analytical and numerical modeling of contact using …
TRANSCRIPT
ANALYTICAL AND NUMERICAL MODELING OF CONTACT
USING ATOMIC FORCE MICROSCOPY IMAGES
By
Kamil Can Bora
A dissertation submitted in partial fulfillment of
the requirements for the degree of
DOCTOR OF PHILOSOPHY
(Engineering Mechanics)
at the
UNIVERSITY OF WISCONSIN-MADISON
2012
Date of final oral examination: 08/23/2012
The dissertation is approved by the following members of the Final Oral Committee:
Michael E. Plesha, Professor, Engineering Physics
Wendy C. Crone, Professor, Engineering Physics
Daniel E. Kammer, Professor, Engineering Physics
Izabela Szlufarska, Associate Professor, Materials Science and Engineering
Robert W. Carpick, Professor, Mechanical Engr. and Appl. Mechanics (U-Penn)
© Copyright by Kamil Can Bora 2012
All rights reserved
i
ABSTRACT
Analytical and Numerical Modeling of Contact Using Atomic Force Microscopy Images
Developments in techniques of surface imagery, like atomic force microscopy (AFM),
permit detection of topography of surfaces at sub-nanometer precision. This opens doors to study
the effects of topography on the contact of surfaces. In this thesis, two different approaches to
model rough surface contact are explained. In the first part, a novel method is presented, which
detects the multiscale features of the surface roughness with analysis of density, height, and
curvature of summits on AFM images of actual silicon micro-electro-mechanical-system
(MEMS) surfaces. The multiscale structure of roughness is then used in a contact model based
on discrete hierarchical length scales and an elastic single asperity contact description. The
contact behavior is shown to be independent of the scaling constant when asperity heights and
radii are scaled correctly in the model. The real contact area estimate is discussed.
In the second part of the study, a numerical finite element contact model is developed to
make use of the high precision surface topography data obtained while minimizing
computational complexity. The model uses degrees of freedom that are normal to the surface,
and uses the Boussinesq solution to relate the normal load to the long-range surface displacement
response. The model for contact between two rough surfaces is developed in a step-by-step
manner, taking into account the far-field effects of the loads developed at asperities that have
come into contact in previous steps. Accuracy of the method is verified by comparison to simple
test cases with well-defined analytical solutions. Agreement was found to be within 1% for a
wide range of practical loads. Applicability of extrapolation from lower precision data is
presented. The real contact area estimates for micrometer-size tribology test machine surfaces
are calculated and convergence behavior with mesh refinement is investigated.
ii
Acknowledgments
I would like to express my gratitude to my advisor Prof. Michael Plesha, previous co-
advisor Prof. Robert Carpick, Final Committee Members, and my other professors, colleagues
and staff at the Department of Engineering Physics at the University of Wisconsin–Madison.
I would like to thank past and present Carpick Research Group members (especially Erin
Flater, Mark Street, Graham Wabiszewski, Mark Hamilton), colleagues at Sandia National
Laboratories (Jim Redmond, Mike Starr and others), who have contributed to part of the study.
I also would like to thank my family (my father M. Nedim Bora, whom we lost last year
and we dearly miss, my mother A. Melek Bora and sister Ceren Bora; who are far away but
always with me), my girlfriend Melissa Ganshert and all other friends in Madison or around the
world who supported me through my journey.
