Bhushan B. Handbook of Micro/Nano Tribology, Second Edition

Подождите немного. Документ загружается.

Tabor, D. (1970), “The Hardness of Solids,” Rev. Phys. Technol. 1, 145–179.

Tangena, A.G. and Hurkx, G.A.M. (1986), “The Determination of Stress-Strain Curves of Thin Layers

Using Indentation Tests,” ASME J. Eng. Mater. Technol. 108, 230–232.

Ternovskii, A.P., Alekhin, V.P., Shorshorov, M. Kh., Khrushchov, M.M., and Skvortsov, V.N. (1973),

“Micromechanical Testing of Materials by Depression,” Zavod. Lab. 39, 1620–1624.

Townsend, P.H., Weihs, T.P., Sanchez, J.E., and Borgesen, P. (eds.) (1993), Thin Films: Stresses and

Mechanical Properties IV, Vol. 308, Materials Research Society, Pittsburgh.

Tsukamoto, Y., Yamaguchi, H., and Yanagisawa, M. (eds.) (1987), “Mechanical Properties of Thin Films:

Measurements of Ultramicroindentation Hardness, Young’s Modulus and Internal Stresses,” Thin

Solid Films 154, 171–181.

Tsukamoto, Y., Kuroda, H., Sato, A., and Yamaguchi, H. (1992), “Microindentation Adhesion Tester and

Its Applications to Thin Films,” Thin Solid Films 213, 220–225.

Valli, J. (1986), “A Review of Adhesion Test Method for Thin Hard Coatings,” J. Vac. Sci. Technol. A 4,

3007–3014.

Vancoille, E., Celis, J.P., and Roos, J.R. (1993), “Mechanical Properties of Heat Treated and Worn TiN,

(Ti, Al)N, (Ti, Nb)N and Ti(C,N) Coatings as Measured by Nanoindentation,” Thin Solid Films

224, 168–176.

Vitovec, F.H. (1986), “Stress and Load Dependence of Microindentation Hardness,” in Microindentation

Techniques in Materials Science and Engineering (P.J. Blau and B.R. Lawn, eds.), STP 889,

pp. 175–185, ASTM, Philadelphia.

Walker, W.W. (1973), The Science of Hardness Testing and its Research Applications (J.H. Westbrook and

H. Conrad, eds.), pp. 258–273, American Society of Metals, Metals Park, OH.

Wang, M., Schmidt, K., Reichelt, K., Dimigen, H., and Hubsch, H. (1992), “Characterization of Metal-

Containing Amorphous Hydrogenated Carbon Films,” J. Mater. Res. 7, 667–676.

Was, G.S. (1990), “Surface Mechanical Properties of Aluminum Implanted Nickel and Co-Evaporated

Ni-Al on Nickel,” J. Mater. Res. 5, 1668–1683.

Was, G.S. and DeKoven, B.M. (1991), “Hardness and Friction of N

and Al

Implanted B

C,” Nucl. Inst.

Meth. Phys. Res. B 59/60, 802–805.

Weihs, T.P., Hong, S., Bravman, J.C., and Nix, W.D. (1988), “Mechanical Deﬂection of Cantilever Micro-

beams: A New Technique for Testing the Mechanical Properties of Thin Films,” J. Mater. Res. 3,

931–942.

Weihs, T.P., Lawrence, C.W., Derby, C.B., and Pethica, J.B. (1992), “Acoustic Emissions during Indentation

Tests,” Thin Films: Stresses and Mechanical Properties III, Symp. Proc. (W.D. Nix, J.C. Bravman, E.

Arzt, and L.B. Freund, eds.), Vol. 239, pp. 361–370, Materials Research Society, Pittsburgh.

Westbrook. J.H. (1957), Proc. Am. Soc. Test. Mater. 57, 873.

Westbrook, J.H. and Conrad, H. (eds.) (1973), The Science of Hardness and Its Research Applications,

American Society of Metals, Metals Park, OH.

