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Acta Materialia 56 (2008) 3577–3584

Contact and friction of single crystal nickel nanoparticlesusing molecular dynamics

T.W. Stone a,*, M.F. Horstemeyer a, Y. Hammi b, P.M. Gullett c

a Department of Mechanical Engineering, Mississippi State University, Starkville, MS, USAb Center for Advanced Vehicular Systems, Mississippi State University, Starkville, MS, USA

c Department of Civil Engineering, Mississippi State University, Starkville, MS, USA

Received 12 November 2007; received in revised form 26 March 2008; accepted 27 March 2008Available online 28 April 2008

Abstract

Molecular dynamics (MD) simulations using the embedded atom method potentials were performed to describe the interparticlebehavior of two single crystal spherical nickel nanoparticles during compaction based on the particle size and contact angle. We analyzedthe evolution of the contact surfaces during the MD simulations and investigated the frictional effects at the contact surfaces. Using mac-roscale contact laws and the Amontons–Coulomb friction law, we compared the MD contact results to previous studies and showedlength scale dependence in friction.� 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Molecular dynamics; Nanocrystalline materials; Granular materials; Friction; Particle deformation

1. Introduction

Since Hall and Petch [1,2] showed that a higher ultimatestress is obtained with decreasing grain size, the relation-ship between a length scale parameter and mechanicalproperties has been investigated by many researchers.Researchers [3–5] have shown that the mechanical proper-ties are significantly different for nanosize particles as com-pared to coarser particles. Siow et al. [3] presented asummary of previous experimental studies on the mechan-ical properties of bulk nanocrystalline copper and nickelillustrating size scale effects as well. This summary revealedthat for grain sizes 10 nm or greater only a slight reductionin elastic modulus was noted; however, the hardness, yieldstress and ultimate tensile stress all increased significantlyas the grain size decreased to 10 nm. In contrast, Zhouet al. [4] studied the elastic modulus for grain sizes less than17 nm in dense materials and observed a noticeabledecrease (up to 14%) from the macroscale values. Experi-

1359-6454/$34.00 � 2008 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2008.03.044

* Corresponding author.E-mail address: tstone@cavs.msstate.edu (T.W. Stone).

mental evidences [3,5] suggest a reverse Hall–Petch relationat the nanoscale, although research differs as to the maxi-mum nanometer grain size (10–20 nm) at which this effectis first observed. To explain this phenomenon, VanSwygenhoven et al. [5] used MD simulations to show grainboundary sliding instead of dislocation movement as thesource of plastic deformation for grain sizes less than10 nm. Similarly, Siow et al. [3] also noted that, for grainsizes below 10 nm, high grain boundary interfacial area/volume ratios resulted in different deformation mechanismsthan observed for larger grain sizes. The observed differ-ences between nanosized and larger-sized grains have ledto an increased interest in the development of nanosizedparticles and to additional research in understanding nano-particle deformation. In addition to material properties, afundamental understanding of friction at the nanoscale isimportant in the application of nanoscale devices.

From the early work of Amontons [6] in 1699 and laterby Coulomb [7] in 1779, the following familiar friction lawfor contact sliding was developed:

F F ¼ lF N ð1Þ

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3578 T.W. Stone et al. / Acta Materialia 56 (2008) 3577–3584

