of a lubricant to sustain an entrapment was characterized as the entrapment width as a function of time. The entrapment widths measured from experiment and predicted by the simulation are shown in Fig. 5. For the experiment, the largest and smallest widths are recorded. This figure reveals two important things. First, the simulation correctly predicts the trend of decreasing width with increasing time and the difference between the Mineral Oil and Heavy PAO entrapment widths. These behaviors can be understood in terms of the model formulation: when the high-pressure viscosity of a fluid is large, the Poiseuille term in Eq. (1) becomes small, resulting in a correspondingly low value of dh/dt and longer entrapment retention. Second, to further probe the effect of highpressure rheology, a model lubricant was created having the same low-pressure behavior as the Heavy PAO, but an artificially low glass transition pressure so that the entrapped liquid shows pressure fragility similar to the Mineral Oil. This model lubricant is referred to as the Low pg PAO. The calculated entrapment behavior of the model Low pg PAO is significantly different from the Heavy PAO as shown in Fig. 5 and approaches the behavior of the Mineral Oil as time progresses.
4. Conclusions Entrapment appears to reduce starting friction. Experimental measurements of the time dependent film thickness in entrapped liquids, measurements of viscosity under pressure, and simulations using realistic pressure–viscosity models have contributed to improved understanding of the mechanisms of entrapment formation and persistence. The ambient viscosity and the pressure–viscosity coefficient affect the entrapment only as much as these are predictors of behavior at much higher pressure and
may result in predictions which contradict actual behavior. The fragility property of lubricants appears to be of overwhelming importance to entrapment.
Acknowledgment The authors were supported by the National Science Foundation under Grant number EEC#0540834. References [1] Glovnea RP, Spikes HA. The influence of lubricant upon EHD film behavior during sudden halting of motion. STLE Tribol Trans 2000;43:731–9. [2] Zhoa J, Sadeghi F. Analysis of EHL circular contact shut down. ASME J Tribol 2003;125:76–89. [3] Chang L. An efficient calculation of the load and coefficient of restitution of impact between two elastic bodies with a liquid lubricant. ASME J Appl Mech 1996;63:347–52. [4] Lee KM, Cheng HS. The pressure and deformation profiles between two normally approaching lubricated cylinders. ASME J Lubr Technol 1973;95: 308–17. [5] Barnocky G, Davis RH. The influence of pressure-dependent density and viscosity on the elastohydrodynamic collision and rebound of two spheres. J Fluid Mech 1989;209:501–19. [6] Ai X, Yu H. A full numerical solution for general transient elastohydrodynamic line contacts and its application. Wear 1988;121:143–59. [7] Liu Y, Wang Q, Wang W, Hu Y, Zhu D. Effects of differential scheme and mesh density on EHL film thickness in point contacts. ASME J Tribol 2006;128: 641–53. [8] Bair S. High-pressure rheology for quantitative elastohydrodynamics. Amsterdam: Elsevier Science; 2007. p. 111–2, 228, 163. [9] Bair S, Roland CM, Casalini R. Fragility and the dynamic crossover in lubricants. Proc Inst Mech Eng Part J, J Eng Tribol 2007;221:801–11. [10] Drozd-Rzoska A, Rzoska S, Roland CM, Imre AR. On the pressure evolution of dynamic properties of supercooled liquids. J Phys Condens Matter 2008;20: 244103. [11] Irving JB, Barlow AJ. An automatic high pressure viscometer. J Phys E 1971; 4:232–6. [12] Ohno N, Yamada S. Effect of high pressure rheology of lubricants upon entrapped oil film behavior at halting elastohydrodynamic lubrication. Proc Inst Mech Eng Part J, J Eng Tribol 2007;221:279–85.