White, R.L., Nelson, J., and Gerberich, W.W. (1993), “Residual Stress Effects in the Scratch Adhesion

Testing of Tantalum Thin Films,” in Thin Films: Stresses and Mechanical Properties IV (P.H.

Townsend, T.P. Weihs, J.E. Sanches, and P. Borgesen, eds.), Vol. 239, pp. 141–146, Materials Research

Society, Pittsburgh.

Whitehead, A.J. and Page, T.F. (1992), “Nanoindentation Studies of Thin Film Coated Systems,” Thin

Solid Films 220, 277–283.

Wierenga, P.E. and Franken, A.J.J. (1984), “Ultramicroindentation Apparatus for the Mechanical Char-

acterization of Thin Films,” J. Appl. Phys. 55, 4244–4247.

Wierenga, P.E., and van der Linden, J.H.M. (1986), “Quasistatic and Dynamic Indentation Measurements

on Magnetic Tapes,” in Tribology and Mechanics of Magnetic Storage Systems, Vol. 3 (B. Bhushan

and N.S. Eiss, eds.), pp. 31–37, SP-21, ASLE, Park Ridge, IL.

Williams, V.S., Landesman, A.L., Shack, R.V., Vukobratovich, D., and Bhushan, B. (1988), “In Situ

Microviscoelastic Measurements by Polarization-Interferometric Monitoring of Indentation

Depth,” Appl. Opt. 27, 541–546.

Wu, T.W. (1990), “Microscratch Test for Ultra-Thin Films,” in Thin Films: Stresses and Mechanical

Properties II (M.F. Doerner, W.C. Oliver, G.M. Pharr, and F.R. Brotzen, eds.), Symp. Proc., Vol. 188,

pp. 191–205, Materials Research Society, Pittsburgh.

Wu, T.W. (1991), “Microscratch and Load Relaxation Tests for Ultra-Thin Films,” J. Mater. Res. 6,

407–426.

Wu, T.W. (1993), “The a.c. Indentation Technique and Its Applications,” Mater. Chem. Phys. 33, 15–30.

Wu., T.W. and Lee, C.K. (1994), “The Micro-Wear Technique and Its Application to Ultrathin Film

Systems,” J. Mater. Res. 9, 805–811.

Wu, T.W., Hwang, C., Lo. J., and Alexopoulos, P. (1988), “Microhardness and Microstructure of Ion-

Beam-Sputtered, Nitrogen Doped NiFe Films,” Thin Solid Films 166, 299–308.

Wu, T.W., Burn, R.A., Chen, M.M., and Alexopoulos, P.S. (1989), “Micro-Indentation and Micro-Scratch

Tests on Sub-Micron Carbon Films,” Symp. Proc., Vol. 130, pp. 117–121, Materials Research Society,

Pittsburgh.

Wu, T.W. Moshref, M., and Alexopoulos, P.S. (1990a), “The Effect of the Interfacial Strength on the

Mechanical Properties of Aluminum Films,” Thin Solid Films 187, 295–307.

Wu, T.W., Shull, A.L., and Lin, J. (1990b), “Microscratch Test on Carbon Films as Thin as 20 nm,” Symp.

Proc., Vol. 188, pp. 207–212, Materials Research Society, Pittsburgh.

Wu, T.W., Shull, A.L., and Berriche, R. (1991), “Microindentation Fatigue Tests on Submicron Carbon

Films,” Surf. Coat. Technol. 47, 696–709.

Yanagisawa, M. and Motomura, Y. (1987), “An Ultramicro Indentation Hardness Tester and Its Applica-

tion to Thin Films,” Lubr. Eng. 43, 52–56.

Yanagisawa, M. and Motomura, Y. (1989), “Apparatus for Determining Microhardness,” U.S. Patent 4,

820,051.

Yeack-Scranton, C.E. (1986), “Novel Piezoelectric Transducer to Monitor Head-Disk Interactions,” IEEE

Trans. Magn. 22, 1011–1016.

