Using JCP format

JOURNAL OF CHEMICAL PHYSICS

VOLUME 114, NUMBER 15

15 APRIL 2001

Nonequilibrium molecular dynamics simulations of confined fluids in contact with the bulk Luzheng Zhang, Ramkumar Balasundaram,a) and Stevin H. Gehrke Department of Chemical Engineering, Kansas State University, Manhattan, Kansas 66506

Shaoyi Jiangb) Department of Chemical Engineering, University of Washington, Seattle, Washington 98195

共Received 28 July 2000; accepted 5 February 2001兲 A nonequilibrium molecular dynamics 共MD兲 simulation study is reported of the structural and rheological properties of confined n-decane between two Au共111兲 surfaces in contact with its bulk under constant normal loads or constant heights. In the constant-load MD simulations, it was observed that fluid molecules were squeezed out of the pore continuously in a single simulation upon compression, whereas fluid molecules in the bulk were soaked into the pore when applied normal load was released. Pore separation depends on bulk pressure under the same normal load and approaches a steady value as normal load increases. In the constant-height simulations, density, velocity, and orientational profiles of the confined film were accumulated along the Z 共perpendicular to the walls兲 and Y 共parallel to the walls and finite due to the bulk兲 directions. These distributions are not uniform not only along the Z direction but also along the Y direction, particularly for weak fluid–wall interactions. The shear-thinning behavior and ‘‘slip’’ boundary conditions were also studied in this work. Even though the shear-thinning behavior was reported by several previous studies, the number of particles was fixed and the bulk condition was unknown in those simulations. The simulation geometry employed in this work is closer to that of surface-force apparatus 共SFA兲 experiments and of engine lubricating systems where confined liquid is in contact with its bulk. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1359179兴

I. INTRODUCTION

recent years due to the development of faster computers and newer experimental techniques such as SFA. Many MD simulations have been performed for confined fluids between two planes in the effort to mimic SFA experiments.7–22 However, direct comparison between simulations and experiments is still difficult. One of the most noticeable issues is the inappropriate configuration of SFA experiments used in MD simulations. All previous MD simulations of SFA experiments except the studies11,24 were carried out under the condition of a constant number of particles. However, the setup of SFA experiments is that of a confined liquid in contact with a surrounding bulk fluid. Furthermore, the confined lubricant film in a car engine is also in contact with its surrounding bulk at extremely high pressures. So far, how the bulk pressure affects the properties of confined films in engine lubricating systems is unknown. Therefore, it is desirable to conduct simulations in an ensemble when confined fluids are in contact with the bulk. Recently, Landman and co-workers24 calculated equilibrium properties of confined thin films in contact with the bulk when pore separation was fixed. Their method combines constant pressure MD with a computational cell containing both bulk and confined fluid regions, allowing systematic investigations of confined liquids in contact with the bulk, and alleviating difficulties associated with the insertion of molecules as in the conventional grand canonical ensemble Monte Carlo 共GCMC兲 method. Earlier, van Swol and co-workers11 carried out a study of confined fluids under shear with grand canonical ensemble MD 共by insertion/deletion of particles兲 simulations

Understanding the atomic processes occurring at the interface of two dry or wet materials when they are brought together or moved with respect to one another is central to many technological problems, including adhesion, lubrication, friction, and wear.1–6 Although our understanding of static interfaces has advanced considerably, very little is known about dynamic interfaces at the molecular level. The classical physics of the continuum has historically provided most of the theoretical and computational tools for engineers. However, technology is now reaching to nanoscale dimensions where the continuum picture is no longer valid. Modern molecular simulation and microscopic experimental techniques provide excellent opportunities to tackle these problems. On the theoretical front, several molecular dynamics 共MD兲 simulations have been carried out to study properties of thin films under shear.7–22 On the experimental front, the surface-force apparatus 共SFA兲 is commonly employed to study both static and dynamic properties of molecularly thin films sandwiched between two molecularly smooth surfaces.23 The rheological behavior of fluids in highly confined geometries is a subject of immense technological importance. Our knowledge of such thin films has increased greatly in a兲

Present address: Molecular Simulations Inc., 9685 Scranton Road, San Diego, CA 92121. b兲 Author to whom correspondence should be addressed. Electronic mail: [email protected] 0021-9606/2001/114(15)/6869/9/$18.00