Contents lists available at ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
The role of fragility in EHL entrapment Ashlie Martini a,, Scott Bair b a b
Purdue University, School of Mechanical Engineering, 585 Purdue Mall, West Lafayette, IN 47907, USA Georgia Institute of Technology, Center for High-Pressure Rheology, George W. Woodruff School of Mechanical Engineering, Atlanta, GA, USA
a r t i c l e in fo
abstract
Article history: Received 13 April 2009 Received in revised form 2 June 2009 Accepted 3 June 2009 Available online 10 June 2009
Experimental measurements of time dependent film thickness in entrapped liquids, measurements of viscosity under pressure, and simulations using realistic pressure–viscosity models contribute to improved understanding of the mechanisms of entrapment formation and persistence. The ambient viscosity and pressure–viscosity coefficient affect entrapment only as much as they are predictors of behavior at much higher pressure. Fragile liquids, such as lubricating oils, experience rapid increase in sensitivity of dynamic properties to temperature and pressure as the glass transition is approached. The fragility property of lubricants appears to be of overwhelming importance to entrapment which experimental evidence indicates will reduce starting friction. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Elastohydrodynamic lubrication Lubricant rheology
1. Introduction There has been lately much interest in transient effects in elastohydrodynamic lubrication (EHL) and one of the most curious artifacts is the formation of a central axisymmetric dimple at the center of a circular contact, a liquid-filled entrapment, which may persist for hours after the motion of the bulk of the rollers has vanished. Entrapment may occur by a sudden halt to rolling/sliding motion [1,2] or by impact [3,4]. Entrapment may be more than merely a curiosity. The captured liquid is compressed to very high pressure and may support a significant portion of the contact load. The friction at the start-up of sliding should therefore be reduced when an entrapment is present if the force to shear the pressurized liquid is less than the force to shear the circular solid conjunction which it replaces. To demonstrate the feasibility of entrapment eased start-up, in a preliminary experiment, friction was measured at the initiation of sliding with and without entrapment as illustrated in Fig. 1. Using the experimental arrangement described below, slow sliding was initiated after a gradual stop to prevent entrapment and also after a sudden stop which formed an entrapment. Both cases tend toward the same steady friction after 2 s, as expected. However, the entrapment apparently results in a lower residual shear force as shown by the data for negative time in Fig. 1 where the contact is stationary. The sliding friction is lower while the entrapment exists as shown by the plateau at 0.5–1.0 s where the friction coefficient is reduced from about 0.066–0.036. This effect can be repeated. After some sliding the entrapment is displaced to the edge whereupon the entrapped liquid drains away and the friction then returns to the entrapment free level after 1.5 s. The
Corresponding author. Tel.: +1765 494 6726.
E-mail address: [email protected] (A. Martini). 0301-679X/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2009.06.006
relatively slow time response of this friction measurement results from the large elastic compliance of the instrument which is designed for steady rolling traction measurements. These results show that entrapment is not only an interesting phenomenon from a scientific standpoint, but may be a means of improving start-up efficiency. However, we cannot take full advantage of this utility until we understand which lubricant properties are active in entrapment formation and persistence. Towards this end, a transient EHL model-based study and qualitative experimental validation are presented with the goal of elucidating the lubricant properties that are important to entrapment. This particular analysis is unique in the use of a realistic description of the pressure-dependence of viscosity. Viscosity has not been adjusted to yield agreement with film thickness measurement. Instead, the viscosities of the experimental liquids have been measured in viscometers and fitted to pressure–viscosity models which are capable of describing pressure– fragility, a property of glass-forming liquids long ignored in EHL. Fragility is found to be a property of overwhelming importance to EHL entrapment.
