Effect of Surface Stiffness on the Friction of Sliding Model

12032

J. Phys. Chem. B 2001, 105, 12032-12045

Effect of Surface Stiffness on the Friction of Sliding Model Hydroxylated r-Alumina Surfaces David J. Mann, Lijuan Zhong, and William L. Hase* Department of Chemistry and Institute for Scientific Computing, Wayne State UniVersity, Detroit, Michigan 48202-3489 ReceiVed: February 28, 2001; In Final Form: May 7, 2001

Molecular dynamics simulations were performed to determine how interfacial properties and friction forces of two model hydroxylated R-alumina surfaces are affected by softening or stiffening the surface potential, without changing the surface-surface intermolecular potential Vinter. The surface-surface vibrational spectrum for the soft surface, in absence of applied load, is influenced by coupling between Vinter and the surface vibrations. For the stiff surface, this coupling becomes important as the applied load is increased. Friction forces for the stiff and soft surfaces are similar at low loads, but at high loads the stiff surface has higher friction forces. This result mirrors the surface-surface vibrational spectra for the soft and stiff surfaces versus load. Increasing the load forces both systems into potential energy wells, requiring additional energy (or external force) to surmount potential energy barriers for sliding, leading to enhanced friction. At high sliding velocity, the instantaneous barriers and friction forces for sliding do not mirror the periodicity of the optimized interfacial potential. As the sliding velocity is decreased, there is a longer time available for structural relaxations and energy transfer between the sliding coordinate and surface modes. As a result, the friction force versus sliding distance begins to exhibit the periodicity of the optimized interfacial potential. Knowledge of the optimized potential along the sliding coordinate (analogous to a reaction path potential) may assist in understanding tribological behavior under applied load and interfacial sliding.

I. Introduction Surface-surface friction arises from lateral interactions between two surfaces, the magnitude of which depends in part upon the extent of vibrational energy relaxation.1-3 As seen from numerous theoretical calculations,4-19 the extent of this relaxation and, hence, the magnitude of the friction force, depends on the sliding velocity as well as the applied load. These conditions influence the regions of the surface-surface intermolecular potential visited by the sliding surfaces and, thus, affect relaxation processes at the interface. Fast moving surfaces are often unable to fully relax into deep potential energy minima, since intermolecular relaxation rates may be small in comparison to the sliding velocity.4-7 When this sliding velocity is low enough, such that the system is capable of sampling a large portion of the local intermolecular potential energy surface, maximum friction may be attained. Accordingly, the friction forces encountered in low sliding velocity simulations are typically much larger than those encountered in high sliding velocity simulations. The rate of vibrational energy relaxation from the surfacesurface interface during sliding depends on the intermolecular vibrational frequencies of the interface, the surface phonons, and the couplings between these interfacial and surface modes.1,2,20-22 The objective of the molecular dynamics (MD) simulation reported here is to examine how varying the surface phonons affects friction and surface-surface interfacial relaxation. More specifically, this study addresses how frictional properties are affected when the vibrational frequencies of the surfaces are raised or lowered (i.e., stiffened or softened) and the surface-surface intermolecular potential is not changed. The

specific model system studied here consists of R-hydroxylated alumina surfaces, with their potentials adjusted to “stiffen” and “soften” their vibrational modes. Using the unadjusted surface potential, this system was investigated in a previous MD simulation23 and the calculated coefficient of friction is in good agreement with the experimental value.24 II. Simulation Method A. Potential Energy Function. The analytic potential energy function used for the work presented here is the same as that used for the previous simulation of sliding R-hydroxylated surfaces.23 This function was derived from available experimental25 and electronic structure data.26-28 An analytic function was first derived for R-alumina from ab initio calculations for an Al4O6 cluster.26 The parameters for the analytic function were chosen to fit the ab initio frequencies. The parameters used to model the O-H groups of the R-hydroxylated alumina surface were chosen to fit the experimental frequency for the O-H stretching mode.25 A sum of curvilinear internal coordinate harmonic stretch and bend potentials, with a Lennard-Jones nonbonded potential between the H atoms, are used to represent the potential energy of the R-hydroxylated alumina surfaces. The harmonic stretch function is represented by

1 V(rij) ) fr(rij - r0)2 2

(1)

where fr is the stretching force constant, rij is the distance between atoms i and j, and r0 is the equilibrium bond length.

10.1021/jp010759c CCC: $20.00 © 2001 American Chemical Society Published on Web 11/07/2001

Friction of Sliding Model Hydroxylated R-Alumina Surfaces TABLE 1: Parameters for the r-Hydroxylated Alumina Potential Energy Function force constant interaction stretcha

O-H Al-O stretcha H-O-Al bendb Al-O-Al bendb O-Al-O bendb

H- - -H LennardJonesc a

coordinate

normal

soft

stiff

r0 ) 0.9482 r0(1) ) 1.8524 r0(2) ) 1.9628 θ0(1) ) 122.732 θ0(2) ) 135.355 θ0(1) ) 93.661 θ0(2) ) 120.262 θ0(3) ) 132.275 θ0(1) ) 79.645 θ0(2) ) 86.339 θ0(3) ) 90.843 θ0(4) ) 101.143 σ ) 2.886

fr ) 7.98 fr ) 3.69

1.995 0.923

31.92 14.76

fθ ) 2.01

0.503

8.04

fθ ) 1.65

0.413

6.60

fθ ) 0.71

0.178

2.84

) 0.044

0.044

0.044

fr in mdyn/Å. b fθ in mdyn-Å/rad2. c in kcal/mol, σ in Å.

TABLE 2: Parameters for the Intermolecular r-Hydroxylated Alumina Potential

a

interaction

parameters

H- - -H H- - -O H- - -Al O- - -O O- - -Al Al- - -Al

) 0.0440, σ ) 2.8860 ) 0.0514, σ ) 3.1782 ) 0.1491, σ ) 3.6033 ) 0.1550, σ ) 3.5330 ) 0.1741, σ ) 3.9682 ) 0.5050, σ ) 4.4990

is in units of kcal/mol and σ is in units of Å.

