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

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Finally, we will examine the relationship of tribological experiments to these more fundamental

atomistic considerations. The primary goals of this section will be to again provide sources for further

study of tribological experiments and to raise critical issues concerning the relationship between basic

surface properties with regard to tribology and the ability of certain classes of experiments to reveal the

underlying interactions. We will attempt to avoid overlapping the material that we present with that

presented by other authors in this publication. This chapter cannot be a complete treatment of the physics

of surfaces due to space limitations. We recommend an excellent text by Zangwill (1988) for a more

thorough treatment. Instead, we concentrate on techniques and issues of importance to tribology on the

nanoscale.

3.2 Geometry of Surfaces

We will now discuss simply from a geometric standpoint what occurs when you create two surfaces by

dividing a solid along a given plane. We limit the discussion to single crystals, since the same arguments

apply to polycrystalline samples except for the existence of many grains, each of which could be described

by a corresponding argument. This discussion will start by introducing the standard notation for describ-

ing crystals given in many solid-state texts (Ashcroft and Mermin, 1976; Kittel, 1986). It is meant to be

didactic in nature and because of length limitations will not attempt to be comprehensive. To establish

notation and concepts we will limit our discussion to two of the possible Bravais lattices, face-centered

cubic (fcc) and body-centered cubic (bcc), which are the structures often found in metals. The unit cells,

i.e., the structures which most easily display the symmetries of the crystals, are shown in Figure 3.1. The

other descriptions that are frequently used are the primitive cells, which show the simplest structures

that can be repeated to create a given structure. In Figure 3.1 we also show the primitive cell basis vectors,

which can be used to generate the entire structure by the relation

(3.1)

where

, and

are integers, and

→

, and

→

are the unit basis vectors.

Since we are interested in describing surface properties, we want to present the standard nomenclature

for specifying a surface. The algebraic description of a surface is usually given in terms of a vector normal

to the surface. This is conveniently accomplished in terms of vectors that arise naturally in solids, namely,

the reciprocal lattice vectors of the Bravais lattice (Ashcroft and Mermin, 1976; Kittel, 1986). This is

FIGURE 3.1

(a) Unit cube of fcc crystal structure with primative cell basis vectors indicated. (b) Unit cube of bcc

crystal structure, with primative cell basis vectors indicated.

rr r

Rnana na=+ +

12233

convenient since these vectors are used to describe the band structure and diffraction effects in the solid.

They are usually given in the form

(3.2)

where

h, k,

and

are integers. The reciprocal lattice vectors are related to the basis vectors of the direct

lattice by

(3.3)

where a cyclic permutation of

i, j, k

are used in the deﬁnition. Typically, parentheses are used in the

deﬁnition of the plane, e.g., (

h,k,l

). The (100) planes for fcc and bcc lattices are shown in Figure 3.2

where dots are used to show the location of the atoms in the next plane down.

This provides the simplest description of the surface in terms of terminating the bulk. There is a rather

nice NASA publication by Bacigalupi (1964) which gives diagrams of many surfaces and subsurface

structures for fcc, bcc, and diamond lattices, in addition to a great deal of other useful information such

FIGURE 3.2

Projection of cubic face (100) plane for (a) fcc and (b) bcc crystal structures. In both cases, smaller

dots represent atomic positions in the next layer below the surface.

rrr

Khbkb lb=++

123

rr r

aa a

=π

()

12 3

as surface density and interplanar spacings. A modern reprinting of this NASA publication is called for.

In many cases, this simple description is not adequate since the surface can reconstruct. The two most

prominent cases of surface reconstruction are the Au(110) surface (Good and Banerjea, 1992) for metals

and the Si(111) surface (Zangwill, 1988) for semiconductors. In addition, adsorbates often form structures

with symmetries different from the substrate, with the classic example the adsorption of oxygen on

W(110) (Zangwill, 1988). Wood (1963) in a classic publication gives the nomenclature for describing

such structures. In Figure 3.3 we show an example of 2

2 structure, where the terminology describes

a surface that has a layer with twice the spacings of the substrate. There are many other possibilities, such

as structures rotated with respect to the substrate and centered differently from the substrate. These are

also deﬁned by Wood (1963).

