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

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continue to increase and as new algorithms are developed, we expect that these more exact methods will

play an increasingly important role in tribological simulations.

11.2.2 Thermodynamic Ensemble

When performing a MD simulation, a choice must be made as to which thermodynamic ensemble to

study. These ensembles are distinguished by which thermodynamic variables are held constant over the

course of the simulation. (For a broader and more rigorous treatment of ensemble averaging, the reader

should consult any statistical mechanics text, (e.g., McQuarrie, 1976)).

Without speciﬁc reasons to do otherwise, it is quite natural to keep the number of atoms (

) and the

volume of the simulation cell (

) constant over the course of a MD simulation. In addition, for a system

without energy transfer, integrating the equations of motion (Equation 11.1) will generate a trajectory

over which the energy of the system (

) will also be conserved. A simulation of this type is thus performed

in the constant-

NVE,

or microcanonical, ensemble.

Systems undergoing sliding friction or indentation, however, require work to be performed on the

system, which raises its energy and causes the temperature to increase. In a macroscopic system, the

environment surrounding the region of tribological interest acts as an inﬁnite heat sink, removing excess

energy and helping to maintain a fairly constant temperature. Ideally, a sufﬁciently large simulation would

be able to model this same behavior. But while the thousands of atoms at an atomic-scale interface are

within reach of computer simulation, the

(10

) atoms in the experimental apparatus are not. Thus, a

thermodynamic ensemble that will more closely resemble reality will be one in which the temperature

(

), rather than the energy, is held constant. These simulations are performed in the constant-

NVT,

canonical, ensemble.

A constant temperature is maintained in the canonical ensemble by using any of a large number of

thermostats, many of which are described in the following section. What is often done in simulations of

indentation or friction is to apply the thermostat only in a region of the simulation cell that is well

removed from the interface where friction is taking place. This allows for local heating of the interface

as work is done on the system, while also providing a means for efﬁcient dissipation of excess heat. These

“hybrid”

NVE/NVT

simulations, although not rigorously a member of any true thermodynamic ensem-

ble, are very useful and quite common in tribological simulations.

A particularly troublesome system for MD simulations is the nonequilibrium dynamics of conﬁned

thin ﬁlms (see Section 11.4.2). In these systems, the constraint of constant atom number is not necessarily

applicable. Under experimental conditions, a thin ﬁlm under shear or tension is free to exchange mole-

cules with a reservoir of bulk liquid molecules, and the total atom number is certainly conserved. But

the number of atoms in the ﬁlm itself is subject to rather dramatic changes. According to some studies,

as many as half the molecules in an ultrathin ﬁlm will exit the interfacial region upon a change in registry

of the opposing surfaces (i.e., with no change in interfacial volume) (Schoen et al., 1989). Changes in

the ﬁlm particle number can be equally large under compression.

The proper conserved quantity in these simulations is not the particle number

but the chemical

potential

. During a simulation performed in the constant-

VT,

or grand canonical, ensemble, the

number of atoms or molecules ﬂuctuates to keep the chemical potential constant. A true grand canonical

MD simulation is too difﬁcult to perform for all but the simplest of liquid molecules, however, due to

the difﬁculties associated with inserting or removing molecules at bulk densities. An alternative chosen

by some authors is to mimic the experimental reservoir of bulk liquid molecules on a microscopic scale.

This involves performing a constant-

NVT

(Wang et al., 1993a,b, 1994) or constant-

NPT

(Gao et al., 1997)

simulation that explicitly includes a collection of molecules that are external to the interfaces (see

Figure 11.14). As liquid molecules drain into or are drawn from the reservoir region, the number of

particles directly between the interfaces is free to change. This method is then an approximation to the

grand canonical ensemble when only a subset of the system is considered. Two drawbacks to this method

are that the interface can extend inﬁnitely (via periodic boundary conditions) in only one dimension

instead of two, and also that a signiﬁcant number of extraneous atoms must be carried in the simulation.

An interesting alternative to the grand canonical ensemble is that chosen by Cushman and co-workers

(Schoen et al., 1989; Curry et al., 1994). They performed a series of grand canonical Monte Carlo

simulations at various points along a hypothetical sliding trajectory. These simulations were used to

calculate the correct particle numbers at a ﬁxed chemical potential, which were in turn used as inputs

to nonsliding, constant-

NVE

MD simulations at each of the chosen trajectory points. Because the system

was fully equilibrated at each step along the sliding trajectory, the sliding speed can be assumed to be

inﬁnitely slow. This offers a useful alternative to continuous MD simulations, which are currently

restricted to sliding speeds of roughly 1 m/s or greater — orders of magnitude larger than most experi-

mental studies.