Similarly, the harmonic bend function is represented by

1 V(θijk) ) fθ(θijk - θ0)2 2

(2)

where fθ is the bending force constant, θijk is the angle between atoms i, j, and k, and θ0 is the equilibrium bond angle. The Lennard-Jones (12-6) potential function is used to describe the nonbonded H-H interactions and is represented by

[( ) ( ) ]

σ V(rij) ) rij

12

σ -2 rij

6

(3)

where rij is the distance between atoms i and j, is the potential energy well depth, and σ is the equilibrium distance between atoms i and j. The parameters for this potential energy function are listed in Table 1. Stiff and soft surface models were constructed, for the work presented here, by increasing and decreasing the surface force constants by a factor of 4, respectively. These scaled force constants are also listed in Table 1. The H-H nonbonded potential was not scaled in forming the soft and stiff potentials. The unscaled potential gives a surface relaxation energy of 2.1 J/m2 for aluminum-terminated R-Al2O3(001) compared to high-level ab initio results of 1.34-1.9 J/m2.26 The intermolecular potential between the two R-hydroxylated alumina surfaces is a sum of Lennard-Jones (12-6) potentials, as in eq 3, for all possible nonbonded interatomic interactions. The parameters for these potentials are not scaled and are the same as those used previously.23 The parameters for the H-H, H-O, H-Al, and Al-Al intermolecular interactions are those for the UFF force field.29 The SPC model was used to represent the O-O interaction.30 These parameters are listed in Table 2. The above potential does not allow rupture of the Al-O and O-H bonds of the surface, and possible dissociation of an Al, O, or H atom, or larger moieties. This is not a serious limitation for the current simulation, since the interest is in how surface

J. Phys. Chem. B, Vol. 105, No. 48, 2001 12033 structural relaxation and changes in the rate of energy transfer, arising from stiffening or softening the surface, may affect friction. However, even if the potential was modified in a realistic way to allow loss of material from the surfaces, this may not actually occur given the simulation times; i.e., the longest time is 4 × 10-10 s (see below). The O-H and Al-O bond strengths are in the range of 400-500 kJ/mol,31 which are much higher than those for dissociations expected to occur on the time scale of the simulations. As discussed below, the largest interfacial temperature found in the simulations is 1243 K. If a unimolecular process, during sliding, is represented by the rate constant k ) υ exp(-Eo/kBT), with υ ∼1014 s-1,32 dissociations with Eo ) 400 kJ/mol have an average lifetime of 1/k ) 645 s at T ) 1243 K. For this temperature, dissociations with Eo ) 119 kJ/mol (28 kcal/mol) occur on a 10-9 s time scale. In the Results Section, it is shown that using a potential in which the Al-O and O-H bonds are represented by Morse functions gives the same result as found with the harmonic potential described above. B. Molecular Dynamics Model. The simulation model consists of two R-hydroxylated alumina surfaces brought into contact at the hydrogen layers and a schematic is given in Figure 1. Periodic boundary conditions (pbc’s), in the x,y-plane parallel to the interface,33 were employed for each surface. Since the unit-cell of an R-hydroxylated alumina surface, in this x,y-plane, is a rhombus with sides 4.74 Å long,34 a rhombohedral geometry, with 5 unit cells on each side (i.e., 5 × 4.74 Å), was employed for the model of the surface. The thickness of the surface model is 12 Å. The resulting primary cell contains 943 atoms, which comprise approximately 36 unit cells. Figure 1 correctly depicts the thickness of surface model and its width in the x-direction, which is the same as that in the y-direction. The top surface in Figure 1 is the sliding surface. To model constant load conditions, the center of mass of the top surface, Zcm, varies dynamically in response to an externally applied load, L, according to the equation4,7

MZ¨ cm ) Fzinter + L

(4)

where M is the mass of the top surface and Fzinter is the total intermolecular force between the two surfaces in the z-direction. To simulate the sliding motion, the upper surface is pulled in the positive x-direction at the shear velocity Vs. This is done by moving the top layer of Al atoms, of the upper surface, in the x-direction at the fixed velocity Vs. The bottom layer of Al atoms of the lower surface are held fixed, and not allowed to move in either the x-, y-, or z-directions. The friction force is the sum of the x-component of the intermolecular Lennard-Jones force between the j atoms of the upper surface and the k atoms of the lower surface, i.e.

-Fx )

∑j ∑k ∂V(rjk)/∂xj

(5)

The reported friction force is not normalized and is the total friction force for the simulation model. Similarly the applied load is not normalized. Equation (5) represents the instantaneous friction force between the two surfaces10 and its integration over the sliding distance does not give the energy acquired by the surfaces as a result of the sliding. The equilibrium coordinates for the two surfaces, before sliding, are shown in Figure 1. The two surfaces are symmetrically above and below each other; i.e., the configuration at which the repulsion between the H atoms of the two surfaces is a maximum and, thus, the attraction between the surfaces is the weakest. With the outer Al atoms of the two surfaces held

12034 J. Phys. Chem. B, Vol. 105, No. 48, 2001

Mann et al.

Figure 1. Schematic of the simulation model.

fixed, the system is then equilibrated35 until it attains a temperature of 300 K for the applied load. Due to the constant applied force, the system is not energy conserved. If the system is not coupled to a dissipative bath, the energy will increase with time and the system will “blow up.” The simplest approach, to overcome this problem, is to couple the entire system to a thermal bath, which will allow the energy added to the system, from the applied load, to dissipate. This model, however, will not accurately describe the nonuniform temperature gradient from the interface to the bulk encountered in a realistic tribology event. A more accurate approach involves coupling outer layer atoms of each surface to a thermal bath, which allows the interface to heat up while at the same time dissipating any built-up energy. In these simulations, the temperature of the oxygen atoms in the layer adjacent to the outer layer of Al atoms, for both the top and bottom surfaces, is kept at 300 K by the widely used36 Berendsen method.35 This is done by scaling the velocities of these atoms according to

Vnew ) Vold i i

() Td Tc

1/2

(6)

where Tc is the calculated temperature and Td is the desired

temperature. The former is determined from N

3NkBTc/2 )

2 m(Vold ∑ i ) /2 i)1

(7)

where N is the number of atoms. The sliding velocity Vs is removed from the x-component of Vold i . Since the dynamics associated with the friction may be different for the two surfaces, different Tc are determined for the upper and lower O atom layers. The trajectories were integrated with the velocity Verlet algorithm using an integration step-size of 0.5 fs. The velocities of the outermost O atom layers were scaled each time-step, as described by eq 6. Similar results were obtained by reducing the integration step-size by a factor of 2 and still scaling the O atom velocities every 0.5 fs, or selecting a different initial condition for the molecular dynamics trajectory. The friction properties calculated here, and given below, are also insensitive to whether this scaling was done each time step or after several time steps, as long as a temperature of 300 K was maintained for each of these O atom layers. These are the appropriate numerical tests for a molecular dynamics simulation.37,38 C. Potential Energy versus Sliding Coordinate. The friction associated with sliding the R-hydroxylated alumina surfaces is expected to depend in part on how the interaction potential