The frictional force FF is the tangential force resistingthe relative motion of two surfaces, which are pressedagainst each other with a normal force FN. The constantof proportionality l is referred to as the coefficient of fric-tion and is dependent on the material and many other fac-tors, such as temperature, strain rate and the environment,and on whether the bodies are at rest (in a state of sticking)ls or in motion (state of sliding) lk. The coefficient of staticfriction ls is proportional to the maximum frictional forcethat the surfaces can sustain without sliding. According tothe Amontons–Coulomb relations, the sliding friction forceis proportional to the normal load but is independent of theapparent or macroscopic contact area Aa and the slidingvelocity V. While the coefficient of friction (COF) is notusually constant, it was considered a constant in the origi-nal Amontons–Coulomb friction law given in Eq. (1).Because the friction between sliding contact surfaces fluctu-ates due to rubbing asperities, the coefficient is a meanvalue. While the Amontons–Coulomb laws of friction arethe most universally applicable friction laws, they are vio-lated at the nanoscale. Previous experiments [8–11] haveshown where the load-independence is violated, which indi-cates that a size-dependence exists in a relationshipbetween the load and apparent contact area. Because thereal area of contact is smaller than the apparent area ofcontact, contact only takes place at the tops of the asperi-ties during sliding. In 1958, Bowden and Tabor [12] pre-sented an adhesion model for friction at the micrometerscale. This model differs from the original Amontons–Cou-lomb model in that it assumes the friction force is propor-tional to both the real area of contact Ar, due to asperities,and shear strength at the interface s:

F F ¼ sAr: ð2ÞIn this formulation, the contact area is dependent on the

applied normal load, the particle radius and the elasticproperties of the material. The shear strength is definedas the shear force per unit area required to cause slidingalong the interface. For macroscale contacts the adhesioneffects are negligible whereas for nanometer scale contactsthe effect of adhesion on the contact area is important,because adhesion is caused by intermolecular forcesbetween the interfaces of materials.

The subject of sliding friction on a smaller scale is ofconsiderable interest. Johnson [13] first looked at the shearstrength being different at the nano- and macroscales. Thescale dependence of the friction stress for single asperitycontacts using a discrete dislocation model was investi-gated by Hurtado and Kim (HK) [14,15]. Adams et al.[16] incorporated the HK model into a multi-asperity, mul-tiscale model for contact and friction. Later, Dominik andTielens [17] studied a theoretical model for sliding frictionbetween two micron sized elastic spheres. Bhushan andNosonovsky [10] used a different approach and developeda model based on the scale-dependent shear strength dueto dislocations nucleating from Frank–Read sources.Using discrete dislocation dynamics, Deshpande et al.

[18] analyzed the initiation of sliding in single crystalsand found that the shear stress is a function of contact size.Kogut and Etsion [19] proposed an elastoplastic frictionmodel with scale-independent plasticity which showed thatthe friction coefficient is dependent on the real contact areaand the normal load.

The surface phenomena associated with frictional slid-ing at the smaller scales has also been experimentally stud-ied. Most research on contact of nanosized particles hasbeen conducted experimentally using nano-indentationtechniques. Experiments focused on elastic and plasticdeformation for nanosized contacts have been performedby Kum [20] and Rekhi et al. [21]. Additionally, using asurface force apparatus, Homola et al. [8] experimentallyobtained the microscale friction stress at the contactbetween two mica surfaces. The development of atomicforce microscopy (AFM) has made direct measurementof nanoscopic friction possible. A few years later usingAFM, Carpick et al. [9] obtained the friction stress betweentwo mica surfaces at the nanoscale.

In the study of friction, analysis of contact behavior isessential. Previous research by several researchers [22–26]on the contact between two spherical particles beingpressed together primarily addressed the macroscopicbehavior. The study of contact between deforming bodieswas first investigated by Hertz [22] in 1882 and remainsthe foundation for most contact analysis today. Addi-tional theoretical studies were based on the assumptionthat the particles were elastic [23], perfectly plastic [24],or nonlinearly plastic with work hardening [25,26]. Mar-tin et al. [27] studied contact behavior using constitutiveequations and the discrete element method (DEM). Theparticles were modeled as an elastoplastic material andfollowed a hardening relation in the plastic regime basedon the work by Storakers et al. [28]. Adhesive effects inthe elastic contact of two spheres under vertical loadingwere included in the model by Johnson et al. [29], whichresulted in an increased contact area and elastic deforma-tion beyond that predicted by the Hertz theory. Adhe-sion was also included in the Derjaguin, Muller andToporov model [30], which assumed that the contactarea does not change due to the attractive surface forces,resulting in the same contact area as that predicted byHertz.

