Correlation between pressure-viscosity coefficient and traction coefficient of the base stocks in traction lubricants: A molecular dynamic approach

Accepted Manuscript Correlation between pressure-viscosity coefficient and traction coefficient of the base stocks in traction lubricants: A molecular dynamic approach Jie Lu, Q. Jane Wang, Ning Ren, Frances E. Lockwood PII:

S0301-679X(19)30076-3

DOI:

https://doi.org/10.1016/j.triboint.2019.02.013

Reference:

JTRI 5605

To appear in:

Tribology International

Received Date: 21 November 2018 Revised Date:

3 February 2019

Accepted Date: 8 February 2019

Please cite this article as: Lu J, Wang QJ, Ren N, Lockwood FE, Correlation between pressureviscosity coefficient and traction coefficient of the base stocks in traction lubricants: A molecular dynamic approach, Tribology International (2019), doi: https://doi.org/10.1016/j.triboint.2019.02.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Correlation between Pressure-Viscosity Coefficient and Traction Coefficient of the Base Stocks in Traction lubricants: A Molecular Dynamic Approach Jie Lu,1 Q. Jane Wang 1,*, Ning Ren 2, Frances E. Lockwood 2 1

Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208 2

Valvoline Inc, Lexington, KY 40509

Corresponding author, E-mail: [email protected]

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Abstract

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The lubricants for traction-continuously variable transmissions (t-CVT) could generate high traction under a high pressure, which requires the design of a special class of base stock molecules. This paper reports the development of a model, based on non-equilibrium molecular dynamics (NEMD) simulations, for estimating traction coefficients to facilitate the design of base stock molecules prior to their synthesis. The pressure-viscosity coefficients (α) of a number of base stocks in traction lubricants are calculated and the results are correlated with the corresponding traction coefficient τ. A linear α - τ correlation is obtained with a coefficient of determination of as high as 0.85.

1. Introduction

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Keyword: Traction lubricant, molecular dynamics, pressure viscosity coefficient.

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Continuously variable transmission (CVT) is a promising solution for powertrain technology in hybrid or small electrical cars [1–5]. Among different types of CVTs, the traction-CVT (t-CVT) are recently developed to transmit power through a glass-transitioned thin film between a roller and a disc [6]. In this application, a high pressure is applied in the contact zone to reduce slip and achieve a desired efficiency. The direct contact of the roller and the disk should be separated by a traction lubricant to avoid seizure and severe component wear. The typical traction lubricant for the t-CVT is the mixture of a base stock and various additives for viscosity index modification, and anti-wear protection, etc. On the contrary to a normal base stock used for gear or bearing lubrication, the base stock of a traction lubricant, to be referred as traction base stock in the following discussion, for t-CVT is expected to generate a high traction under a relatively high pressure [7–9], 0.6-2.5GPa, and a wide range of temperature [10,11], from room temperature to 140°C; it should render a 10 % higher torque capacity than does such a general-purpose base stock as polyalphaolefin (PAO) [8]. The most common type traction base stock has two cyclic carbon rings (normally it is a cyclohexane ring) connected by a short alkylene chain [12–14], but a recent discovery also reveals a different molecule called longifolene [15]. Representative molecular structures are

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shown in Figure 1(a), where a, b, n and m indicate the length of different functional groups. Generally, n is less than 4. Short methyl side chains are often grafted onto the ring or on the short alkylene chain to reduce the potential of self-rotation.

Figure 1. Formula of: (a) common traction base stock; (b) octane; (c) squalane; (d) cyclohexane.

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Special designs and optimizations of such unique structures have been conducted intensively, most of which were done by correlating the traction coefficient with relevant property/behaviour parameters, such as the melting point [16], pour point [17], solubility, alkylene chain length, rotational energy barrier [10,18,19], free volume [20], etc. However, the synthesis and characterization of traction base stocks can be costly. Moreover, even “pure” base stocks are still mixtures of different isomers, and it is difficult to determine the effects of individual components simply via experiment.