Harrison, J.A. et al. “Atomic-Scale Simulation of Tribological and Related...”

Handbook of Micro/Nanotribology.

Ed. Bharat Bhushan

Boca Raton: CRC Press LLC, 1999

Atomic-Scale

Simulation of

Tribological and

Related Phenomena

Judith A. Harrison, Steven J. Stuart,

and Donald W. Brenner

11.1 Introduction

11.2 Molecular Dynamics Simulations

Interatomic Potentials • Thermodynamic Ensemble •

Temperature Regulation

11.3 Nanometer-Scale Material Properties: Indentation,

Cutting, and Adhesion

Indentation of Metals • Indentation of Metals Covered by

Thin Films • Indentation of Nonmetals • Cutting of

Metals • Adhesion

11.4 Lubrication at the Nanometer Scale: Behavior of

Thin Films

Equilibrium Properties of Conﬁned Thin Films • Behavior of

Thin Films under Shear

11.5 Friction

Solid Lubrication • Friction in the Presence of a Third

Body • Tribochemistry

11.6 Summary

Acknowledgments

References

11.1 Introduction

Understanding and ultimately controlling friction and wear have long been recognized as important to

many areas of technology. Historical examples include the Egyptians, who had to invent new technologies

to move the stones needed to build the pyramids (Dowson, 1979); Coulomb, whose fundamental studies

of friction were motivated by the need to move ships easily and without wear from land into the water

(Dowson, 1979); and Johnson et al. (1971), whose study of automobile windshield wipers led to a better

understanding of contact mechanics, including surface energies. Today, the development of microscale

(and soon nanoscale) machines continues to challenge our understanding of friction and wear at their

most fundamental levels.

Our knowledge of friction and related phenomena at the atomic scale has rapidly advanced over the

last decade with the development of new and powerful experimental methods. The surface force apparatus

(SFA), for example, has provided new information related to friction and lubrication for many liquid

and solid systems with unprecedented resolution (Israelachvili, 1992). The friction force and atomic force

microscopes (FFM and AFM) allow the frictional and mechanical properties of solids to be characterized

with atomic resolution under single-asperity contact conditions (Binnig et al., 1986; Mate et al., 1987;

Germann et al., 1993; Carpick and Salmeron, 1997). Other techniques, such as the quartz crystal

microbalance, are also providing exciting new insights into the origin of friction (Krim et al., 1991; Krim,

1996). Taken together, the results of these studies have revolutionized the study of friction, wear, and

mechanical properties, and have reshaped many of our ideas about the fundamental origins of friction.

Concomitant with the development and use of these innovative experimental techniques has been the

development of new theoretical methods and models. These include analytic models, large-scale molec-

ular dynamics (MD) simulations, and even ﬁrst-principles total-energy techniques (Zhong and Tomanek,

1990). Analytic models have had a long history in the study of friction. Beginning with the work of

Tomlinson (1929) and Frenkel and Kontorova (1938), through to recent studies by McClelland and Glosli

(1992), Sokoloff (1984, 1990, 1992, 1993, 1996), and others (Helman et al., 1994; Persson, 1991), these

idealized models have been able to break down the complicated motions that create friction into basic

components deﬁned by quantities such as spring constants, the curvature and magnitude of potential

wells, and bulk phonon frequencies. The main drawback of these approaches is that simplifying assump-

tions must be made as part of these models. This means, for example, that unanticipated defect structures

may be overlooked, which may strongly inﬂuence friction and wear even at the atomic level.

Molecular dynamics computer simulations, which are the topic of this chapter, represents a compro-

mise between analytic models and experiment. On the one hand, this method deals with approximate

interatomic forces and classical dynamics (as opposed to quantum dynamics), so it has much in common

with analytic models (for a comparison of analytic and simulation results, see Harrison et al., 1992c).

On the other hand, simulations often reveal unanticipated events that require further analysis, so they

also have much in common with experiment. Furthermore, a poor choice of simulation conditions, as

in an experiment, can result in meaningless results. Because of this danger, a thorough understanding

of the strengths and weaknesses of MD simulations is crucial to both successfully implementing this

method and understanding the results of others.