The objective of this research is to use molecular dynam-ics (MD) simulations to study the applicability of macro-scopic contact and friction laws at the nanoscale and toprovide a comprehensive analysis of frictional effects onthe interparticle behavior of two spherical nickel nanopar-ticles. As shown by Derjaguin et al. [30], the frictional forcemay be rewritten as

F F ¼ lF N þ F adh ð3Þto account for the adhesion effects present at the nanoscale.Quantified frictional effects at the nanoscale will lead to abetter understanding of nanoparticle behavior and can beused to understand the multiscale aspects of friction.

T.W. Stone et al. / Acta Materialia 56 (2008) 3577–3584 3579

2. Computer simulations

2.1. Molecular dynamics

For the MD simulations we used EAM potentials devel-oped by Daw and Baskes [31]. The EAM potentialdescribes the bonding of an atom in terms of the local elec-tronic density and incorporates electrostatic and repulsiveinteractions between atoms. The total energy is given bythe following equation:

E ¼X

i

F iXi6¼j

qiðrijÞ !

þ 1

2

Xij

uijðrijÞ ð4Þ

This equation predicts that the total energy is equal tothe embedding energy F times the electron density q dueto neighboring atoms plus the addition of the potentialenergy term u. From the energy calculation, the dipoleforce tensor, b, is determined for each atom and is givenby

bimk ¼

1

Xi

XN

jð6¼iÞf ij

k rijm ð5Þ

where N is the number of nearest neighbor atoms, f is theforce vector, r is the displacement vector, and X is theatomic volume. Thus, b is analogous to the stress tensorat the atomic site. The global stress over the continuumis interpreted as a volume average over the specimen,

rmk ¼1

N

XN

bimk ð6Þ

where N is the total number of atoms in the specimen.The MD code performs the energy, force, and stress cal-

culations based on the chosen potential. In addition, theMD code calculates the centrosymmetry parameter [32].The centrosymmetry parameter of a given atom providesa measure of the level of disturbance of that atom’s envi-ronment from the symmetric crystal structure. The formulafor the parameter for an fcc crystal is [28]:

Cfcca ¼

X6

b¼1

jra;b þ ra;bþ6j2 ð7Þ

Fig. 1. (a) Model setup with c = 0�, 30� and 60�, D = 3.52, 7.04, 10 and 14.08with 30� contact angle after 10 ps of temperature relaxation.

where a and b are atom indices and r is the distance be-tween the atoms. The summation is taken over the six pairsof opposing neighbors of an atom. For an atom in a perfectfcc structure, the centrosymmetry parameter is zero. Byplotting the atoms with a centrosymmetry parameter largerthan some cutoff value (2.0 for this study), we can visualizethe dislocation structure of the deforming material.

2.2. Model configuration

In this study, uniaxial compression simulations wereperformed on single crystal spherical nickel nanoparticles.As shown in Fig. 1, the atomistic models were created bydefining two nanoparticles of various sizes, contact angles,and crystal orientations to analyze the effect of theseparameters on deformation behavior. The particle diame-ters used in the models were 3.52 nm (2123 atoms per par-ticle), 7.04 nm (16,757 atoms per particle), 10 nm (48,000atoms per particle) and 14.08 nm (134,000 atoms per parti-cle). The contact angle, which is the angle between the y-axis and the normal axis of the contact point, varied from0� to 60�, as can be observed in Fig. 1. The particles had a<100> crystal lattice orientation, with origins at the parti-cle centers. Because the model consisted of a perfect latticestructure, there were no initial defects in the model. A halfsphere of atoms was added to each end of the model to pro-vide a fixed surface for the boundary conditions. Bound-aries were defined as free surfaces in the x- and z-directions. In each model, a few planes of atoms at thetop and bottom (xz planes at the +y and �y extrema) ofthe half spheres were fixed on their perfect lattice sites.Thermal velocities of the interior (active, non-fixed) atomswere then initialized using a Boltzmann distribution at achosen temperature of 300 K. The temperature wasallowed to equilibrate for 10 ps at 300 K to accommodateany surface relaxation in the system. At the completionof the equilibrium phase, a compressive load, using a linearvelocity profile