Friction of Sliding Model Hydroxylated R-Alumina Surfaces

J. Phys. Chem. B, Vol. 105, No. 48, 2001 12035

Figure 2. Intermolecular Vinter and total Vtotal potential energy curves for the top surface sliding across the bottom surface. In the upper two plots the surfaces are held rigid. The middle and lower plots are for the stiff and soft surface models. For these plots the positions of the atoms in the surfaces are optimized to find the minimum in Vtotal versus the x-direction sliding distance ds (Å).

between the two surfaces varies as the top surface slides across the bottom surface. Both the intermolecular Vinter and total potential Vtotal arising from the interaction between the sliding surfaces, are shown in Figure 2 for three models of the surfaces. For each of the models, the lower surface’s outer layer of Al atoms is rigid and does not move. The upper surface is pulled across the lower surface by moving the upper surface’s outer layer of Al atoms in the x-direction, with no movement in the y-direction and the relative positions of these atoms kept fixed.

This outer layer moves up and down in the z-direction to minimize Vtotal for each value of the x-direction sliding distance ds. For the potential energy curves of the rigid surface model, the relative positions of the surface atoms are held fixed in their equilibrium geometries, for the surfaces separated. The atoms in the lower surface do not move and those in the upper surface only move in the x and z directions. These rigid surfaces have a potential energy minimum of -303 kcal/mol and a barrier


Mann et al.

Figure 3. Intermolecular potential (left side) and friction force (right side) versus sliding distance for several possible situations: (a) periodic Vinter and no transfer of energy between the sliding coordinate and the surface modes; (b) periodic Vinter, with energy transfer from the sliding coordinate to the surface modes occurring at the same rate for the slide and slip steps; (c) nonperiodic Vinter, arising from structural changes at the surfacesurface interface, and no energy transfer between the sliding coordinate and the surface modes; and (d) same as (c), except there is energy exchange between the sliding coordinate and the surface modes, flowing in both directions.

for sliding of 77 kcal/mol. Since there are 25 unit cells on the surface of the primary cell (see Section II.B), per unit cell surface area the attraction between the rigid surfaces and barrier for sliding are -12.1 and 3.1 kcal/mol, respectively. The positions of the peaks in the barrier for sliding and the minima in the attractive potential reflect the 4.74 Å width of the unit cell. The small barrier for sliding, midway between the principal barriers, occurs when ds is increased by one-half of the unit cell width and the H atoms of the surfaces are close to, but not on top of each other. The middle and lower plots of Vinter and Vtotal in Figure 2 are for the stiff and soft surface models described above. For these plots, the positions of all the atoms in the upper and lower surfaces are optimized, except those for the outer layers of Al atoms, as the upper surface moves across the lower one. The

potential curves for the stiff surface model are very similar to those for the rigid model. This is a result of very little movement of the surface atoms from their equilibrium positions and negligible structural changes of the surface for the stiff model. Per unit cell, the potential energy minimum and barrier for sliding are -12.8 and 2.9 kcal/mol, respectively, for the stiff model. The potential energy curves for the soft surface model are significantly different from those for the rigid and stiff models. Though the potential energy is substantially more attractive for the soft surfaces, the barrier for sliding is much smaller. Per unit cell, these potential energies are -14.2 and 0.8 kcal/mol, respectively. The soft surface relaxes giving rise to a Vinter term significantly more negative Vtotal. The increased attraction from Vinter is greater than the increased positive potential arising from



Figure 4. Plots of the surface-surface vibrational spectrum, with no sliding, obtained by Fourier transforming the coordinate ∆Zcm. (a), (b), (c), and (d) are for the stiff surface with a load of 0, 1, 5, and 10 nN, respectively, and (e), (f), (g), and (h) are for the soft surface with the same respective loads.

the displacement of the surface atoms from their equilibrium positions. The increased flexibility of the soft surface makes its barrier for sliding lower. The difference in the potential energy curves of the stiff and soft surface systems is explained by the potential energy parameters in Table 2. The most attractive atomic interactions are Al- - -Al, Al- - -O, and O- - -O, whereas the weakest interactions are for H- - -H and H- - -O. In order for the system to feel the most attractive portion of the potential, the Al and O atoms in the outermost layers of the two surfaces must come close together. This implies that the H atoms must be able to relax, as well as the first layer of O atoms. For an optimized surface the O-H bonds are oriented between 20 and 30° from the surface normal. In order for this relaxation to occur the O-H bonds should be flexible enough to orient themselves further

from the surface normal making the H atom layer closer to the O and Al atom layers directly beneath. This relaxation occurs easily for the soft surface system, since the O-H stretch and Al-O-H bend force constants are very small requiring little energy to relax. For the stiff surface, significantly more energy is needed for the O-H bonds to orient themselves away from the surface normal enhancing surface relaxation. It is for this reason that the soft surface system is more attractive than the stiff surface system. Finally, it should be noted that the orientation of the sliding surfaces is expected to affect their friction force.39 For the simulation model used here the surfaces are identically oriented, have the same periodicity, and the upper surface is pulled so that its unit cell moves across that of the lower surface. The commensurability and periodicity of the interfacial potential will


Mann et al.