On the surface, atomistic computer simulations appear rather straightforward to carry out: given a

set of initial conditions and a way of describing interatomic forces, one simply integrates classical

equations of motion using one of several standard methods (Gear, 1971). Results are then obtained from

the simulations through mathematical analysis of relative positions, velocities, and forces; by visual

inspection of the trajectories through animated movies; or through a combination of both (Figure 11.1).

However, the effective use of this method requires an understanding of many details not apparent in this

simple analysis. To provide a feeling for the details that have contributed to the success of this approach

in the study of adhesion, friction, and wear as well as other related areas, the next section provides a

brief review of MD techniques. For a more detailed overview of MD simulations, including computer

algorithms, the reader is referred to a number of other more comprehensive sources (Hoover, 1986;

Heermann, 1986; Allen and Tildesley, 1987; Haile, 1992).

The remainder of this chapter presents recent results from MD simulations dealing with various aspects

of mechanical, frictional, and wear properties of solid surfaces and thin lubricating ﬁlms. Section 11.2

summarizes several of the technical details needed to perform (or understand) an MD simulation. These

range from choosing an interaction potential and thermodynamic ensemble to implementing tempera-

ture controls. Section 11.3 describes simulations of the indentation of metals and nonmetals, as well as

the machining of metal surfaces. The simulations discussed reveal a number of interesting phenomena

and trends related to the deformation and disordering of materials at the atomic scale, some (but not

all) of which have been observed at the macroscopic scale. Section 11.4 summarizes the results of

simulations that probe the properties of liquid ﬁlms conﬁned to thicknesses on the order of atomic

dimensions. These systems are becoming more important as demands for lubricating moving parts

approach the nanometer scale. In these systems, ﬂuids have a range of new properties that bear little

resemblance to liquid properties on macroscopic scales. In many cases, the information obtained from

these studies could not have been obtained in any other way, and is providing unique new insights into

recent observations made by instruments such as the SFA. Section 11.5 discusses simulations of the

tribological properties of solid surfaces. Some of the systems discussed are sliding diamond interfaces,

Langmuir–Blodgett ﬁlms, self-assembled monolayers, and metals. The details of several unique mecha-

nisms of energy dissipation are discussed, providing just a few examples of the many ways in which the

conversion of work into heat leads to friction in weakly adhering systems. In addition, simulations of

molecules trapped between, or chemisorbed onto, diamond surfaces will be discussed in terms of their

effects on the friction, wear, and tribology of diamond. A summary of the MD results is given in the

ﬁnal section.

11.2 Molecular Dynamics Simulations

Atomistic computer simulations are having a major impact in many areas of the chemical, physical,

material, and biological sciences. This is largely due to enormous recent increases in computer power,

FIGURE 11.1

Flow chart of an MD simulation.

increasingly clever algorithms, and recent developments in modeling interatomic interactions. This last

development, in particular, has made it possible to study a wide range of systems and processes using

large-scale MD simulations. Consequently, this section begins with a review of the interatomic interac-

tions that have played the largest role in friction, indentation, and related simulations. For a slightly

broader discussion, the reader is referred to a review by Brenner and Garrison (1989). This is followed

by a brief discussion of thermodynamic ensembles and their use in different types of simulations. The

section then closes with a description of several of the thermostatting techniques used to regulate the

temperature during an MD simulation. This topic is particularly relevant for tribological simulations

because friction and indentation do work on the system, raising its kinetic energy.

11.2.1 Interatomic Potentials

Molecular dynamics simulations involve tracking the motion of atoms and molecules as a function of

time. Typically, this motion is calculated by the numerical solution of a set of coupled differential

equations (Gear, 1971; Heermann, 1986; Allen and Tildesley, 1987). For example, Newton’s equation of

motion,

(11.1)

where

is the force on a particle,

is its mass,

its acceleration,

its velocity, and

is time, yield a set

of 3

(where

is the number of particles) second-order differential equations that govern the dynamics.