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The molecular dynamics (MD) simulation is a cost-effective alternative. It can be used to calculate lubricant properties at extreme conditions, for example, high shear rate, high pressure and low temperature that are difficult to be established in experiments [21,22]. A confined model under the EHL (elastohydrodynamic lubrication) configuration [23–27] were commonly used. Similarly, the traction coefficient τ can be studied by such confined model, as the ratio between the horizontal resistance force and the normal force. This is a direct approach to calculate τ, but certain restrictions are still applied. A high shear rate (>108 s-1) is needed in the MD simulation to overcome the thermal noise within a reasonable calculation time, which could be much higher than the normal shear rate applied in a realistic test (<106 s1 ). Moreover, a direct simulation can only consider a limited number of traction molecules, the confined film (~10nm) [28] thus modelled could be much thinner than a real EHL film (~100nm or thicker). If a realistic film thickness is considered [29], tremendous computational resources should be required, which is too slow to screen several traction molecules with potential. On the other hand, the MD simulation method could be applied indirectly if there exists a strong correlation between two properties. If one property can be calculated by MD, the other one can be acquired through correlation using an empirical equation [30]. In the EHL regime, pressure-viscosity coefficient α dominates the traction performance of a lubricating fluid. Muraki [31] showed that α values had a great impact on traction coefficient by means of a full EHL line contact model. Bair [32] measured the α values, in the pressure up to 1.4GPa, of several traction base stocks and found that α data could be ranked with respect to their traction coefficients. Generally speaking, it is more difficult to measure α than other lubricant properties, and most of the available data in literature were for the base stocks of linear

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molecules only. Many researchers have shown that α of several traditional base stocks, such as n-octane [33], 9-octylheptadecane [34], poly-α-olefin 4 [21], could be calculated via MD simulations, but very few of them dealt with traction molecules. It will be necessary to examine whether the pressure-viscosity coefficient α of a traction base stock can be reasonably estimated purely by the MD simulation, and if the calculated α values of several traction base stock molecules can be correlated with their traction coefficient. An accurate force field is needed for simulating the behaviours of traction base stock molecules. One can find several well established force fields from literature but they need to be validated first. Moreover, calculating the Newtonian viscosity at a high pressure could be challenging because of lubricant molecule shear-thinning. The shear rate applied in the MD simulation should be carefully selected for a reasonable balance between efficiency and accuracy.

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2. Targeted base traction fluids and simulation details

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The structures of several traction base stocks are shown in Table 1. The traction coefficient, τ, of c1 to c11 are acquired from literatures [10,13]. They have a similar molecular structure, e.g. two “bulky” sides (ring with 6 to 10 carbons) connected by a short carbon chain with less than 4 carbons. Molecule c11 is a recently developed base-stock traction molecule with enhanced performance. High purity sample c12 was purchased from Sigma-Aldrich, and its traction coefficient was measured by authors in the Valvoline laboratory. The traction coefficients of all 12 candidates range from 0.042 to 0.082 at 140°C under the maximum Hertzian contact pressure of 1.1GPa.

τ (T=140 °C, Pmax=1.1GPa)

c7 0.073

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c1-c10 are selected from reference 13

c1 0.055

c2 0.064

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Table 1. Traction base stock molecules and their measured traction coefficients.

c8 0.046

c3 0.069

c4 0.054

c5 0.082

c6 0.064

c9 0.05

c10 0.058

c11* 0.081

c12** 0.042

* τ data for c11 was the average from the values of different isomers in reference 9. ** τ data of c12 was measured in the Valvoline lab. The material was purchased from Sigma-Aldrich, tested under the same condition reported in reference 14 with a mini traction machine from PCS Instruments. The united atom transferable potentials for phase equilibria force field (UA-TraPPE) [35] was used to model the traction base stocks. This forcefield has been found to work well at describing the dynamics of linear and branched hydrocarbons, and it is also good at capturing the relative difference of their physical properties [33], although in general it may still

ACCEPTED MANUSCRIPT underestimate viscosity. The UA-TraPPE model is applied in such a way that each CH3, CH2, and CH group is simplified as a single pseudoatom. The forcefield parameters for the carbon rings are from cyclohexane [35] and those of the middle alkylene chain are from linear and branched hydrocarbon [36].

