Davim J.P. Tribology for Engineers: A Practical Guide

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1.5 Multiscale characterization of

surface topography

The deviation of a surface from its mean plane is assumed to

be a random process, which is characterized by the statistical

parameters such as the variance of the height, the slope and

the curvature. But, it has been observed that surface topography

is a non-stationary random process. It means the variance of

the height distribution is related to the sampling length and

hence is not unique for a particular surface. Rough surfaces

are also known to exhibit the feature of geometric self-

similarity and self-afﬁ nity, by which similar appearances of

the surface are seen under the various degrees of magniﬁ cation

as quantitatively shown in Fig. 1.11. Since increasing amounts

of detail in the roughness are observed at decreasing length

scale, the concepts of slope and curvature, which inherently

assume the smoothness of the surface, cannot be deﬁ ned.

So the variances of slope and curvature depend strongly on

the resolution of the roughness-measuring instrument or some

other form of ﬁ lter and are therefore not unique (Ling, 1990;

Majumdar and Bhushan, 1990; Ganti and Bhushan, 1995;

Sahoo and Roy Chowdhury, 1996). In contemporary literature

such a large number of characterization parameters occur that

the term ‘parameter rash’ is aptly used. The use of instrument-

dependent parameters shows different values for the same

Qualitative description of statistical self-afﬁ nity

for a surface proﬁ le

Figure 1.11

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Surface topography

surface. Thus, it is necessary to characterize rough surfaces

by intrinsic parameters, which are independent of all scales

of roughness. Since this ‘one-scale’ characterization provided

by statistical functions and parameters is insufﬁ cient to

describe the multiscale nature of tribological surfaces, new

‘multiscale’ characterization methods need to be developed.

Recent developments in this area have been concentrated

mostly on four different approaches: Fourier transform

methods; wavelet transformation methods; fractal methods;

and the hybrid fractal-wavelet method.

Fourier transform methods basically decompose the

surface data into complex exponential functions of different

frequencies. The Fourier methods are used to calculate the

power spectrum and the autocorrelation function in order to

obtain the surface topography parameters. However, the

difﬁ culty with the application of these methods is that they

provide results which strongly depend on the scale at which

they are calculated, and hence they are not unique for a

particular surface. This is because the Fourier transformation

provides only the information whether a certain frequency

component exists or not. As the result, the surface parameters

calculated fail to provide information about the scale at

which the particular frequency component appears.

Wavelet methods decompose the surface data into different

frequency components and characterize it at each individual

scale. While applying wavelets, the surfaces are ﬁ rst

decomposed into roughness, waviness and form. Then the

changes in surface peaks, pits and scratches, together with

their locations, are obtained at different scales. However,

there are still major problems in extracting the appropriate

surface texture parameters from wavelets.

The fractal method incorporates fractal dimension which is

an intrinsic property of multiscale roughness of surface

characterization. It is invariant with length scales and is closely

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linked to the concept of geometric self-similarity. The self-

similarity or self-afﬁ nity of rough surfaces implies that as the

unit of measurement is continuously decreased, the surface

area of the rough surface (a two-dimensional measure) tends to

inﬁ nity and the volume (a three-dimensional measure) tends

to zero. Here, self-similarity implies the property of equal

magniﬁ cation in all directions while self-afﬁ nity refers to

unequal scaling in different directions. Thus, the Hausdorff or

fractal dimension, D + 1, of rough surfaces is a fraction between

2 and 3. The proﬁ le of a rough surface Z(x), typically obtained

from stylus measurements, is assumed to be continuous even at

the smallest scales. This assumption breaks down at atomic

scale. But for engineering surfaces the continuum is assumed to

exist down to the limit of a zero-length scale. Since repeated

magniﬁ cations reveal the ﬁ ner levels of detail, the tangent at

any point cannot be deﬁ ned. Thus the surface proﬁ le is

continuous everywhere but non-differentiable at all points.

This mathematical property of continuity, non-differentiability

and self-afﬁ nity is satisﬁ ed by the Weierstrass-Mandelbrot

(W-M) fractal function, which is thus used to characterize and

simulate such proﬁ les. The W-M function has a fractal

dimension D, between 1 and 2, and is given by

Z(x)



L(G / L)

D1

兺

∞

cos 2

πγ

(x / L)

, 1

1 [1.15]

nn

(2D)n

where G is a scaling constant. The parameter n

corresponds

to the low cut-off frequency of the proﬁ le. Since surfaces are

non-stationary random process the lowest cut-off frequency

depends on the length L of the sample and is given by

= 1/L. The W-M function has the interesting mathematical

property that the series for Z(x) converges but that for dZ/dx

diverges. It implies that it is non-differentiable at all points.