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© 2001 American Institute of Physics

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J. Chem. Phys., Vol. 114, No. 15, 15 April 2001

Zhang, Balasundaram, and Jiang

TABLE I. Intramolecular interaction parameters. Bond length K ␪ 共bending兲 ␪0 V0 V1 V2 V3 V4 V5

TABLE II. Lennard-Jones potential parameters. 1.545 Å 62545.1 K/radian2 114.6° 1037.76 K 2426.07 K 81.64 K ⫺3129.46 K ⫺163.28 K ⫺252.73 K

in which fluid particles between two infinite plates were allowed to interact stochastically with the bulk at a given chemical potential. The method was later developed into the dual control volume grand canonical ensemble MD 共DCVGCMD兲 simulation technique by Heffelfinger and co-workers25 for the studies of gradient diffusion of mixtures through a membrane. In DCV-GCMD simulations of binary mixtures, instead of inserting particles of both components into the control volumes, only one component was inserted while keeping bulk fluids in the control volume at constant pressure.26 Such a method is reduced to the one proposed by Landman and co-workers24 if applied to a pure component. Alternatively, Monte Carlo simulations were carried out in a grand isostress27 or isoforce28 ensemble in order to mimic SFA experiments. However, the dynamics response of confined liquid films cannot be obtained from MC simulations. In this work, nonequilibrium MD simulations were carried out to study various properties 共including ‘‘squeezed out’’ and ‘‘soaked in’’ phenomena in a single simulation兲 of confined chain molecules in contact with the bulk under shear. Simulations were performed under constant number of particles, temperature, bulk pressure, and normal load 共or height兲. In Sec. II, we describe potential models used for chain molecules and walls, and the implementation of the constant-load or constant-height nonequilibrium MD method for fluid particles in both confined and bulk regions. Results and discussion on confined n-decane (C10H22) are given in Sec. III. The work is summarized in Sec. IV.

Group

␴共Å兲

␧/k 共K兲

Mass 共u.m.a.兲

CH3 CH2 Au atom

3.905 3.905 2.655

59.38 88.06 529.28

15.0 14.0 197.0

count between nearest neighbors 共1–2 interactions兲 and nextnearest neighbors 共1–3 interactions兲. The masses and Lennard-Jones 共LJ兲 interaction parameters for different groups are given in Table II. The cross parameters for the interactions between different groups are calculated using the Lorentz–Berthelot combination rules. The walls are modeled using the explicit gold atoms forming two 共111兲 planes. To maintain a well-defined solid structure with a minimum number of solid atoms, each wall atom was attached to a lattice site with a spring.30 The spring constant controls the thermal roughness of the wall and its responsiveness to the fluid. It is adjusted so that the meansquared displacement 共MSD兲 about the lattice sites is less than the Lindemann criterion for melting.31 However, we found that the dynamics wall did not significantly affect the simulation results such as density profiles, velocity profiles, shear viscosity, etc. To reduce the computation time, we treat wall atoms as static in all MD simulations. The interactions between wall atoms and fluids are described by the LJ potential. The parameters of the LJ potential for the Au 共111兲 wall atoms are also listed in Table II. The solid walls consist of a total of 1728 gold atoms forming an Au 共111兲 crystalline structure with the lattice constant of 2.881 Å. B. Simulation details

An illustration of the simulation box is shown in Fig. 1. The origin is located at the center of the box. Fluid 共ndecane兲 molecules are confined between the two walls. Top and bottom walls have opposite shear direction and the shear direction is along the X axis. Two control volumes containing bulk fluids are placed on both sides of the confined region along the Y axis. Three layers of gold atoms are used for each of the walls placed along the Z axis. The solid walls are

II. MODELS AND SIMULATION DETAILS A. Models

The united atom 共UA兲 model29 with bond constraints is adopted to describe n-decane molecules. The bond angle potential is described by a cosine harmonic potential E angle⫽ 21 k ␪ 共 cos ␪ i ⫺cos ␪ 0 兲 2 .

共1兲

The torsional potential is represented by 5

E torsion⫽

兺

m⫽0

V m 共 cos ␾ 兲 m .

