Majumdar, A. et. al. “Characterization and Modeling of Surface...”
Handbook of Micro/Nanotribology.
Ed. Bharat Bhushan
Boca Raton: CRC Press LLC, 1999
© 1999 by CRC Press LLC© 1999 by CRC Press LLC

4

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


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

F

f

is the frictional force and

F

n

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,
·


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=
F
F
f
n

© 1999 by CRC Press LLC

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


–9

m
(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.

© 1999 by CRC Press LLC

any particular structural pattern (Thomas, 1982). The

randomness

and the

multiple
roughness scales

both
contribute to the

complexity


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

F

f

. 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.

© 1999 by CRC Press LLC

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

,

n

(

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

,

∆

(

a

i

,a

j

), of contact spots
where

∆

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



a

,

a

L

is the area of the largest contact spot, and

n

(

a

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

V

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


ν

(

a

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

a

. It can be seen that as long as the
size distribution

n

(

a

) 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

a

L

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

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,

n

(

a

),
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,


y

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)

g

(

z

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

z


g

(

z

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

A

r

, to the apparent area,

A

a

, can be written as
(4.3.1)
where

d

is the separation between the flat surface,

σ

is the standard deviation of the surface heights, and



n

(

a

). 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

–

z

m

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

=
()
=
()
∞∞
∫∫
σ
σ
gz
zz
z
m
()
=
π
−
−
()








−∞< <∞
1
2
2

(4.3.5)
which has a variance of 2ν and a maximum height
–
z
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
–
z.
z
LL
zx y dxdy
NN
zx y
m
xy
LL
xy
ij
j
N
i
N
yx
y
x

zx y z
xy
m
LL
xy
ij m
j
N
i
N
yx
y
x
,,
gz z z ez z z z
()
=
()
()
−
()
−
()
−∞< <
()
−
ν
ν
ν
ν

of two regions of lengths L
1
and L
2
can be added up as
(4.3.7)
They gathered roughness measurements of a wide range of surfaces to show that the surfaces follow the
nonstationary behavior of Equation 4.3.6. However, Berry and Hannay (1978) suggested that the variance
can be represented in a more general way as follows:
(4.3.8)
where n varies between 0 and 2.
If the exponent n in Equation 4.3.8 is not equal to zero of a particular surface, then the standard
deviation or the rms height, σ, is scale dependent, thus making a rough surface a nonstationary random
process. This basically arises from the multiscale structure of surface roughness where the probability
distribution of a small region of the surface may be different from that of the larger surface region as
depicted in Figure 4.4b. If the larger segment follows the normal distribution, then the magnified region
may or may not follow the same distribution. Even if it does follow the normal distribution, the rms σ
can still be different.
Other statistical parameters that are also used in tribology (Nayak, 1971, 1973) are the rms slope, σ′
x
,
and rms curvature, σ″
x
, defined as
(4.3.9)
*The rms values (of height, slope, or curvature) are related to the corresponding standard deviation, σ, of a surface
in the following way: rms
2
= σ
2

=
∂
()
∂








=
−
()
−
()








∫
∑
+
=
−

the statistical sample size or the instrument resolution. Figure 4.6 shows the rms data for a magnetic tape
surface (Bhushan et al., 1988; Majumdar et al., 1991). Along the ordinate is plotted the ratio of the rms
value at a magnification, β, to the rms value at magnification of unity. The magnification β = 1 corre-
sponds to an instrument resolution of 4 µm and scan size of 1024 × 1024 µm containing 256 × 256
roughness data points. The roughness data in the range 1 < β < 10 were obtained by optical interferometry
(Bhushan et al., 1988), whereas for β > 10, the data were obtained by atomic force microscopy (Majumdar
et al., 1991; Oden et al., 1992). An increase in β corresponds to an increase in instrument resolution with
the highest being equal to 1 nm. The data clearly show that the rms height does not change over five
decades of length scales and can therefore be considered scale independent over this range of length scales.
However, the rms slope increases with magnification as β
1
and the rms curvature increases as β
2
. Figure 4.7
shows similar variations for a polished aluminum nitride surface where the roughness data was obtained
by atomic force microscopy. In this case, the rms height σ reduces with decreasing sample size but does
not follow the trend σ ≈ as suggested by Sayles and Thomas