The power spectrum of this W-M function can be expressed

by a continuous function as

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Surface topography

) 

2(D1)

1 [1.16]

2ln

52D

When this equation is compared with the power spectrum of

a surface, the dimension D is related to the slope of the

spectrum on a log-log plot against

. The constant G is

the roughness parameter of a surface, which is invariant

with respect to all frequencies of roughness and determines

the position of a spectrum along the power axis. In this

characterization method both G and D are independent of

the roughness scales of the surface and hence intrinsic

properties. The constants of the W-M function, G, D, and

, form a complete and fundamental set of scale-independent

parameters to characterize a rough surface.

The drawback of fractals is that they characterize surfaces

at all scales while the wavelets provide a description at any

particular scale. A possible solution to these problems is to

use a combination of fractals and wavelets. Recently, a hybrid

fractal-wavelet technique, based on the combination of fractal

and wavelet methods, has been developed allowing for the

3-D characterization of often complex tribological surfaces

with a unique precision and accuracy, without the need for

any parameters. First, the surface topography features are

broken down into individual scale components by wavelets

and then fractals are applied to provide a surface topography

description over the ﬁ nest achievable range of scales.

1.6 Surface roughness measurement

The surfaces of any engineering component contain a vast

number of peaks and valleys and it is not possible to measure

the height and location of each of the peaks. So measurements

are taken from a small and representative sample of the surface

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so chosen that there is a high probability for the surface lying

outside the sample to be statistically similar to that lying

within the sample. Over the years many different methods

have been devised to study the topography of surfaces and a

brief outline of some of them is presented here.

1.6.1 Surface proﬁ lometer

The most common method of studying surface texture

features is the stylus proﬁ lometer, the essential features of

which are illustrated in Fig. 1.12. A ﬁ ne, very lightly loaded

stylus is dragged smoothly at a constant speed across the

surface under examination. The transducer produces an

electrical signal, proportional to displacement of the stylus,

which is ampliﬁ ed and fed to a chart recorder that provides a

magniﬁ ed view of the original proﬁ le. But this graphical

representation differs from the actual surface proﬁ le because

of difference in magniﬁ cations employed in vertical and

Component parts of a typical stylus surface-

measuring instrument

Figure 1.12

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horizontal directions. Surface slopes appear very steep on

a proﬁ lometric record though they are rarely steeper than

10° in actual cases. The shape of the stylus also plays a vital

role in incorporating error in measurement. The ﬁ nite tip

radius (typically 1 to 2.5 microns for a diamond stylus) and

the included angle (of about 60° for pyramidal or conical

shape) results in preventing the stylus from penetrating fully

into deep and narrow valleys of the surface and thus some

smoothing of the proﬁ le is done. Some error is also introduced

by the stylus in terms of distortion or damage of a very

delicate surface because of the load applied on it. In such

cases a non-contacting optical proﬁ lometer having optical

heads replacing the stylus may be used. Reﬂ ection of infrared

radiation from the surface is recorded by arrays of photodiodes

and analysis of these in a microprocessor results in the

determination of the surface topography. Vertical resolution

of the order of 0.1 nm is achievable though maximum height

of measurement is limited to a few microns. This method is

clearly advantageous in cases of very ﬁ ne surface features.

1.6.2 Optical microscopy

In this method, the surface of interest is held to reﬂ ect a beam

of visible light and then these are collected by the objective of

the optical microscope. An image of the surface is produced

and is analysed at very high rates of resolution (up to

0.01 microns) by optical interferometers. Depth of ﬁ eld

achievable is up to 5 microns. But success of the method depends

on the reﬂ ective property of the material, which limits its use.

Optical methods may be divided into two groups: geometrical

methods and physical methods. Geometrical methods include

light-sectioning and taper-sectioning methods, while physical

methods include specular reﬂ ection, diffuse reﬂ ection, speckle

pattern and optical interference.

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In the light-sectioning method, the image of a slit is thrown

onto the surface at an incident angle of 45°. The reﬂ ected

image appears as a straight line if the surface is smooth and as

an undulating line if the surface is rough. In the taper-sectioning

method, a section is cut through the surface to be examined at

an angle of θ, thus effectively magnifying the height variation

by a factor of cot θ, and is subsequently examined by an

optical microscope. The surface is supported with an adherent

coating that prevents smearing of the contour during the

sectioning process, while the taper section is lapped, polished

and lightly heat tinted to provide good contrast for optical

examination. This process suffers from disadvantages like

destruction of test surface and tedious specimen preparation.

In the specular reﬂ ection method, gloss or specular

reﬂ ectance that is a surface property of the material and a

function of reﬂ ective index and surface roughness, is measured

by gloss meter. Surface roughness scatters the reﬂ ected light

and affects the specular reﬂ ectance, thus a change in specular

reﬂ ectance provides a measure for surface roughness.

