Bhushan B. Nanotribology and Nanomechanics: An Introduction

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13 Computer Simulations of Nanometer-Scale Indentation and Friction 687

200

Z sample position (Å)

Friction (nN)

200

400

800

1000

100 0 100 200 100 0 100 200 100 0 100

ì = 0.6

ì = 0.7

ì = 0.3

MMM

b) d) f)

200

Z sample position (Å)

Load (nN)

200

100

100

100 0 100 200 100 0 100 200 100 0 100

a) c) e)

Fig. 13.28a–f. Measured values for friction and load as an atomic-force microscope tip is

scanned across a 30 Å-thick sample of perﬂuoropolyether on Si(100). (a)and(b) The un-

bonded polymer with unreactive end groups. (c)and(d) The unbonded polymer with alcohol

end groups. (e)and(f) A bonded polymer. After [161] with the permission of the APS (1992)

mechanical and atomic-scale frictional properties of DLC coatings. MD simulat-

ions with bond-order potentials by Sinnott et al. [166] examined the diﬀerences in

indentationbehaviorof a hydrogen-terminateddiamond tipon hydrogen-terminated

single-crystaldiamond surfacesand diamondsurfaces coveredwith DLC. In the for-

mer case, the tip goes through shear and twist deformations at low loads that change

to plastic deformation and adhesion with the surface at high loads. When the sur-

face is covered with the DLC ﬁlm, the tip easily penetrates the ﬁlm, as illustrated in

Fig. 13.29, which “heals” easily when the tip is retracted so that no crater or other

evidence of the indentation is left behind.

MD simulations by Glosli et al. [167] of the indentation of DLC ﬁlms that are

about 20nm-thick give similar results. In this case a larger, rigid diamond tip was

used in the indentations and was also slid across the surface. During sliding, the tip

plows the surface, which causes some changes to the ﬁlm not seen during inden-

tation. However, because the tip is perfectly rigid, adhesion between the ﬁlm and

surface is not allowed which inﬂuences the results.

This section shows that repulsive interactions between surfaces covered with

molecular ﬁlms and proximal probe tips are minimized relative to interactions be-

tween bare surfaces and indentationtips. The lubricationpropertiesof polymers and

SAMs can vary with chain length, the rigidity of the tip, and the chemical proper-

ties of the end groups. In some cases, indentations can disrupt the initial ordering of

polymers and SAMs, which aﬀects their responses to nanoindentation and friction.

688 Susan B. Sinnott et al.

Fig. 13.29. Snapshot from

a molecular dynamics simu-

lation where a pyramidal

diamond tip indented an

amorphous carbon thin ﬁlm

that is 20 layers thick. The

simulation took place at room

temperature and the carbon

atoms in the ﬁlm were 21%

-hybridized and 58% sp

hybridized (the remaining

atoms were on the surface

and were not counted). Af-

ter [166] with the permission

of the AIP (1997)

13.4 Friction and Lubrication

Work is requiredto slide two surfaces againstone another.When the work of sliding

is converted to a less ordered form, as required by the ﬁrst law of thermodynamics,

friction will occur. For instance, if the two surfaces are strongly adhering to one

another, the work of sliding can be converted to damage that extends beyond the

surfaces and into the bulk. If the adhesive force between the two surfaces is weaker,

the conversion of work results in damage that is limited to the area at or near the

surface and produces transfer ﬁlms or wear debris [169,170]. While the thermody-

namic principles of the conversion of work to heat are well known, the mechanisms

by which this takes place at sliding surfaces are much less well established despite

their obvious importance for a wide variety of technological applications.

Atomic-scale simulations of friction are therefore important tools for achieving

this understanding. They have consequently been applied to numerous materials in

a wide variety of structures and conﬁgurations, including atomically ﬂat and atomi-

cally rough diamond surfaces [171–173], rigid substrates covered with monolayers

of alkane chains [174], perﬂuorocarboxylic acid and hydrocarboxylic Langmuir–

Blodgett (LB) monolayers [175], between contacting copper surfaces [168, 176],

between a silicon tip and a silicon substrate [116,140], and between contacting di-

amond surfaces that have organic molecules absorbed on them [177]. These and

several other studies are discussed below.

13 Computer Simulations of Nanometer-Scale Indentation and Friction 689

13.4.1 Bare Surfaces

Sliding friction that takes place between two surfaces in the absence of lubricant is

termed “dry” friction even if the process occurs in an ambient environment. Simple

modelshavebeen developedto modeldrysliding friction that,for example,consider

the motion of a single atom over a monoatomic chain [178]. Results from these

models reveal how elastic deformation of the substrate from the sliding atom aﬀects

energy dissipation and how the average frictional force varies with changes in the

force constant of the substrate in the direction normal to the scan direction. Much

of the correct behavior involved in dry sliding friction is captured by these types

of simple models. However, more detailed models and simulations, such as MD

simulations, are required to provide information about more complex phenomena.