iii
Table of Contents
page
Abstract i
Acknowledgments ii
Table of Contents iii
Chapter 1 Introduction
1.1 Contact and Friction 1
1.2 Contact and Friction at the Small Scales 2
1.3 Purpose and Scope the Study 3
1.3.1 Multiscale Analysis and Modeling 3
1.3.2 Direct Utilization of Surface Images and the Elastic Substrate Model 4
References
5
Chapter 2 Literature Review
2.1 Contact and Friction 6
2.2 Single Asperity Contact 7
2.2.1 The Analytical Solution 7
2.2.2 Finite Element Analysis and Other Numerical Solutions 9
2.2.3 Adhesion and Related Modifications to the Hertz Contact Case 10
2.3 Statistical Models of Multi-Asperity Contact 13
2.4 Multiscale Properties of Surfaces and Fractal Models 16
2.4.1 Multiscale Properties 16
2.4.2 Fractal Models 18
2.5 Numerical Models for Multi-Point Contact 22
References
24
Chapter 3 Multiscale Roughness and Modeling of MEMS Interfaces
3.1 Introduction 29
3.2 Analysis of Surfaces 33
3.3 Contact Model 44
3.3.1 Constraints on Smaller Scale Roughness Features 49
3.4 Conclusions 54
Acknowledgements 56
References 56
iv
Chapter 4 A Numerical Contact Model Based on Real Surface Topography
4.1 Introduction 58
4.2 Description of the Model 62
4.2.1 The Substrate 64
4.2.2 The Interface 69
4.2.3 Contact Between Surfaces 70
4.3 Algorithmic Considerations 71
4.3.1 Memory and Speed Considerations 71
4.3.2 The Algorithm 72
4.4 Verification Examples 74
4.4.1 Rigid Cylindrical Punch Pressed into an Elastic Half-Space 74
4.4.2 Rigid Square Punch Pressed into an Elastic Half-Space 79
4.4.3 Rigid Spherical Surface Pressed into an Elastic Half-Space 82
4.5 AFM Surface: Experiments with Resolution 86
4.6 Conclusion 94
References 95
Chapter 5 Conclusions and Future Directions
5.1 Refining the Multiscale Model and the Surface Analysis Technique 97
5.1.1 Introduction of Adhesion and Plasticity to the Model 97
5.1.2 Further Investigation of the Multiscale Properties of the Surfaces 98
5.2 Possible Improvements to the Boussinesq Finite Element Analysis Model 99
5.2.1 Addition of Plasticity 100
5.2.2 Possible Changes to the Program to Improve Accuracy and Speed 100
5.2.3 Other Functionalities That Can Be Introduced to the Model 103
5.3 Final Remarks 104
References 105
Appendix A Contact Area and Length Scales 106
Appendix B Structure of the Program Described in Chapter 4 108
1
Chapter 1
Introduction
1.1 Contact and Friction
Mechanical systems consisting of more than one component usually involve interaction
between these components through mechanical contact. This interaction is used to remit
mechanical forces from one part of the system to the other. The forces, which can be divided into
normal and tangential direction components, materialize as resistance to relative motion in the
respective directions. When the surfaces are pressed against each other, compressive (or bearing)
stresses are created, in a direction perpendicular to the surface plane. Also, at locations where the
surfaces are in close proximity tensile stresses develop due to adhesion, which pulls the surfaces
towards each other. While a normal stress is present, and these components are undergoing
relative motion in a direction parallel to the contact plane, there also exists lateral resistance, in
other words, friction. Tribology is the study of interacting surfaces in relative motion.
Understanding and controlling contact and friction has been the goal of many engineering
studies. Contact and friction are necessary for the different parts of a mechanical system to work
together. At the same time, friction is the main cause of unrecoverable energy losses. Friction
experiments carried out at macro scales yield empirical relations that are repeatable, but not fully
explained. One reason of this is that the inherent roughness of engineering surfaces results in
multi-point contacts. It is difficult to exactly map out these individual contacts and their effects
2
in an actual experiment. This makes analytical or numerical modeling necessary while studying
contact and friction.
1.2 Contact and Friction at the Small Scales
The behavior of surfaces at micro/nano scales (with feature sizes measured with
micrometers/nanometers) differ from macro scales. One reason is that there is more “surface” at
the small scales (surface to bulk ratio increases for smaller objects). Thus, the attractive surface
forces (i.e., surface adhesion, capillary forces due to surface humidity and electrostatic forces)
acquire a bigger role compared to otherwise prominent mechanical/structural forces. In macro-
scale contact, the effect of the tensile forces developing at locations of the two surfaces in close
proximity is minimal when compared to the other mechanical forces. When the objects get
smaller, the importance of this attraction is amplified. Initial test devices built with nanometer
sized features fail due to high adhesion (stiction) or surface failure due to friction. To develop
reliable devices at these small scales, further studies are necessary. Although empirical, the linear
relation between the normal force and the friction force, commonly known as Amontons’ law or
Coulomb’s law, works from macro scales down to small scales where there is more than a few
single points of contact. When the contact occurs at only a few points, the behavior diverges
from linear, due to the surface forces [1.1].