Figure 5. Snapshots of the interface of the R-hydroxylated alumina surfaces during sliding: (a) stiff surfaces; and (b) soft surfaces.

change if the upper surface is shifted or rotated from the orientation used here. If the upper surface was shifted in the y-direction, the periodicity arising from the upper unit cell moving across the lower unit cell would be lost. Rotating the upper surface, should give rise to an incommensurate interfacial potential that retains some periodicity, but is much different than the one in Figure 2.39 III. Dynamic Friction Preliminaries Before the simulation results for the sliding R-hydroxylated alumina surfaces are presented, it is of interest to consider the range of possible dynamic friction for this system versus load and fixed sliding velocity Vs. Some idealized possibilities of the surface-surface intermolecular potential Vinter and friction force, versus sliding distance are depicted in Figure 3. Figures 3a and 3b illustrate two possible friction force traces for an attractive and periodic Vinter. Such a potential may arise from either stiff surfaces that do not undergo structural changes during the sliding or sliding that is so slow that the surfacesurface intermolecular potential finds its minimum at each position along the sliding coordinate. The latter is akin to the dynamics of protein folding or any macromolecular, many-atom system finding its potential energy minimum. The potential energy barrier for sliding may be either large or small. In Figure 3a the time τVr for relaxation of the vibrational kinetic energy from the sliding coordinate is much longer than the characteristic time τs for sliding, so that the magnitude of the forces at constant Vs required to pull the system up the barrier and to restrain it as it moves down the barrier are the same. In Figure 3b, τVr ∼ τs, so there is energy transfer from the sliding coordinate. Furthermore, it is assumed that the rate of energy transfer from the

sliding coordinate is the same when the system moves up the barrier as when it moves down.40 For this case, the forces required to pull the system up the barrier are then larger than those required to restrain it as it moves down. The average friction force is zero in Figure 3a, but negative in 3b. More complicated friction traces are depicted in Figure 3, parts c and d, for cases where structural changes occur in the surfaces during the sliding, giving rise to a nonperiodic Vinter. The nature of this nonperiodicity depends on both the applied load and the sliding velocity. In Figure 3c, it is assumed that relaxation of vibrational kinetic energy from the sliding coordinate is slow compared to the characteristic sliding time and this energy relaxation may be separated from the surface structural changes. For this situation there is a clear relationship between Vinter and the friction force versus sliding distance. The plots in Figure 3d depict a substantially more complex situation than that in Figure 3c. Vinter is the same in Figure 3, parts d and c, but the friction forces are not. This is because energy transfer between the sliding coordinate and the surface modes cannot be separated from the surface relaxation and, as a result, the rate of this relaxation may depend on the instantaneous surface structure and differ for the slip and slide steps. To further complicate the dynamics, energy transfer between the sliding coordinate and the surface may have statistical, chaotic attributes, with energy transfer from the sliding coordinate to the surface modes as well as in the opposite direction. The latter transfer of energy to the sliding coordinate will have the effect of increasing the friction force for the slip step while decreasing it for the slide step.40 The above portrays the wide range of dynamic friction possible for sliding surfaces. The types of dynamic friction,



Figure 6. Plots of the friction force versus sliding distance at a sliding velocity of 50 m/s. Plots in the top and bottom rows are for a load of 5 and 10 nN, respectively. The upper surface is pulled 20 Å.

observed in the actual simulations of the sliding R-hydroxylated alumina surfaces, are described in the next section. IV. Simulation Results A. Surface-Surface Vibrational Spectrum. Any noticeable difference between the results of the stiff and soft surface simulations may arise from differences in the rate of energy transfer between the interface and the surfaces. This relaxation rate depends on the surface phonons and the surface-surface vibrational frequency ωss. For this reason it is of interest to study ωss for the stiff and soft surfaces, and to see if differences in the ωss values may be related to any differences in the friction forces for the two surfaces. The frequency ωss is determined from the power spectrum41 computed by Fourier transforming the coordinate ∆Zcm from eq 4. Simulations are performed with no load and applied loads of 1, 5, and 10 nN for both the stiff and soft surface systems, with no sliding, for a total of 100 ps. The resulting spectra are given in Figure 4. The frequency ωss, obtained from the location of the largest peak in the spectrum, is 13.48, 13.03, 13.98, and 14.63 cm-1 for the stiff surface and 3.25, 3.26, 3.30, and 3.36 cm-1 for the soft surface at loads of 0, 1, 5, and 10 nN, respectively. Of particular interest is the much smaller ωss for

the soft surface in the absence of an applied load in comparison to what is found for the stiff surface. For the soft surface, ωss is nearly independent of load, but increases with load for the stiff surface. This suggests there is increased coupling between the surface-surface vibration and the surface phonons for the stiff surface as the load is increased. B. Simulations with a 50 m/s Sliding Velocity. 1. Friction Forces. Simulations for the stiff and soft potentials, with a 50 m/s sliding velocity, were performed at applied loads of 5 and 10 nN. Of interest from these simulations are the maximum and average friction forces. Snapshots of the interfaces of the soft and stiff surfaces, during sliding, are shown in Figure 5. For the soft surfaces there is more disorder both at the interface and within the surfaces, and the surfaces are closer together. Plots of the friction force versus sliding distance are given in Figure 6 for simulations in which the top surface was pulled a total of 20 Å. At 5 nN the friction traces for the stiff and soft surfaces are similar, giving rise to similar friction forces as shown in Table 3. The maximum negative friction force for the stiff and soft surfaces is -4.8 and -5.3 nN, respectively, while their respective average negative friction forces are -1.13 and -0.98 nN. For the load of 5 nN, the ratio of the maximum negative friction force, RmF ) Fx(stiff)/Fx(soft), and the ratio of


Mann et al.

TABLE 3: Friction Forcesa maximum frictiond load

b

vs

c

5

50

10

50

10

2

10

50

surface positive

average frictione

negative positive negative

total

stiff soft stiff soft stiff soft

Sliding Distance of 20 Å 4.1 -4.8 0.94 2.7 -5.3 0.70 7.2 -14.3 1.52 3.2 -5.8 0.79 12.5 -9.7 1.78 5.2 -7.0 1.04

-1.13 -0.98 -2.28 -1.22 -2.13 -1.20

-0.33 -0.38 -0.86 -0.50 -0.53 -0.19

stiff soft

Sliding Distance of 200 Å 11.7 -14.4 1.13 5.9 -6.2 0.74

-1.45 -1.04

-0.33 -0.37

a The friction force is in units of nN. The sign of the friction force is defined in Equation 5. b The load is in units of nN. c The sliding velocity is in units of m/s. d Maximum friction force observed in the simulation. e Average friction force of the simulation.