MD simulations have been used to study the sliding of metal tips across clean

metal surfaces by numerous groups [168,179–183]. An illustrative case is shown in

Fig. 13.30 for a copper tip sliding across a copper surface [168]. Adhesion and wear

occur when the attractive force between the atoms on the tip and the atoms at the

surface becomes greater than the attractive forces within the tip itself. Atomic-scale

stick and slip can occur through nucleation and subsequent motion of dislocations,

and wear can occur if part of the tip gets left behind on the surface (Fig. 13.30). The

simulations can further provide data on how the characteristic ‘stick-slip’ friction

motion candepend on theareaof contact,the rate of sliding, and thesliding direction

(Fig. 13.31).

An additional study of stick-slip in the sliding of much larger, square-shaped

metal tips across metal surfaces was carried out by Li et al. [184] using EAM poten-

tials. The initial structure of a NiAl tip and surface system is shown in Fig. 13.32.

This study predicted that collective elastic deformation of the surface layers in re-

sponse to sliding is the main cause of the stick-slip behavior shown in Fig. 13.33.

The simulations also predicted that stick-slip produces phonons that propagate

through the surface slab.

Large-scale simulations using pairwise Morse potentials that are similar in form

to (13.6) were used to study the wear of metal surfaces caused by metal tips that

Fig. 13.30. Snapshot from

a molecular dynamics simu-

lation of a copper tip sliding

across a Cu(100) surface.

A connective neck between

the two is sheared during the

sliding, leading to wear of

the tip. The simulation was

performed at a temperature

of 0 K. After [168] with the

permission of the APS (1996)

690 Susan B. Sinnott et al.

2

Distance (Å)

(nN)

2 4 6 8 10

(nN)

300 K, 2 m/s

12 K, 2 m/s

12 K, 10 m/s

2

Fig. 13.31a–c. Plots of the

lateral force versus distance

from a simulation similar

to that shown in Fig. 13.30.

The plots illustrate the de-

pendence of the force on

temperature and sliding ve-

locity. After [168] with the

permission of the APS (1996)

Fig. 13.32. Starting conﬁguration of sliding NiAl tip on a NiAl surface. After [184] with the

permission of the AIP (2001)

plowthe surface, asillustrated in Fig. 13.34.Theyprovideinsight into the wear track

dependence of the sliding rate [186] and how variations in the scratching force, fric-

tion coeﬃcient, and other quantities depend on the scratch depth [185],as illustrated

in Fig. 13.35.

On the whole,the results of experimentalstudies show good agreement with the

resultsof the computationalstudies describedabove.Thisis truedespite the factthat

all of these MD simulations use empirical potentials that do not include electronic

eﬀects and thus eﬀectively assume that the electronic contributions to friction on

metalsurfacesare negligible.However,experimentshavemeasured anon-negligible

contribution of conduction electrons to friction [188]. Thus, future simulations of

metal-tip–metal-substrateinteractionsusingmore sophisticatedtight-bindingor ﬁrst

principles methods that include electronic eﬀects are encouraged.

13 Computer Simulations of Nanometer-Scale Indentation and Friction 691

5

Time step

X displacement (Å)

30 00010 000 20 000

» Lattice constant of NiAl

F = 3.5×10

6

(dyn)

Fig. 13.33. A structured

curve of frictional dynamics

of an atom in the upper right

corner that is indicative of

stick-slip. After [184] with

the permission of the AIP

(2001)

Fig. 13.34a–d. Snapshots of the scratching of an aluminium surface with a rigid tip at a depth

of 0.8 nm. After [185] with the permission of the APS (2000)

Layered ceramics, such as mica, graphite and MoS

, that have structures that

include strongly bound layers that interact with one another through weak van der

Waals bonds, have long been known to have good lubricating properties because of

the ease with which the layers slide over one another. They have, therefore, been the

focus of some of the earliest experimental studies of nanometer-scale friction [19,

189]. The results of these early studies lead researchers to hypothesize that at high

loads measured friction forces were related to “incipient sliding” [190,191] caused

by a small ﬂake from the surface becoming attached to the end of the tip. If true, this

would mean that all measured interactions were between the surface and the ﬂake,

692 Susan B. Sinnott et al.

Scratch depth (nm)

Specific energy (GPa)

0.2 0.4 0.6 0.8 1

Force ratio (F

)

Force/unit width (N/mm) 3×10

0.2

0.4

0.6

0.8

0.5

1.5

2.5

Scratch force (F

)

Resultant force (F

)

Nominal force (F

)

Fig. 13.35. Variationin(a)

the scratching force, the nor-

mal force, and the resultant

force, and (b) the friction co-

eﬃcient, and (c) the speciﬁc

energy during scratch pro-

cesses similar to those shown

in Fig. 13.34 at scratch depths

ranging from 0.8nmtoal-

most 0 nm. After [185] with

the permission of the APS

(2000)

which has a larger contact area than the clean tip. However, subsequent simulations

of constant force AFM images of graphite by Tang et al. [192] showed that there

is no need for the assumption of a graphite ﬂake under the tip to reproduce the

experimental images of a graphite surface.