Development of new technology which allows us to investigate objects at micro/nano
scales creates new possibilities to study the contact and friction at these scales. For example,
atomic force microscopy (AFM) is used to image surfaces with precisions close to atomic
spacing dimensions. In this method, a cantilever with a sharp tip is moved over a sample surface
measuring the resistance to the motion. The contact is only a few atoms wide when particularly
3
sharp tips are used, and this contact can usually be idealized as a single point contact. The
similarity of this case to individual contact points in a multi-point contact case may be used to
help pin-point the mechanical laws that underlie friction.
1.3 Purpose and Scope of the Study
Nominally flat, dry surfaces are considered in this study. Due to the inherent roughness
of the surfaces, when two flat surfaces are in contact, the real contact area is only a fraction of
the apparent contact area. The topography of the surfaces governs the real contact area and the
pressure distribution, and it is the main aspect that ties the small single point contact problem to
the large multi-point problem. Atomic force microscopy can provide valuable information with a
precision of a few nanometers in the axes parallel to the surface and less than 0.1 nm precision in
the normal direction. Smaller details are perhaps not needed for mechanics purposes, as the
elasticity solutions are generally not defined to be applicable at distances less than a few atomic
spacings (an atomic spacing is in the order of 0.25-0.5 nm.) The purpose of the currently
presented work is to find efficient ways of analyzing the effect of topography on the real contact
area, while using to its fullest extent the high resolution information obtained from the AFM
experiment.
1.3.1 Multiscale Analysis and Modeling
Several researchers have studied the topographical effects on multi-asperity contact with
different methods and with varying degrees of simplification. A brief overview of these may be
found in Chapter 2. Some commonly referred works include statistical methods to analyze the
surfaces. The pioneering work by Greenwood and Williamson [1.2] assumes all the contacts to
4
occur at peaks with the same geometrical shape (an averaged radius of curvature), and an
analysis of the peak heights which follow an exponential or a normal distribution. More detailed
models consider changes in the radii of peaks along with other geometrical properties [1.3, 1.4].
It has been found that the statistical properties that are commonly used in the contact models
depend on the sampling size of the topography data. This is due to the fact that engineering
surfaces often have multiscale features. That is, when a section of a rough surface is magnified,
smaller scales of roughness appear. Furthermore, roughness at smaller scales has been shown to
be similar to that at larger scales, but usually with a different scaling of length and height, which
is a property known as self-affinity [1.5, 1.6, 1.7].
The first part of this work (Chapter 3) describes a novel method to analyze peaks by
directly comparing pixels of AFM images with their neighbors and investigate the multiscale
properties of the surfaces. A method for analyzing these peaks by comparing feature heights,
radii and numbers at varying length scales is presented. To model the contact within a scale, the
bumps on the surface are replaced with spheres for which there are established mechanical
models. Then the multiscale structure is extrapolated to smaller and smaller scales to investigate
the real contact area. We give upper and lower bound estimates for elastic contact areas at the
smallest length scales where it is still possible to employ the macro-scale (bulk material)
mechanics techniques.
1.3.2 Direct Utilization of Surface Images and the Elastic Substrate Model
In the second part of the study (Chapter 4), a numerical finite element contact model is
developed to make use of the high precision surface topography data obtained while minimizing
computational complexity, and considering long-range surface displacement response. Normal
5
direction degrees of freedom (d.o.f.) are used. These are coupled to each other by the Boussinesq
solution [1.8, 1.9], which relates the normal load to surface displacements. Application of this
simplifies the solution from a conventional, three dimensional finite element model, while
preserving accuracy.
In Chapter 2, a brief overview of the previous work has been presented. Chapters 3 and 4
describe the two different methods proposed. The final chapter includes a discussion of possible
ways to improve the presented methods.
References:
[1.1] R.W. Carpick, D.F. Ogletree, and M. Salmeron, “A general equation for fitting contact
area and friction vs. load measurements,” J. Colloid Interface Sci. 211 (1999) 395–400.