The diffuse reﬂ ection method is particularly suitable for

on-line roughness measurement during manufacture since it

is continuous, fast, non-contacting and non-destructive. This

method employs three varieties of approaches. In the total

integrated scatter (TIS) approach, the total intensity of the

diffusely scattered light is measured and used to generate the

maps of asperities, defects and particles rather than micro-

roughness distribution. The diffuseness of the scattered light

(DSL) approach measures a parameter that characterizes

the diffuseness of the scattered radiation pattern and relates

this to surface roughness. In the angular distribution (AD)

approach, the scattered light provides roughness height,

average wavelength or average slope. With rougher surfaces,

this may be useful as a comparator for monitoring both

amplitude and wavelength surface properties.

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In the speckle pattern method, surface roughness is related

to speckle, which is basically the local intensity variation

between neighbouring points in the reﬂ ected beam when a

surface is illuminated with partially coherent light.

The optical interference technique involves looking at the

interference fringes and characterizing the surface with

suitable computer analysis. Common interferometers include

the Nomarski polarization interferometer and Tolanski

multiple beam interferometer.

1.7 Advanced techniques for surface

topography evaluation

A further improvement in the resolution of surface

topographic examination is possible by the use of electron

microscopes. Two basic types of electron microscopes are

available: scanning electron microscopes and transmission

electron microscopes. In scanning electron microscopy

(SEM) a focused beam of high-energy electrons is incident

on the surface at a point resulting in the emission of secondary

electrons. These are then collected and fed to an ampliﬁ er to

send an electric signal to a cathode ray tube (CRT). The

electron beam is scanned over the surface to have a complete

picture and the CRT screen gives a topographical image of

the entire area of interest. Depth of ﬁ eld is up to 1,000 microns,

which acts as a primary advantage of this method over the

optical method, but one drawback is the requirement on size

of the specimen to be placed within the vacuum chamber of

the instrument. This can be overcome by preparing a replica

of the surface.

In transmission electron microscopy (TEM), the focused

beam of high-energy electrons is made to transmit through a

very thin specimen and the deﬂ ection and scattering of the

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electrons is recorded to analyse surface topography. Preparation

of a specimen thin enough to transmit electrons plays a vital

role and sometimes a replica of the surface retaining all the

texture features but of a material having greater electron

transparency is produced for the same purpose.

Recently, a different type of electron microscopy called

scanning tunnelling electron microscopy (STM) has been

introduced. It incorporates the electron-tunnelling phenomenon

through an insulating layer separating two conductors. The

sharp pointed tip of a probe forms one electrode and the

surface of the specimen the other. The probe is moved by a

highly precise positional controller to keep the tunnelling

current at a steady value and provides an image of the surface

under examination. The method is superior to the earlier ones

in that it does not require any vacuum, but the one disadvantage

is the poor design of the controller mechanism. The principle

of the STM is very simple. Just like in a record player, the

instrument uses a sharp needle, referred to as the tip, to

investigate the shape of the surface, but the STM tip does not

touch the surface. The schematic of the method is shown in

Fig. 1.13. A voltage is applied between the metallic tip and the

specimen, typically ranging between a few milli-volts and

several volts. The tip touching the surface of the specimen

results in a current and when the tip is far away from the

surface, the current is zero. The STM operates in the regime of

extremely small distances between the tip and the surface of

only 0.5 to 1.0 nm, which are typically 2 to 4 atomic diameters.

At these distances, the electrons can jump from the tip to the

surface or vice versa. This jumping is necessarily a quantum

mechanical process, known as ‘tunnelling’ and hence the name

‘scanning tunnelling microscope’. The STMs usually operate

at tunnelling currents between a few pico-Amperes (pA) and a

few nano-Amperes (nA). The tunnelling current depends

critically on the precise distance between the last atom of the

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tip and the nearest atom or atoms of the underlying specimen.

When this distance is increased a little bit, the tunnelling

current decreases heavily. As a thumb rule, for each extra atom

diameter that is added to the distance, the current becomes a

factor of 1,000 lower. Thus the tunnelling current provides a

highly sensitive measure of the distance between the tip and the

surface. The STM tip is attached to a piezo-electric element,

which changes its length a little bit when it is put under an

electrical voltage. The distance between the tip and the surface

can be regulated by adjusting the voltage on the piezo element.

In most STMs, the voltage on the piezo elements is adjusted in

a manner that the tunnelling current always has the same value,

say 1 nA. Thus the distance between the last atom on the tip

and the nearest atoms on the surface is kept constant. Using so

called electronics, the distance regulation is done automatically.

The feedback electronics continually measure the deviation of

the tunnelling current from the desired value and accordingly

adjust the position of the tip. While this feedback system is

Schematic of working of STM

Figure 1.13

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