2. Methodology 2.1. Experiment Viscosities were measured in falling body viscometers which apply a shear stress sufficiently low (o100 Pa) so that the viscosities can be considered to be the limiting low shear values. Two types of film thickness experiments were performed, sudden halting and impact. The quiescent loaded contact conditions were the same for either case: A sapphire disc is loaded against a steel ball of 7.6 mm radius with a force of 11.3 N by a weight suspended from the disc. The contact surface of the disc
ARTICLE IN PRESS 278
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Nomenclature h hc K0 K 00 K00 p pg pN pN p+, p
r T t
film thickness, m central film thickness, m isothermal bulk modulus at p ¼ 0, Pa pressure rate of change of isothermal bulk modulus at p¼0 K0 at zero absolute temperature, Pa pressure, Pa glass transition pressure, Pa pressure at which viscosity diverges, Pa tension at which the associated viscosity term vanishes, Pa parameters in the Bair and Kottke pressure–viscosity model, Pa
radius, m temperature, K time, s local pressure–viscosity coefficient, Pa1 conventional pressure–viscosity coefficient, Pa1 reciprocal asymptotic isoviscous pressure coefficient, Pa1 pressure–viscosity coefficient of the liquid at the glass transition, Pa1 parameters of the Irving and Barlow model, Pa1 temperature coefficient of K 0 , K1 limiting low-shear and Newtonian viscosity, Pa s viscosity at the glass transition, Pa s mass density, kg/m3 mass density, at p ¼ 0, kg/m3
a a0 a* ag an, ap, bk
m mg r r0
has a semi-transparent metal coating to enhance optical interference so that the details of the film may be observed. The contact is illuminated with light filtered to have a narrow distribution of wavelength centered at 600 nm. The micrographs were calibrated with a stage micrometer. For film thickness measurement, the refractive index was assumed to be equal to 1.5 and the phase change on reflection was adjusted to yield a power-law relation between rolling velocity and film thickness. Sliding motion results in the formation of a film about 300 nm thick. Slowly reducing the sliding speed to rest results in uniform contact. However, using a mechanical stop to rapidly bring the sliding to rest can result in a central entrapment. This method was applied to entrap a Mineral Oil and a Heavy polyalphaolefin (PAO). Entrapments can also be formed by normal motion of the surfaces. The disc with attached weight were lifted 1.8 mm above the ball and released in efforts to entrap a Light PAO and a polyol ester (POE).
coordinates can be written as [5] ! 3 @ rh @p @ðrhÞ r ¼ 12r m @r @r @t
(1)
where h is film thickness, r is radial position, p is pressure, m is viscosity, r is density, and t is time. The model includes elastic deformation of the solids as described by the Boussinesq approximation, load balance, the Tait pressure–density relationship, and multiple pressure–viscosity models described in the next section. The general solution approach is similar to that given in [6]. Some of the numerical methods described in [7] and references therein were employed for accurate and efficient solution of these equations. The computational domain is 0.22 mm square subdivided into 128 discrete units. The transient iteration scheme is solved using time steps of 0.716, 1.791, and 3.581 ms which were found to be small enough to capture even nanometer scale effects of lubricant properties on entrapment behavior. The larger step sizes were used only after (and if) the dynamics of the system slowed sufficiently to allow accurate numerical solution. The Tait pressure–density model [8] was employed because it is known to be accurate both at moderate pressures and when extrapolated to very high pressures such as those required for
2.2. Simulation Entrapment formation was modeled using a full numerical transient solution for EHL point contact using only the Poiseuille and squeeze terms of the Reynolds equation which in spherical
0.15
0.1
Friction Coefficient
With Entrapment Without Entrapment
0.05
0 -1.5 -0.05
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Time / seconds
-0.1
-0.15 Fig. 1. Friction coefficient at start up with and without entrapment illustrating the potential utility of entrapment as a means of improving start up efficiency.
