Molecular dynamics characterization of thin film viscosity for EHL simulation

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1 Tribology Letters, Vol. 21, No. 3, March 2006 (Ó 2006) 217 DOI: /s x Molecular dynamics characterization of thin film viscosity for EHL simulation A. Martini a, *, Y. Liu a, R.Q. Snurr b and Q.J. Wang a a Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA b Department of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208, USA Received 18 October 2005; accepted 10 January 2006; published online 30 April 2006 Molecular simulations were used to characterize changes in lubricant viscosity that may occur during thin film elastohydrodynamic lubrication (EHL). Molecular dynamics simulations were performed at variable wall speed and film thickness such that the effects of both parameters could be evaluated. Using this approach it was found that the viscosity of thin films under large shear is subject to both shear thinning and oscillation with film thickness. A composite model was developed that incorporated both effects. The expected impact that this model might have on an EHL interface was evaluated using a continuum simulation. An overall decrease in viscosity with some oscillation near the interface edges was predicted due to the molecularly modeled thin film effects. KEY WORDS: nanotribology, EHL with non-newtonian lubricants, viscosity 1. Introduction Many tribological applications operate some or all of the time in a regime where the lubricant film thickness is comparable to the fluid molecular size. This type of lubrication exists in micro and nano-scale engineering applications, such as magnetic storage devices and high performance gear systems. In addition, it may occur at asperity contacts, as well as during the starting or stopping of hydrodynamic lubricated interfaces. Design of applications that operate in the thin film elastohydrodynamic lubrication (EHL) regime is currently difficult because there are few proven formulas and experimental testing is still challenging [1 3]. Thin layers of lubricating film have been found to differ from their bulk counterparts in several ways. Viscosity is one of the most significant of these differences. Thin film lubricants are subject to large shear rates even under moderate operating speeds. Therefore, the effect of shear thinning is often significant. Many research efforts have focused on characterization of viscosity as a function of shear rate [4 7]. Typically, the shear rate is varied by modulating the wall speed and maintaining constant film thickness. However, the viscosity predicted by these models at a given shear rate may not be applicable to a different combination of film thickness and wall speed that correspond to the same shear rate. One reason for this *To whom correspondence should be addressed. a-martini@northwestern.edu is that viscosity has also been found to oscillate as a function of film thickness [8]. Therefore, constant film thickness studies cannot capture the oscillatory effect. This research considers the effects of both shear rate and film thickness, which enables inclusion of both the shear thinning and oscillatory behaviors in the resultant viscosity model. The viscosity characterization presented in this research is obtained using molecular simulation. Molecular simulation has been shown to be an effective tool for studying thin film lubrication [9,10]. Viscosity will be calculated from simulations run at a range of film thicknesses and wall speeds. A composite thin film viscosity model is developed from the simulation results that incorporate the effects of both shear thinning and oscillation. This model is directly applicable to a thin film EHL interaction in which both the wall speed and film thickness may vary. The expected effect that the composite thin film viscosity model would have on the EHL interface is evaluated. This analysis is performed using the film thickness and pressure distributions predicted by a continuum EHL simulation. 2. Molecular simulation details 2.1. Lubricant and wall models The lubricant modeled in this work was n-decane. This popularly modeled fluid was chosen so that results obtained in the initial phases of the research could be /06/ /0 Ó 2006 Springer Science+Business Media, Inc.