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In order to test the accuracy of the MD simulation model thus constructed, the densities of c10, c12, and 1,2-dicyclohexylethane were calculated under the isothermal–isobaric ensemble (NPT) at the equilibrium state for 20 ns via LAMMPS [37,38]. The MD results were compared in Figure 2 with the data from experimental measurements at the Valvoline laboratory (for c10, c12) and from reference [39] (for 1,2-dicyclohexylethane) at atmosphere pressure. One can see that the applied UA model is able to capture the temperature-density trend with minimal errors. Therefore, this model will be utilized to capture the viscositypressure trend.

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Figure 2. Simulated temperature-density trends using UA-TraPPE compared with experimental results (a) (b) from Valvoline laboratory and (c) from reference [39].

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In order to calculate the shear viscosity by means of the non-Equilibrium Molecular Dynamics (NEMD) simulation, the system containing 100 molecules was initially run to reach the equilibrium stage at the designated temperature and pressure. This equilibrium stage was set to be at 20ns (Figure 3(b)), which was sufficiently long for the small molecules (<20 pseudo atoms) considered in this work; it provided the initial state for the next shearing stage (Figure 3(c)). Next, the molecules underwent a planar Couette flow modelled using the SLLOD equations of motion [40,41] for 100ns. The Lees-Edwards periodic boundary conditions were applied, and the Nosé/Hoover thermostat [42,43] was coupled for achieving energy conservation of the system. This procedure was suggested by Tuckerman et al [44]. The viscosity is then calculated via Eq. (1), where σ xy is the average of the xy-component

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of the stress tensor, divided by the imposed shear rate, . The obtained viscosity data were averaged in the last 10-20ns after the system was stabilized. The timestep for the entire simulation is 1fs.

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Figure 3. A C10 molecule (a); Snapshots from equilibrium (b) and shearing (c) stage.

(1)

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3.

Simulation results and discussion

a)

Simulated α of two linear molecules

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The pressure-viscosity coefficient simulations started with two linear molecules, octane and squalane to further examine the accuracy of the UA model and the NEMD method. Their reference values in the pressure up to ~300MPa were found in literature [33,45]. In order to acquire α, Newtonian viscosity η at the selected pressure were calculated first.

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When pressure increases, fluid tends to shear thin more easily since the dynamics of liquid molecules are suppressed by the higher pressure. Thus, a proper shear rate needs to be selected, which should be 1) large enough to overcome the thermal noise of the molecules but 2) smaller than the critical shear rate to avoid shear-thinning. Here, the shear rate of 2x108 s-1 was used because Bair et al [46] showed that at P=316 MPa and T=99 °C, squalane (containing 30 carbons) did not shear thin. Therefore, if P<300MPa and T>100°C, it should not shear thin either. Moreover, a previous work [21] suggested that at P=400MPa and T=100°C, the critical shear rate of 1-decene trimer (which is a highly branched molecule containing 30 carbons) is about 2x108 s-1. This shear rate should be applicable to octane (8 carbons) because its thermal dynamic motion should be quicker than squalane’s.

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Figure 4 plots η vs. p for these two molecules and their reference results. The simulation result agree well with those from references [33,45] although both calculated curves are lower than the experimental measurement results. The η-p trend is well captured, so that the α values fitted from the exponential form , = ,0 are very close to each other, with about 10% relative error compared to the experimental value. This equation is selected based on its simplicity and functionality. It should be noted that fitting of α is closely related to the pressure range, and the fitted values are only effective in the selected datum range. Figure 4 shows that our results matches the literature, and that even though absolute η is underestimated as expected, the η-p trend can be well captured. o

Octane 75 C

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Squalane 140 C 100 This work Exp. Ref. 9.98p

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Figure 4. η vs. p for (a) octane at 75 °C and (b) squalane at 140 °C.

b) Simulated α of two cyclic molecules

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Next, cyclohexane and 1,1'-bicyclohexyl (c12) are examined to show how effective the UA-TraPPE model is to capture the η-p trends for the structures with cyclic rings. Again, the shear rate of 2x108 s-1 was applied in all cases because these two molecules have fewer number of carbon atoms and no branch, which makes them more difficult to shear thin as compared to octane and squalane. The values of viscosity for the pressure up to ~200MP at different temperatures were found in literatures [47–50].

