Tài liệu Handbook of Micro and Nano Tribology P11 docx

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

© 1999 by CRC Press LLC

11

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

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(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

© 1999 by CRC Press LLC

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.

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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 F is the force on a particle, m is its mass, a its acceleration, v its velocity, and t is time, yield a set

of 3n (where n 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 1

/25

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 F 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

F a v = = m m

d

dt ,

© 1999 by CRC Press LLC

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 Eb can be written as a sum over pairs of atoms,

(11.2)

The indexes i and j are atom labels, rij is the scalar distance between atoms i and j, and Vpair(rij) is an

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

(11.3)

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

expression are the Morse potential (X = e –βrij), and the Lennard–Jones (LJ) “12-6” potential (X = (σ/rij)6

).

(β and σ 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/r6

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/r 6

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 qi. These point charges then interact with a pair potential

Because the 1/r 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

E Vr b ij

i j i

= ( )

( ) >

∑∑ pair .

V r D XX pair( )ij =⋅ − ( )1 ,

V r q q

r ij

i j

ij

pair ( ) = .

EV V V

i k j i l j kji

b body body body = + + +… ∑ ∑ ∑ −− − ∑ ∑ ∑ ∑∑∑ 1

2

1

3

1

4 23 4 ! ! .

© 1999 by CRC Press LLC

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 Φ(rij) represents the core–core repulsion, F is the embedding function, and ρ(rij) is the

contribution to the electron density at site i from atom j. 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).

E E b i

i

=∑ ,

E rF r i ij

j

ij

j

= ( ) + ( ) 











 ∑ ∑ 

1

2 Φ ρ .