2 218 A. Martini et al./molecular dynamics characterization of thin film viscosity for EHL simulation compared to previously published results. The united atom (UA) model was used to model the n-decane molecules. The UA model combines each carbon atom with its bonded hydrogen atoms into a single composite atom in order to improve computational efficiency [11]. Flexibility of the fluid molecules was modeled in the simulation by bond bending and torsion potentials. The bond length was held constant. The bond bending was modeled by a cosine harmonic potential, U bending ¼ 1=2k h ðcos h i cos h equilibrium Þ; with constants of k h ¼ 518:816 kj/mol and h equilibrium ¼ 114 [12]. Torsional flexibility was modeled by a cosine expansion potential, U torsion ¼ P3 V n ðcos /Þ n ; where the expansion n¼0 constants are V 0 =8.395, V 0 =16.781, V 2 =1.134, and V 3 =) kj/mol [12]. The walls were modeled as gold atoms in a facecentered-cubic (fcc) lattice structure with an initial nearest neighbor distance of 0.29 nm. Each wall had four layers consisting of 512 gold atoms and was approximately 3.5 nm 2. Although both wall structure [13] and roughness [14] have been found to affect viscosity in MD simulations of Couette flow, these effects will not be considered here. An ideal lattice with consistent structural properties will be used in all cases to isolate the simulation results from the effect of changing these parameters. In order to maintain wall solidity during the simulation, the wall atoms were tethered to their fcc lattice sites by a spring potential [15,16]. U spring ¼ 1 2 kðr r latticeþ 2 ð1þ In this expression, k is the spring constant that controls how tightly the walls atoms are bound to their lattice sites. The spring constant is critical in this type of simulation because it must be small enough to allow sufficient thermal motion but large enough to maintain wall solidity. In this work, the optimal value of k was found to be 1 nm/ps 2. The non-bonded interactions between two fluid atoms in different molecules, two fluid atoms in the same molecule separated by at least three bonds, and between a fluid atom and a wall atom were modeled by the Lennard Jones (LJ) potential [17]. For each atom type, the atomic mass and the LJ interaction parameters are given in table 1. The potential parameters for interactions between different atom types were calculated using the Lorentz Berthelot mixing rules [11] Simulations All molecular simulations were performed with a modified version of the multipurpose simulation code (Music), which was developed in-house [18]. The distance between the upper and lower walls was varied from 0.6 to 3.0 nm in 0.1 nm increments. Wall separation was defined as the distance between atomic centers in the upper and lower walls. The initial placement of the fluid molecules between the semi-rigid walls was performed using a grand canonical Monte Carlo (GCMC) simulation. In the grand canonical ensemble the chemical potential, the volume, and the temperature are constant. The fixed volume was the area of the walls multiplied by the wall separation, the temperature was 300 K, and the chemical potential corresponded to atmospheric pressure. The GCMC simulation performs configuration biased insert, delete, cut and re-grow, and translate molecular moves until an equilibrium density and configuration is obtained [11,12,19]. A GCMC simulation was run for each wall separation and the resultant set of equilibrium fluid configurations was then used as input into molecular dynamics (MD) simulations. Thin film lubrication was simulated using non-equilibrium MD. A snapshot of the simulation cell as viewed from the y direction is shown in figure 1. The simulation cell is periodic in the x and y directions and the confining walls form the boundary in the z direction. In order to impose shear on the fluid, the upper and lower walls are moved in opposite directions at a constant velocity each timestep [2,16,20,21]. Three different wall speeds were evaluated: 1, 5, and 10 m/s. The simulations were run in the ensemble of constant number of particles, volume, and temperature. However, only the wall temperature was held constant while the fluid temperature was allowed to increase due to viscous heating. This commonly used approach was employed because it is analogous to the experimentally observed conduction of frictional heat out of a lubricating fluid through solid Table 1. Modeled atomic masses and interaction potential parameters. Group r (nm) e/k B (K) Mass (amu) CH 3 [12] CH 2 [12] Au [22] Figure 1. Snapshot of the simulation cell illustrating the coordinate axes, a typical configuration of the walls and fluid, and the direction of the constant wall velocity imposing shear on the fluid.