[1.2] J. A. Greenwood and J. B. P. Williamson, “Contact of Nominally Flat Surfaces,” Proc.
Roy. Soc. London A295 (1966) 300-319.
[1.3] A.W. Bush, R.D. Gibson and G.P. Keogh, “Strongly anisotropic rough surfaces,” ASME
Journal of Lubrication Technology 101 (1979) 15–20.
[1.4] J.I. McCool, “Comparison of Models for the Contact of Rough Surfaces,” Wear 107
(1986) 37-60.
[1.5] R.S. Sayles and T.R. Thomas, “Surface topography as a nonstationary random process,”
Nature 271 (1978) 431-434.
[1.6] J.F. Archard, “Elastic Deformation and the Laws of Friction,” Proc. Roy. Soc. London
A243 (1957) 190-205.
[1.7] J.A. Greenwood and J. J. Wu, “Surface roughness and contact: an apology,” Meccanica
36 (2001) 617-630.
[1.8] J. Boussinesq, Application des potentiels à l'étude de l'équilibre et du mouvement des
solides élastiques (Application of potentials to the study of equilibrium and motion of
elastic solids), (Gauthier Villars, Paris, 1885).
[1.9] A.E.H. Love, “The stress produced in a semi-infinite solid by pressure on part of the
boundary,” Proc. Roy. Soc. London A228 (1929) 377-420.
6
Chapter 2
Literature Review
2.1 Contact and Friction
When two surfaces are brought against each other, mechanical contact occurs and normal
forces are developed. When the two surfaces undergo relative motion in a direction parallel to
the surface, there is a resistance to this motion. The behavior up to the point where gross motion
starts is named static friction. After the relative displacement starts, the behavior is called kinetic
friction, which typically has a smaller value than the static friction. While static friction is mostly
elastic, kinetic friction is the primary source of energy loss in mechanical systems. The
difference between the static and kinetic friction is usually explained by the following: i) the
initial bonding between the two surfaces needs to be broken, ii) as the hills and valleys on the
two surfaces would be somewhat conforming, the initial tangential interaction creates a higher
resistance than the interaction once the two surfaces are in motion.
Tribology has been a subject of scientific and engineering studies since ancient times.
Basic treatment of friction in mechanics is empirical. Equation (2.1) shows the linear relation
between the friction force, F and the normal force N. The coefficient of friction μ depends on the
materials and surface properties.
NF (2.1)
7
This relation, known as Amontons’ Law, simply states that the resistance to relative
displacement is proportional to the normal force between the two surfaces in contact, and not
related to the apparent area of contact. This formula is a result of basic experimentation and
observation.
It is important to note that the full apparent area is never completely in contact, as the
surfaces are not perfectly flat. Actual contact occurs at separate spots or patches, which is usually
about 0.1 to 10% of the apparent area [2.1]. In fact, the inherent roughness is the main reason
that gives the friction relation its empirical character. Calculation of the real contact area has
been the primary aim for researchers studying contact and friction. Numerous studies in literature
attempt to accurately calculate the real contact area. These studies employ different mathematical
models of surfaces and various assumptions for simplification.
2.2 Single Asperity Contact
Individual spots of contact are commonly simplified as interaction of two spheres. A
classical elasticity solution is summarized here, followed by a list of more recent studies
applying finite element analysis.
2.2.1 The Analytical Solution
Contact of two elastic half-spaces at a single point or along a line is calculated in the
Hertz theory. In this theory, the elastic deformation energy is minimized to calculate the contact
deformations, pressures and areas. The normal contact model of two elastic solids assumes that
the surfaces are smooth and non-conforming [2.2, 2.3].
8
An equivalent modulus E* can be defined for the contact with the following equation
where E1 and E2 are the Young’s moduli and υ1 and υ2 are the Poisson’s ratios for the two elastic
half-spaces.
