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

Подождите немного. Документ загружается.

Rose, J. H., Ferrante, J., and Smith, J. R. (1981), “Universal Binding Energy Curves for Metals and

Bimetallic Interfaces,” Phys. Rev. Lett. 47, 675–678.

Rose, J. H., Smith, J. R., and Ferrante, J. (1983), “Universal Features of Bonding in Metals,” Phys. Rev. B

28, 1835–1845.

Rous, P. J. and Pendry, J. B. (1989), “Applications of Tensor LEED,” Surf. Sci. 219, 373–394.

Shinjo, K. and Hirano, M. (1993), “Dynamics of Friction: Superlubric State,” Surf. Sci. 283, 473–478.

Smith, J. R. and Ferrante, J. (1985), “Materials in Intimate Contact,” Mater. Sci. Forum 4, 21–38.

Smith, J. R. and Ferrante, J. (1986), “Grain-Boundary Energies in Metals from Local-Electron-Density

Distributions,” Phys. Rev. B 34, 2238–2245.

Sokoloff, J. B. (1990), “Theory of Energy Dissipation in Sliding Crystal Surfaces,” Phys. Rev. B. 42, 760–765.

Sokoloff, J. B. (1992), “Theory of Atomic Level Sliding Friction between Ideal Crystal Interfaces,” J. Appl.

Phys. 72, 1262–1269.

Sokoloff, J. B. (1995), “Theory of the Contribution to Sliding Friction from Electronic Excitations in the

Microbalance Experiment,” Phys. Rev. B 52, 5318–5322.

Sokoloff, J. B., Krim, J., and Widom, A. (1993), “Determination of an Atomic-Scale Frictional Force Law

through Quartz-Crystal Microbalance Measurements,” Phys. Rev. B 48, 9134–9137.

Thomas, G. and Goringe, M. J. (1979), Transmission Electron Microscopy of Materials, John Wiley & Sons,

New York.

Tsukizoe, T., Tanaka, S., Nishizaki, K., and Ohmae, N. (1985), “Field Ion Microscopy of Metallic Adhesion

and Friction,” Proc. JSLE Int. Tribol. Conf., Tokyo, pp. 121–126.

Van Hove, M. A., Moritz, W., Over, H., Rous, P. J., Wander, A., Barbieri, A., Materer, N., Starke, U., and

Somorjai, G. A. (1993), “Automated Determination of Complex Surface Structures by LEED,” Surf.

Sci Rep. 19, 191–229.

Vlachos, D. G., Schmidt, L. D., Aris, R. (1993), “Kinetics of Faceting of Crystals in Growth, Etching, and

Equilibrium,” Phys. Rev. B 47, 4896–4909.

Wheeler, D. R. (1976), “Effect of Adsorbed Chlorine and Oxygen on the Shear Strength of Iron and

Copper Junctions,” J. Appl. Phys. 47, 1123–1130.

Wheeler, D. R. (1978), “X-Ray Photoelectron Spectroscopic Study of Surface Chemistry of Dibenzyl

Disulﬁde on Steel under Mild and Severe Wear Conditions,” Wear 47, 243–254.

Widom, A. and Krim, J. (1986), “Q Factors of Quartz Oscillator Modes as a Probe of Submonolayer-ﬁlm

Dynamics, “Phys. Rev B 34, 1403–1404.

Williams, E. D. (1994), “Surface Steps and Surface Morphology: Understanding Macroscopic Phenomena

from Atomic Observations,” Surf. Sci. 299/300, 502–524.

Wilson, R. G., Stevie, F. A., and Magee, C. W. (1989), Secondary Ion Mass Spectroscopy, Wiley, New York.

Wood, E. A. (1963), “Vocabulary of Surface Crystallography,” J. Appl. Phys. 35, 1306–1312.

Zalm, P. C. (1995), “Ultra Shallow Doping Proﬁling with SIMS,” Rep. Prog. Phys. 58, 1321–1374.