These can be solved with ﬁnite-time-step integration methods, where time steps are on the order of

of a vibrational frequency (typically tenths to a few femtoseconds) (Gear, 1971). Most current simulations

then integrate for a total time of picoseconds to nanoseconds. The evaluation of these equations (or any

of the other forms of classical equations of motion) requires a method for obtaining the force

between

atoms.

Constraints on computer time generally require that the evaluation of interatomic forces not be

computationally intensive. Currently, there are two approaches that are widely used. In the ﬁrst, one

assumes that the potential energy of the atoms can be represented as a function of their relative atomic

positions. These functions are typically based on simpliﬁed interpretations of more general quantum

mechanical principles, as discussed below, and usually contain some number of free parameters. The

parameters are then chosen to closely reproduce some set of physical properties of the system of interest,

and the forces are obtained by taking the gradient of the potential energy with respect to atomic positions.

While this may sound straightforward, there are many intricacies involved in developing a useful potential

energy function. For example, the parameters entering the potential energy function are usually deter-

mined by a limited set of known system properties. A consequence of that is that other properties,

including those that might be key in determining the outcome of a given simulation, are determined

solely by the assumed functional form. For a metal, the properties to which a potential energy function

might be ﬁt might include the lattice constant, cohesive energy, elastic constants, and vacancy formation

energy. Predicted properties might then include surface reconstructions, energetics of interstitial defects,

and response (both elastic and plastic) to an applied load. The form of the potential is therefore crucial

if the simulation is to have sufﬁcient predictive power to be useful.

The second approach, which has become more useful with the advent of powerful computers, is the

calculation of interatomic forces directly from ﬁrst-principles (Car and Parrinello, 1985) or semiempirical

(Menon and Allen, 1986; Sankey and Allen, 1986) calculations that explicitly include electrons. The

advantage of this approach is that the number of unknown parameters may be kept small, and, because

the forces are based on quantum principles, they may have strong predictive properties. However, this

does not guarantee that forces from a semiempirical electronic structure calculation are accurate; poorly

chosen parameterizations and functional forms can still yield nonphysical results. The disadvantage of

this approach is that the potentials involved are considerably more complicated, and require more

computational effort, than those used in the classical approach. Longer simulation times require that

==mm

both the system size and the timescale studied be smaller than when using more approximate methods.

Thus, while this approach has been used to study the forces responsible for friction (Zhong and Tomanek,

1990), it has not yet found widespread application for the type of large-scale modeling discussed here.

The simplest approach for developing a continuous potential energy function is to assume that the

binding energy

can be written as a sum over pairs of atoms,

(11.2)

The indexes

and

are atom labels,

is the scalar distance between atoms

and

pair

(

) is an

assumed functional form for the energy. Some traditional forms for the pair term are given by

(11.3)

where the parameter

determines the minimum energy for pairs of atoms. Two common forms of this

expression are the Morse potential (

–

), and the Lennard–Jones (LJ) “12-6” potential (

= (

)

(

and

are arbitrary parameters that are used to ﬁt the potential to observed properties.)

The short-range exponential form for the Morse function provides a reasonable description of repulsive

forces between atomic cores, while the 1/

term of the LJ potential describes the leading term in long-

range dispersion forces. A compromise between these two is the “exponential-6,” or Buckingham, poten-

tial. This uses an exponential function of atomic distances for the repulsion and a 1/

form for the

attraction. The disadvantage of this form is that as the atomic separation approaches zero, the potential

becomes inﬁnitely attractive.

For systems with signiﬁcant Coulomb interactions, the approach that is usually taken is to assign each

atom a fractional point charge

. These point charges then interact with a pair potential

Because the 1/

Coulomb interactions act over distances that are long compared to atomic dimensions,

simulations that include them typically must include large numbers of atoms, and often require special

attention to boundary conditions (Ewald, 1921; Heyes, 1981).