vy ¼ _ey ð8Þwhere _e is the strain rate and was applied to the atomsalong the y-direction. A linear velocity profile was used

nm, and v = 0.22 nm ps�1 up to 20% strain; (b) 7.04 nm particle diameter

3580 T.W. Stone et al. / Acta Materialia 56 (2008) 3577–3584

in order to avoid an initial shock wave, which otherwisewould result from the instantaneous application of theboundary conditions on the �y surfaces. The imposedvelocity of 0.22 nm ps�1 resulted in very high strain rateson the order of 108 s�1. The applied strain rates were highbecause the atomistic simulations start with an atomic fre-quency in which the periods are on the order of femtosec-onds. Because straining via moving the frozen planes addsconsiderable energy to the active atoms, a Nose–Hooverthermostat [33] was used during the molecular dynamicssimulations to keep the active atoms at constant tempera-ture. The thermostat applies a damping factor to the activeatoms based on the difference between their current tem-perature and the desired temperature of 300 K. With thevelocity profile being superimposed on the thermal veloci-ties, the models were compressed for up to 20% true strain.

3. Results and discussion

The MD simulations were run using Warp, a moleculardynamics code [34], and were performed using multipleparallel processors on IBM Linux Superclusters. The glo-bal stress tensor (Eq. (6)) and the centrosymmetry param-eter (Eq. (7)) from the MD simulations were used to studythe evolution of the contact surface for the nanoparticlemodels.

3.1. Contact deformation

Because indentation involves plastic deformation, in thecurrent study we applied a macroscopic plastic contactmodel proposed by Storakers et al. [28] that included theeffects of strain hardening. The power law form of strainhardening is represented by the constitutive equation

r ¼ r0em ð9Þ

where r0 is the flow stress, m is the strain hardening expo-nent, and r and e are the stress and strain in the uniaxialcase. For linear elasticity, m = 1 and r0 = E, while for per-fect plasticity, m = 0 and r0 = ry, where ry is the yield

Fig. 2. Indentation vs. measured contact radius fo

strength of the material. By analyzing the stress–strain re-sponse from the MD simulations and correlating the re-sults with the constitutive relation in Eq. (9), we obtainedr0 and m for the nickel particles. For two contacting mono-sized particles of radius r, denoted by subscripts 1 and 2,the relative radius rc and contact modulus Ec were definedas

rc ¼r1r2

r1 þ r2

ð10Þ

and

Ec ¼1� v2

1

E1

þ 1� v22

E2

� ��1

ð11Þ

where E is the elastic modulus and v is Poissons ratio.The normal displacement or indentation h of the two

particles was due primarily to deformation in the contactregion. In the model by Storakers et al. [28], a finite elementanalysis was performed for the spherical indentation ofviscoplastic solids and the following relation was presentedfor the indentation h between two contacting particles andfor the contact radius a:

hðaÞ ¼ a2

2c2ðmÞrc

ð12Þ

where

c2ðmÞ ¼ 1:43 expð�0:97ðmÞÞ ð13Þis related to the contact area size and piling up of materialat the contact. The c2(m) parameter ranges from 0.5 forlinear elasticity to 1.43 for perfectly plastic behavior [35].By assuming a circular contact area and measuring thewidth and depth of the circular overlap of the two nano-particles during the simulation, we obtained the contactradius and indentation from the MD model. By plottingthe measured h and a values from the MD simulations withthe calculated h values using Eq. (13), as shown in Fig. 2,we confirmed the validity of the macroscopic relation forindentation with our nanoscale model. By setting m =0.42 in the work hardening model, while also assuring that

r 7.04 nm dia. particles at a 0� contact angle.