11.2.3 Temperature Regulation

As was discussed in the previous section, some method of controlling the system temperature is required

in simulations involving friction or indentation. In this section, we discuss some of the many available

thermostats that are used for this purpose. For a more formal discussion of heat baths and the trajectories

that they produce, the reader is referred to Hoover (1986).

The most straightforward method for controlling heat production is simply to rescale intermittently

the atomic velocities to yield a desired temperature (Woodcock, 1971). This approach was widely used

in early MD simulations, and is often effective at maintaining a given temperature during the course of

a simulation. However, it has several disadvantages that have spurred the development of more-sophis-

ticated methods. First, there is little formal justiﬁcation. For typical system sizes, averaged quantities,

such as pressure, do not correspond to those obtained from any particular thermodynamic ensemble.

Second, the dynamics produced are not time reversible, again making results difﬁcult to analyze in terms

of thermodynamic ensembles. Finally, the rate and mode of heat dissipation are not determined by system

properties, but instead depend on how often velocities are rescaled. This may inﬂuence dynamics that

are unique to a particular system.

A more-sophisticated approach to maintaining a given temperature is through Langevin dynamics.

Originally used to describe Brownian motion, this method has found widespread use in MD simulations.

In this approach, additional terms are added to the equations of motion, corresponding to a frictional

term and a random force (Schneider and Stoll, 1978; Hoover, 1986; Kremer and Grest, 1990). The

equations of motion (see Equation 11.1) are given by

(11.9)

where

are the forces due to the interatomic potential, the quantities

and

are the particle mass and

velocity, respectively,

is a friction coefﬁcient, and

(

) represents a random “white noise” force. The

friction kernel is deﬁned in terms of a memory function in formal applications; kernels developed for

harmonic solids have been used successfully in MD simulations (Adelman and Doll, 1976; Adelman,

1980; Tully, 1980).

As with any thermostat, the atom velocities are altered in the process of controlling the temperature.

It is important to keep this in mind when using a thermostat, because it has the potential to perturb any

dynamic properties of the system being studied. To help avoid this problem, one effective approach is to

add Langevin forces only to those atoms in a region away from where the dynamics of interest occurs.

In this way, coupling to a heat bath is established away from the important action, and simpliﬁed

approximations for the friction term can be used without unduly inﬂuencing the dynamics produced by

the interatomic forces. For heat ﬂux via nuclear (as opposed to electronic) degrees of freedom in solids,

it has been shown that a reasonable approximation for the friction coefﬁcient

is 6/

times the Debye

frequency

(Adelman and Doll, 1976). Lucchese and Tully (1983) have shown that with this approxi-

mation and a sufﬁciently large reaction zone, the vibrational modes of atoms away from the bath atoms

are well described by the interatomic potential.

mmRtaF v== +

()

ξ ,

The random force can be given by a Gaussian distribution where the width, which is chosen to satisfy

the ﬂuctuation–dissipation theorem, is determined from the equation

(11.10)

The function

is the random force in Equation 11.9,

is the particle mass,

is the desired temperature,

is Boltzmann’s constant,

is time, and

is the friction coefﬁcient. The random forces are uncoupled

from those at the previous steps (as denoted by the delta function), and the width of the Gaussian

distribution from which the random force is chosen depends on the temperature.

The simpliﬁed Langevin approach outlined above does not require any feedback from the current

temperature of the system; instead, the random forces are determined solely from Equation 11.10. A

slightly different expression has been developed that eliminates the random forces and replaces the

constant-friction coefﬁcient with one that depends on the ratio of the desired temperature to current

kinetic energy of the system (measured as a temperature) (Berendsen et al., 1984). The resulting equations

of motion are

(11.11)

where

is the force due to the interatomic potential,

and

are the desired and actual temperatures,

respectively, and

v is again the particle velocity. The advantage of this approach is that it requires no

evaluation of random forces, which can be expensive for a large number of bath atoms. One disadvantage

in practice is that if the system is not pre-equilibrated to populate properly the vibrational modes, or if

nonrandom external forces are applied to the system, it can be slow to fully equilibrate the system. For

example, a simulated indentation of a solid surface requires that the bottom layers of the solid be held

rigid (Section 11.3). Compression of the surface during indentation may cause sound waves to propagate

into the bulk, reﬂect from the bottom layers, and continue to reﬂect between the surface and rigid layers.