Other forms of pair potentials have been explored, and each has its strengths and weaknesses. However,

the approximation of a pairwise-additive binding energy is so severe that in most cases no form of pair

potential will adequately describe every property of a given system (an exception might be rare gases).

This does not mean that pair potentials are without use — just the opposite is true! A great many general

principles of many-body dynamics have been gleaned from simulations that have used pair potentials,

and they will continue to ﬁnd a central role in computer simulations. As discussed below, this is especially

true for simulations of the properties of conﬁned ﬂuids.

A logical extension of the pair potential is to assume that the binding energy can be written as a many-

body expansion of the relative positions of the atoms

(11.4)

Normally, it is assumed that this series converges rapidly so that four-body and higher terms can be

ignored. Several functional forms of this type have had considerable success in simulations. Stillinger

and Weber (1985), for example, introduced a potential of this type for silicon that has found widespread

EVr

bij

()

∑∑

pair

Vr DXX

ijpair

()

=⋅ −

()

pair

()

= .

EV V V

jikjilkji

b body body body

=+ + +…

−− −

∑∑∑∑∑∑∑∑∑

23 4

use, an example of which is discussed in Section 11.3.3. Another example is the work of Murrell and co-

workers (1984) who have developed a number of potentials of this type for different gas-phase and

condensed-phase systems.

A common form of the many-body expansion is a valence force ﬁeld. In this approach, interatomic

interactions are modeled with a Taylor series expansion in bond lengths, bond angles, and torsional

angles. These force ﬁelds typically include some sort of nonbonded interaction as well. Prime examples

include the molecular mechanics potentials pioneered by Allinger and co-workers (Allinger et al., 1989;

Burkert and Allinger, 1982). A common variation of the valence-force approach is to assume rigid bonds,

and allow only changes in bond angles. Because the angle bends generally have smaller force constants,

there tend to be larger variations in angles than in bond lengths, so this approximation often gives accurate

predictions for the shapes of large molecules at thermal energies. The advantage of this approximation

is that because the bending modes have lower frequencies than those involving bond stretching modes,

time steps may be used that are an order of magnitude larger than those required for ﬂexible bonds, with

no larger numerical errors in the total energy.

One method of including many-body effects in Coulomb systems is to account for electrostatic

induction interactions. Each point charge will give rise to an electric ﬁeld, and will induce a dipole

moment on neighboring atoms. This effect can be modeled by including terms for the atomic or molecular

dipoles in the interaction potential, and solving for the values of the dipoles at each step in the dynamics

simulation. An alternative method is to simulate the polarizability of a molecule by allowing the values

of the point charges to change directly in response to their local environment (Streitz and Mintmire,

1994; Rick et al., 1994). The values of the charges in these simulations are determined by the method of

electronegativity equalization and may either by evaluated iteratively (Streitz and Mintmire, 1994) or

carried as dynamic variables in the simulation (Rick et al., 1994).

Several potential energy expressions beyond the many-body expansion have been successfully devel-

oped and are widely used in MD simulations. For metals, the embedded atom method (EAM) and related

methods have been highly successful in reproducing a host of properties, and have opened up a range

of phenomena to simulation (Finnis and Sinclair, 1984; Foiles et al., 1986; Ercollessi et al., 1986a,b). These

have been especially useful in simulations of the indentation of metals, as discussed in Section 11.3. This

approach is based on ideas originating from effective medium theory (Norskov and Lang, 1980; Stott

and Zaremba, 1980). In this formalism, the energy of an atom interacting with surrounding atoms is

approximated by the energy of the atom interacting with a homogeneous electron gas and a compensating

positive background. The EAM assumes that the density of the electron gas can be approximated by a

sum of electron densities from surrounding atoms, and adds a repulsive term to account for core–core

interactions. Within this set of approximations, the total binding energy is given as a sum over atomic sites

(11.5)

where each site energy is given as a pair sum plus a contribution from a functional (called an embedding

function) that, in turn, depends on the sum of electron densities at that site:

(11.6)

The function

(

) represents the core–core repulsion,

is the embedding function, and

(

) is the

contribution to the electron density at site

from atom

. For practical applications, functional forms are

assumed for the core–core repulsion, the embedding function, and the contribution of the electron

densities from surrounding atoms. For a more complete description of this approach, including a much

more formal derivation and discussion, the reader is referred to Sutton (1993).