共2兲

The valence parameters of the force field used in this work are listed in Table I. The inter- and intramolecular nonbond interactions are described by the shifted-force potential truncated at r c ⫽2.5␴ f . The shifted-force potential is used to remove the discontinuities in the potential and force. The intramolecular nonbond interactions are not taken into ac-

FIG. 1. Illustration of the simulation box. Fluid particles are filled in both confined and bulk regions. Both walls are kept at either constant height or constant normal load while the bulk pressure is kept constant for all simulations. The top and bottom walls move in opposite directions along the X axis at constant velocity U * . Z and Y directions are finite due to confined walls and bulk fluids, respectively, while the X direction is infinite through the periodic boundary condition.



Confined fluids in contact with the bulk

infinite in the X direction and finite in the Y direction. The size of the solid walls in the Y direction was chosen so that fluid molecules in the middle of the wall behave in the same way as in the case of fluids confined between two infinitely large walls. Three-dimensional periodic boundary conditions were applied to the entire simulation cell except for confined fluids in the Z direction. The pore size was either kept constant in constant-height simulations or varied with the specified normal loads in constant-load simulations. During simulations, fluid molecules in both confined and bulk regions can exchange. When the system 共confined fluid ⫹ its bulk兲 reaches steady state, there is a flow along the X direction, but not the Y and Z directions if shear is applied along the X direction. Using this strategy one could overcome the difficulty of inserting or deleting molecules, especially for longer molecule chains.24,25 For those simulations with fluids confined between infinitely large plates, one does not know the state of the simulated systems exactly. Thus, simulations in the ensemble in which the confined fluid is in contact with its bulk are much closer to the situation of SFA experiments and of engine lubricating systems. To maintain the bulk pressure in MD 共both constant-load and constant-height兲 simulations, we assume that there is a virtual piston contacting with the bulk on each side of the simulation cell. We added the following equation of motion for the piston: y¨ piston⫽

共 P b ⫺ P 0 兲 A xz , Mb

共3兲

where y¨ piston is the acceleration of the piston, P b and P 0 are the current and target bulk pressures, respectively, A xz is the cross section of the simulation cell on the x – z plane, and M b is the mass of the piston, an adjustable parameter to maintain the bulk pressure. The oscillation of the bulk pressure will be large if the mass is large. By testing through simulations, the mass was set to 10.0 in u.m.a. It was shown that this value was good enough to maintain the bulk pressure for both constant-height and constant-load MD simulations. The uncertainty is less than ⫾1.0%. Based on the configuration of the fluids in both bulk regions, P b is determined by the equations given in Ref. 32. The y coordinates of bulk-phase molecules are rescaled at each time step to maintain constant bulk pressure as the y dimension of the cell is rescaled. The same scaling factors are used for the two bulk regions due to the symmetry of the simulation box. Since the scaling factors are applied only to the bulk regions and are quite small, they have negligible effect on the dynamics of confined fluids. The normal pressure P⬜ can be calculated by using either the method of plane 共MOP兲33 or the forces which are exerting on the walls. We have found that the calculations are the same when the plane in MOP is set to the position of the walls. To simplify, all the normal loads P⬜ in this work are calculated by the forces that each of the fluid particles exerts on the walls. Two additional terms (␧p zi and ␰ p zi ) are introduced into the equations of motion to maintain a constant normal load on the solid blocks and a constant system temperature, respectively. The basic equations are p˙ zi ⫽ f zi ⫺␧p zi ⫺ ␰ p zi ,