The next consideration is that the interplanar spacing can vary, and slight shifts in atomic positions

can occur several planes from the free surface. A recent paper by Bozzolo et al. (1994) presents the results

for a large number of metallic systems and serves as a good review of available publications. Figure 3.4

shows some typical results for Ni(100). The percent change given represents the deviation from the

equilibrium interplanar spacing. The drawing in Figure 3.4 exaggerates these typically small differences

to elucidate the behavior. Typically, this pattern of alternating contraction and expansion diminishing as

the bulk is approached is found in most metals. It can be understood in a simple manner (Bozzolo et al.,

1994). The energy for the bulk metal is a minimum at the bulk metallic density. The formation of the

surface represents a loss of electron density because of the missing neighbors for the surface atoms.

Therefore, this loss of electron density can be partially offset by a contraction of the interplanar spacing

between the ﬁrst two layers. This construction causes an electron density increase between layers 2 and

3, and thus the energy is lowered by a slight increase in the interplanar spacing. There are some exceptions

FIGURE 3.3

Representation of fcc (110) face with an additional “2

2” layer, in which the species above the surface

atoms have twice the spacing of the surface. Atomic positions in the next layer below the surface are presented by

smaller dots.

FIGURE 3.4

Side view of nickel (100) surface. On the left, the atoms are positioned as if still within a bulk fcc

lattice (“unrelaxed”). On the right, the surface planes have been moved to minimize system energy. The percent

change in lattice spacing is indicated, with the spacing in the image exaggerated to illustrate the effect. (From Bozzolo,

G. et al. (1994),

Surf. Sci.

315, 204–214. With permission.)

to this behavior where the interplanar spacing increases between the ﬁrst two layers due to bonding

effects (Needs, 1987; Feibelman, 1992). However, the pattern shown in Figure 3.4 is the usual behavior

for most metallic surfaces. There can be similar changes in position within the planes; however, these

are usually small effects (Rodriguez et al., 1993; Foiles, 1987). In Figure 3.5, we show a side view of a

gold (110) surface (Good and Banerjea, 1992). Figure 3.5a shows the unreconstructed surface and Figure

3.5b shows a side view of the (2

1) missing row reconstruction. Such behavior indicates the complexity

that can arise even for metal surfaces and the danger of using ideas which are too simplistic, since more

details of the bonding interactions are needed in this case and those of Needs (1987) and Feibelman

(1992).

Crystal surfaces encountered typically are not perfectly oriented nor atomically ﬂat. Even “on-axis”

(i.e., within a fraction of a degree) single-crystal low-index faces exhibit some density of crystallographic

steps. For a gold (111) face tilted one half degree toward the (011) direction, evenly spaced single atomic

height steps would be only 27 nm apart. Other surface-breaking crystal defects such as screw and edge

dislocations may also be present, in addition to whatever surface scratches, grooves, and other polishing

damage which remain in a typical single-crystal surface. Surface steps and step kinks would be expected

to show greater reactivity than low-index surface planes. During either deposition or erosion of metal

surfaces, one expects incorporation into or loss from the crystal lattice preferentially at step edges. More

generally on simple metal surfaces, lone atoms on a low-index crystal face are expected to be most mobile

(i.e., have the lowest activation energy to move). Atoms at steps would be somewhat more tightly bound,

and atoms making up a low-index face would be least likely to move. High-index crystal faces can often

be thought of as an ordered collection of steps on a low-index face. When surface species and even

interfaces become mobile, consolidation of steps may be observed. Alternating strips of two low-index

crystal faces can then develop from one high-index crystal plane, with lower total surface energy but

with a rougher, faceted topography. Much theoretical and experimental work has been done over the last

decade on nonequilibrium as well as equilibrium surface morphology (e.g., Redﬁeld and Zangwill, 1992;

Vlachos et al., 1993; Conrad and Engel, 1994; Bartelt et al., 1994; Williams, 1994; Kaxiras, 1996).

Semiconductors and insulators generally behave differently. Unlike most metals for which the electron

gas to some degree can be considered to behave like a ﬂuid, semiconductors have strong directional

bonding. Consequently, the loss of neighbors leaves dangling bonds which are satisﬁed in ultrahigh

vacuum by reconstruction of the surface. The classic example of this is the silicon (111) 7

7 structure,

where rebonding and the creation of surface states gives a complex structure. Until STM provided real-

space images of this reconstruction (Binnig et al., 1983) much speculation surrounded this surface.