3 A. Martini et al./molecular dynamics characterization of thin film viscosity for EHL simulation 219 walls [2,15,16,22]. Simulating the effect of conduction is particularly important in this research because conduction is thought to be the primary mechanism for heat transfer out of a thin EHL film [23]. The temperature regulation in the walls was performed using a Nose Hoover thermostat [17]. Due to the effect of viscous heating, the steady-state temperature of the fluid increased to, and then fluctuated about, a value equal to or greater than the constant wall temperature. The magnitude of this increase was a function of the applied wall speed. For speeds of 1, 5, and 10 m/s, the average fluid temperature was 0, 6, and 18 K greater than the fixed wall temperature. The equations of motion were integrated using a 6th order Gear predictor corrector algorithm [17] with a timestep of ps. The simulation was considered to have obtained steady-state when the temperature of the fluid was equilibrated, and the rate of viscous heating in the fluid equaled the rate at which heat was removed from the walls by the thermostat [20] Viscosity calculation Viscosity was calculated from the MD simulation as the ratio of shear stress, s xz, to shear strain rate, _c. g MD xz ¼ hs xzi _c ð2þ The shear stress for each set of operating conditions was averaged over several simulations run to a total duration of between 1 and 2 ns (slower shear rates run longer to improve statistical accuracy). Researchers use several different methods for calculating shear stress from MD simulation. One of the most frequently used expressions is the average shear force of the lubricant atoms on the wall atoms divided by the area of the walls. that this assumption is not always applicable to thin films under shear. However, effective viscosity is still frequently used as a measure of the viscous behavior of thin films [3,20 22,26]. Use of effective viscosity is common not only because it is readily obtained from MD simulation, but because its calculation method is consistent with that used in viscosity measurements taken using a surface force apparatus [22]. Effective viscosity was used in this research for consistency with results published by other researchers, and so that the shear rate calculation in the molecular model was the same as that in the continuum EHL simulation. 3. Viscosity characterization 3.1. Molecular simulation results Simulations were run at film thicknesses from 0.6 to 3.0 nm with wall speeds of 1, 5, and 10 m/s. These operating conditions correspond to applied shear rates of between and /s. It is important to note that the shear rate is varied in this research by changing both the film thickness and the wall speed. The resultant viscosity is illustrated as a function of shear rate in figure 2 and of film thickness in figure 3. Two primary trends can be observed from these results. First, the viscosity decreases with increasing shear rate. And second, the viscosity oscillates as a function of film thickness. Decreasing viscosity with increasing shear, or shear thinning, is a well-researched phenomenon. However, oscillatory behavior is typically not observed in these studies. This may be attributed to the fact that studies of thin film behavior are often performed on cases where the film thickness is either not changed, or the difference between film thicknesses in consecutive test is much larger than the 0.1 nm used in this research. s xz ¼ XN Wall i XN Fluid j F ij x =A ð3þ This method of stress calculation has been found to yield the same results as the Method of Planes [24] if the plane is chosen to be at the position of the walls [22,25]. Another popular method is the Irving Kirkwood relationship. The Irving Kirkwood expression and that given in Equation 3 were found to yield similar results both in this research and others [21]. Therefore, to improve the accuracy of results, both methods were used and the results averaged. The shear strain, _c, was considered to be the applied shear rate (i.e. wall speed divided by wall separation). Viscosity calculated using the applied shear rate is often termed the effective viscosity. Effective viscosity assumes that the velocity of the fluid layer next to the walls is the same as that of the wall (i.e. no-slip). It has been found Figure 2. Molecular dynamics viscosity results at wall speeds of 1, 5, and 10 m/s as a function of shear rate.

4 220 A. Martini et al./molecular dynamics characterization of thin film viscosity for EHL simulation Figure 3. Molecular dynamics viscosity results at three different walls speeds as a function of film thickness. Exact data points connected by a smooth curve to illustrate oscillatory behavior. In either of these cases, the high frequency oscillation observed in figure 3 may not be observable. Both the shear thinning and oscillatory effects will be analyzed and then incorporated into a composite thin film viscosity model. The phenomenon of enhanced viscosity, or solidification, in very thin films was not observed in this research. Solidification has been observed both experimentally [27,28] and using MD simulation [9,10]. However, solidification is typically exhibited in unmoving or slowly moving fluids. Fluids under large, continuous shear, such as those studied in this research, are not expected to exhibit solidification [3,29] Shear thinning The phenomenon of shear thinning has been studied extensively both experimentally [4,30] and via molecular simulation [2,5 7,21,26]. Although the mechanism behind shear thinning on different length scales may not be the same, it has been found to follow consistent, length scale-independent behavior using the time temperature superposition principle [31]. Shear thinning is of particular interest in thin film lubrication because a very small film thickness may correspond to a very large shear rate even under moderate operating speeds. The shear rate above which shear thinning begins is known as the critical shear rate. The effect of shear rate on viscosity is often described using a power law relationship. One such model used frequently is