the average negative friction force, RaF ) 〈Fx(stiff)〉 〈Fx(soft)〉, are near unity and 0.94 and 1.15, respectively. However, at 10 nN the friction forces are remarkably different for the two surfaces, with much higher friction forces for the stiff surface (see Table 3). For the stiff and soft surfaces, the maximum negative friction forces are -14.3 and -5.8 nN, respectively, with RmF ) 2.5. The average negative friction forces for the two surfaces are -2.28 and -1.22 nN, with RaF ) 1.9. These results indicate the friction force of the soft surface changes very little in going from a load of 5 to 10 nN, while that of the stiff surface increases by more than a factor of 2. For the above simulations, the top surface is pulled for 20 Å. To determine if this distance is sufficient to establish the sliding dynamics, both the stiff and soft surfaces were pulled for 200 Å at Vs ) 50 m/s and the higher load of 10 nN. The friction forces for these simulations are plotted in Figure 7. A comparison of Figures 6 and 7 shows that the differences in the friction of the stiff and soft surfaces is recovered by pulling the upper surface for only 20 Å. For the 200 Å pull, the maximum negative friction force for the stiff and soft surfaces is -14.4 and -6.2 nN, respectively, and the average negative friction forces for these two surfaces are -1.45 and -1.04 nN. These results are similar to those given above for the 20 Å pull. The friction forces for the 200 Å pull are summarized in Table 3. Simulations were performed in which the stiff surface was pulled for 200 Å at Vs ) 50 m/s and a load of 10 nN, but with Morse potentials for the Al-O and O-H stretches instead of the harmonic potentials used for the other calculations reported here and described in Section IIA. The Morse parameters are De ) 400 kJ/mol and βe ) 3.33 Å-1 for Al-O and De ) 500 kJ/mol and βe ) 4.38 Å-1 for O-H. The friction force versus sliding distance curve for the Morse function potential is shown in Figure 8, and is similar (if not statistically the same) to that in Figure 7 for the harmonic potential and gives nearly the same average friction forces. The average total friction, negative friction, and positive friction forces for the Morse potential model are -0.45, -1.50, and 1.10 nN, respectively, and are in excellent agreement with the values of -0.33, -1.45, and 1.13 nN in Table 3 for the harmonic potential model. No bond ruptures were observed for the simulations based on the Morse function potential and this result is expected as described in Section IIA. The similar friction force versus sliding distance curves, for the Morse and harmonic function potentials, results from similar energy transfer probabilities between the interfacial intermolecular modes and surface intramolecular modes for the two potentials. This is expected, since the highest average energy

Figure 7. Plots of the friction force versus sliding distance at a sliding velocity of 50 m/s and load of 10 nN. The upper surface is pulled 200 Å.

Figure 8. Same as Figure 7 for the stiff surface, except the Al-O and O-H bonds have Morse potentials.

in the bonds is RT ∼ 8 kJ/mol for the O-H bonds, which is much less than the bond energy of 500 kJ/mol. As a result, the average O-H vibrational frequency for the Morse potential is nearly identical to that for the harmonic potential. The vibrational frequency νM for a Morse bond, with energy E and bond dissociation energy D, is νM ) νH(1 - E/D)1/2, where νH is the harmonic frequency.42 Using a value for E of 8 kJ/mol, the average energy in the O-H bonds, gives an average νM which is only 1% smaller than νH. Thus, unless energy transfer between the surface modes is controlled by very, very narrow resonances, which is very unusual,43 the Morse and harmonic bond potentials will give very similar results. Therefore, a potential energy



Figure 9. Plots of the distance between the centers-of-mass of the two surfaces versus sliding distance at a sliding velocity of 50 m/s. Plots in the top and bottom rows are for a load of 5 and 10 nN, respectively.

function, with harmonic internal coordinate potentials for the surface degrees of freedom, is expected to be an accurate model for the 10-10 s or shorter simulations reported here. Of course, for much longer time simulations, which result in bond dissociations (see Section IIA), the Morse and harmonic potentials will give different results. A harmonic model was also found to be an accurate representation of the surface modes in a recent study of energy transfer in collisions of Ne atoms with an alkanethiolate selfassembled monolayer.44a Harmonic and Morse function potential models give identical energy transfer results in collisions of Ar atoms with highly excited methane molecules.44b Though the internal coordinate potential function for the hydroxylated R-alumina surface is harmonic, there are nonlinear terms in the internal coordinate kinetic energy expression which makes the motion of the surface modes nonseparable,43 giving rise to energy transfer between these modes and layers of the surface. It is of interest to note that the similarity between the simulations with the Morse and harmonic bond potentials is representative of the accuracy of the simulations. Introducing the small perturbation on the Hamiltonian, by replacing the harmonic bond potentials with Morse potentials, results in a new trajectory which creates an ensemble similar to that created by the unperturbed trajectory. This perturbation on the trajectory is not dissimilar from that resulting from a different initial condition for the trajectory (see Section IIB).

2. Surface-Surface Separation. Plots of the distance between the center-of-mass of the two surfaces, ∆Zcm, for the simulations at a sliding velocity of 50 m/s and at loads of 5 and 10 nN, for both the stiff and soft surface systems, are given in Figure 9. The averages of ∆Zcm over the entire simulation, along with the standard deviations, for the soft surface are 13.6 ( 0.12 and 13.2 ( 0.11 Å at 5 and 10 nN, respectively. Average ∆Zcm for the stiff surface is 14.2 ( 0.08 Å at both 5 and 10 nN. Following the above discussion in Section IIC, smaller ∆Zcm values are expected for the soft surface, since its weaker potential allows the surface-surface intermolecular potential to pull the surfaces closer together. For the soft surface model, the surfaces come closer together when the load is increased. However, this is not the case for the stiff surface system where the average surface distances are the same and plots of ∆Zcm versus sliding distance are nearly identical for the 5 and 10nN loads. Figure 9 shows there are more fluctuations in ∆Zcm for the stiff surface than for the soft surface. This is expected since, from Figure 4, the surface-surface intermolecular vibrational frequency is larger for the stiff surface system. 3. Intermolecular Potential Energy. Plots of the surfacesurface intermolecular potential energy Vinter, during the sliding, are given in Figure 10 for the stiff and soft systems at a load of 10 nN. The Vinter are measured relative to the potential energy minimum found for both the stiff and soft systems with no applied load, i.e. -321 and -355 kcal/mol, respectively (see


Figure 10. Plots of the intermolecular potential energy, relative to the minimum without applied load, versus sliding distance at a sliding velocity of 50 m/s and load of 10 nN. The upper surface is pulled 200 Å.