T.W. Stone et al. / Acta Materialia 56 (2008) 3577–3584 3581

the constitutive relation for strain hardening in Eq. (9) wasupheld, a good correlation for indentation was obtainedbetween the MD simulation results (measured indentation)and the plasticity model with work hardening (calculatedindentation from Eq. (12)). The hardening parameterc2(m) was determined to be 0.95, which lies between line-arly elastic and perfectly plastic deformation behavior.

The elastoplastic relation for indentation was also eval-uated with respect to particle diameter and contact angle.Similar results were obtained for the 3.25, 10 and 14 nmdiameter particle models as those shown for the 7.04 nmdiameter particle model in Fig. 2. As shown in Fig. 3, byplotting indentation based on contact radius, we observedthat although the contact size changed with contact anglefor the same particle size, Eq. (12) accurately predicts theindentation measured from the MD simulation results.

Next we evaluated the normal contact force from theMD simulations using the formulation by Storakers et al.[28], who used the work hardening power law andexpressed the normal load as a function of indentationby the relation

F N ¼ 3pr0ð2m=2Þð3�mÞcðmÞ2þmðr1�m=2c Þðh1þm=2Þ ð14Þ

Due to the variation in contact angle between the parti-cles, a tangential loading was also applied in the simulationmodels having a contact angle greater than 0�. The tangen-tial force at the contact surface was determined from theaverage tangential stress of the spherical particles, calcu-lated by stress transformation of the global stresses fromthe simulation. Since a constant velocity was applied tothe model and the weight of the nanoparticles was negligi-ble, the magnitude of the friction force was equal to thetangential force.

3.2. Interparticle friction

From the normal and tangential contact forces, we ana-lyzed the coefficient of friction between the nanoparticles.First, we tested the validity of the Amontons–Coulombfriction law,

Fig. 3. EAM MD comparisons of indentation vs. contact radius for7.04 nm two particle compression at various contact angles.

l ¼ F T

F N

ð15Þ

by plotting FT/FN vs. the apparent contact area Aa, asshown in Fig. 4 for a 60� contact angle. Fig. 4 clearly showsan increased dependence of the coefficient of friction onapparent contact area with decreased particle size. Thus,the load independence of the Amontons–Coulomb law isviolated. Because of the nanometer scale contacts, we ex-pected the effect of adhesion on the contact area to be sig-nificant due to the intermolecular forces between theinterface of the particles. Thus, we plotted the tangentialforce vs. the normal force to obtain a relation for the coef-ficient of friction as a function of the normal load and theadhesion force. These results are shown in Fig. 5 for the 3and 14 nm nanoparticle models, with similar results ob-tained for the 7 and 10 nm nanoparticle models. We deter-mined an average constant coefficient of friction byapplying a linear approximation to the results, whichyielded the following formulation for the frictional force:

F F ¼ lF N þ F adh ð16ÞBased on the direction of the applied strain and crystal

orientation, different slip systems are activated in a mate-rial. Early experiments on fcc single crystals as a functionof orientation were performed by Nabarro et al. [36]. Morerecently, Horstemeyer et al. [37] used MD modeling toexamine crystal orientation on the stress–strain relation-ship showing the Schmid law operated at the nanoscale.The particles in the current study were oriented with con-tact angles of 0�, 30� and 60� such that a different numberof glide planes would be available for crystallographic slipduring the MD simulations. As shown in Fig. 6, the coeffi-cient of friction at the various contact angles and particlesizes was calculated for both a normal indentation andfor an oblique indentation, which was assumed to be circu-lar in this analysis. With applied strain along the [010], asfor the 0� contact model, there were initially eight activeslip systems based on the <100> lattice orientation. Fora 0� contact angle, the coefficient of friction was zero,and only particle deformation (no sliding or rotation) with

Fig. 4. EAM MD plot of FT/FN vs. apparent contact area for variousparticle sizes at 60� contact angle.