∑

ErFr

iij

()













∑∑

Φρ.

For close-packed metals, it has been found that the electron densities in the solid can be adequately

approximated by a pairwise sum of atomic-like electron densities. With this approximation, the com-

puting time required to evaluate an EAM potential scales with the number of atoms in the same way as

a pair potential. The quantitative results of the EAM, however, are dramatically better than those obtained

with pair potentials. Energies and structures of solid surfaces and defects in metals, for example, match

experimental results (or more-sophisticated calculations) reasonably well despite the relatively simple

analytic form. Because of this, EAM potentials have been used extensively in studies of indentation.

Schemes similar to EAM but based on other levels of approximation have also been developed. For

example, three-body terms in the electron-density contributions have been studied for modeling a range

of metallic and covalently bonded systems (Baskes, 1992). Another example is the work of DePristo and

co-workers, who have studied various hierarchies of approximation within effective medium theory by

including additional correction terms to Equation 11.6 (Raeker and DePristo, 1991), as have Norskov

and co-workers (Jacobson et al., 1987).

A potential that is similar to the EAM but based on bond orders has also been developed (Abell, 1985).

Originally adapted by Tersoff (1986) to model silicon, the approach has found use in computer simula-

tions of a wide range of covalently bonded systems (Tersoff, 1989; Brenner, 1989a, 1990; Khor and Das

Sarma, 1988). Like the embedded atom potentials, Tersoff potentials begin by approximating the binding

energy of a system as a sum over sites:

(11.7)

In this case, however, each site energy is given by an expression that resembles a pair potential,

(11.8)

The functions

(

) and

(

) represent pairwise-additive attractive and repulsive terms, respectively,

and

represents an empirical bond-order expression that modulates the attractive pair potential. This

bond-order term is where the many-body effects are introduced. As in the other potentials, functional

forms for the bond order and the pair terms are ﬁt to a range of properties for the systems of interest.

Applications of this approach have assumed functional forms for the bond order that decrease its value

as the number of nearest-neighbors of a given pair of bonded atoms increases. Physically, the attractive

pair term can be envisioned as bonding due to valence electrons, with the bond order destabilizing the

bond as the valence electrons are shared among more and more neighbors. Tersoff, Abell, and others

have shown that this simple expression can capture a range of bond energies, bond lengths, and related

properties for group IV solids. Also, if properly parameterized, the potential yields Pauling’s bond-order

relations (Abell, 1985; Khor and Das Sarma, 1988; Tersoff, 1989). These properties have been shown to

make this expression very powerful for predicting covalently bonded structures and, therefore, useful for

predicting new phenomena through MD simulations.

Although they are based on different principles, the EAM and Tersoff expressions are quite similar. In

the EAM, binding energy is deﬁned by electron densities through the embedding function. The electron

densities are, in turn, deﬁned by the arrangement of neighboring atoms. Similarly, in the Tersoff expres-

sion the binding energy is deﬁned directly by the number and arrangement of neighbors through the

bond-order expression. In fact, the EAM and Tersoff approaches have been shown to be identical for

simpliﬁed expressions provided that angular interactions are not used (Brenner, 1989b).

Although not all potential energy expressions widely used in MD simulations have been covered, this

section provides a brief introduction to those that have found the most use in the simulations of friction,

wear, and related phenomena. As mentioned above, methods that explicitly incorporate semiempirical

and ﬁrst-principles electronic structure calculations have not been discussed. As computer speeds

∑

EVrBVr

i R ij ij A ij

()

+⋅

()

[]

()

∑