Zangwill, A. (1988), Physics at Surfaces, Cambridge University Press, Cambridge, U.K.

Zhong, W. and Tomanek, D. (1990), “First-Principles Theory of Atomic-Scale Friction,” Phys. Rev. Lett.

64, 3054–3057.

Zhu, X. Y., Hermanson, J., Arlinghaus, F. J., Gay, J. G., Richter, R., and Smith, J. R. (1984), “Electronic

Structure and Magnetism of Ni(100) Films: Self-Consistent Local-Orbital Calculations,” Phys.

Rev. B 29, 4426–4438.

Majumdar, A. et. al. “Characterization and Modeling of Surface...”

Handbook of Micro/Nanotribology.

Ed. Bharat Bhushan

Boca Raton: CRC Press LLC, 1999

Characterization and

Modeling of Surface

Roughness and

Contact Mechanics

Arun Majumdar and Bharat Bhushan

4.1 Introduction

4.2 Why Is Surface Roughness Important?

How Rough Is Rough? • How Does Surface Roughness

Inﬂuence Tribology?

4.3 Surface Roughness Characterization

Probability Height Distribution • rms Values and Scale

Dependence • Fractal Techniques • Generalized Technique

for Fractal and Nonfractal Surfaces

4.4 Size Distribution of Contact Spots

Observations of Size Distribution for Fractal Surfaces •

Derivation of Size Distribution for Any Surface

4.5 Contact Mechanics of Rough Surfaces

Greenwood–Williamson Model • Majumdar–Bhushan

Model • Generalized Model for Fractal and Nonfractal

Surfaces • Cantor Set Contact Models

4.6 Summary and Future Directions

References

Appendix 4.1

Appendix 4.2

Appendix 4.3

Abstract

Almost all surfaces found in nature are observed to be rough at the microscopic scale. Contact between

two rough surfaces occurs at discrete contact spots. During sliding of two such surfaces, interfacial

forces that are responsible for friction and wear are generated at these contact spots. Comprehensive

theories of friction and wear can be developed if the size and the spatial distributions of the contact

spots are known. The size of contact spots ranges from nanometers to micrometers, making tribology

a multiscale phenomena. This chapter develops the framework to include interfacial effects over a

whole range of length scales, thus forming a link between nanometer-scale phenomena and macro-

scopically observable friction and wear. The key is in the size and spatial distributions, which depend

not only on the roughness but also on the contact mechanics of surfaces. This chapter reexamines the

intrinsic nature of surface roughness as well as reviews and develops techniques to characterize rough-

ness in a way that is suitable to model contact mechanics. Some general relations for the size distri-

butions of contact spots are developed that can form the foundations for theories of friction and wear.

4.1 Introduction

Friction and wear between two solid surfaces sliding against each other are encountered in several day-

to-day activities. Sometimes they are used to our advantage such as the brakes in our cars or the sole of

our shoes where higher friction is helpful. In other instances such as the sliding of the piston against the

cylinder in our car engine, lower friction and wear are desirable. In such cases, lubricants are often used.

Friction is usually quantiﬁed by a coefﬁcient µ, which is deﬁned as

(4.1.1)

where

is the frictional force and

is the normal compressive force between the two sliding bodies.

The basic problem in all studies on friction is to determine the coefﬁcient µ. With regard to wear, it is

necessary to determine the volume rate of wear,

, and to establish the conditions when catastrophic

failure may occur due to wear.

Despite the common experiences of friction and wear and the knowledge of its existence for thousands

of years, their origins and behavior are still not well understood. Although the effects of friction and wear

can sometimes be explained post-mortem, it is normally very difﬁcult to predict the value of µ

and the

wear characteristics of the two surfaces. One could therefore characterize tribology as a ﬁeld that is perhaps

in its early stages of scientiﬁc development, where phenomena can be observed but can rarely be predicted

with reasonable accuracy. The reason for this lies in the extreme complexity of the surface phenomena

involved in tribology. Three types of surface characteristics contribute to this complexity: (1) surface

geometric structure; (2) the nature of surface forces; and (3) material properties of the surface itself.