Because the Berendsen thermostat uses a frictional force that depends on the average kinetic energy, it

would only reduce the total kinetic energy of the system, and not help dissipate a traveling wave. On the

other hand, the Langevin approach using a random force on each atom does not require feedback from

the system, and thermostats each atom individually. Thus, the random forces are much more efﬁcient

in eliminating these nonphysical reﬂecting waves.

Nonequilibrium equations of motion have also been developed to maintain a constant temperature

(Hoover, 1986). Like the Berendsen thermostat, this approach adds a friction to the interatomic forces.

However, it is derived from Gauss’ principle of least constraint, which maintains that the sum of the

squares of any constraining forces on a system should be as small as possible. Using a Lagrange multiplier,

a frictional force on each atom i of the form

(11.12)

where

(11.13)

can be derived that maintains a constant temperature. The quantity m

is the mass of atom i, v

is its

velocity, and F

is the total force on atom i due to the interatomic potential. Note that there is no target

temperature in this expression; instead, the temperature of the system when the constraint is initiated is

R R t mkT t02

()

⋅

()

ξδ .

aF v=+ −













iii

friction

=−ς ,

ς= ⋅

()

∑∑

Fv v

maintained for all time. This approach has several obvious advantages. First, it does not rely on an

approximated input such as the Debye frequency as in the simpliﬁed Langevin or Berendsen thermostats.

Heat loss and gain are therefore determined only by implicit system properties. Second, because a random

force is not required, it does not signiﬁcantly increase computational time. Third, the equations of motion

are time reversible. Finally, by differentiating total energy with respect to time, the heat loss (or gain)

due to the thermostat can be calculated directly (this is also true of the Langevin thermostat). However,

as with the Berendsen thermostat, coupling of the frictional to global properties of the system may be

slow to randomize nonphysical vibrational disturbances.

A thermostat that corresponds rigorously to a canonical ensemble has been developed by Nosé

(1984a,b). This signiﬁcant advance also adds a friction term, but one that maintains the correct distri-

bution of vibrational modes. It achieves this by adding a new dimensionless variable to the standard

classical equations of motion that can be thought of as a large heat bath, which couples to each of the

physical degrees of freedom. The actual effect of the variable, however, is to scale the coordinates of either

time or mass in the system. The dynamics of the expanded system correspond to the microcanonical

ensemble, but when projected onto only the physical degrees of freedom they generate a trajectory in

the canonical ensemble. Sampling problems associated with very small or very stiff systems can be

overcome by attaching a series of these Nosé-Hoover thermostats to the system (Martyna et al., 1992).

The resulting equations of motion are time reversible, and the trajectories can be analyzed exactly with

well-established statistical mechanical principles (Martyna et al., 1992). For a complete description of

the Nosé thermostat, its relation to other formalisms for generating classical equations of motion, and

a comparison of the dynamics generated with this approach and the others that have been reviewed here,

the reader is referred to Hoover (1986).

Each of the potential energy functions, thermodynamic ensembles, and thermostats outlined above

has advantages and disadvantages. The optimum choice depends strongly on the particular system and

process being simulated as well as on the type of information in which one is interested. For example,

general principles related to liquid lubrication in conﬁned areas may be most easily understood and

generalized from simulations that use pair potentials and may not require a thermostat. On the other

hand, if one wants to study the wear or indentation of a surface of a particular metal, then EAM or other

semiempirical potentials, together with a thermostat, may yield more reliable results. Even more-detailed

studies, including the evaluation of electronic degrees of freedom, may require interatomic forces derived

from some level of electronic structure calculation. The best way to make this choice is to understand

carefully the strong points of each of these approaches, decide what one wishes to learn from the

simulation, and form conclusions based on this careful understanding.