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279
entrapment. The Tait model for the effect of pressure on density, r, is r0 1 p ¼1 ð1 þ K 00 Þ (2) 0 ln 1 þ r K0 1 þ K0 where r0 is the low-pressure density, and K0 and K 00 are the bulk modulus and the pressure rate of change of the bulk modulus, respectively, at ambient pressure. The bulk modulus is a temperature dependant parameter given by K 0 ¼ K 00 exp bK T . The compressibilities of the test liquids have not been experimentally determined, therefore following [8], we use K00 ¼ 9 GPa, bK ¼ 6.5e3 1 K1, and K 00 ¼ 11 as representative of general lubricants. The elastic modulus and Poisson ratio for the steel ball and sapphire flat surface were E0 ¼ 210 and 365 GPa and n ¼ 0.29 and 0.25, respectively. The ball radius was 7.6 mm and the applied load 11.3 N. The temperature (used in the density calculation) was 25 1C for the Mineral Oil and Heavy PAO, and 50 1C for the Light PAO and POE. The assumed initial conditions were Hertz (hemispheric) pressure distribution (pHertz ¼ 0.97 GPa) and a uniform film thickness of 300 nm for all cases. The intention is to qualitatively simulate the condition of the interface either immediately after a sliding motion has been abruptly stopped or at some instant after a dropped ball impacts a lubricant layer. Note that, while there is certainly some surface roughness that contributes to experimental results, the ball and disk are modeled as ideally smooth. 2.3. Rheology Models are necessary to describe the pressure-dependence of viscosity in the Reynolds equation. For glass-forming liquids, the temperature and pressure dependences of the viscosity become greater as the glass transition is approached from conditions of lower viscosity. This effect has been termed ‘‘fragility’’. Fragility is therefore a property of liquid lubricants [9] which must be included in high-pressure calculations. Pressure fragility may be quantified by the local pressure–viscosity coefficient at the glass pressure, j@ ln m=@pjp¼pg . However, the most reasonable metric for pressure fragility is still a matter of debate [10]. An early model which captures fragile behavior in the pressure dependence was introduced by Irving and Barlow in 1971 [11]. a þ ap a0 a0 an exp m ¼ m0 exp n expðap pÞ expðan pÞ
ap
ap
(3) This model is, however, incapable of reproducing the pressure versus log viscosity inflection when the inflection pressure is very high, as is the case for polyalphaolefins. In fact, the inflection that is present in the data for Heavy PAO at about 0.5 GPa in Fig. 2 is absent in the Irving and Barlow equation fitted to the data. Bair and Kottke [8] found a relation with one more parameter which removes that restriction. p pþ pþ p exp (4) m ¼ m0 exp þ p1 p1 p1 p p p1 The entrapment formation capabilities of four lubricants designated Heavy PAO (Mobil SHF403), Mineral Oil (Hygold L750), Light PAO (polyalphaolefin described in [8]), and POE (polyol ester described in [8]) are evaluated here. The viscosity data for the Mineral Oil and Heavy PAO given in Table 1 were fitted to Eq. (3), and the data for the POE and the Light PAO from Ref. [8] were fitted to Eq. (4). In addition, a hypothetical liquid was constructed from Eq. (4) which has the low-pressure character of the Heavy PAO but with a low glass pressure, that is to say, an inflection occurs at a low pressure (0.2 GPa) that is more characteristic of a Mineral Oil. This model lubricant is referred
Fig. 2. Variation of viscosity with pressure both measured (data points) and predicted by empirical models (curves).
Table 1 High pressure viscometer measurements for the Mineral Oil and Heavy PAO. Pressure (MPa)
Viscosity (Pa s) Mineral Oil
0.1 25 50 100 150 200 250 300 400 500 600 700 800
Heavy PAO
0.383 0.886 1.87 9.26 40.7 183 1012 5580
0.752 2.17 5.68 30.1 134 480 1850 6350 2300 9040
Table 2 Pressure–viscosity model parameters for five lubricants considered in this study.
m0 (Pa s) a* (GPa1) a0 (GPa1) ap (GPa1) an (GPa1) pN (GPa) pN (GPa) p+ (GPa) p (GPa) pg (GPa) m0 a* (ns) m0 a0 (ns)
Mineral Oil
Heavy PAO
Low pg PAO
Light PAO
POE
0.383 32.3 37.63 1.8203 16.016 – – – – 0.55 12.4 13.5
0.752 20.0 26.293 0.1 12.417 – – – – Insufficient data 15.0 19.4
0.752 20.1 23.2 – – 1.329 0.2715 14.112 1.1366 0.83 15.1 17.4
0.0201 16.4 20.5 – – 3.38 0.422 37.51 3.011 2.12 0.33 0.41
0.0169 14.4 17.6 – – 3.38 0.381 71.29 1.503 1.71 0.24 0.30
A dash indicates that the lubricant viscosity was not described by the model corresponding to that parameter.