Figure 2). The reason for the negative Vinter is that the local Vinter are dependent upon the applied load. When a load is applied, the system is capable of sampling lower Vinter values (i.e., the system becomes “squeezed”). The plots of Vinter for a load of 5nN are very similar to those for the 10 nN load, except with the lighter load the Vinter values are not quite as negative. The intermolecular potential energy plots in Figure 10 show that, for the stiff surface system, fluctuations about the corresponding potential energy minimum are much larger than for the soft surface system. Values ranging from approximately -40 to 80 kcal/mol are observed for the stiff surface model and -30 to 40 kcal/mol for the soft model. The Vinter plots versus sliding distance in Figure 10 indicate higher barriers for the stiff surface. The estimated barriers for sliding as determined from the plots in Figure 10 are as large as 100 kcal/mol for the stiff system and 40 kcal/mol for the soft system. The smaller barrier for the soft surface is consistent with the “smoother” sliding and lower friction forces for the soft surface, as shown in Figure 7. Despite the slightly smaller (more negative) intermolecular potential of -40 kcal/mol observed at one point along the sliding coordinate for the stiff surface simulation, as compared to the approximate value of -30 kcal/mol for the soft system, the potential averaged over the duration of the simulation is 30 and 0 kcal/mol for the stiff and soft systems, respectively. This finding is what one might expect, since the soft surfaces can more easily relax and deform, resulting in a greater ability to sample deeper portions of the attractive surface (i.e., become squeezed). It is noteworthy that the plots of Vinter versus sliding distance in Figure 10 do not show the periodic potentials, which arise

Mann et al.

Figure 11. Plots of the friction force versus sliding distance for the stiff and soft surface systems at a load of 10 nN and a sliding velocity of 2 m/s.

when sliding the rigid surface or stiff and soft surfaces with optimized coordinates (see Figure 2). These random fluctuations in Vinter are consistent with nonperiodic relaxations at the interface giving rise to instantaneous barriers for sliding higher than those in Figure 2. The Vinter plots versus sliding distance in Figure 10 indicate higher barriers for the stiff surface which is consistent with the larger force for pulling the stiff surface. Also, as discussed in Section III, differences in the rate and direction of energy transfer between the sliding coordinate and the surface modes may contribute to the differences in the friction for the soft and stiff surfaces. C. Simulations with a 2 m/s Sliding Velocity. 1. Friction Forces. In previous simulations23 of sliding R-hydroxylated alumina surfaces with an unscaled potential, a transition from stick-slip to near smooth sliding was observed between a 20 to 50 m/s sliding velocity for a load of 5 nN. The plots in Figure 5 for a sliding velocity of 50 m/s, show near smooth sliding for the soft surface at a load of 5 nN, but a transition to stick-slip at the higher load of 10 nN. For the stiff surface, stick-slip sliding becomes stronger in going from a load of 5 to 10 nN. To observe smooth sliding for the stiff surface at a sliding velocity of 50 m/s, a load smaller than 5 nN may be required. In addition to the nature of the friction for the soft and stiff surfaces versus load, there is also an interest in how their friction is affected by the sliding velocity. Simulation results for a sliding velocity of 2 m/s are presented in this section, to compare with the above results for a sliding velocity of 50 m/s. In Figure 11 are plots of the friction force versus sliding distance for both the stiff and soft surface systems with an applied load of 10 nN and a sliding velocity of 2 m/s. The maximum and average friction forces of these plots are listed


Figure 12. Plots of the intermolecular potential energy, relative to the minimum without applied load, versus sliding distance for the stiff and soft surface systems at a load of 10 nN and a sliding velocity of 2 m/s.

in Table 3. A comparison of these friction forces with those in Figures 6 and 7 and Table 3, for the same load but higher sliding velocity of 50 m/s, shows that at both 2 and 50 m/s the friction forces are larger for the stiff surface. However, there are differences in the details of the friction forces for the two sliding velocities. At the slower sliding velocity, the friction force versus sliding distance begins to exhibit the periodicity of the optimized interfacial intermolecular potential, particularly for the stiff surface which has the higher barrier for sliding (see Figure 2). The average negative friction force for both the stiff and soft surface is nearly independent of sliding velocity. However, the average positive friction forces at the 2 m/s sliding velocity are somewhat larger than those found at 50 m/s, which gives rise to a less negative average total friction force at 2 m/s for both surfaces. With the slower sliding velocity of 2 m/s, as compared to 50 m/s, the sliding surfaces begin to access deeper intermolecular potential wells (see presentation below). In the absence of strong fluctuations in the energy relaxation between the sliding coordinate and the surface modes, the friction force versus sliding distance may manifest the periodicity of the interfacial intermolecular potential. As discussed above, this appears to be the case for the stiff surface. The higher positive friction forces at 2 m/s as compared to 50 m/s do suggest deeper Vinter at 2 m/s. However, such a picture does not emerge from the average negative friction force which is nearly the same at 2 and 50 m/s. Obviously there are other properties, besides the depth of the intermolecular potential, which affect the friction force. As discussed in Section III and depicted in Figure 3c, a