11.3 Nanometer-Scale Material Properties: Indentation,

Cutting, and Adhesion

Understanding material properties at the nanometer scale is crucial to developing the fundamental ideas

needed to design new coatings with tailor-made friction and wear properties. One of the ways in which

these properties is being characterized is through the use of the AFM. This technique is proving to be a

very versatile tool that can provide a rich variety of atomic-scale information pertaining to a given

tip–sample interaction (Burnham and Colton, 1993). For example, when an AFM tip (the radius of

curvature of AFM tips typically ranges from 100 Å to 100 µm) is rastered across a sample substrate and

the force on the tip perpendicular to the substrate is measured at each point, a force map of the surface

is obtained that can be related to the actual topography of the surface (Meyer et al., 1992). Rastering the

AFM tip across a substrate in the same way, but measuring the deﬂection of the tip in the lateral direction

instead, produces a friction map of the surface (Germann et al., 1993). Finally, by moving the tip

perpendicular to the surface of the substrate, the AFM can be used as a nanoindenter that probes the

mechanical properties of various substrates and thin ﬁlms (Burnham and Colton, 1989; Burnham et al.,

1990).

Adhesive contact can be examined by gradually moving the tip closer to the substrate until the two

come into contact. After contact, the tip is retracted from the surface, and any differences between the

force curves for contact and retraction reﬂect characteristics of adhesion. A plot of normal force on the

tip vs. tip–sample separation (i.e., a force curve) is typically made to record this sequence of events. A

force curve for the adhesive contact of a Ni tip with an Au substrate (denoted Ni/Au) is shown in

Figure 11.2a. This is a typical curve possessing the same qualitative features as most AFM force curves.

For instance, as the tip and the sample begin to interact, a small attractive well, due to long-range forces

(Burnham and Colton, 1993) is apparent in the force curve at large tip–sample separations (the well is

centered at a tip–sample separation of approximately 17 nm). The distance between the tip and the

sample is gradually decremented until the tip comes into contact with the sample. After contact the tip

is retracted, and adhesion between the tip and the substrate manifests itself as a hysteresis in the force

curve.

After contact, if the tip is moved farther toward the substrate, rather than away from the substrate,

indentation of the sample by the tip results. This indentation is reﬂected as a dramatic increase in force

as the tip is moved farther into the substrate (Figure 11.2b). This region of the force curve is known as

the repulsive wall region (Burnham and Colton, 1993), or when considered without the rest of the force

curve, an indentation curve. Retraction of the tip subsequent to indentation results in an enhanced

adhesion, therefore, in a larger hysteresis in the force curve. The origin of this enhanced adhesion is

discussed later.

Many types of adhesion at a tip–substrate interface are possible. Adhesion might result from the

formation of covalent chemical bonds between the tip and the sample. Alternatively, real surfaces usually

have a layer of liquid contamination on the surface that can lead to capillary formation and adhesion.

FIGURE 11.2 Experimentally measured force vs. tip-to-sample distance curves for an Ni tip interacting with an

Au substrate for contact followed by separation in (a) and contact, indentation, then separation in (b). These curves

were derived from AFM measurements taken in dry nitrogen. (From Landman, U. et al. (1990), Science 248, 454–461.

With permission.)

In metallic systems a different sort of wetting is possible; speciﬁcally, the sample can be wet by the tip

(or vice versa). The result of this wetting is the formation of “connective neck” of metallic atoms between

the tip and the sample and a consequent adhesion. Finally, entanglement of molecules that are anchored

on the tip with molecules anchored on the sample could also be responsible for an observed hysteresis.

Molecular dynamics simulations of indentations were ﬁrst employed in an effort to shed light on the

physical phenomena that are responsible for the qualitative shape of AFM force curves. In addition to

succeeding at this task, MD simulations have revealed an abundance of atomic-scale phenomena that

occur during the indentation process. In the remainder of this section, MD simulations related to

indentation and related processes are discussed.

11.3.1 Indentations of Metals

Landman et al. (1990) were among the ﬁrst groups to use MD to simulate the indentation of a metallic

substrate with a metal tip. In an early simulation, a Ni tip was used to indent a Au(001) substrate. The

tip was originally arranged as a pyramid and contained 1400 dynamic atoms and 1176 rigid atoms used

as a holder. The substrate was composed of 11 layers of Au atoms containing 450 atoms each. These

constant-temperature simulations were carried out at 300 K. The forces governing the interatomic motion

of the system were derived from EAM potentials for Ni and Au.

After equilibration of the tip and substrate to 300 K, the tip was brought into contact with surface by

moving the tip holder 0.25 Å closer to the surface every 1525 fs. This rate (or a tip velocity of approximately

16 m/s), while fast compared with experiment, is much smaller than the speed of sound in Au, and

allowed the system to evolve dynamically such that only natural ﬂuctuations of system properties

occurred. This process was continued throughout the indentation. By calculating the force on the rigid

layers of the tip while moving the tip closer to the sample, a plot of force vs. tip sample separation was

generated (Figure 11.3).