2
2
1
1
*
111
EEE
(2.2)
The contact between two spheres can be approximated as a contact of a single sphere with a rigid
flat substrate. For the case of the two spheres, the equivalent radius of curvature R for the contact
can be found using the two sphere radii as:
21
111
RRR (2.3)
When one of the surfaces is flat, its radius of curvature approaches infinity, and the equivalent
contact radius of curvature equals to the radius of the single sphere. Using these equivalent
parameters, the contact area A and the elastic deflection due to the normal load magnitude L can
be calculated as:
3/2
3/2
*4
3L
E
RA
(2.4)
3/1
2*
2
16
9
RE
L (2.5)
The contact area and the elastic strains are assumed to be small when compared to the radius of
curvature of the contact.
9
2.2.2 Finite Element Analysis and Other Numerical Solutions
Finite element analysis (FEA) is used in many cases to further study the behavior of a
single asperity in contact. Akyuz and Merwin [2.4] apply FEA to analyze the 2-D indentation
problem using Prandtl-Reuss equations of plastic deformations. Komvopoulos [2.5] analyzes a
2D cylindrical indentation problem on a layered half-space with a commercial FEM software
ABAQUS in the elastic mode, and documents the effects of the layer thickness, contact friction,
and investigates creation of micro cracks on the coating layer. This study is later expanded to
elastic-plastic contact [2.6], where the effects of mesh properties and layer thickness on pressure
are explained. Kral and Komvopoulos [2.7] further the model to 3-D, to model a rigid spherical
indenter on a layered medium. Elastic-plastic contact and sliding of the indenter is investigated.
Kral et al. [2.8], study the effects of repeated loading of a sphere on an elastic-plastic half space.
Kogut and Etsion [2.9] develop an elastic-plastic finite element model of a deformable
sphere on rigid flat with frictionless contact. They use a constitutive law that is appropriate for
both elastic and plastic modes of deformation, to provide continuity between the modes. They
later develop an analytical multi-point contact model [2.10] that is based on their FEM results for
single asperity elastic-plastic behavior.
Jackson and Green [2.11] develop a 2-D axisymmetric contact model of a sphere using
constitutive relations that are continuous between elastic and plastic modes. They show that the
hardness of the contact changes with the evolving contact geometry and the material. Jackson et
al. [2.12] investigate the residual stresses at the tip of a contact sphere.
Luan and Robbins [2.13], among others, show using atomistic modeling, that the initial
distribution of surface atoms near the contact location greatly affects the actual pressure
distribution. This is an indication that considering small deviations from the spherical geometry
10
is important for accurate contact modeling. Szlufarska et al. [2.14] provide a comprehensive
review of singe asperity solutions using FEA and atomistic models. Mo, Turner and Szlufarska
[2.15] and Mo and Szlufarska [2.16] use large scale molecular dynamics simulations to study
contact at small length scales. They show that macro-scale roughness models (which are
discussed next) can be used to model the effect of small scale roughness that occurs within the
nominally smooth spherical regions that many models employ.
2.2.3 Adhesion and Related Modifications to the Hertz Contact Case
The attractive and repulsive interactions between the atoms of the contacting bodies
create competing forces. An equilibrium separation distance z0 can be defined as in Fig. 2.1,
where at separation distances smaller than z0 the two surfaces repel each other, and at separation
distances larger than z0 they attract each other. This discussion leads to a more detailed definition
of contact: When the atoms of one surface are within a sufficient proximity to the atoms of the
other surface, these two surfaces are said to be in contact.
Figure 2.1. Force vs. separation distance [2.17]
z0
11
The Hertz contact model can be generalized to include the effects of adhesion. The
Johnson-Kendall-Roberts (JKR) [2.18] method balances the elastic strain energy with the
adhesive interfacial energy to determine the contact area. Griffith’s concept of brittle fracture is
used. The model assumes that the attractive intermolecular surface forces cause elastic
deformation beyond that predicted by the Hertz theory, and produces a subsequent increase of
the contact area. It disregards the spatial attraction when the materials are not in direct contact.
The attractive forces are confined to the contact area and are zero outside the contact area. This
approach is reasonable only for relatively compliant, strongly adhering materials exhibiting
short-range attraction. The contact area is calculated as in the following formula.