J. Phys. Chem. B, Vol. 105, No. 48, 2001 12043 likely candidate is energy transfer between the sliding coordinate and surface modes, which should be more extensive at the slower sliding velocity of 2 m/s. The larger positive friction forces at 2 m/s than 50 m/s is consistent with efficient energy transfer from the surface modes to the sliding coordinate during the time of these simulations. 2. Surface-Surface Separation. The averages of ∆Zcm over the duration of the simulations, with a sliding velocity of 2 m/s and load of 10 nN are 14.1 ( 0.06 and 13.2 ( 0.09 for the stiff and soft systems, respectively. These values are identical to those for the same load but higher sliding velocity of 50 m/s. However, the deviations from the mean are slightly smaller, as is expected, since with a lower sliding velocity the system has more time to vibrationally relax and so the fluctuations will be smaller. 3. Intermolecular Potential Energies. Plots of the intermolecular potential energy relative to the corresponding minimum for the stiff and soft surface systems, at an applied load of 10 nN and a sliding velocity of 2 m/s, are given in Figure 12. As for simulations described above for sliding at a faster velocity of 50 m/s, the fluctuations in the intermolecular potential and corresponding barriers for sliding are larger for the stiff surface model than for the soft surface. Another interesting finding is that there are fewer fluctuations in Vinter versus sliding distance for the stiff surface at a sliding velocity of 2 m/s than 50 m/s (see Figure 10). The average of the intermolecular potential energy sampled from these simulations at 2 m/s are 2 and -17 kcal/mol for the stiff and soft surface systems, respectively. These values are lower than the values at 30 and 0 kcal/mol given above for these respective systems at 50 m/s. This result agrees with our earlier prediction23 that, at lower interfacial sliding velocities, the two layers in contact have more time to vibrationally relax and sample deeper portions of the potential energy wells, giving rise to stick-slip behavior. V. Comparison with Experiment The simulations reported here illustrate how the softness/ stiffness of sliding surfaces, with identical intermolecular potentials, affects structure relaxations at the interface, the instantaneous barriers for sliding and the friction force for sliding. There are several experimental studies45-51 which are either directly or indirectly related to these simulations. The frictional properties of alkylthiol and alkylsilane monolayers have been studied as a function of the alkyl chain length.45 Monolayers with long chains are solidlike, while those with short chains are more liquidlike. It is proposed that the increased friction for the short chain monolayers results from the disorder of their liquidlike character, which favors the increase of the number and type of low-energy modes (kinks, bends, distortions) that are available for excitation and energy dissipation. The barriers and friction forces for sliding two identical rigid surfaces may be changed by rotating the surfaces.46-48 When the surfaces are oriented so that they are commensurate with their atoms aligned, there may be high barriers for sliding. Incommensurate surfaces resulting from their rotation, may have much smaller barriers for sliding. The softening of the hydroxylated alumina surfaces, as modeled here, enhances the ability of the surface interface to distort from its commensurate structure and, thus, find lower barriers for sliding. This is somewhat similar to the above effect of rotating two rigid surfaces from their commensurate alignment. Experimental studies have shown the same effect of surface stiffness on friction as the simulations reported here.49-51 Comparisons of the tribology of thin films of perfluoropolyalkyl ethers have shown that the shear stress is highest for those most

12044 J. Phys. Chem. B, Vol. 105, No. 48, 2001 likely to solidify.49 Temperature affects the friction of hydrocarbon monolayers by modifying their stiffness.50 Increasing the temperature makes the films more liquidlike. This increases the interdigitation across the interface, but the friction energy dissipation is low, since the chains are easy to disentangle because of their high mobility. At much lower temperatures, friction is also low because the chains freeze and do not interdigitate. However, at an intermediate temperature there is maximum friction because the rate of molecular interdigitation is high in relation to the slower disentanglement rate. Combined theoretical/experimental studies of traction fluids by Hata and Tsubouchi51 reveal that stiff traction fluids, i.e., molecules with large torsional barriers, yield enhanced traction coefficients. They measured the traction coefficient for three monocyclic compounds and calculated and compared the corresponding torsional barriers for internal rotation. They found a strong correlation between the rotational barriers with the experimentally determined traction coefficients, and concluded that stiff molecules enhance the friction as a result of their inability to deform under high pressure and high shear. Our findings from this study, employing two model soft and stiff surface systems, agree well with their conclusions. We conclude that the tribology of fluidic and crystalline materials alike is strongly correlated with the softness and stiffness of the material under investigation, and an understanding of a system’s ability to deform under stress is critical for gaining insight into a systems tribological properties. VI. Summary This work reports simulations of sliding model R-hydroxylated alumina surfaces, a system with a relatively strong surface-surface intermolecular potential as compared to that for other systems such as diamond surfaces. The study focuses on how interfacial and dynamic frictional properties of these surfaces vary as the potentials of the surfaces are softened or stiffened, without changing the surface-surface intermolecular potential Vinter. The following results were obtained: 1. The surface-surface vibrational spectrum, without sliding, not only depends on Vinter but on the forces for the surfaces. The surface-surface vibrational frequency ωss is much smaller for the soft surface, presumably as a result of increased coupling between Vinter and the surface forces. The frequency ωss is nearly independent of load for the soft surface, but increases with load for the stiff surface. 2. Sliding friction forces are similar for the soft and stiff alumina surfaces with low applied loads. However, with higher applied loads the friction forces are higher for the stiff surface. Increasing the load forces the stiff surfaces into deeper interfacial potential energy wells, giving rise to higher barriers for sliding and, thus, higher friction forces. 3. The above results suggest that the dependence of the transition from stick/slip to smooth sliding on applied load will differ for the soft and smooth surfaces. For the soft surface, the critical sliding velocity Vc required for this transition will only weakly increase with applied load. Since the soft and stiff surfaces acquire similar friction forces as the load is decreased, at low load the critical velocity required for smooth sliding is expected to be the same for the two surfaces. However, with increased load this critical velocity becomes larger for the stiff surface. 4. When the sliding velocity is decreased, the surface-surface interface samples deeper intermolecular potential energy wells as expected, since there are now longer times available for the interface to undergo the necessary structural changes to find