共4兲

z¨ ia ⫽

f zia ⫹ 共 ␧˙ ⫹␧ 2 兲 z i ⫺ ␰␯ zia , m ia

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共5兲

where f zi is the force exerted on the molecule i in the Z direction, p˙ zi the first derivative of the Z-component momentum p zi of the molecule i with respect to time, z¨ ia the Z-component acceleration of the segment a in the molecule i, and ␧ and ␰ the friction parameters that control normal load and temperature, respectively. The thermostat is applied to the Y and Z components of the velocities of the fluid particles. When fluid is under applied shear, work done on the system is converted into heat. Appropriate coupling to a thermal reservoir 共thermostat兲 is needed to remove the heat. Two approaches for shear flow simulations could be used: in one case, the sheared fluid is not thermostated and only the confining plates are maintained at a constant temperature, while in the other a thermostat is employed to keep the entire mass of the fluid at a constant temperature. Khare et al.17 showed that in the first case the sheared fluid undergoes significant viscous heating at shear rates studies, consistent with experimental observation. Our previous results19 showed that it is advantageous to thermostat the plates, particularly for large pore sizes, high shear rates, and strong wall–fluid interactions. However, our previous simulations19 along with others9,16 showed that velocity profile and shear viscosity obtained by thermostating fluids or surfaces are the same for small pore sizes such as the one used in this work. This was also confirmed in this work. The first derivative ␧˙ in Eq. 共5兲 is a link between the instantaneous normal load and target one, P⬜ and P N ␧˙ ⫽

共 P⬜ ⫺ P N 兲 V

␶ 2p kT

,

共6兲

where ␶ p is the relaxation time that determines the time evolution of ␧. The equations of motion of the methylene segments were solved by the velocity Verlet algorithm with bond constraints 共RATTLE兲34,35 using a time step of 1.4 femtoseconds 共fs兲. Padilla used a similar algorithm, but integrated the equations of motion using a leap-frog-like scheme.9 The initial configuration of the system with 338 n-decane molecules for the very first run was set up by the commercial software CERIUS2 共Molecular Simulations Inc.兲. For the rest of the simulations, the initial configurations were taken from the previous run. The temperature of all simulations was set to 481 K and the bulk pressure was set to 0.5 共in reduced unit兲 for constant-height simulations, 0.5 and 1.0 for constant-load simulations. For constant-height simulations, systems were equilibrated for 10 000 steps and properties were accumulated over another 10 000 configurations were discarded. Velocities, orientations, and positions of n-decane chains were recorded every 500 steps after the system reached steady state. The parameters used in this work were reduced as follows: length was reduced with ␴ f ( ␴ CH2), while energy with ␧ f (␧ CH2) and mass with m CH3 . The reduced length, temperature, pressure, and density, and time are L * ⫽L/ ␴ f , T * ⫽kT/␧ f , P * ⫽ P ␴ 3f /␧ f , ␳ * ⫽ ␳␴ 3f , and t * ⫽t(␧ f /m ␴ 2f ) 1/2, respectively.


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III. RESULTS AND DISCUSSION

MD simulations were previously carried out to study confined fluids in contact with the bulk.24 However, only equilibrium properties such as density profiles were studied under constant heights. In all previous simulation studies of rheological properties the number of confined fluid particles was fixed. Here, we present both constant-load and constantheight MD simulation studies of various properties such as density profile, velocity profile, orientation, and shear viscosity of n-decane confined between two Au 共111兲 surfaces under shear at 481 K. A. Constant-load MD simulations

In SFA experiments, when normal load increases, molecules in the confined region will be squeezed out and the number of the fluid layers inside the pore will decrease.36,37 As mentioned in Sec. I, our simulation geometry is much closer to the SFA experiments. In this work we performed constant-load MD simulations to mimic the ‘‘squeezed-out’’ phenomenon in SFA experiments. For a constant-load simulation, normal load and bulk pressure are fixed, whereas pore separation is allowed to oscillate to keep the normal load constant. When the load of the system is lower than that of the target value, pore separation will be reduced and fluid particles confined between walls will be squeezed out of the pore region into the bulk 共or the number of layers of the confined films is decreased兲. The virtual pistons will move out to release the bulk pressure so that the bulk pressure will remain constant. When the system reaches steady state, the normal load will reach its target value. One example of the squeezed-out phenomenon is shown in Fig. 2 from our simulation. At the beginning of the simulation, there are seven layers of fluid particles inside the pore along the Z direction 共pore separation is 31.24 Å兲. During the course of MD simulation, molecular chains are squeezed out of the confined region into the bulk. The number of fluid layers decreases while the bulk region becomes larger. The number of fluid layers reaches two at t⫽33.6 ps and stays at two layers after a long simulation run. The two snapshots at t⫽33.6 and 42.0 ps are similar, indicating that the system approaches steady state. Similar squeezed-out phenomena have been observed in SFA experiments by both Tonck et al.36 and Kelin et al.37 for various molecular chains. To the best of our knowledge, this is the first time that a squeezed-out phenomenon is observed continuously in a single MD simulation run. When the normal load was decreased at the final state, as shown in Fig. 2共f兲, fluid molecules in the bulk regions were ‘‘soaked in’’ the pore. A sequence of snapshots for the case in which the applied normal load was reduced to 2.0 is shown in Fig. 3. It is interesting to see from this figure that the pore separation reached 17.97 Å, instead of the original value of 31.24 Å. This is because normal load oscillates as pore separation increases 共solvation force vs distance curve兲 and a normal load corresponds to several pore sizes. Thus, the pore separation does not necessarily resume to its original value as the normal load is decreased. In fact, the process of squeezed out and soaked in is not necessarily reversible. However, by perturbing the system 共e.g., by varying bulk pressure兲, the system could overcome the barrier and return to its original pore