the Carreau equation [5,30,32]. g ShearThinning ð_cþ ¼ g 0 ½1 þð_c=_c c Þ 2 Š n=2 ð4þ In this expression, g 0 is the viscosity of the fluid subject to zero or low shear, _c c is the critical shear rate, and n is the slope of the log _c log gð _cþ curve in the shear thinning region. In the present simulation, the fluid was allowed to heat up and therefore the effect of temperature must also be considered. Both the zero shear viscosity and the critical shear rate have been found to be functions of temperature [32]. Although an exact expression relating critical shear rate to temperature is not available, it has been observed that increasing temperature corresponds to increasing the critical shear rate [32]. The relationship between zero shear viscosity and temperature is better understood, and there are several empirical and theoretical expressions available. One of the most accurate of these is the Vogel equation [23]. g 0 b expðc=t dþ ð5þ In this expression, b, c, and d are fluid dependant, empirical constants. For bulk n-decane, experimental viscosity-temperature data [33] can be used to obtain b=0.03 cp, c=787 K, and d=69 K. The simulations reported here were run at three different wall speeds, which resulted in three different average fluid temperatures. The Vogel equation was used to calculate the corresponding zero shear viscosities of 0.92 cp at 300 K, 0.83 cp at 306 K, and 0.71 cp at 318 K. Since the fluid temperature was different at each wall speed, and both the zero shear viscosity and the critical shear rate are functions of that temperature, the Carreau shear thinning model must be fit to the simulation data at each wall speed independently. This approach was used to obtain an average value for the exponent, n, and to determine the relationship between critical shear rate and temperature. The resultant fit for each wall speed is illustrated in figure 4. It was found that the critical shear rate does in fact increase with temperature. The critical shear rate increased _c max C asymptotically towards ¼ 1: /s. The shear thinning exponent, n, has been found to be relatively constant by many different researchers using a variety of simulation and experimental techniques. In addition, it has been found to be independent of temperature [32]. Typically, its value is reported to be between 1/2 and 1. Larger values are found to correspond to extreme conditions such as high load, small film thickness, or large shear rate [3]. The average value calculated from the three wall speeds considered in the present work was 0.91±0.10. Other MD simulation studies of n-decane under wall imposed shear reported a value of 0.56 [20]. Differences in reported values for this exponent can be attributed to the effect of simulation parameters such as wall surface corrugation, wall fluid strength, and applied normal load. In addition, the shear rate was varied in the present research by changing the film thickness at constant wall speed. This differs

5 A. Martini et al./molecular dynamics characterization of thin film viscosity for EHL simulation Viscosity oscillation As illustrated in figure 3, viscosity was observed to oscillate as a function of film thickness where the frequency of oscillation was independent of wall speed. Viscosity has been found to oscillate with film thickness in thin films by researchers studying fluids in equilibrium [8]. In that research, the oscillatory viscosity behavior was partially validated using an analysis of solvation pressure. Solvation pressure arises from the force that acts between two walls with a very thin layer of fluid separating them. This force has been observed both experimentally [34,35] and via molecular simulation [1,8] to oscillate as a function of film thickness. The solvation pressure, p zz, is calculated from molecular simulation as the average force in the direction perpendicular to the walls divided by the wall area. The solvation pressure and viscosity as functions of film thickness from the 10 m/s wall speed simulation are illustrated in figure 5. It can be observed that the frequency of oscillation is approximately the same for the viscosity and solvation pressure. This suggests that the viscosity oscillation has a physical origin. The relationship between solvation pressure and viscosity illustrated for the 10 m/s wall speed case is representative of all three wall speeds. The relationship between film thickness and solvation pressure has been modeled using a sinusoidal expression [1]. p zz ðhþ ¼ A p e h=r cos 2ph ð6þ k In this expression, A p, r, and k are constants that can be fit from experimental or simulation data. Physically, these constants represent the amplitude, A p, the rate of decay, r, and the oscillation wavelength, k. Experimental research on octamethylcyclotetrasiloxane produced values for these constants as A p =172 MPa Figure 4. Viscosity as a function of shear rate at three wall speeds. Exact data points (hollow shapes) fit with the Carreau shear thinning model (solid line). Temperature-dependent zero shear viscosity (horizontal dashed lines) and critical shear rate (vertical arrow) indicated. from the approach used by other researchers in which the shear rate was modulated by varying the wall speed at constant film thickness [20]. Figure 5. Solvation pressure (solid diamonds) and viscosity (hollow triangles) as functions of film thickness. Consistent frequency of oscillation illustrated by shaded bars.