The shape of the computer-generated force curve for the indentation of the Au substrate by the Ni tip

showed qualitative agreement with the experimentally derived force curve (Figure 11.2b) with two

exceptions. First, there was no attractive well at large tip–sample separations in the computer-generated

force curve. This was due to the lack of long-range attractive interactions, such as dispersion forces, in

the simulations. Second, the computer-generated force curve contained a ﬁne structure not present in

the force curve generated from experimental data.

FIGURE 11.3 Computationally derived force F

vs. tip-to-sample distance d

curves for approach, contact, inden-

tation, then separation using the same tip–sample pair as in Figure 11.2. These data were calculated from an MD

simulation. (From Landman U. et al. (1990), Science 248, 454–461. With permission.)

One advantage of simulations is that the shape of the force curve, along with its ﬁne structure, can be

related to speciﬁc atomic-scale events. For instance, Landman et al. (1990) reported the observation of

a jump to contact (JC) that corresponded to that region of the force curve where there was a precipitous

drop in the force just prior to tip–substrate contact (Figure 11.3, point D). The JC phenomenon was

previously observed by Pethica and Sutton (1988) and by Smith et al. (1989). In this region of the force

curve, the gold atoms “bulged up” to meet the tip. Deformation occurred in the gold substrate because

its modulus is much lower than that of the nickel tip. This deformation occurred in a short time span

(approximately 1 ps) and was accompanied by wetting of the Ni tip by several Au atoms. Landman et al.

(1990) concluded that the JC phenomenon in metallic systems is driven by the tendency of the interfacial

atoms of the tip and the substrate to optimize their embedding energies while maintaining their individual

material cohesive binding.

Advancing the tip past the JC point caused indentation of the gold substrate accompanied by the

characteristic increase in force with decreasing tip–substrate separation (Figure 11.3, points D to M).

This region of the computer generated force curve had a maximum not present in the force curve

generated from experimental data (Figure 11.3, point L). The origin of this variation in force was tip-

induced ﬂow of the Au atoms. This ﬂow caused “piling-up” of Au atoms around the edges of the Ni

indenter. *

The force curve was completed by reversal of the tip motion (Figure 11.3, points M to X). The hysteresis

in this force curve was due to adhesion between the tip and the substrate. As the tip was retracted from

the sample, a “connective neck” of atoms between the tip and the substrate formed (Figure 11.4). While

this connective neck of atoms was largely composed of Au atoms, some Ni atoms did diffuse into the

neck. Retraction of the tip caused the magnitude of the force to increase (i.e., become more negative)

until, at some critical force, the atoms in adjacent layers of the connective neck rearranged so that an

FIGURE 11.4 Illustration of atoms in the MD simulation of an Ni tip being pulled back from an Au substrate.

This causes the formation of a connective neck of atoms between the tip and the surface. Red spheres represent Ni

atoms. The ﬁrst layer of Au substrate is colored yellow, the second blue, the third green, the fourth yellow, and so

on. (From Landman U. et al. (1990), Science 248, 454–461. With permission.)*

* Color reproduction follows page 16.

additional row of atoms formed in the neck. These rearrangement events were the essence of the

elongation process, and they were responsible for the ﬁne structure (apparent as a series of maxima)

present in the retraction portion of the force curve. These elongation and rearrangement steps were

repeated until the connective neck of atoms was severed.

When the Ni tip was coated with an epitaxial gold monolayer (Landman et al., 1992) and the inden-

tation of an Au(001) substrate repeated, the adhesive contact between the tip and the substrate was

reduced. The JC instability, formation of an adhesive contact, and hysteresis during subsequent retraction

were all observed. In contrast to the Ni/Au study, complete separation of the tip and substrate resulted

in the transfer of a smaller number of substrate atoms to the tip when the connective neck of atoms,

composed entirely of Au, was severed. Because the tip was covered with Au, the interaction between the

tip and the substrate was composed mostly of Au–Au interactions, and Au possesses less of a tendency

to wet itself than it does to wet Ni. This accounted for the insigniﬁcant number of substrate atoms

transferred to the tip.

When a hard Ni tip indented a soft Au substrate, the substrate sustained most of the damage. Con-

versely, damage was predominately done to the tip when a soft tip was used to indent a hard substrate.