The lack of predictability in tribology lies in the convolution of effects of surface chemistry, mechanical

deformations, material properties, and complex geometric structure. It is very difﬁcult to say which of

these is more dominant than the others. Physicists and chemists normally focus on the surface physics

and chemistry aspects of the problem, whereas engineers study the mechanics and structural aspects. In

a real situation of two macroscopic surfaces sliding in ambient conditions, it is very difﬁcult to separate

or isolate the different effects and then study their importance. They can indeed be isolated under

controlled conditions such as in ultrahigh vacuum, but how those results relate to real situations is not

clearly understood. It is, therefore, of no surprise that the tribology literature is replete with different

theories of friction and wear that are applicable at different length scales — macroscopic to atomic scales.

It is also of no surprise that a single unifying theory of tribology has not yet been developed.

In this chapter we will examine only one aspect of tribology and that is the effect of surface geometric

structure. It will be shown that friction and wear depend on surface phenomena that occurs over several

length scales, starting from atomic scales and extending to macroscopic “human” scales, where objects,

motion, and forces can be studied by the human senses. Atomic-scale studies focus on the nature of

surface forces and the displacements that atoms undergo during contact and sliding of two surfaces. This

book has several chapters devoted to this important new ﬁeld of

nanotribology

. One must, however,

remember that friction is still a macroscopically observable phenomena. Hence, there must be a link or

a bridge between the atomic-scale phenomena and the macroscopically observable motion and measur-

able forces. This chapter takes a close look at surface geometric structure, or surface roughness, and

attempts to formulate a methodology to form this link between all the length scales.

µ=

First, the inﬂuence of surface roughness in tribology is established. Next, the complexities of the surface

microstructure are discussed, and then techniques to quantify the complex structures are developed. The

ﬁnal discussion will demonstrate how to combine the knowledge of surface microstructure with that of

surface forces and properties to develop comprehensive models of tribology. The reader will ﬁnd that

the theories and models are not all fully developed and much research remains to be done to understand

the effects of surface roughness, in particular, and tribology, in general. This, of course, means that there

is tremendous opportunity for contributions to understand and predict tribological phenomena.

4.2 Why Is Surface Roughness Important?

Solid surfaces can be formed by any of the following methods: (1) fracture of solids; (2) machining such

as grinding or polishing; (3) thin-ﬁlm deposition; and (4) solidiﬁcation of liquids. It is found that most

solid surfaces formed by these methods are not smooth. Perfectly ﬂat surfaces that are smooth even on

the atomic scale can be obtained only under very carefully controlled conditions and are very rare in

nature. Therefore, it is of most practical importance to study the characteristics of naturally occurring

or processed surfaces which are inevitably rough. However, the ﬁrst question that the reader may ask is

how rough is rough and when does one call something smooth?

4.2.1 How Rough Is Rough?

Smoothness and roughness are very qualitative and subjective terminologies. A polished metal surface

may appear very smooth to the touch of a ﬁnger, but an optical microscope can reveal hills and valleys

and appear rough. The ﬁnger is essentially a sensor that measures surface roughness at a lateral length

scale of about 1 cm (typical diameter of a ﬁnger) and a vertical scale of about 100 µm (typical resolution

of the ﬁnger). A good optical microscope is also a roughness sensor that can observe lateral length scales

of the order of about 1 µm and can distinguish vertical length scales of about 0.1 µm. If the polished

metal surface has a vertical span of roughness (hills and valleys) of about 1 µm, then the person who

uses the ﬁnger would call it a smooth surface and the person using the microscope would call it rough.

This leads to what one may call a “roughness dilemma.” When someone asks whether a surface is rough

or smooth, the answer is — it depends!! It basically depends on the length scale of the roughness

measurement.