FIG. 2. Snapshots from a constant-load MD simulation of n-decane confined between Au 共111兲 surfaces at T⫽481 K, P⬜* ⫽10.0, U * ⫽2.0, and P b* ⫽1.0. Fluid particles are squeezed out of the confined region and the number of layers inside the pore decreases upon compression.

size. For example, for the case we discussed above, by running simulation at higher bulk pressure 共e.g., 1.3 in reduced unit兲 first and then at its normal bulk pressure 共i.e., 1.0兲 the pore separation became 31.34 Å, which is very close to 31.24 Å at the initial state before compression. To study how pore separation is affected by normal load and bulk pressure, two sets of constant-load MD simulations at different bulk pressures ( P b* ⫽0.5 and 1.0兲 were carried out. For each set of simulations, pore separation is calculated at a series of normal loads and plotted in Fig. 4 as a function of normal load. In all these simulations, the reduced density in the bulk is 0.1404, very close to the experimental value of 0.1439 at 481 K. As shown in Fig. 4, pore separation reaches a constant value at high loads for both cases. This findings is consistent to that observed in SFA experiments,36,38 in which two substrates did not have direct contact and some molecule chains 共either polyisobutenesuccinimide or polybutadiene兲 still remained inside the confined region even though a large load was applied on the substrates. At extremely high pressures when fluid particles are totally squeezed out, friction increases dramatically and damages to the surfaces will oc-




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FIG. 4. Pore separation versus normal load from constant-load MD simulations at two bulk pressures at T⫽481 K and U * ⫽2.0. Pore separation depends on bulk pressure under the same normal load and approaches a steady value as normal load increases.

FIG. 3. Snapshots from a constant-load MD simulation of n-decane confined between Au 共111兲 surfaces at T⫽481 K, U * ⫽2.0, and P * b ⫽1.0 when the applied normal load P⬜* of the system shown in Fig. 2共f兲 was decreased from 10.0 to 2.0. Fluid particles are soaked into the confined region from the bulk region.

cur. Furthermore, as shown in Fig. 4, separation is larger for higher bulk pressure at the same normal load. This is because more fluid particles in bulk regions are forced into the confined region at higher bulk pressure, leading to a larger pore size. Our simulation results suggests that how pore separation changes with normal load is affected by bulk pressure. Therefore, attention should be paid to the effect of bulk pressure on normal load 共or pore separation兲, especially for engine-related applications, in which bulk pressure could be very high.

two control volumes are placed along the Y direction in our simulations, the walls are no longer infinite along the Y direction. Thus, density profile is a function of both Z and Y directions. In this work, the density profile was computed by dividing the confined region into a number of small rectangles 共0.39 Å⫻1.95 Å兲 in the Z and Y directions. Our results shoed that density, velocity, and orientation profiles were quite symmetric. The averaged density over time and over the X direction for each area was calculated during simulations by assuming the symmetry of the density profile to improve statistics. A segment density profile along the Z and Y directions is shown in Fig. 6共a兲. Figure 6共b兲 gives segment density profiles in the Z direction at two particular Y positions in the middle (y * ⫽0.25) and near the edge (y * ⫽5.75) of the pore. The behavior of density distribution along the Z direction is similar to that when fluids are confined between infinitely large walls in both X and Y directions. The sharp peaks near both solid surfaces are clearly seen, indicating that the chains tend to wet the surfaces. Even near the ends of the finite Y direction the peaks in the contact layer are still very high due to strong fluid–wall interactions. Fluid density in the middle of the pore along the Z direction is close to that of the bulk. Figure 7 shows density profiles along the Y direction for three fluid layers (z * ⫽3.1– 3.2, 1.9–2.2, and 0.7–1.0兲. From Fig. 7, it can be seen that fluid density changes along the Y direction and reach a maximum in the middle of the pore. The similar density profiles of the