Surprisingly strong localized ﬂuctuations in atomic-scale friction are displayed

by layered ceramics [187, 194–196]. For instance, square-well signals with sub-

angstrom lateral width are obtained in FFM scans on MoS

(001) in the direction

across the scan direction, while sawtooth signals are detected along the scan direc-

tion, as shown in Fig. 13.36. This ﬁnding can be explained by a stick-slip model

by Mate [19] and Erlandsson [189] that assumes that the tip does a zigzag walk

along the scan. Measured variations in the frictional force with the periodicity of

cleavage planes [189] are consistent with the results of this simple model. How-

ever, additional experiments indicate a more complex tip–surface interaction, such

as changes in the intrinsic lateral force between the substrate and the AFM tip [197]

or sliding-induced chemistry between the tip and the surface [198,199].

Crystalline ceramics diﬀer from layered ceramics in that they are held together

by relatively strong covalent or ionic bonds. In the case of ionic systems, Shluger

et al. [193] used a mixture of atomistic and macroscopic modeling methods to study

the interaction of a MgO tip and a LiF surface. In particular, the tip–surface in-

teraction was treated atomistically and the cantilever deﬂection was treated with

a macroscopic approach. The results, shown in Fig. 13.37, show that if the tip is

13 Computer Simulations of Nanometer-Scale Indentation and Friction 693

9.2 Å

a) b) c)

3.16 Å

6.9Å

4.6Å

1.58 Å

6.9 Å

2.74 Å

25 Å 25 Å

2.74Å

1.58 Å

3.16 Å

Stick-point Slip-motion

Fig. 13.36a–c. Displacement data from a scan across a MoS

(001) surface. The data in (a)

and (b) are form scans along the x -and y-directions, respectively, on the surface shown in

(c). After [187] with the permission of the APS (1996)

charged and in hard contact with the surface, tip and surface distortions are possible

that can lead to motion of the surface ions within the surface plane and the transfer

of some of the ions onto the tip.

In the case of covalently bound ceramics, there is extensive literature related to

friction of diamond [200,201] because, while it is the hardest material known, it

also exhibits relatively low friction. The “ratchet mechanism” has been proposed

for energy dissipation during friction on the macroscale in diamond, where energy

is released by the transfer of normal force from one surface asperity to another. The

elastic mechanismis anothermechanism that has been proposed, where the released

energycomesfrom elastic strain in an asperity.Atomic-scale frictionhas been meas-

ured experimentally [20] for diamond tips with near atomic-scale radii sliding over

hydrogen-terminated diamond surfaces. These experiments are sensitive enough to

detect the 2×1 reconstruction on the diamond (100) surface. Furthermore, the aver-

age friction coeﬃcient determined with an AFM on H-terminated diamond (111)

surfaces is about two orders of magnitude smaller than the value measured on bare,

2×1 diamond (111) surfaces, indicating greater adhesion in the latter case [202].

More recently, the friction between a tungsten carbide tip and hydrogen-terminated

diamond (111) was examined with AFM in UHV by Enachescu et al. [203]. The

friction between these two hard surfaces was shown to obey Derjaguin–Muller–

Toporov or DMT [204] contact mechanics and the shear strength of the interface

was determined to be 246MPa.

Extensive MD simulations have been carried out by Harrison and cowork-

ers that examine the friction between hydrogen-terminated diamond (111) sur-

faces [171, 208] and diamond (100) surfaces [205] in sliding contact. The simu-

lations of sliding between the diamond (111) surfaces reveal that the potential

energy,load,and frictionare allperiodicfunctionsof theslidingdistance(Fig.13.38).

Maxima in thesequantities occur when the hydrogenatoms on opposingsurfacesin-

694 Susan B. Sinnott et al.

1

5

Lateral tip displacement along the (100)axis (Å)

2.5

2.5

Lateral force (nN)

0 123456789

Li F Li F Li

2

Fig. 13.37. Top: The lat-

eral force calculated for

a MgO tip scanning in the

001-direction on LiF(001).

Bottom: A view of the side of

the surface plane along the

scan direction. The surface

and F

−

atoms are seen to

relax to relieve the frictional

energy and this relaxation

motion is indicated in the

ﬁgure by the category lines.