Mann et al. the deeper potential wells. In addition, with the slower sliding, there is also more energy transfer between the sliding coordinate and surface vibrational modes. The simulations presented here suggest both of these effects influence the nature of the friction force versus sliding velocity. However, they do not identify either the relative importance of these two effects or how it may vary versus sliding velocity. The ability of an interface to sample deeper potential energy minima, as the sliding velocity is decreased, depends on the surfaces’ vibrational frequencies and stiffness. Similarly, the efficiency of energy transfer from the sliding coordinate to the surface modes, for a particular sliding velocity, depends on the surfaces’ vibrational frequencies. In future studies it would be of interest to investigate the effects of films at the solid-solid interface and how the stiffness of the film affects the friction forces. With QM/MM52 direct dynamics simulations it will be possible to study chemical reactions at the interface of the sliding surfaces. Acknowledgment. This research was supported by the Office of Naval Research and the Institute for Manufacturing Research at Wayne State University. The authors thank Professor Kihyung Song for his assistance in preparing the manuscript. References and Notes (1) McClelland, G. M. In Adhesion and Friction, Springer Series of Surface Sciences, Vol. 17; Grunze, M., Kreuzer, H. J., Eds.; SpringerVerlag: Berlin, 1989. (2) McClelland, G. M.; Glosli, J. N. In NATO ASI Proceedings on Fundamentals of Friction: Macroscopic and Microscopic Processes; Singer, I. L., Pollock, H. M., Eds.; Kluwer Academic Publishers: Dordrecht, 1992. (3) Krim, J. Sci. Am. 1996, October, 74. (4) Landman, U.; Luedtke, W. D.; Nitzan, A. Surf. Sci. 1989, 210, L177. (5) Gao, J.; Luedtke, W. D.; Landman, U. Science 1995, 270, 605. (6) Landman, U.; Luedtke, W. D.; Gao, J. Langmuir 1996, 12, 4514. (7) Gao, J.; Luedtke, W. D.; Landman, U. J. Phys. Chem. B 1998, 102, 5033. (8) Thompson, P. A.; Robbins, M. O. Science 1990, 250, 792. (9) Robbins, M. O.; Thompson, P. A.; Grest, G. S. MRS Bull. 1993, May, 45. (10) Cieplak, M.; Smith, E. D.; Robbins, M. O. Science 1994, 265, 1209. (11) He, G.; Mu¨ser, H.; Robbins, M. O. Science 1999, 284, 1650. (12) Harrison, J. A.; White, C. T.; Colton, R. J.; Brenner, D. W. Phys. ReV. B 1992, 46, 9700. (13) Harrison, J. A.; White, C. T.; Colton, R. J.; Brenner, D. W. MRS Bull. 1993, May, 50. (14) Harrison, J. A.; White, C. T.; Colton, R. J.; Brenner, D. W. J. Phys. Chem. 1993, 97, 6573. (15) Harrison, J. A.; White, C. T.; Colton, R. J.; Brenner, D. W. Thin Solid Films 1995, 260, 205. (16) Perry, M. D.; Harrison, J. A. Thin Solid Films 1996, 290, 211. (17) Perry, M. D.; Harrison, J. A. J. Phys. Chem. B 1997, 101, 1364. (18) Harrison, J. A.; Perry, M. D. MRS Bull. 1998, June, 27. (19) Tupper, K. J.; Brenner, D. W. Thin Solid Films 1994, 253, 185. (20) Tomlinson, G. A. Philos. Mag. 1929, 7, 905. (21) Frenkel, F. C.; Kontorova, T. Zh. Eksp. Teor. Fiz. 1938, 8, 1340. (22) Sokoloff, J. B. Phys. ReV. B 1990, 42, 760. (23) Mann, D. J.; Hase, W. L. Tribology Lett. 1999, 7, 153. (24) Berman, A.; Steinberg, S.; Campbell, S.; Ulman, A.; Israelachvili, J. Tribology Lett. 1998, 4, 43. (25) Coustet, V.; Jupille, J. Surf. Sci. 1994, 307-309, 1161. (26) Wittbrodt, J. M.; Hase, W. L.; Schlegel, H. B. J. Phys. Chem. B 1998, 102, 6539. (27) Bolton, K.; Bosio, S. B. M.; Hase, W. L.; Schneider, W. F.; Hass, K. C. J. Phys. Chem. B 1999, 103, 3885. (28) Sawilowsky, E. F.; Meroueh, O.; Schlegel, H. B.; Hase, W. L. J. Phys. Chem. A 2000, 104, 4920. (29) Rappe, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard, W. A., III; Skiff, W. M. J. Am. Chem. Soc. 1992, 114, 10024. (30) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Hermans, J. In Intermolecular Forces; Pullman, B., Ed.; Reidel: Dordrecht, 1981. (31) Handbook of Chemistry and Physics, 81st ed; Lide, D. R., Ed.; CRC: New York, 2000. (32) Baer, T.; Hase, W. L. Unimolecular Reaction Dynamics. Theory and Experiments; Oxford, New York, 1996.

Friction of Sliding Model Hydroxylated R-Alumina Surfaces (33) Mann, D. J. M.S. Thesis, Wayne State University, 2000. (34) de Sainte Claire, P.; Hass, K. C.; Schneider, W. F.; Hase, W. L. J. Chem. Phys. 1997, 106, 7331. (35) Haile, J. M. Molecular Dynamics Simulations; New York: Wiley: 1997. (36) Faeder, J.; Ladanyi. B. M. J. Phys. Chem. B 2000, 104, 1033. Tielemann, D. P.; van der Spoel, D.; Berendsen, H. J. C. J. Phys. Chem. B 2000, 104, 6380. Rabinovich, A. E.; Ripatti, P. O.; Balaber, N. K. J. Biol. Phys. 1999, 25, 245. Del Corte, A.; Garrison, B. J. J. Phys. Chem. B 2000, 109, 6785 and references therein. (37) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; New York: Oxford, 1989; p 77. (38) Frenkel, D.; Smit, B. Understanding Molecular Simulation; New York: Academic Press, 1996; p 61. (39) He, G.; Mu¨ser, M. H.; Robbins, M. O. Science 1999, 284, 1650. (40) Regardless of the relationship between the τVr for the slip and slide steps, if there is only energy transfer from the sliding coordinate to the surface modes for sliding with constant Vs, then |Fslide| > |Fslip| for both τVr (slip) < τVr (slide) and τVr (slide) < τVr (slip).

J. Phys. Chem. B, Vol. 105, No. 48, 2001 12045 (41) Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. J. Chem. Phys. 1977, 67, 404. (42) Oxtoby, D. W.; Rice, S. A. J. Chem. Phys. 1976, 65, 1676. (43) Intramolecular and Nonlinear Dynamics. AdVances in Classical Trajectory Methods, Vol. 1; Hase, W. L., Ed.; JAI: London, 1992. (44) (a) Yan, T.-Y.; Hase, W. L. J. Phys. Chem. A 2001, 105, 2617. (b) Hu, X.; Hase, W. L. J. Phys. Chem. 1988, 92, 4040. (45) Lio, A.; Charych, D. H.; Salmeron, M. J. Phys. Chem. B 1997, 101, 3800. (46) (a) Krim, J.; Solina, D. H.; Chiarello, R. Phys. ReV. Lett. 1991, 66, 181. (b) Mak, C.; Krim, J. Phys. ReV. B 1998, 58, 5157. (47) Yoshizawa, H.; Israelachvili, J. Thin Solid Films 1994, 246, 71. (48) Ruths, M.; Granick, S. Langmuir 2000, 16, 8368. (49) Ruths, M.; Granick, S. Tribology Lett. 1999, 4, 161. (50) Yamada, S.; Israelachvili, J. J. Phys. Chem. B 1998, 102, 234. (51) Hata, H.; Tsubouchi, T. Tribology Lett. 1998, 5, 69. (52) Warshel, A.; Levitt, M. J. Mol. Biol. 1976, 103, 227.

Effect of Surface Stiffness on the Friction of Sliding Model

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