B. Constant-height MD simulations

A typical configuration of a constant-height MD simulation (H * ⫽8.0 and U * ⫽3.5) for the confined n-decane at 481 K is shown in Fig. 5. This figure clearly displays fluid layers between two walls. For hydrocarbon chain molecules, the density profile can be obtained based on the center of mass or the methylene segments. The density distribution of the methylene segments shows more structure than that of the center of mass of the chain molecules as a whole. Since

FIG. 5. A typical configuration of a constant-height MD simulation for n-decane confined between Au 共111兲 surfaces at T⫽481 K, H * ⫽8.0, U * ⫽3.5, and P * b ⫽0.5.


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FIG. 7. Segment density profiles of confined n-decane along the Y direction at three Z locations at T⫽481 K, H * ⫽8.0, U * ⫽3.5, and P * b ⫽0.5. The thick lines represent the average values.

The orientation order parameters of the chains in the pore along the Z and Y directions and along the Z direction at two Y locations (y * ⫽0.25 and 5.75兲 are given for H * ⫽8 and U * ⫽3.5 in Figs. 9共a兲 and 9共b兲, respectively. The order parameter is defined as S⫽

FIG. 6. Segment density profiles 共a兲 along the Y and Z directions and 共b兲 along the Z direction at two Y locations for confined n-decane at T⫽481 K, H * ⫽8.0, U * ⫽3.5, and P b* ⫽0.5. Density oscillates along the Z direction and is nonuniform along the Y direction.

contact layer 共first layer兲 under different wall strengths are shown in Fig. 8. For the weakest wall strength (␧⫽0.1␧ w) studied, it can clearly be seen that fluid density in the middle is quite different from that near the mouth of the pore. This difference indicates that the bulk does have an effect on density distribution along the Y direction, particularly for weaker wall strengths. It is expected that fluid density near the end of the pore along the Y direction will approach the bulk density if the wall strength continues to decrease. In nearly all simulations reported previously, thin films were confined between two infinitely large plates in both X and Y directions and layer-by-layer fluid structure between two walls were observed. The unique density behavior along the Y direction and the effect of bulk fluid on density distribution in the pore can only be seen and studied in the simulation geometry used in this work. Therefore, our simulation results suggest that density distribution of thin films in SFA experiments or of lubricants in engines is not uniform not only in the Z direction due to fluid–wall interactions, but also in the Y direction due to the existence of bulk fluid.

3 具 cos2 ␪ 典 ⫺1 , 2

共7兲

where 具 . . . 典 indicates that the ensemble average and ␪ is the angle between the chain-end-to-end vector and the Z axis perpendicular to the wall surfaces. By definition, S⫽⫺0.5 if molecules lie parallel to the surface and S⫽0 if molecules are randomly oriented. The orientation profile in Fig. 9共a兲 shows that chain molecules in the contact layer lie parallel to the surfaces while the orientation of the chains is more or less random in the inner layers. Figure 10 gives the top view of a molecular configuration in the first layer near the surface. There are three distinct regions—bulk, confined, and bulk regions. The methylene subunits in the confined region tend to form entire molecular chains, while the segments in the bulk are from different chains. Similar to density profile, the orientation of fluids in the middle of the Y direction is different from that near the edge of the pore, particularly for the inner layers along the Z direction as shown in Fig. 9共b兲.

FIG. 8. Segment density profiles of confined n-decane along the Y direction for the contact layer (z * ⫽3.0– 3.3) at T⫽481 K, H * ⫽8.0, U * ⫽3.5, and P b* ⫽0.5 with three different fluid–wall interaction strengths. The thick lines represent the average values.