6 222 A. Martini et al./molecular dynamics characterization of thin film viscosity for EHL simulation and r=k=1 nm [1]. In this research on n-decane, these constants were found to be A p =325 MPa, r=1.45 nm, and k=0.37 nm. All of these constants were found by other researchers [35] and in the present work to be independent of temperature. The comparison between solvation pressure and viscosity in figure 5 indicates that the wavelength and decay rate of the viscosity and solvation pressure curves are approximately the same. Therefore, it is expected that the oscillation in viscosity can be described using a sinusoidal expression similar to that for solvation pressure where only the amplitude is different. This expression describing the oscillation of viscosity with film thickness is g Oscillation ðhþ ¼ A g e h=r cos 2ph ð7þ k Here, the constants r and k are the same as in the expression for solvation pressure (i.e. the decay rate and wavelength are the same). But the amplitude, A g, is different. The value of the viscosity amplitude was found to be A g =0.084 cp. The accuracy of this fit to the molecular simulation data will be evaluated in the next section as part of the analysis of the composite thin film viscosity model. Figure 6. Comparison of the simulation results (solid squares connected by dotted line) with viscosity predicted by the composite thin film viscosity model (solid line) The inaccuracy is due primarily to the low shear data. It was found here and in other research that shear stress (and therefore viscosity) calculations from non-equilibrium MD simulations are less accurate at lower shear rates [30] Composite viscosity model In the previous two sections, viscosity was characterized in terms of oscillation with film thickness and shear thinning separately. A composite viscosity model would contain contributions from both. A proposed expression of this form is g gð _c; h; TÞ ¼ 0 ðtþ ½1 þð_c= _c c ðtþþ 2 Š A ge h=r cos 2ph ð8þ n=2 k The values of the constants corresponding to the simulation parameters used in this research are summarized in table 2. Using Equation 8 with the constants reported in table 2, the current simulation results could be predicted as a function of shear rate and film thickness. The exact simulation data points are compared to the composite thin film viscosity model in figure 6. The overall RMS measure of fit accuracy is 4. Thin film EHL 4.1. EHL Simulation The viscosity model developed in the previous sections is directly applicable to EHL simulations in which both wall speed and film thickness may vary. The effect that the viscosity model would have on a thin film EHL interface was analyzed using a continuum EHL simulation. This simulation is based on the full numerical solution given by Zhu and Hu [36]. The simulation predicts interface pressure and film thickness by numerical solution of the Reynolds equation, an elasticity equation for the solids, and expressions relating fluid viscosity and density to pressure. The Barus and Dowson Higginson models are used to describe the pressure effect on viscosity and density, respectively [36]. The EHL case simulated was that of point contact between ideally smooth surfaces. The operating condi- Table 2. Composite thin film viscosity model constants, physical meaning, and either approximate value or function form. Constant Physical Meaning Value/Function Units g 0 Zero shear viscosity Vogel: b exp (c/(t)d)); b=0.03 cp, c=787 K, d=69 K cp _c c Critical shear rate Asymptotic: c c (c max c ); c max c = /s 1/s N Shear thinning exponent 0.91 A g Oscillation amplitude cp R Oscillation decay rate 1.29 nm k Oscillation wavelength 0.37 nm

7 A. Martini et al./molecular dynamics characterization of thin film viscosity for EHL simulation 223 Table 3. Summary of the EHL simulation operating conditions and material properties. Parameter Symbol Value Units Reduced modulus E 117 GPa Reduced radius R x 2 lm Mean sliding speed U 10 (0.1) m/s (nm/ps) Load W 0.1 mn Ambient temperature T K tions and material properties are summarized in table 3. This geometry can be viewed as an idealized single asperity interaction. The radius corresponds to the size of the smallest significant asperity and the applied load is the load on that asperity. The Hamrock Dowson [23] predicted average and minimum film thickness values for this case are 1.7 and 2.7 nm, respectively. These correspond to effective shear rates of )9 and )9 1/s. It was expected that the composite thin film viscosity model would have an effect for a contact with film thickness and shear rate of these orders of magnitude Interface area viscosity The EHL simulation was run initially without considering the effects of shear rate and film thickness on viscosity. Then, the wall speed and temperature input into the simulation, and the output film thickness distribution across the interface were substituted into the composite thin film viscosity model (Equation 8) in order to predict the corresponding viscosity change. The interface area film thickness and corresponding predicted change in viscosity due to thin film effects are illustrated in figure 7. The film thickness is normalized by the Hertz contact radius (137 nm) and the viscosity is normalized by the low shear, bulk viscosity of n-decane at 318 K (g 0 =0.71 cp). Analyses of the film thickness and viscosity contour plots indicate that the overall effects of shear rate and film thickness are to decrease viscosity in the interface area. The predicted viscosity change due to shear rate and film thickness can be evaluated using two-dimensional distributions across the interface centerlines. The film thickness and corresponding predicted change