(1978). Nevertheless, the variation does
make the surface a nonstationary random process. The rms slope and the rms curvature, on the other
hand, increase with the instrument resolution, as observed in Figure 4.7.
Although Figures 4.6 and 4.7 show statistics for specific surfaces, the trends are typical for most rough
surfaces that have been examined. The following can be concluded from these trends. The rms height,
FIGURE 4.6 Variation of rms height, slope, and curvature of a magnetic tape surface as a function of magnification,
β, or instrument resolution. The vertical axis is the ratio of an rms quantity at a magnification β to the rms quantity
at magnification of unit, which corresponds to an instrument resolution of 4 µm. Roughness measurements of β <
10 were obtained by optical interferometry (Bhushan et al., 1988), whereas that for β > 10 were obtained by atomic
force microscopy (Oden et al., 1992).
′′
=

∫
∑
+
+
=
−
σ
x
x
L
ij ij
ij
ii
N
L
zx y
x
dx
N
zx y zx y
zx y
x
x
11
2
2
2
2
0
2

FIGURE 4.8 Illustration of roughness measurements at different instrument resolution τ. As τ is reduced, the
surface that is measured is quite different, as qualitatively shown. The average slope and the average curvature of the
profile is higher for smaller τ.
© 1999 by CRC Press LLC
4.3.3 Fractal Techniques
4.3.3.1 A Primer for Fractals
The self-repeating nature of surface roughness has not only been found in surfaces but also in several
objects found in nature. In his classic paper, Mandelbrot (1967) showed that the coastline of Britain has
self-similar features such that the more the coastline is magnified, the more features and wiggliness are
observed. In fact, the answer to the question — “How long is the coastline of Britain?” — is it depends
on the unit of measurement and is not unique. This is shown in Figure 4.9 for several coastlines and also
for a circle. The fundamental problem of this scale dependence is that “length” as measured by a ruler
or a straightedge is a measure of only one-dimensional objects. No matter how small a unit you take for
the measurement, the length would still come out the same. In other words, if you take a straight line,
then the length would be the same whether you take 1 mm or 1 µm as the unit of measurement. The
reason for the scale independence at a very minute scale is that the line or the curve is made up of smooth
and straight line segments. However, if an object is never smooth no matter what length scale you choose,
then repeated magnifications will reveal different levels of wiggliness as shown in Figure 4.10. Large units
of measurement fail to measure the small wiggliness of the curve, whereas the small units of measurement
will measure them. In other words, different units of measurement will measure only some levels of the
wiggliness but not all levels. Thus, one would get a different number for the length of the object as the
unit of measurement is changed.
Since objects of the dimension unity are defined to have their lengths independent of the unit of
measurement, an object with scale-dependent length is not one-dimensional. Similarly, if the area of a
surface depends on the unit of measurement, then it is not a two-dimensional object.
FIGURE 4.9 Dependence of the length of different coastlines and curves on the unit ε of measurement. Note the
power law dependence of the length on ε.
© 1999 by CRC Press LLC
One of the properties of naturally occurring wiggly objects is that if a small part of the object is
enlarged sufficiently, then statistically it appears very similar to the whole object. For example, if you

L
D
≈
−
ε
1
© 1999 by CRC Press LLC
where D is called the fractal dimension of the coastline. If D = 1, then the length is independent of ε and
it can be called a one-dimensional object. It is observed that this power law behavior remains unchanged
over several decades of length scales such that the value of D, which in some sense measures the wiggliness
of the curve, remains constant and independent of ε. Therefore, D is one parameter that can be used to
characterize a coastline. Another way of looking at this behavior is the following — although the coastline
seems a rather convoluted and complex geometric structure, the power law behavior represents a pattern
or order in this chaotic structure.
4.3.3.2 Fractal Characterization of Surface Roughness
The same concept can be used to characterize a rough surface. However, there is a difference between a
coastline and a rough surface. To show the self-similarity of a coastline, one needs to take a small part
and enlarge it equally in all directions to resemble the full coastline statistically, as qualitatively shown
in Figure 4.9. However, for a small region of a rough surface to statistically resemble* a larger region, the
enlargement should be done unequally in the vertical (z) and lateral (x and y) directions. Such objects,
which scale differently in different directions, are called self-affine (Mandelbrot, 1982, 1985; Voss, 1988).
To characterize a self-affine object one cannot use the length of the surface profile or the area of the
surface as a measure