This problem of scale-dependent roughness is very intrinsic to solid surfaces. If one uses a sequence

of high-resolution microscopes to zoom in continuously on a region of a solid surface, the results are

quite dramatic. For most solid surfaces it is observed that under repeated magniﬁcation, more and more

roughness keeps appearing until the atomic scales are reached where roughness occurs in the form of

atomic steps (Williams and Bartlet, 1991). This basic nature of solid surfaces is shown graphically for a

surface proﬁle in Figure 4.1. Therefore, although a surface may appear very smooth to the touch of a

ﬁnger, it is rough over all lateral scales starting from, say, around 10

–4

m (0.1 mm) to about 10

–9

(1 nm). In addition, the roughness often appears random and disordered, and does not seem to follow

FIGURE 4.1

Appearance of surface roughness under repeated magniﬁcation up to the atomic scales, where atomic

steps are observed.

any particular structural pattern (Thomas, 1982). The

randomness

and the

multiple

roughness scales

both

contribute to the

complexity

of the surface geometric structure. It is this complexity that is partly

responsible for some of the problems in studying friction and wear.

The multiscale structure of surface roughness arises due to the fundamentals of physics and thermo-

dynamics of surface formation, which will not be discussed in this chapter. What will be discussed is the

following. Given the complex multiscale roughness structure of a surface, (1) how does it inﬂuence

tribology; (2) how does one quantify or characterize the structure; and (3) how does one use these

characteristics to understand or study tribology?

4.2.2 How Does Surface Roughness Inﬂuence Tribology?

Consider two multiscale rough surfaces (belonging to two solid bodies), as shown in Figure 4.2a, in

contact with each other without sliding and under a static compressive force of

. Since the surfaces are

not smooth, contact will occur only at discrete points which sustain the total compressive force. Figure 4.3

shows a typical contact interface which is formed of contact spots of different sizes that are spatially

distributed randomly over the interface. The spatial randomness comes from the random nature of

surface roughness, whereas the different sizes of spots occur due to the multiple scales of roughness. For

a given load, the size of spots depends on the surface roughness and the mechanical properties of the

contacting bodies. If this load is increased, the following would happen. The existing spots will increase

in size, new spots will appear, and two or more spots may coalesce to form a larger spot. This is depicted

by computer simulations of real surfaces in Figure 4.3. The surface in this case has isotropic statistical

properties; that is, it does not have any texture or bias in any particular direction. It is evident that even

for an isotropic surface the shapes of the contact spots are not isotropic and can be quite irregular and

complex. In addition, when the load is increased, there are no set rules that the contact spots follow.

Thus, the static problem itself is quite difﬁcult to analyze. But one must, nevertheless, attempt to do so

since these contact spots play a critical role in friction as explained below.

Consider the two surfaces to slide against each other. To do so, one must overcome a resistive tangential

or frictional force

. It is clear that this frictional force must arise from the force interactions between

the two surfaces that act only at the contact spots, as shown in Figure 4.2b. Since the normal load-bearing

FIGURE 4.2

(a) Schematic diagram of two surfaces in static contact against each other. Note that the contact takes

place at only a few discrete contact spots. (b) When the surfaces start to slide against each other, interfacial forces

act on the contact spots.

capacity depends on the contact spot size, it is reasonable to assume that the tangential force is also size

dependent. Therefore, to predict the total frictional force, it is very important to determine the

size

distribution

(

), of the contact spots such that the number of spots between area

and

is equal

(

)