Fig. 5. EAM MD plot of tangential force vs. normal force for (a) 3 nmand (b) 14 nm particle diameter at 60� contact angle.

Fig. 6. MD simulation results for the coefficient of friction vs. particlediameter for 0�, 30� and 60� contact angles.

3582 T.W. Stone et al. / Acta Materialia 56 (2008) 3577–3584

dislocation pile-ups at the grain boundaries was observedat the interface region. For the 30� model, the coefficientof friction ranged from 0.03 to 0.06 for all of the particlesizes analyzed. Dislocation motion was observed at theparticle interface, but no slippage was observed. For the60� model, the coefficient of friction ranged from 0.06 to0.19, with an increased coefficient of friction observed withincreasing particle size. With a 60� contact, only a few dis-location pile-ups were observed at the contact, but slippage

was prominent along crystallographic slip planes. Becausethe atomic attraction was very large at this small size scale,the slippage occurred during the MD simulation along spe-cific slip planes based on the lattice orientation and thecontact angle.

Martin et al. [27] conducted a similar study relatingcoefficient of friction to contact angle using DEM. TheDEM approximation is limited up to a maximum coeffi-cient of friction of 0.192 (up to 65�). Martin et al. [27]found that beyond this threshold the coefficient of frictionincreased too rapidly with increased contact angle. Asshown in Fig. 7, the results of the MD simulation com-pared reasonably well. In the MD simulations, the coeffi-cient of friction increased linearly as the contact angleincreased between the particles, up to 14 nm, when the fric-tion increased exponentially. The MD results showed amore gradual increase in the coefficient of friction for the60� contact than the increase shown in the DEM study.

Also, the dependence of the coefficient of friction on parti-cle size is evident in the 60� model, with the friction coeffi-cient increasing with particle size. In addition, for theparticles with 60� contacts, we observed sticking at the con-tact and sliding along slip planes. As the contact angleincreased, the particle rotation increased, with very littledislocation movement at the contact. For contacts of 30�or less, increased deformation and increased dislocationpile-ups at the interface were observed with decreasing con-tact angle.

The maximum coefficient of friction obtained from theMD simulations was 0.19, which is less than half the mac-roscopic coefficient of friction of 0.5 typically measured fornickel on nickel [38]. These MD simulation results qualita-tively compared favorably to multiscale experimental stud-ies on wear [39,40] for electrodeposited pure nickel coatingswith average grain sizes ranging from 8 nm to 61 lm. In theexperiments the steady-state friction coefficient rangedfrom 0.12 to 0.32 for the nanocrystalline (<100 nm) sur-faces with averages grain sizes of 8–30 nm. Friction coeffi-cient measurements of 0.2 for the ultrafine (100–900 nm)surface [39] and 0.3–0.6 for the microcrystalline (>1 lm)surfaces [39,40] were also obtained. Reduced friction coef-ficient with decreased grain size suggest a length scale effectof friction as observed by other researchers [9,14–18]. TheMD simulation in the current study captured this effect,as well as showed the dependence of an adhesion force atthe nanoscale. Nanocrystalline materials have a high grainboundary interfacial area/volume ratio, which has an effecton the coefficient of friction. Because contact area is depen-dent on grain size, the interfacial surface area and volumewere calculated based on grain size and compared to thecoefficient of friction, as shown in Fig. 8. With decreasinggrain size, the surface-to-volume ratio increases and sur-face effects, such as friction, start to dominate. Thus,Fig. 8 shows a length scale effect of friction for nickel, rang-ing from the nanoscale through the microscale [39,40] tothe macroscale [41], and compares MD simulation resultsto previous experimental work.

Fig. 7. Comparison of coefficient of friction vs. contact angle using EAM MD simulations and DEM.

Fig. 8. Coefficient of friction for nickel vs. grain boundary interfacialsurface area per volume comparing EAM MD simulations and experi-ments.