(Mandelbrot, 1985). There are two other ways to characterize it — the power
spectrum P(ω) and the structure function, S(τ).
4.3.3.2.1 Power Spectrum
Consider a surface profile, z(x) in the x-direction. The power spectrum of the profile can be found by
the relation (Blackman and Tuckey, 1958; Papoulis, 1965):
(4.3.12)

() ( )
∫
1
0
2
exp
σωω
ω
ω
=
()
∫
Pd
l
h
′
=
()
∫
σωωω
ω
ω
2
Pd
l
h
′′
=
()
∫

(2–D)
;
σ′ = ω
h
(D–1)
; σ″ = ω
h
D
. It is evident that the rms values depend either on the low-frequency or high-
frequency cutoff and are therefore scale dependent. Figures 4.6 and 4.7 confirm this experimentally and,
in fact, show the decrease in exponent by 1 as we go from the rms curvature to the rms slope and finally
to the rms height. The only difference that one finds in the rms quantities is that the rms slope and the
curvature depend on the high-frequency cutoff, whereas the rms height depends on the low-frequency
cutoff. The relation σ ≈ L
(2-D)
is exactly the same as suggested in Equation 4.3.8 with n = 2(2 – D). In
fact, the relation suggested by Sayles and Thomas (1978) in Equation 4.3.6 is a special case when D = 1.5.
The variance of the height distribution, σ
2
, is equal to the area under the power spectral curve as
mathematically shown in Equation 4.3.13. When the variance (or the rms height) is independent of the
sample size or any length scale, as demonstrated in Figure 4.6, the area under the power spectrum must
be constant and independent of ω
l
and ω
h
. Therefore, the fractal power law variation of the spectrum in
Equation 4.3.16 is clearly not valid for such a case since it always leads to ω
l
-dependence of the rms

frequencies in the range ω > ω
n
into the range ω < ω
n
. The problem comes about due to the discreteness
of the roughness measurement. To overcome this problem, we have found that the structure function
can yield more accurate estimation of D and C.
4.3.3.2.2 Structure Function
The structure function (Mandelbrot, 1982; Voss, 1988) is defined as
(4.3.17)
The summation on the right-hand side can be used for calculation of a measured surface profile con-
taining N points. As one can see, the structure function is easy to calculate since it does not involve any
transformation but simple height differences and averages. It is sometimes used in experimental and
theoretical analysis of velocity and scalar fluctuations in turbulent fluid dynamics

(Kolmogoroff, 1941).
In turbulence, the fluctuating quantity varies with time and space, whereas for rough surface, the same
varies with space. The problems are quite similar since in turbulence, too, the power spectrum of the
fluctuations is broadband and follows the power law behavior of Equation 4.3.16.
It is interesting to note that in some ways the structure function and the variance, σ
2
, of height in
Equation 4.3.4 are similar since both involve finding the average of the square of height differences.
However, the structure function uses height differences with points separated by a distance τ, whereas
for the variance, the height differences are with the mean height z
m
. The structure function yields much
more information than the rms height since by varying τ, one can study the roughness structure at
different length scales. This is, of course, not possible for the variance, σ
2

τ
τ
()
=+
()
−
()
[]
=
−
()
+
()
−
()
[]
∫
∑
=
−
()
11
2
0
2
1
∆
∆
′
()