Zangwill (1988) shows both the terminated bulk structure of Si(111) and the relaxed 7

7 structure. It

is clear that viewing a surface as a simple terminated bulk can lead to severely erroneous conclusions.

The relevance to tribology is clear since the nature of chemical reactions between surfaces, lubricants,

and additives can be greatly affected by such radical surface alterations.

There are other surface chemical state phenomena, even in ultrahigh vacuum, just as important as the

structural and bonding states of the clean surface. Surface segregation often occurs to metal surfaces and

interfaces (Faulkner, 1996, and other reviews cited therein). For example, trace quantities of sulfur often

segregate to iron and steel surfaces or to grain boundaries in polycrystalline samples (Jennings et al.,

1988). This can greatly affect results since sulfur, known to be a strong poisoning contaminant in catalysis,

can affect interfacial bond strength. Sulfur is often a component in many lubricants. For alloys similar

geometric surface reconstructions occur (Kobistek et al., 1994). Again, alloy surface composition can vary

dramatically from the bulk, with segregation causing one of the elements to be the only component on

a surface. In Figure 3.6 we show the surface composition for a CuNi alloy as a function of bulk composition

with both a large number of experimental results and some theoretical predictions for the composition

FIGURE 3.5

Side view of gold (110) surface: (a) unrecon-

structed; (b) 1

2 missing row surface reconstruction.

(From Good, B. S. and Banerjea, A. (1992),

Mater. Res. Soc.

Symp. Proc.,

278, 211–216. With permission.)

(Good et al., 1993). In addition, nascent surfaces typically react with the ambient, giving monolayer ﬁlms

and oxidation even in ultrahigh vacuum, producing even more pronounced surface composition effects.

In conclusion, we see that even in the most simple circumstances, i.e., single-crystal surfaces, the situation

can be very complicated.

3.3 Theoretical Considerations

3.3.1 Surface Theory

We have shown how the formation of a surface can affect geometry. We now present some aspects of the

energetics of surfaces from ﬁrst-principles considerations. For a long time, calculations of the electronic

structure and energetics of the surface had proven to be a difﬁcult task. The nature of theoretical

approximations and the need for high-speed computers limited the problem to some fairly simple

approaches (Ashcroft and Mermin, 1976). The advent of better approximations for the many body effects,

namely, for exchange and correlation, and the improvements in computers have changed this situation

in the not too distant past. One aspect of the improvements was density functional theory and the use

of the local density approximation (LDA) (Kohn and Sham, 1965; Lundqvist and March, 1983). Difﬁ-

culties arise because in the creation of the surface, periodicity in the direction perpendicular to the surface

is lost. Periodicity simpliﬁes many problems in solid-state theory by limiting the calculation to a single

unit cell with periodic boundary conditions. With a surface present the wave vector perpendicular to the

surface,

→

⊥

, is not periodic, although the wave vector parallel to the surface,

→



, still is.

FIGURE 3.6

Copper (111) surface composition vs. copper-nickel alloy bulk composition: comparison between the

experimental and theoretical results for the ﬁrst and second planes. (See Good et al., 1993, and references therein.)

The process usually proceeds by solving the one-electron Kohn–Sham equations (Kohn and Sham,

1965; Lundqvist and March, 1983), where a given electron is treated as though it is in the mean ﬁeld of

all of the other electrons. The LDA represents the mean ﬁeld in terms of the local electron density at a

given location. The Kohn–Sham equations are written in the form (using atomic units where the constants

appearing in the Schroedinger equation along with the electron charge and the speed of light,



e = c

= 1).

(3.4)

where

and



are the one-electron wave function and energy, respectively, and

(3.5)

where

[

(

→

)] is the exchange and correlation potential,

(

→

) is the electron density (the brackets

indicate that it is a functional of the density), and

(

→

) is the electrostatic potential given by

(3.6)

in which the ﬁrst term is the electron–electron interaction and the second term is the electron–ion

interaction,

is the ion charge, and the electron density is given by

(3.7)

where occ refers to occupied states. The calculation proceeds by using some representation for the wave

functions such as the linear mufﬁn tin orbital approximation (LMTO), and iterating self-consistently.