to as the Low pg PAO. Fitted parameters are given in Table 2 and both the experimental data and model predictions are illustrated in Fig. 2. Fragility is apparent in Fig. 2 from the upturn in the log viscosity at high pressure. Eq. (4) is unbounded for p ¼ p1 and the glass transition is accommodated by truncating the viscosity at 109 Pa s. Also reported in Table 2 are the glass transition pressure, pg, defined as the pressure for viscosity equal to 109 Pa s, and two pressure–viscosity coefficients. The initial or conventional
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Fig. 3. Images taken from experiment 3 s after impact (top), simulation-predicted film thickness (middle) and pressure (bottom) distributions varying with time for the Light PAO (left) and POE (right).
pressure–viscosity coefficient is dðln mÞ a0 ¼ dp p¼0
(5)
and the reciprocal asymptotic isoviscous pressure coefficient is Z 1 mðp ¼ 0Þdp 1 a ¼ (6) mðpÞ 0
3. Results and analysis It has been observed before that the low-pressure viscosity and pressure–viscosity coefficient affect the entrapment capability of a liquid [1,12]. Here the effect on entrapment of the pressure– viscosity response of liquids is investigated over a large range of pressure with specific focus on the glass transition pressure and fragility. To that end, two groups of lubricants, low viscosity (Light PAO and POE) and high viscosity (Heavy PAO and Mineral Oil), were investigated. Within each group, the fluids vary significantly only in their high-pressure rheology.
Consider first the low viscosity lubricants, Light PAO and POE. Neither is able to sustain an entrapment for more than a minute. This may be expected based on their relatively small low-pressure viscosities (0.0201 and 0.0169 Pa s, respectively) and pressure–viscosity coefficients (16.4 and 14.4 or 20.5 and 17.6 GPa1 depending on definition). Fig. 3 provides a qualitative comparison between simulation results (lower figures) and experimental observations (upper figures) for these lubricants. The experimental images captured 3 s after impact reveal that the Light PAO and POE entrapment behavior is quite different. The film thickness difference between fully developed light and dark fringes is about 100 nm. From these images, it can be observed that the Light PAO may exhibit at most a 50 nm thick entrapment at this time while the POE supports a small central dimple with steep sides about 200 nm thick. This is consistent with the simulation results which are illustrated as 2-dimensional film thickness profiles in Fig. 3. The model clearly predicts that, in a timeframe on the order of a few seconds, the POE will be able to support an entrapment while the Light PAO will not. This behavior is attributable directly to the high-pressure characteristics of the lubricant; the POE
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Fig. 4. Experimental images from the rapid stop tests with Mineral Oil and Heavy PAO at times from 14 to 2430 s after the motion is mechanically stopped.
Fig. 5. Entrapment width as a function of time measured experimentally (solid data points indicate maximum and minimum measurements) and predicted by the EHL simulation (hollow points).
viscosity increases more with pressure than the Light PAO at the high pressures as shown in Fig. 2. Or, in another way of looking at the effect, the glass pressure is reached over a greater portion of the contact for the POE. Low-pressure comparisons of these lubricants give just the opposite rheological picture. Both the ambient pressure viscosity and the two pressure–viscosity coefficients are greater for the Light PAO than for the POE, reinforcing the importance of the high-pressure behavior. The calculated pressure distributions are plotted at the bottom of Fig. 3. The shear stress in the vicinity of the pressure spike exceeds 5 MPa. These liquids should therefore shear-thin [8] in these regions, reducing the sharpness of the spike. Next, the behavior of entrapments formed by high viscosity fluids is evaluated. These lubricants, Mineral Oil and Heavy PAO,