T.W. Stone et al. / Acta Materialia 56 (2008) 3577–3584 3583

Other MD simulations [42–45] have revealed that as thespecimen size decreased, the stress state correspondinglyincreased as a result of dislocation nucleation. Clearly,the results of this study show that friction is length scaledependent and the mechanisms that produce a size effecton the deformational stresses from dislocation nucleationplay a role in interparticle contact friction at the nanoscaleas well.

4. Conclusions

A study of contact behavior for the uniaxial compres-sion of two single crystal nickel nanoparticles using MDsimulations was presented. The MD simulations were per-formed using an EAM potential for nickel. The effects ofparticle size and contact angle on the coefficient of frictionwere examined, quantified and compared to previous stud-ies. Particle indentation and contact forces were consideredin the analysis. From the indentation, the normal contactforce was determined using an equation for a normal plas-tic contact force based on the power law. The tangential

loading was calculated from the MD simulations by trans-forming the average shear stress of the two particles alongthe contact interface direction. Then, using the Amontons–Coulomb friction law and calculating the slope of the curvefor the tangential force vs. the normal force, we determinedthe average coefficient of friction between the particles.

The measured indentation from the MD simulationsbased on particle size correlated nicely with the predictedindentation, which is dependent upon particle size. Thisagreement is important since the normal plastic force cal-culation is a function of indentation. The angle of con-tact between the nanoparticles showed little effect onindentation.

Next we determined the relation between particle sizeand friction, as well as between contact angle and friction.No dependence of the friction coefficient on particle sizewas observed for the 0� contact model. In general, no slid-ing was observed at the contact, only a state of sticking andparticle deformation. However, we found that the contactangle between the particles had the most significant effecton the coefficient of friction. For contacts of 45� or less,the coefficient of friction was between 0.03 and 0.06, withlittle variation noted between the particle sizes. As the con-tact angle decreased, increased deformation and disloca-tion pile-ups at the interfaces were observed. Forparticles with 60� contact angles, the coefficient of frictionranged from 0.06 for the 3 nm particles to 0.19 for the10 nm particles, and we observed increased particle rota-tion with increased contact angle, as well as sticking andvery little dislocation movement at the contact. The nano-scale coefficient of friction values were less than the typicalmacroscopic friction values, which range between 0.5 and0.8. However, the results of the MD simulation were con-sistent with previous nanoscale experiments and comparedreasonably well with previous work using DEM. This studydemonstrated the applicability of macroscale contact lawsto describe nanoscale deformation behavior and the need

3584 T.W. Stone et al. / Acta Materialia 56 (2008) 3577–3584

to use a modified Amontons–Coulomb friction law thatincludes a length scale parameter to account for the effectsof adhesion in modeling nanoscale friction.

Acknowledgements

This research is based upon work supported by aNational Science Foundation (NSF) Graduate ResearchFellowship. The authors would like to acknowledge addi-tional funding from the Center for Advanced VehicularSystems at Mississippi State University and from USAMPAMD 410. The assistance of Bohumir Jelinek, MississippiState University, on the modification of the WARP code isgreatly appreciated.

References

[1] Hall EO. Proc Phys Soc B 1951;64:747.[2] Petch NJ. J Iron Steel Inst 1953;174:25.[3] Siow KS, Tay AO, Oruganti P. Mater Sci Technol 2004;20:285.[4] Zhou Y, Erb U, Aust KT, Palumbo G. Z Metallkd 2003;94(10):1157.[5] Van Swygenhoven H, Spaczer M, Caro A, Farkas D. Phys Rev B

1999;60:22.[6] Amontons G. Mem Acad R Sci 1699:206.[7] Coulomb CA de. Mem Math Phys Acad R Sci 1781:161. Reprinted

Paris: Bachelier; 1821.[8] Homola AW, Israelachvili JN, McGuiggan PM, Gee ML. Wear