in viscosity along the x and y direction centerlines are illustrated in figure 8. The centerline viscosity distributions indicate that the most significant changes in viscosity due to shear rate and film thickness are expected to occur near the perimeter of the interface area. At these locations, both shear thinning (normalized viscosity less than one) and oscillation may occur. The largest predicted viscosity change was approximately a 50% decrease that was observed at the sides of the interface. However, the viscosity at the inlet area may be of more importance because it is this area that is critical in Figure 7. Interface film thickness normalized by the Hertzian contact radius (upper) and corresponding predicted change in viscosity due to shear rate and film thickness normalized by the bulk, low shear value (lower). The direction of motion, x, is from left to right. forming an EHL film [23]. Therefore, changes in viscosity due to shear rate and film thickness at the inlet may have a significant effect on the overall EHL film thickness. For the operation conditions and material properties considered here, the composite thin film viscosity model predicts a viscosity decrease of approximately 15% as well as oscillatory behavior at the inlet. It is expected that these changes, although small compared to the edge effects, will have the most significant impact on the EHL film Integrated simulation approach In the previous section, EHL film thickness was simulated without consideration of the effects of shear rate and film thickness. The next step is to integrate the composite thin film viscosity model presented in this

8 224 A. Martini et al./molecular dynamics characterization of thin film viscosity for EHL simulation Figure 8. X-direction (left) and Y-direction (right) interface area centerline distributions of normalized film thickness (solid line) and normalized predicted thin film viscosity (dashed line). paper into an EHL simulation. It is proposed that this be done with an approach similar to those typically used for developing non-newtonian EHL models. Non-Newtonian EHL models integrate the effect of shear thinning into a traditional EHL model in order to develop a modified Reynolds equation. The details for one such integration can be found in another publication [37]. Only a brief description of the approach will be presented here. First, the rheological model is extended to two-dimensional vector form. Then, linear shear forces are expressed along the film thickness based on force balance. Next, the expressions for shear flow are integrated in accordance with the rheological constitutive equation. Applying speed boundary conditions, shear forces at the central layer in two directions are determined. And finally new flow rate factors are calculated for the modified Reynolds equation. The composite (i.e. shear thinning and oscillation) viscosity model developed in this work will be integrated into the Reynolds equation using a similar approach. The primary difference is that the oscillatory behavior is a function of film thickness. Therefore, both film thickness and shear rate will have to be considered in the derivation of the modified Reynolds equation. 5. Conclusions Molecular simulation was used to characterize the change in viscosity that occurs in thin film lubricants. Simulations were performed at variable wall speed and film thickness such that the effects of both parameters could be evaluated. It was found that the viscosity of thin films is subject to both shear thinning and oscillation with film thickness. A composite model was developed that incorporated both effects. The expected impact that this model would have on an EHL interface was evaluated using a continuum simulation. For the EHL parameters considered here, an overall decrease in viscosity with some oscillation near the interface perimeter was predicted due to thin film effects. Due to the importance of the inlet area in formation of an EHL film, it is anticipated that the inlet area viscosity changes will have the most significant impact on an EHL film thickness calculation. To validate these predictions, a proposed approach for development of a modified Reynolds equation that captures the composite thin film viscosity effect was outlined. Acknowledgments The authors would like to express their sincere gratitude for the support of the US National Science Foundation IGERT Program, Office of Naval Research, and Department of Energy. This research was also supported in part by the National Science Foundation through TeraGrid resources provided by NCSA. References [1] M.F. Abd-AlSamich and H. Rahnejat, Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 215 (2001) [2] S.T. Cui, P.T. Cummings and H.D. Cochran, J. Chem. Phys. 111 (1999) [3] Y.Z. Hu and S. Granick, Tribol. Lett. 5 (1998) 81. [4] S. Bair, Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol. 215 (2001) 223. [5] L.I. Kioupis and E.J. Maginn, J. Phys. Chem. B 104 (2000) [6] C. McCabe, S. Cui, P.T. Cummings, P.A. Gordon and R.B. Saeger, J. Chem. Phys. (2001) [7] G. Luengo, J. Isrealachvili and S. Granick, Wear 200 (1996) 328. [8] J.C. Wang and K.A. Fichthorn, Colloid. Surf. 206 (2002) 267. [9] Y.Z. Hu, H. Wang, Y. Gao and L.Q. Zheng, Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol. 216 (1998) 165. [10] Y.R. Jeng, C.C. Chen and S.H. Shyu, Tribol. Lett. 15 (2003) 293. [11] A.R. Leach, Molecular Modeling Principles and Applications 2nd ed. (Prentice Hall, Harlow, 2001). [12] M.D. Macedonia and E.J. Maginn, Mol. Phys. 96 (1999) [13] A. Jabbarzadeh, J.D. Atkinson and R.I. Tanner, J. Chem. Phys. 110 (1999) 2612.

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