. In addition to the interactions at each spot, there could be tangential force interactions between

two or more contact spots. This is because the contact spots cannot operate independently of each other

since they are connected by the solid bodies that can sustain some elastic or plastic deformation. So one

can imagine the contact spots to be connected by springs whose spring constant depends on the elastic-

ity/plasticity of the contacting materials. Because the number of contact spots is very large, the mesh of

contact spots and springs thus forms a very complicated dynamic system. The deformations of the springs

are usually localized around the contact spot and so the proximity of two spots inﬂuences their dynamic

interactions. Therefore, it is also important to determine the

spatial distribution

∆

(

), of contact spots

where

∆

is equal to the average closest distance between a contact spot of area

and a spot of area

In other words, the frictional force,

, is a cumulative effect that arises due to force interactions at each

spot and also dynamic force interactions between two or more spots. This can be written in a mathe-

matical form as

(4.2.1)

where

(

) is the shear stress on a contact spot of area

is the area of the largest contact spot, and

(

) is the size distribution of contact spots. Similarly, the total volume rate

of wear that is removed

from the surface can be written as

(4.2.2)

FIGURE 4.3

Qualitative illustration of behavior of contact

spots (dark patches) on a contact interface under different

loads. (a) At very light loads only few spots support the load;

(b) at moderate loads the contact spots increase in size and

number; (c) at high loads the contact spots merge to form

larger spots and the number further increases.

Fanaada

() ()

{}

∫

dynamic interaction terms

V vanada

() ()

∫

where

(

) is the volume rate of wear at the microcontact of area

. It can be seen that as long as the

size distribution

(

) is known, tribological phenomena can be studied at the scale of the contact spots.

Let us concentrate on the ﬁrst term on the right-hand side of Equation 4.2.1. This term adds up the

tangential force on each contact spot starting from areas that tend to zero to the upper limit

, which

is the area of the largest contact spot. Recent studies have shown that the shear stress

(

) is not a constant

and can be size dependent. In other words, the frictional phenomena at the nanometer scale can be quite

different from that at macroscales (Mate et al., 1987; Israelachvili et al., 1988; McGuiggan et al., 1989;

Landman et al., 1990). In addition, the shear stress is strongly inﬂuenced by the different types of surface

forces

(Israelachvili, 1992). Some of the chapters in this book have concentrated on studying the nature

(

) when

is at the atomic or nanometer scales.

The second term on the right hand side of Equation 4.2.1 represents the dynamic spring–mass inter-

actions between the contact spots. Although this depends on the spatial and size distributions, it is unclear

what the functional form would be. However, it is not insigniﬁcant since collective phenomena such as

onset of sliding and stick-slip depend upon these types of interactions. Recently, there has been some

interest in studying this as a percolation or a self-organized critical phenomena

(Bak et al., 1988). The

onset of sliding friction can be pictured as follows. When an attempt is made to slide one surface against

another, the force on a contact spot can be released and distributed among neighboring spots. The forces

in at least one of these spots may exceed a critical level creating a cascade or an avalanche. The avalanche

may turn out to be limited to a small region or become large enough so that the whole surface starts

sliding. During this process, the interface evolves into a self-organized critical system insensitive to the

details of the distribution of initial disorder. This type of analysis has been used to provide a physical

interpretation of the Guttenberg–Richter relation between earthquake magnitude and its frequency

(Sornette and Sornette, 1989; Knopoff, 1990; Carlson et al., 1991).

In summary, the basic problem of tribology can be divided as follows: (1) to determine the size,

(

and spatial distribution of contact spots which depends on the surface roughness, normal load, and

mechanical properties; (2) to ﬁnd the tangential surface forces at each spot; (3) to determine the dynamic

interactions between the spots; and, ﬁnally, (4) to ﬁnd the cumulative effect in terms of the frictional

force,

4.3 Surface Roughness Characterization

A rough surface can be written as a mathematical function:

(

x,y

), where

is the vertical height and

and

are the coordinates of a point on the two-dimensional plane, as shown in Figure 4.4a. This is

typically what can be obtained by a roughness-measuring instrument. The surface is made up of hills

and valleys often called surface asperities of different lateral and vertical sizes, and are distributed

randomly on the surface as shown in the surface proﬁles in Figure 4.4b. The randomness suggests that

one must adopt statistical methods of roughness characterization. It is also important to note that because

of the involvement of so many length scales on a rough surface, the characterization techniques must be

independent of any length scale. Otherwise, the characterization technique will be a victim of the “rough

or smooth” dilemma as discussed in Section 4.2.1.