2
SG
DD
ττ
()
=
−
()
−
()
2122
SPidτ ω ωτ ω
()
=
() ( )
−
[]
−∞
∞
∫
exp 1
© 1999 by CRC Press LLC
(4.3.21)
such that the factor C of the power spectrum is related to the scaling constant G of the structure function as
(4.3.22)
Berry and Blackwell (1981) follow a slightly different definition of a fractal surface — a surface profile
is said to be a self-affine fractal when
(4.3.23)
where the parameter G is called “topothesy” following the term coined by Sayles and Thomas


roughness measured by different techniques.
4.3.3.3.1 Stylus Profilometry
The roughness of machined (lapped, ground, and shape turned) stainless steel surfaces was measured by
a contact stylus profiler

(Majumdar and Tien, 1990). The instrument used a diamond stylus of radius
2.5 µm and had a vertical resolution of 0.5 nm. The scan lengths ranged from 50 to 30 mm with each
scan having 800 to 1000 evenly spaced points. Figure 4.12 shows the roughness profile of a lapped stainless
steel surface.
The structure functions, S(τ), of these surface profiles are plotted on a log–log plot in Figure 4.13.
Also shown is the straight line, S(τ) ≈ τ
1
, which corresponds to a fractal dimension of D = 1.5. It is
S
C
D
D
D
D
ττ
()
=
−
()
π−
()







−
()
−
()
2
23
2
23
21
sin Γ
SG
DD
ττ τ
()
=→
−
()
−
()
2122
0 for
S
G
DD
τσ
τ
σ
()

of S(τ) for the rougher surfaces leads to a higher value of G. The structure function for the lapped-4*
FIGURE 4.12 Profile of lapped stainless steel surface measured by a stylus profilometer. Fractal simulation of the
profile was conducted by the Weierstrass–Mandelbrot function. (From Majumdar, A. and Tien, C. L. (1990), Wear
136; 313–327. With permission.)
FIGURE 4.13 Structure function of machined stainless steel surfaces whose roughness was measured by stylus
profilometry.
*The number associated with each process is the rms roughness produced in microinches. One microinch is equal
to 25.4 nm.
© 1999 by CRC Press LLC
profile departs from the D = 1.5 behavior at about 30 µm and those of ground-8 and lapped-8 profile
depart at about 100 µm. This behavior is probably due to the following reason. For any machining process,
there exists a critical length scale below which the surface remains unaffected during machining. For
grinding, this length scale is the grain size of the abrasive material, whereas for turning it is the tool
radius. Below this scale, the surface is formed by a natural process such as fracture. This natural process
seems to lead to the same type of surface fractal behavior with D = 1.5. At length scales larger than the
critical one, the machining processes flattens the surface and thus reduces the height differences between
two points on the surface. Thus, the structure function decreases at larger scales. As shown earlier, the
rms height depends on the total length, L, of the roughness sample as σ ≈ ω
l
D–2
= L
2–D
. Although the
structure function of the different surface profiles at small length scales are nearly the same, their rms
heights are quite different. This is because at larger length scales, which control the value of σ, the
structure functions are different with smoother surfaces having smaller values of S(τ).
4.3.3.3.2 Atomic Force Microscopy
Oden et al. (1992) measured the surface roughness of magnetic tape A (Bhushan et al., 1988) at four
different resolutions by atomic force microscopy. Figure 4.14 shows the image of the tape obtained from
a 0.4 × 0.4 µm scan and 2.5 × 2.5 µm scan. The accicular magnetic particles, typically 0.1 µm in diameter

0
in Figure 4.4.
The scale independence of the rms height, and the general behavior of the structure function data of
the magnetic tape, suggests that this surface is a perfect example of the model proposed by Berry and
Blackwell

(1981), given in Equation 4.3.24 — power law behavior of D = 1.39 as τ→0 and a saturation
behavior as τ→∞.
FIGURE 4.14 Images of a magnetic tape A (Bhushan et al., 1988) surface obtained by atomic force microscopy.
Note the magnetic particles, which are oblong in shape with aspect ratio 10 and a diameter of about 100 nm.
C C C
© 1999 by CRC Press LLC
The surface topography of several magnetic thin-film rigid disks was also studied by atomic force
microscopy