For example, Landman and Luedtke (1991) used a pyramidal Au(001) tip to indent a Ni(001) substrate.

These constant-temperature indentations were carried out in the same manner described above for the

Ni/Au system. Force curves generated from the indentation of the Ni substrate by the Au tip had the

same qualitative shape as the Ni/Au force curves. However, there were differences in the ﬁne structure

of these force curves, therefore, in the atomic-scale events responsible for this structure. For instance, in

this case the tip bulged toward the substrate during the JC, rather than the substrate bulging toward the

tip. Thus, the softer material (i.e., the one with the lower modulus) was displaced during the JC. The

adhesive contact between the tip and the substrate caused large structural rearrangements in the interfacial

region of the Au tip. The closest three or four Au layers to the Ni substrate exhibited a marked tendency

toward a (111) reconstruction, consistent with an increase in interlayer spacing. In fact, this reconstruc-

tion persisted throughout the separation process.

When the tip was pushed farther toward the substrate subsequent to the JC, it became ﬂattened (or

compressed) increasing its contact area. This ﬂattening involved structural rearrangements of the outer

layers of the tip that reduced the number of crystalline layers, leaving an interstitial-layer defect in the

core of the tip. The interstitial defect was annealed away upon further compression of the tip. Upon

separation, a connective neck of Au atoms was formed due to adhesion between the Au and the Ni. This

connective neck of atoms underwent a series of elongation events, as in the Ni/Au study, until the

tip–sample distance was such that the neck became thin and broke.

Other metallic tip–substrate systems were examined with interesting results. For instance, Tomagnini

et al. (1993) studied the interaction of a pyramidal Au tip with a Pb(110) substrate using MD. These

constant-energy simulations were carried out at approximately room temperature and again at temper-

atures high enough to initiate surface melting of the Pb substrate (600 K). The forces were calculated

using a many-body potential, called the glue model, that is very similar to the EAM potentials (Ercolessi

et al., 1988).

At room temperature (300 K), when the Au tip was brought into close proximity to the Pb substrate,

a JC was initiated by a few Pb atoms wetting the Au tip. The connective neck of atoms between the tip

and the surface was composed almost entirely of Pb. The tip became deformed because the inner-tip

atoms were pulled more toward the sample surface than toward atoms on the tip surface. Because these

were constant-energy simulations, the energy released due to the wetting of the tip caused an increase

in temperature of the tip (of approximately 15 K). Extensive structural rearrangements in the tip occurred

when the tip–sample distance was decremented further. Results for the retraction of the tip from the Pb

substrate were not reported.

Increasing the substrate temperature to 600 K caused the formation of a liquid Pb layer (approximately

four layers thick) on the surface of the substrate. During the indentation, the distance at which the JC

occurred increased by approximately 1.5 Å. Due to the high diffusivity of Pb surface atoms at this

temperature, the contact area also increased. Eventually, the Au tip dissolved in the liquid Pb atom “bath.”

This liquid-like connective neck of atoms followed the tip upon retraction. As a result, the liquid–solid

interface moved farther back into the bulk Pb substrate, increasing the length of the connective neck.

Similar elongation events have been observed experimentally. For example, scanning tunneling micros-

copy (STM) experiments on the same surface demonstrated that the neck can elongate approximately

2500 Å without breaking.

Similar atomic-scale phenomena were observed for an Ir tip indenting a Pb substrate (Rafﬁ-Tabar

et al., 1992). These constant-temperature MD simulations made use of long-range, many-body Finnis-

Sinclair potentials formulated to model the interatomic forces between atoms in face-centered-cubic (fcc)

metallic alloys (Rafﬁ-Tabar and Sutton, 1991). The method developed by Nosé (1984a) was used to

regulate the temperature of the simulation. This simulation was unique in that periodic boundary

conditions were also employed in the indentation direction. Therefore, images of the substrate–tip system

were located in the cells above and below the computational cell. Indentation of the substrate by the tip

was achieved by decrementing the computational cell length normal to the substrate surface.