Self-consistency is obtained when either the output density or potential agree to within some speciﬁed

criterion with the input. These calculations are not generally performed for the semi-inﬁnite solid.

Instead, they are performed for slabs of increasing thickness to the point where the interior atoms have

essentially bulk properties. Usually, ﬁve planes are sufﬁcient to give the surface properties. The values of



(

→



) give the surface band structure and surface states, localized electronic states created because of

the presence of the surface.

The second piece of information given is the total energy in terms of the electron density, as obtained

from density functional theory. This is represented schematically by the expression

(3.8)

where

is the kinetic energy contribution to the energy,

is the electrostatic contribution,

is the

exchange correlation contribution, and the brackets indicate that the energy is a functional of the density.

Thus, the energy is an extremum of the correct density. Determining the surface energy accurately from

such calculations can be quite difﬁcult since the surface energy, or indeed any of the energies of various

structures of interest, are obtained as the difference of big numbers. For example, for the surface the

energy would be given by

(3.9)

where

is the distance between the surfaces (a = 0 to get the surface energy) and

is the cross-sectional

area.

−∇+

()

[]

()

() ( )

Vr k r k k r

iii

ΨΨ



,,

Vr r V r

rr r

()

[]

Φρ

rr r

rdr

()

′

()

−

′

−∑

−

∫

rkr

()

ΣΨ

occ 

EE E Eρρρρ

[]

ke es xc

Ea E

surface

()

−∞

()

The initial and classic solutions of the Kohn–Sham equations for surfaces and interfaces were accom-

plished by Lang and Kohn (1970) for the free surface and Ferrante and Smith for interfaces (Ferrante

and Smith, 1985; Smith and Ferrante, 1986). The calculations were simpliﬁed by using the jellium model

to represent the ionic charge. In the jellium model the ionic charge is smeared into a uniform distribution.

Both sets of authors introduced the effects of discreteness on the ionic contribution through perturbation

theory for the electron–ion interaction and through lattice sums for the ion–ion interaction. The jellium

model is only expected to give reasonable results for the densest packed planes of simple metals.

In Figures 3.7 and 3.8 we show the electron distribution at a jellium surface for Na and for an

Al(111)–Mg(0001) interface (Ferrante and Smith, 1985) that is separated a small distance. In Figure 3.7

we can see the characteristic decay of the electron density away from the surface. In Figure 3.8 we see

the change in electron density in going from one material to another. This characteristic tailing is an

indication of the reactivity of the metal surface.

In Figure 3.9 we show the electron distribution for a nickel (100) surface for the fully three-dimensional

calculations performed by Arlinghouse et al. (1980) and that for a silver layer adsorbed on a palladium

(100) interface (Smith and Ferrante, 1985) using self-consistent localized orbitals (SCLO) for approxi-

mations to the wave functions. First, we note that for the Ni surface we see there is a smoothing of the

surface density characteristic of metals. For the adsorption we can see that there are localized charge

transfers and bonding effects indicating that it is necessary to perform three-dimensional calculations in

order to determine bonding effects. Hong et al. (1995) have also examined metal–ceramic interfaces and

the effects of impurities at the interface on the interfacial strength.

In Figure 3.10 we schematically show the results of determining the interfacial energies as a function

of separation between the surfaces with the energy in Figure 3.10a and the derivative curves giving the

interfacial strength. In Figure 3.11 we show Ferrante and Smith’s results for a number of interfaces of

jellium metals (Ferrante and Smith, 1985; Smith and Ferrante, 1986; Banerjea et al., 1991). Rose et al.

(1981, 1983) found that these curves would scale onto one universal curve and indeed that this result

applied to many other bonding situations including results of fully three-dimensional calculations. We

show the scaled curves from Figure 3.11 in Figure 3.12. Somewhat surprisingly because of large charge

transfer, Hong et al. (1995) found that this same behavior is applicable to metal–ceramic interfaces. Finnis

(1996) gives a review of metal–ceramic interface theory.