(Bhushan and Blackman, 1991). The manufacturing process for these disks are discussed by
Bhushan and Doerner (1989) and are summarized in Table 4.1. Figure 4.16 is an example of an AFM
image of magnetic disk C (Bhushan and Doerner, 1989) for which the surface is composed of columnar
grains of about 0.1 to 0.2 µm width which form during sputter-deposition. Since the substrate was
untextured, the roughness of the films appeared quite isotropic. This is a 2.5 × 2.5 µm image which has
a resolution of 12.5 nm. To check whether or not surface roughness appears at even smaller scales, a
0.4 × 0.4 µm scan, having a resolution of 2 nm, was obtained for the same surface. The structure function
of the surface profiles for both scans revealed that roughness does appear fractal at nanometer scales as
shown in Figure 4.17. The power law behavior of S(τ) ~ τ
1.49
suggests a fractal dimension of D = 1.26
for the surface profiles. The structure function deviates from this power law behavior at about 0.2 µm.
It is interesting to note that this corresponds to the size of the columnar grains that are visible in the
atomic force microscopy image. Therefore, this power law behavior corresponds to intergranular surface
roughness. It is difficult to obtain any meaningful information for larger length scales when τ is compa-

goes across all the quasi-periodic texture marks, which leads to oscillations in the structure function.
Such oscillations cannot be modeled by fractals and must be handled by a more general technique, as
discussed in Section 4.3.4. Figure 4.20 shows the structure function of the textured magnetic rigid disk
D, whereas Figure 4.21 shows that of the untextured rigid disk E. In both these cases the magnetic thin
films were electroless plated on to the substrate. Note that the data levels off for τ > 50 nm, which is
probably a characteristic length scale for the plating process.
It is clear from the structure function data that there normally exists a transition length scale, l
12
,
which demarcates two regimes of power law behavior. At scales smaller than l
12
, the fractal power law
behavior is generally followed for all of the surfaces. At larger length scales, the structure function of the
polished (or untextured) disks either saturates such that the Berry–Blackwell model can be easily applied
or, in some cases, it follows a different power law behavior that can be characterized by another fractal
dimension. If the surface is textured, however, the structure function at larger length scales oscillates and
does not follow a scaling power law behavior. Such nonfractal behavior cannot be characterized by the
fractal techniques and a more general method is needed. This is discussed in detail in Section 4.3.4. The
transition length scale, l
12
, usually corresponds to a surface machining or growth process. For polycrys-
talline surfaces this may be the grain size, whereas for machining it is the characteristic tool size.
Recent experiments by Ganti and Bhushan (1995) showed that when a surface is imaged with atomic
force microscopy and an optical profiler, values of D of a wide variety of surfaces fall in a close range
but the values of G can vary a lot for the same surface. This is in contrast with the data presented above.
However, the check for reliable data is to see whether or not the structure functions of the roughness
measured at different resolutions and by different instruments overlap over common length scales.
Inspection of their data showed that although the structure functions of the atomic force microscopy
measurements of different scan sizes for the same surface seem to overlap over the common length scales,
there was large discrepancy between the structure functions obtained from atomic force microscopy and

the concepts of rough and smooth are too crude to distinguish between amplitude variations and
frequency variation (or jaggedness) and so it is difficult to say which can be called rougher or smoother.
It could be a combination of both, but at present it is unknown what this combination is.
FIGURE 4.21 Structure function of untextured magnetic rigid disk E (Bhushan and Doerner, 1989) in which the
magnetic thin films were electroless plated on a polished substrate. The triangles are for a 0.4 × 0.4 µm atomic force
microscopy scan containing 200 × 200 points. The circles are for a 2.5 × 2.5 µm atomic force microscopy scan
containing 200 × 200 points.
*These are fractal simulations of rough surfaces obtained by using the Weierstrass–Mandelbrot function. Details
of the simulation procedure is discussed in detail elsewhere

(Voss, 1988; Majumdar and Bhushan, 1990; Majumdar
and Tien, 1990).