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A cyclohexane molecule has a carbon ring with no branch, whose molecular length is much smaller than its linear alternative. Thus, its dynamic response should be faster than that of a linear molecule, which could result in a larger underestimation of viscosity η via the UATraPPE model. This limitation is also found for 1,1'-bicyclohexyl (c12) in literature [50] that compared the calculated η and experimental η side by side. A similar trend is found here since the simulated η was underestimated by a larger amount when the pressure was higher, as shown in Figure 6, the 100°C case. As expected, the α values in Figure 5 and Figure 6 are underestimated compared to the trend fitted from experimental data. The underestimation of α for cyclohexane is around 20%, which is smaller than that for 1,1'-bicyclohexyl with two rings included. The traction fluid molecules in Table 1 have a similar structure as 1,1'bicyclohexyl so that their calculated α should also be underestimated by largely the same amount (40%). Moreover, since the underestimation reduces when the temperature is higher, the following simulations conducted at 140°C can more or less increase the precision of the α prediction.

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50 C 10 This work Exp. Ref.

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Figure 5. η vs. p for cyclohexane at (a) 50 °C (experimental data is from Tanaka et al.[48]) and (b) 140 °C ((experimental data is from Rajagopal et al. [49]).

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Figure 6. η vs. p for c12 at (a) 40 °C, (b) 60°, (c) 80°C (experimental data is from Kuss et al. [47]) and (d) 100°C (experimental data is from Gordon et al. [50]).

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%=(αRef-αMD)/αRef

squalane octane cyclohexane 1,1'-bicyclohexyl

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Figure 7. Relative errors for four molecules, which slightly reduce with temperature.

c) Simulated α of selected traction molecules

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The α of candidates in Table 1 were then calculated at 140 °C in accordance to the experiment conditions [10]. The η- curve of c10 at pressure 600MPa is shown in Figure 8 to assist the careful selection of the applied shear rate. It shows that no shear-thinning occurs even at shear rate 2x108 s-1. Therefore, at a lower pressure, e.g. 400MPa, 2x108 s-1 should be sufficient to calculate the Newtonian viscosity.

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Figure 8(b) shows the η vs. t curves when the molecules are under constant shearing. All five cases gradually reach a constant value after 60ns. The η vs. p curves of all candidates were calculated and three representatives are shown in Figure 9 for clarity. Each curve is normalized by its η at the atmosphere pressure, so that the relative difference suggests how fast η increases with p. Each viscosity calculation was repeated for three times, and the error bars are included accordingly (If a bar is not clearly seen, it should be within the symbol size.). One can clearly see from Figure 9 that, τ of the three representatives are correctly ranked by the trends of the curves, which confirms that a correlation between α and τ exists.

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Figure 8. (a) η- of c10 at 600MPa. No shear-thinning is observed up to 2x108 s-1; (b) η vs. t of c10 under different pressures. A 100ns shearing time is sufficiently long to reach the equilibrium.

η (cP)

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Figure 9. Normalized η vs. p for molecule c1, c2, and c12.

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The α value of each candidate is then fitted to = . Fitted data with 95% confidence intervals (CI) is shown in Table 2, and α' is used here to indicate that it is a numerically obtained pressure-viscosity coefficient. The coefficient of determination (R2) ranges from 0.97 to 0.99, indicating a satisfactory fit of the η-p relationship. More η-p points can be obtained to further reduce the width of the 95% CI and improve the precision of the modelled pressure-viscosity coefficient, α'. Here, considering the balance between calculation time and accuracy, five η-p datum points for each candidate (12 in total) can well reflect the trend, while a single datum point was repeated for three times. Table 2. Calculated pressure-viscosity coefficient α' for all traction fluid candidates. The numbers in parentheses are the width of 95% confidence interval.

c1

c2

c3

c4

α' ±95% CI

7.36 (0.368)

8.05 (0.41)

9.58 (0.46)

8.07 (1.051)