(a)HowaF

−

iononthe

surface can be moved into an

interstitial site by the tip and

then it returns to its original

position. (b) How the relax-

ation of the surface atoms is

reversible. After [193] with

the permission of Elsevier

(1995)

teract strongly. Recent ab initio studies by Neitola and Park of the friction between

hydrogen-terminated diamond (111) surfaces also show that the potential energy is

periodic with sliding distance (Fig. 13.39) [206]. Because the results of the ab initio

studies and the MD simulations are in good agreement, Neitola and Park conclude

that the potential model used in the MD studies is accurate.

As mentioned previously, the maxima in the load and the friction values dur-

ing sliding are caused by the interactions of hydrogen atoms on opposing surfaces.

When sliding in the [11

2] direction, the H atoms “revolve” around one another,

thus decreasing the repulsive interaction between the sliding surfaces because the

hydrogen atoms are not forced to pass directly over one another [171]. Increasing

the load causes increased stress at the interface. The opposing hydrogen atoms be-

come “stuck”. Once the stress at the interface becomes large enough to overcome

the hydrogen–hydrogeninteraction between opposingsurfaces, the hydrogenatoms

“slip” past one another with the same “revolving” motion observed at low loads.

This phenomenon is known as atomic-scale stick-slip and has the periodicity of the

diamond lattice. It should be noted that due to the alignment of the opposing sur-

13 Computer Simulations of Nanometer-Scale Indentation and Friction 695

Sliding distance / unit cell length

Force (nN/atom)

0.5

1 2 3 4 5 6

y or [112



]

Fig. 13.38. Calculated frictional force (lower lines) and normal force (upper lines)feltby

a hydrogen-terminated (111) surface as it slides against another hydrogen-terminated dia-

mond (111) surface in a MD simulation. The sliding direction is given in the legend. The

sliding speed is 1Å/ps and the simulation temp is 300 K. The two plots show how the simu-

lated stick-slip motion changes as a function of the applied load. The load is high and low in

the upper and lower panels, respectively. After [205] with the permission of the ACS (1995)

Sliding distance (Å)

Distance between surfaces (Å)

3.2

3.4

3.6

3.8

4.2

4.4

4.6

0.8 nN

1.7 nN

3.3 nN

5.0 nN

6.6 nN

8.3 nN

10.0 nN

Normal load

246810

Fig. 13.39. Distance between hydrogen-terminated (111) crystals as a function of sliding

distance. After [206] with the permission of the ACS (2001)

faces, the hydrogen atoms are directly in line with each other when sliding in the

[11

2] direction. However, the hydrogen atoms are not “aligned” with each other

when sliding in the [11

0], so the friction in this direction is lower than in the [112]

696 Susan B. Sinnott et al.

direction. It should be noted, however, that experimentally all initial alignments are

likely to be probed.

Harrison and coworkershavefurther shown that the peaks in the frictional force

are correlated with peaks in the temperature of the atoms at the interface when

two hydrogen-terminated diamond (111) surfaces are in sliding contact [208]. Fig-

ure 13.40 shows the vibrational energy (or temperature) between diamond layers as

a function of sliding distance. These data clearly showthat layers close to the sliding

interface can be vibrationally excited during sliding. When the hydrogen atoms are

“stuck” or interacting with each other strongly, the stress and friction force at the

interface build up. When the hydrogen atoms “slip” past one another, the stress at

the interface is relieved and the energy is transferred to the diamond in the form of

vibration or heat. Thus, the peaks in the temperature occur slightly after the peaks

in the frictional force.

It should be noted that atomic-scale stick slip is observed in other systems.

Harrison and coworkers used MD simulations to demonstrate that two hydrogen-

terminated diamond (100) (2 × 1) surfaces in sliding contact also exhibit stick-

slip [205]. In addition, it was shown that the shape of the friction versus sliding

distance curves is inﬂuenced slightly by the speed of the sliding, with features in the

curves becoming more pronounced at slower speeds. Stick-slip behavior was also

observed in AFM studies of diamond (100) (2×1) surfaces [202]. However, in this

Sliding distance / unit cell

Vibrational energy (K)

500

1000

500

1000

500

1000

2468

Fig. 13.40. Average vibrational energy of oscillators between diamond layers as a function

of sliding distance. These energies are derived from a molecular dynamics simulation of the

sliding of a hydrogen-terminated diamond (111) surface over another hydrogen-terminated

diamond (111) surface. The vibrational energy between the ﬁrst and second layers of the

lower diamond surface is shown in the lower panel, between the second and third layers in

the middle panel, and between the third and fourth layers in the upper panel. After [207] with

the permission of Elsevier (1995)