The complexities that we described earlier with regard to surface relaxations and complex structures

can also be treated now by modern theoretical techniques. Often in these cases it is necessary to use

“supercells” (Lambrecht and Segall, 1989). Since these structures are extended, it would require many

atoms to represent a defect. Instead, in order to model a defect and take advantage of the simplicities of

periodicities, a cell is created selected at a size which will mimic the main energetics of the defects. In

conclusion, we can see that theoretical techniques have advanced substantially and are continuing to do

so. They have and will shed light on many problems of interest experimentally.

3.3.2 Friction Fundamentals

Friction, as commonly used, refers to a force resisting sliding. It is of obvious importance since it is the

energy loss mechanism in sliding processes. In spite of its importance, after many centuries friction

surprisingly has still avoided a complete physical explanation. An excellent history of the subject is given

in a text by Dowson (1979). In this section we will outline some of the basic observations and give some

recent relevant references treating the subject at the atomic level, in keeping with the theme of this chapter,

and since the topic is much too complicated to treat in such a small space.

There are two basic issues, the nature of the friction force and the energy dissipation mechanism.

There are several commonplace observations, often considered general rules, regarding the friction force

as outlined in the classic discussions of the subject by Bowden and Tabor (1964):

1. The friction force does not depend on the apparent area of contact.

2. The friction force is proportional to the normal load.

3. The kinetic friction force does not depend on the velocity and is less than the static friction force.

FIGURE 3.7

Electron density at a jellium surface vs. position for a Na(011)–Na(011) contact for separations of 0.25, 3.0, and 15.0 au. (From Ferrante, J. and Smith, J. R.

(1985),

Phys. Rev. B

31, 3427–3434. With permission.)

Historically, Coulomb (Bowden and Tabor, 1964; Dowson, 1979), realizing that surfaces were not

ideally ﬂat and were formed by asperities (a hill-and-valley structure), proposed that interlocking asper-

ities could be a source of the friction force. This model has many limitations. For example, if we picture

a perfectly sinusoidal interface there is no energy dissipation mechanism, since once the top of the ﬁrst

asperity is attained the system will slide down the other side, thus needing no additional force once set

in motion. Bowden and Tabor (1964), recognizing the existence of interfacial forces, proposed another

mechanism based on adhesion at interfaces. Again, recognizing the existence of asperities, they propose

that adhesion occurs at asperity surfaces and that shearing occurs on translational motion. This model

explains a number of effects such as the disparity between true area of contact and apparent area of

contact and the tracking of friction force with load, since the asperities and thus the true area of contact

change with asperity deformation (load). The actual arguments are more complex than indicated here

and require reading of the primary text for completeness. These considerations also emphasize the basic

topic of this chapter, i.e., the important effect of the state of the surface and interface on the friction

process. Clearly, adsorbates, the differences of materials in contact, and lubricants greatly affect the

interaction.

We now proceed to brieﬂy outline some models of both the friction force and frictional energy

dissipation. As addressed elsewhere in this book, there have recently been a number of attempts to model

theoretically the friction interaction at the atomic level. The general approaches have involved assuming

a two-body interaction potential at an interface, which in some cases may only be one dimensional, and

FIGURE 3.8

Electron number density

and jellium ion charge density for an aluminum (111)–magnesium (0001)

interface. (From Ferrante, J. and Smith, J. R. (1985),

Phys. Rev. B

31, 3427–3434. With permission.)

allowing the particles to interact across an interface, allowing motion of internal degrees of freedom in

either one or both surfaces. Hirano and Shinjo (1990) examine a quasi-static model where one solid is

constrained to be rigid and the second is allowed to adapt to the structure of the ﬁrst, interacting through

a two-body potential as translation occurs. No energy dissipation mechanism is included. They conclude

that two processes occur, atomic locking where the readjusting atoms change their positions during

sliding, and dynamic locking where the conﬁguration of the surface changes abruptly due to the dynamic

process if the interatomic potential is stronger than a threshold value. The latter process they conclude

is unlikely to happen in real systems. They also conclude that the adhesive force is not related to the

FIGURE 3.9

(a) Electronic charge density contours at a nickel (100) surface. (From Arlinghaus, F. J. et al. (1980),

Phys. Rev. B

21, 2055–2059. With permission.) (b) Charge transfer of the palladium [100] slab upon silver adsorption.

(From Smith, J. R. and Ferrante, J. (1985),

Mater. Sci. Forum,

4, 21–38. With permission.)