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FIG. 12. Velocity profiles along the Y direction for two layers of confined n-decane at T⫽481 K, H * ⫽8.0, U * ⫽3.5, and P b* ⫽0.5.

calculated from experimental measurements of shear stress ( ␶ zx ) and the applied shear rate 共␥兲 by this equation:

␩ eff⫽

␶ zx ␶ zx ⫽ . ␥ U/H

共8兲

The applied shear rate is determined using the measured pore separation 共H兲 and assuming a linear profile between plates. Other simulations such as those by Robbins et al.7 used the similar definition of effective viscosity. The effective viscosity versus shear rate curve is given in Fig. 14, from which one can see a plateau at lower shear rates and shear-thinning behavior at higher shear rates. The exponent of the power law for this case is 0.55. Although the shear-thinning and slip behavior was studied extensively before, it should be pointed out that most of the previous simulations were carried out with constant number of particles in the confined region and the bulk condition is unknown. The results from this study are closer to those of SFA experiments and of engine lubricating systems where confined liquid is in contact with its bulk. IV. SUMMARY AND CONCLUSIONS

We carried out constant-load and constant-height nonequilibrium MD simulations of ultrathin film 共n-decane兲 confined between Au 共111兲 surfaces in contact with the bulk under shear. This simulation geometry is much closer to that

FIG. 13. Velocity profiles of confined n-decane films along the Z direction and averaged over Y at T⫽481 K and H * ⫽8.0 with three different shear velocities U * ⫽1.5, 2.5, and 3.5.

FIG. 14. Effective viscosity versus shear rate curve for confined n-decane at T⫽481 K, H * ⫽8.0, and P * b ⫽0.5.

of SFA experiments and of engine lubricating systems. In this work various properties of the confined film, including density profiles, orientational parameters, velocity profiles, and shear viscosity’s, were calculated and compared with those of the film confined by infinitely large walls 共in both X and Y directions兲. Several new results are obtained. For constant-load simulations, it was observed 共Fig. 2兲 in a single run of constant-load MD simulation that fluids were squeezed out of the pore continuously upon compression. Fluid particles were soaked into the pore from the bulk region when applied normal load was decreased 共Fig. 3兲. The process of squeezed out and soaked in is not necessarily reversible. Pore separation depends on bulk pressure at the same normal load and approaches a steady value at a sufficiently high load. Thus, attention should be paid to the effect of bulk pressure on normal load 共or pore separation兲, especially for engine-related applications, in which bulk pressure could be very high. For constant-height simulations under shear, density, velocity, and orientation profiles of confined film were accumulated along the Z and Y directions. These profiles along the Z direction are similar to those in infiniteplate simulations. However, these profiles exhibit nonuniform characteristics along the Y direction, particularly for weak fluid–wall interaction strengths 关Figs. 7, 8, 9共b兲, and 12兴. Slip was found in all constant-height simulations. The degree of slip increases as shear velocity increases 共Fig. 13兲. The calculated effective viscosity versus shear rate curve 共Fig. 14兲 shows the shear-thinning behavior of confined n-decane at high shear rates. Although the observed shearthinning and slip boundary conditions were reported previously in infinite-plate simulations, it should be pointed out that the number of particles was fixed and the bulk condition was unknown in those simulations. Our simulation results are closer to those of surface-force apparatus 共SFA兲 experiments and of engine lubricating systems where confined liquid is in contact with its bulk. As pointed out earlier, direct comparison between simulation and experiment is still difficult. One of the most noticeable issues is the inappropriate configuration of SFA experiments. Recent simulation work by Landman et al.24 on equilibrium properties and this work on rheological properties of confined films in contact with the bulk move one step closer towards simulations of SFA experiments and of en-



gine lubricating systems. As all other simulations, however, current MD simulations were carried out at a scanning rate several orders of magnitude higher than that of SFA experiments. Recently, we proposed a ‘‘hybrid’’ simulation method to extend the simulation time scale to that of atomic force microscopy 共AFM兲.39 The method combined a ‘‘dynamic element’’ model for the tip–cantilever system in AFM and a molecular dynamics relaxation approach for the molecular sample, such as self-assembled monolayers 共SAMs兲 on gold. The relaxation time of chain molecules could be quite different from that of SAMs, particularly in the nonNewtonian region. In future work, we will evaluate the possibility of applying the hybrid method originally developed for AFM to extend the time scale of MD simulations of SFA experiments by incorporating the relaxation time of the fluid molecules. ACKNOWLEDGMENTS

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