11.50 (0.39)

9.33 (0.262)

c7

c8

c9

c10

c11

c12

10.45 (0.15)

6.75 (0.72)

9.94 (0.737)

4.98 (0.506)

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α' ±95% CI

8.05 (0.713)

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7.23 (0.968)

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Finally, the α'-τ relationship is given in Figure 10 for calculated α' and experimentally measured τ. The measured τ of all 12 candidates are regressed with their α' values, and the R2 value of 0.85 indicates a strong linear trend. Considering the fact that the measured τ data were from real traction base stocks that most likely are mixtures of isomers, which may even contain light impurities left from chemical synthesis, and that the MD modelling involves only pure molecule, this correlation should be fairly accurate. The slope, k, is 0.0067, which is close to k=0.0051 from Bair’s results 25 (fitted results by author from a pool of base stocks) despite of the temperature and chemical differences. This is another evidence that shows the current estimation is reasonable. The p-value of slope k, acquired from the linear regression hypothesis test, is about 1.7x10-5, which means that the correlation between α' and τ is statistically significant. In fact, the α'-τ relationship can be well explained by the molecular structure of the traction fluid molecule [10,18,19], such as chain length, quantity and position of the side chains, and rotation energy barrier. When shear is applied, the stearic hindrance between two bulky rings should increase the in-layer resistance, the so-called “interlocking.” However, if a traction molecule starts to rotate by itself, e.g. the relative out-of-plane rotation of one ring with respect to the other, interlocking will be broken, thus intermolecular slip may happen. Since a longer linear carbon chain can bend more easily, a shorter carbon chain between two carbon rings can lead to reduction of self-rotation, which contributes to a higher τ. This is the reason why most of the traction molecules shown in Table 1 have up to 4 carbon bonds between the two rings. If there are side chains grafted on to the short linkages, it is more difficult for the traction fluid molecule to rotate by itself because the short-term repulsive

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α'-τ

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Figure 10. Calculated α' vs. measured τ for candidates from Table 1 at 140 °C. A relatively high R2 of 0.85 is achieved.

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4. Conclusions

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The linear equation given in Figure. 9 suggests that one can use the UA-TraPPE model to calculate the α' of a potential traction fluid molecule at 140 °C and then obtain an estimated τ. On the other hand, because α' is underestimated, one can precisely measure the real α of a traction base stock and use the simulated slope k to reproduce such a line. In all, the above linear equation can be useful for designing new traction fluid molecule because it finds a way to predict traction coefficient.

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This work has developed a molecular dynamics (MD) method to build the relationship between pressure-viscosity coefficient α' and traction τ. The relationship of α'-τ for 12 traction molecules has been obtained using the UA-TraPPE model via a NEMD simulation with a relatively high confidence of R2 of 0.85. The results indicate that this MD method can correctly capture the relationship between viscosity η and pressure p for linear molecules like octane and squalane. For the traction molecules with 1 to 2 carbon rings, it can still well capture the relative difference of traction coefficient from a molecular structure point of view, although the calculated absolute value of α is lower than the experimental results in the higher-pressure regime. This suggests a promising method to estimate the trend of the traction coefficient variation with molecular structures without conducting high-pressure viscosity measurement experiment. For the traction fluid molecules with similar structures, i.e. two or more rings connected by a short carbon chain, the linear equation τ =0.0067α+d obtained from correlating

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Acknowledgments

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The authors would like to express sincere gratitude to the financial support from Valvoline Inc, and the generosity of providing the test materials and data. The calculations were supported by the computational resources provided for the Quest high-performance computing facility at Northwestern University that is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.

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Highlights: This paper reports the novel work on

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1. A molecular dynamics method developed to capture the relationship between viscosity η and pressure p for the traction molecules with 1 to 2 carbon rings; 2. The relationship of pressure-viscosity coefficient and shear stress, α'-τ, for 12 traction fluid molecules, via a NEMD simulation, with a relatively high confidence (R2 ~ 0.85); 3. A simple formula, τ =0.0067α+d, obtained from correlating the calculated α' and experimentally measured τ for designing new traction fluid molecule prior to its synthesis.

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