1990;136:65.[9] Carpick RW, Agrait N, Ogletree DF, Salmeron M. J Vacuum Sci

Technol B 1996;14:1289.[10] Bhushan B, Nosonovsky M. Acta Mater 2003;51:4331.[11] Zhang LC, Johnson KL, Cheong WCD. Tribol Lett 2001;10:23.[12] Bowden FP, Tabor D. The friction and lubrication of sol-

ids. Oxford: Clarendon Press; 1950.[13] Johnson KL. Proc R Soc London 1997;A453:163.[14] Hurtado JA, Kim KS. Proc R Soc London 1999;A455:3363.[15] Hurtado JA, Kim KS. Proc R Soc London 1999;A455:3385.[16] Adams GG, Muftu SS, Mohd Azhar N. ASME J Tribol 2003;125: 700.[17] Dominik C, Tielens AGGM. Philos Mag A 1996;73:1279.[18] Deshpande VS, Needleman A, Vander Giessen E. Acta Mater

2004;52:3135.

[19] Kogut L, Etsion I. ASME J Tribol 2004;126:34.[20] Kum O. Mol Simul 2005;31(2–3):115.[21] Rekhi S, Saxena K, Ahuja R, Johansson B, Hu J. J Mater Sci

2001;36:4719.[22] Hertz H. Uber die beruhrung fester elasticher korper (On the contact

of elastic solids). In: Jones, Schott. editors. Miscellaneous Papers byH. Hertz. London: Macmillan; 1896.

[23] Morrison HL, Richmond O. Powder Technol 1976;14(1):153.[24] Matthews JR. Acta Mater 1980;28:311.[25] Johnson KL. Contact mechanics. 1st ed. Cambridge: Cambridge

University Press; 1987.[26] Li H, Saigal S, Wang PT. Acta Mater 1996;44(7):2591.[27] Martin CL, Bouvard D, Shima S. J Mech Phys Solids 2003;51(4):667.[28] Storakers B, Biwa S, Larsson PL. Int J Solids Struct 1997;34:3061.[29] Johnson KL, Kendall K, Roberts AD. Proc R Soc London

1971;A324:301.[30] Derjaguin BV, Muller VM, Toporov YP. J Colloid Interface Sci

1975;53:314.[31] Daw MS, Baskes MI. Phys Rev B 1984;29(12):6443.[32] Kelchner CL, Plimpton SJ, Hamilton JC. Phys Rev B 1998;58(17):

11085.[33] Frenkel D, Smit B. Understanding molecular simulation: from

algorithms to applications. 2nd ed. San Diego, CA: Academic Press;2002.

[34] Gullett PM, Wagner G, Slepoy A. Numerical tools for atomisticsimulations. SAND 2003-8782; January 2004.

[35] Larsson J, Storakers B. Eur J Mech A/Solids 2000;19:565.[36] Nabarro FRN, Basinski ZS, Holt DB. Adv Phys 1964;13:193.[37] Horstemeyer MF, Baskes MI, Godfrey A, Hughes DA. Int J Plast

2002;18:203.[38] Avallone EA, Baumeister III T. Marks’ standard handbook for

mechanical engineers. 9th ed. New York: McGraw-Hill; 1987.[39] Hanlon T, Chokshi AH, Manoharan M, Suresh S. Int J Fatigue

2005;27:1159.[40] Mishra R, Basu B, Balasubramaniam R. Mater Sci Eng A

2004;373:370.[41] Surender M, Basu B, Balasubramaniam R. Tribol Int 2004;37:743.[42] Horstemeyer MF, Baskes MI. J Eng Mater Technol Trans ASME

1998;121:114.[43] Horstemeyer MF, Baskes MI, Plimpton SJ. Theor Appl Fracture

Mech 2001;37(1–3):49.[44] Horstemeyer MF, Plimpton SJ, Baskes MI. Acta Mater 2001;49:4363.[45] Gerberich WW, Tymak NI, Grunlan JC, Horstemeyer MF, Baskes

MI. J Appl Mech 2002;69(4):443.