During the indentation process, the Pb substrate wetted the Ir tip subsequent to the JC. Signiﬁcant

structural rearrangement of the Pb substrate was brought about when the tip was pushed closer to the

substrate after the JC. This structural rearrangement led to a “piling up” of the Pb atoms around the

edges of the tip and was brought about by the local diffusional ﬂow of the Pb atoms in much the same

way as it was when an Ni tip was used to indent a soft Au substrate (Landman et al., 1990). In both the

Ni/Au and the Ir/Pb systems, the tip (Ni and Ir) retained most of its shape because it was harder than

the substrate. Plastic ﬂow in the Ir/Pb system resulted in a hysteresis in the force curve upon retraction

of the tip. The nonmonotonic features in the force curves generated from the Ir/Pb simulation were

associated with discrete, local, atomic movements; however, the precise nature of these movements was

not elucidated.

The large-scale indentation of approximately 70,000-atom Cu and Ag(111) surfaces with a rigid,

triangular-shaped tip has been simulated by Belak and co-workers (Belak and Stowers, 1992; Belak et al.,

1993). The forces between metal atoms in these large-scale MD simulations were derived from EAM

potentials and interaction between the tip and the metal substrate was modeled by an LJ potential.

Indentations were performed by moving the tip closer to the substrate at constant velocities of 1, 10, and

100 m/s. A Nosé thermostat (1984a) was used to control the temperature of the simulation.

For the 1 m/s indentation of Cu(111), the load increased linearly with indentation depth until the tip

indented approximately 0.6 nm into the substrate (Figure 11.5). At this point, the surface yielded plas-

tically and the force dropped suddenly. This plastic yielding was concomitant with a single atom “pop-

ping” out onto the surface of the substrate from beneath the tip. Continued indentation caused several

of these events to occur. At the maximum indentation depth (1.7 nm) atoms from the substrate were

“piled up” around the edges of the tip (Figure 11.6) and plastic deformation was limited to a few lattice

spacings surrounding the tip. The piling up of atoms is typical in cases where a hard tip is used to indent

FIGURE 11.5 Computationally derived load vs. indentation

depth curve for the indentation of Cu(111) with a rigid tip.

These data were calculated from an MD simulation. (From

Belak, J. and Stowers, I. F. (1992), in Fundamentals of Friction:

Macroscopic and Microscopic Processes (I. L. Singer and H. M.

Pollock, eds.), 511–520, Kluwer, Dordrecht. With permission.)

a soft substrate. Retracting the tip from the substrate caused the load to drop quickly to zero with only

small oscillations, presumably due to some plastic events at the surface. Both the pile up of substrate

atoms and the plastic nature of the indentation were apparent from analysis of the Cu(111) substrate

subsequent to indentation (Figure 11.6).

Plastic deformation in this system during the 1 m/s indentation occurred via the motion of point

defects. The formation of defects or dislocations was driven by the need to release stored elastic energy.

Plots of the energy required to create a dislocation and the elastic energy vs. length of the radius of the

indenter cross at length values of a few nanometers. In contrast, at the faster indentation rates of 10 m/s,

the system did not have time to relax completely and the stored elastic energy was much greater.

Eventually, the surface yielded and a small dislocation loop was observed on the surface.

Even though the tip was much sharper for the indentation of the Ag(111), the force curves generated

from that indentation were similar to those generated from the indentation of Cu(111). For the 1 m/s

indentation, the initial plastic events corresponded to the “popping” of single atoms out of the substrate.

The hardness value obtained from this simulation, estimated from the load divided by contact area, was

approximately 4 GPa, or approximately four times larger than the experimentally determined hardness

value of approximately 1 GPa (Pharr and Oliver, 1989).

11.3.2 Indentation of Metals Covered by Thin Films

Using simulation procedures similar to those in their earlier work, Rafﬁ-Tabar and Kawazoe (1993) used

an Ir tip to indent an Ir substrate that was covered with a monolayer Pb ﬁlm. As the tip approached the

Pb/Ir substrate, the Pb atoms directly below the tip strained upward to wet the Ir tip and a JC was

observed. The disruption of the Pb monolayer caused by the JC also resulted in local deformation of the

Ir substrate beneath the monolayer. Further indentation resulted in penetration of the Pb ﬁlm at only

FIGURE 11.6 Illustration of Cu(111) substrate atoms in an MD simulation after indentation by a rigid tip at 1 m/s.

The piling-up of the substrate atoms (gray spheres) around the edge of the tool tip after indentation is evident in

this picture. (From Belak, J. and Stowers, I. F. (1992), in Fundamentals of Friction: Macroscopic and Microscopic

Processes (I. L. Singer and H. M. Pollock, eds.), 511–520, Kluwer, Dordrecht. With permission.)