4.3.1 Probability Height Distribution

One of the characteristics of a rough surface is the probability distribution

(Papoulis, 1965)

(

), of the

surface heights such that the probability of encountering the surface between height

and

is equal

(

)

. Therefore, if a rough surface contacts a hard perfectly ﬂat surface* and it is assumed that the

*Although hard ﬂat surfaces are rarely found in nature, we make the assumption because contact between two

rough surfaces can be reduced to the contact between an equivalent surface and a hard ﬂat surface (see Section 4.4).

distribution

(

) remains unchanged during the contact process, then the ratio of real area of contact,

, to the apparent area,

, can be written as

(4.3.1)

where

is the separation between the ﬂat surface,

is the standard deviation of the surface heights, and

–

is the nondimensional surface height. The real area of contact,

, is usually about 0.1 to 10%

of the apparent area and is the sum of the areas of all the contact spots. Therefore, the probability

distribution,

(

), can be used to determine the sum of the contact spot areas but does not provide the

crucial information on the size distribution,

(

). In addition, it contains no information concerning

the shape of the surface asperities.

It is often found that the normal or Gaussian distribution ﬁts the experimentally obtained probability

distribution quite well

(Thomas, 1982; Bhushan, 1990). In addition, it is simple to use for mathematical

calculation. The bell-shaped normal distribution

(Papoulis, 1965) which has a variance of unity is given as

(4.3.2)

where

–

is the nondimensional mean height. The mean height and the standard deviation can be found

from a roughness measurement

(

x,y

) as

FIGURE 4.4

(a) Schematic diagram of a rough surface whose surface height is

(

x, y

) at a coordinate point (

x, y).

(b) A vertical cut of the surface at a constant y gives surface proﬁle z(x) with a certain probability height distribution.

gzdz gz dz

()

∞∞

∫∫

()

−

()













−∞< <∞

exp

(4.3.3)

(4.3.4)

Here, L

and L

are the lengths of surface sample, whereas N

and N

are the number of points in the x

and y lateral directions, respectively. The integral formulation is for theoretical calculations, whereas the

summation is used for calculating the values from ﬁnite experimental data.

Although used extensively, the normal distribution has limitations in its applicability. For example, it

has a ﬁnite nonzero probability for surface heights that go to inﬁnity, whereas a real surface ends at a

ﬁnite height, z

max

, and has zero probability beyond that. Therefore, the normal distribution near the tail

is not an accurate representation of real surfaces. This is an important point since it is usually the tail of

the distribution that is signiﬁcant for calculating the real area of contact. Other distributions, such as

the inverted chi-squared (ICS) distribution, ﬁt the experimental data much better near the tail of the

distribution

(Brown and Scholz, 1985). This is given for zero mean and in terms of nondimensional

height,

–

z, as

(4.3.5)

which has a variance of 2ν and a maximum height

–

max

= . The advantage of the ICS distribution

is it has a ﬁnite maximum height, as does a real surface, and has a controlling parameter ν, which gives

a better ﬁt to the topography data. The Gaussian and the ICS distributions are shown in Figure 4.5. Note

that as ν increases, the ICS distribution tends toward the normal distribution. Brown and Scholz (1985)

FIGURE 4.5 Comparison of the Gaussian and the ICS distributions for zero mean height and nondimensional

surface height

–

zx y dxdy

zx y

()

∫∫

∑∑

σ=

()

−

[]

()

−

[]

∫∫

∑∑

z x y z dx dy

zx y z

ij m

gz z z ez z z z

()

−

()

−

()

−∞< <

()

−

max max max

ν 2⁄