are expected to be able to retain entrapments for longer times based on their relatively large low-pressure viscosities (0.383 and 0.752 Pa s, respectively) and pressure–viscosity coefficients (32.3 and 20.0 or 37.6 and 26.3 GPa1 depending on definition). This is indeed what is observed experimentally as illustrated in Fig. 4. The non-circular form of the entrapment is believed to result from small scratches on the ball surface. Although both lubricants retain entrapments for more than a minute, the Mineral Oil-based entrapment is thicker, larger in diameter, and persists for a much greater time than the Heavy PAO. This is in contradiction to predictions based upon the lubricant parameter [12], defined as the product of the ambient viscosity, m0, and the pressure viscosity coefficient. The lubricant parameter is listed in Table 2 using both definitions, Eqs. (5 and 6), of the pressure–viscosity
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coefficient. In both cases, the lubricant parameter is greater for the Heavy PAO than for the Mineral Oil. The EHL simulation was used to evaluate the effect of rheology with high viscosity fluids. The ability of a lubricant to sustain an entrapment was characterized as the entrapment width as a function of time. The entrapment widths measured from experiment and predicted by the simulation are shown in Fig. 5. For the experiment, the largest and smallest widths are recorded. This figure reveals two important things. First, the simulation correctly predicts the trend of decreasing width with increasing time and the difference between the Mineral Oil and Heavy PAO entrapment widths. These behaviors can be understood in terms of the model formulation: when the high-pressure viscosity of a fluid is large, the Poiseuille term in Eq. (1) becomes small, resulting in a correspondingly low value of dh/dt and longer entrapment retention. Second, to further probe the effect of highpressure rheology, a model lubricant was created having the same low-pressure behavior as the Heavy PAO, but an artificially low glass transition pressure so that the entrapped liquid shows pressure fragility similar to the Mineral Oil. This model lubricant is referred to as the Low pg PAO. The calculated entrapment behavior of the model Low pg PAO is significantly different from the Heavy PAO as shown in Fig. 5 and approaches the behavior of the Mineral Oil as time progresses.
4. Conclusions Entrapment appears to reduce starting friction. Experimental measurements of the time dependent film thickness in entrapped liquids, measurements of viscosity under pressure, and simulations using realistic pressure–viscosity models have contributed to improved understanding of the mechanisms of entrapment formation and persistence. The ambient viscosity and the pressure–viscosity coefficient affect the entrapment only as much as these are predictors of behavior at much higher pressure and
may result in predictions which contradict actual behavior. The fragility property of lubricants appears to be of overwhelming importance to entrapment.
Acknowledgment The authors were supported by the National Science Foundation under Grant number EEC#0540834. References [1] Glovnea RP, Spikes HA. The influence of lubricant upon EHD film behavior during sudden halting of motion. STLE Tribol Trans 2000;43:731–9. [2] Zhoa J, Sadeghi F. Analysis of EHL circular contact shut down. ASME J Tribol 2003;125:76–89. [3] Chang L. An efficient calculation of the load and coefficient of restitution of impact between two elastic bodies with a liquid lubricant. ASME J Appl Mech 1996;63:347–52. [4] Lee KM, Cheng HS. The pressure and deformation profiles between two normally approaching lubricated cylinders. ASME J Lubr Technol 1973;95: 308–17. [5] Barnocky G, Davis RH. The influence of pressure-dependent density and viscosity on the elastohydrodynamic collision and rebound of two spheres. J Fluid Mech 1989;209:501–19. [6] Ai X, Yu H. A full numerical solution for general transient elastohydrodynamic line contacts and its application. Wear 1988;121:143–59. [7] Liu Y, Wang Q, Wang W, Hu Y, Zhu D. Effects of differential scheme and mesh density on EHL film thickness in point contacts. ASME J Tribol 2006;128: 641–53. [8] Bair S. High-pressure rheology for quantitative elastohydrodynamics. Amsterdam: Elsevier Science; 2007. p. 111–2, 228, 163. [9] Bair S, Roland CM, Casalini R. Fragility and the dynamic crossover in lubricants. Proc Inst Mech Eng Part J, J Eng Tribol 2007;221:801–11. [10] Drozd-Rzoska A, Rzoska S, Roland CM, Imre AR. On the pressure evolution of dynamic properties of supercooled liquids. J Phys Condens Matter 2008;20: 244103. [11] Irving JB, Barlow AJ. An automatic high pressure viscometer. J Phys E 1971; 4:232–6. [12] Ohno N, Yamada S. Effect of high pressure rheology of lubricants upon entrapped oil film behavior at halting elastohydrodynamic lubrication. Proc Inst Mech Eng Part J, J Eng Tribol 2007;221:279–85.