3/22
3/2
*363
4
3
RRLRL
E
RA (2.6)
In the Derjaguin-Müller-Toporov (DMT) model [2.19], Hertz contact is assumed and
adhesion is added as an additional effect. The theory describes behavior of weakly adhering,
relatively hard materials with long range forces. The model assumes that the contact
displacement and stress profiles remain the same as in the Hertz theory. In this model the
attractive forces are assumed to act only outside of the contact area.
3/2
3/2
*2
4
3
RL
E
RA (2.7)
Intermediate theories exist, where a non-dimensional physical parameter can be
introduced to describe the cases in between DMT and JKR. [2.17,2.20]. Figure 2.2 shows the
12
force vs. distance plots of Hertz, DMT, JKR cases compared to the actual force distance curve
described in Fig. 2.2.
De Boer, et al. [2.21] describe an alternative method where adhesion between two
nominally flat surfaces is modeled based on Van der Waals force interaction. The following
equation shows the relation between adhesion energy per unit area Г in terms of the Hamaker
constant AH and mean separation do between the two nominally flat surfaces.
2
012/~ dA
H (2.8)
Figure 2.2. The force vs. distance plots of Hertz, DMT, JKR cases
compared to the actual response [2.17]
13
2.3 Statistical Models of Multi-Asperity Contact
Statistical properties are commonly used to describe the random roughness structure of
surfaces. When a surface is assumed to be a stationary random process, a surface sample is
assumed to be representative of the entire rough surface [2.1].
Figure 2.3. Description of the Greenwood-Willamson Model [2.1]
The Greenwood-Williamson (GW) model [2.22] assumes the roughness of the surfaces to
be represented by hemispherical asperities with the same radius of curvature R, as shown in Fig.
2.3. The summit heights have a Gaussian distribution given by
2*
2
*2/12
exp)2(
1
h
h
hP
h
(2.9)
where h* is the root-mean square (RMS) amplitude of the summit heights. RMS of summit
heights is slightly smaller than the RMS of the surface heights. Modeling the summit heights to
follow a Gaussian distribution does not necessarily mean that the surface heights follow a
Gaussian distribution. The summits are assumed to be distributed on the surface uniformly in the
14
horizontal plane. One of the main assumptions of the model is that the elastic interactions
between contact regions can be neglected.
In a contact problem between the rough surface and a flat substrate, a separation distance
d is defined. An asperity with height h larger than d makes contact with the flat surface, whereas
asperities with lower heights do not make contact. Using the Hertz contact theory at each sphere
making contact, the ratio of the real area of contact to the apparent area of contact A0 can be
calculated as
hPdhRn
A
A
d
h
rd)(
0
0
(2.10)
The normal force squeezing the rough surface and the flat can be calculated as
(2.11)
Using equations (2.10) and (2.11), the real area of contact is found to follow a power law
Ar ≈ FN0.95
. In 1940, Zhuravlev [2.23], making assumptions similar to the G-W model, finds a
similar dependence, namely A α L0.91
. These relations are very close to linear, and give clues for
an empirical relation for the macroscopic contact behavior.
Nayak [2.24], following the previous work on statistical geometry by Longuet-Higgins
[2.25, 2.26], uses variance of distributions of the profile heights, slopes and curvatures to
completely characterize the Gaussian height distribution. Bush et al. [2.27] and McCool [2.28],
in following studies use this method and relax the assumptions in the G-W model by using
anisotropic surfaces, elliptical contact points, and a random distribution of asperity radii.
hPRdhnE
F
d
hNd)(
)1(3
4 2/12/3
02
15
However, a common criticism to the models still remains: The radius of curvature R, and the
height distribution information is obtained using only a single length scale. Due to the multiscale
structure of engineering surfaces, the results of the statistical models are only partially useful
estimates, at the length scales where the statistical properties are obtained. This is because the
RMS height, slope and curvature depend on the instrument resolution and sampling size [2.1].
Chang, Etsion and Bogy [2.29] present a method to use an elastic-plastic asperity contact
method with volume conservation to extend the statistical methods. Later they include adhesion
in their model [2.30] using the DMT model [2.19] for contacting asperities and Lennard-Jones
potential between non-contacting asperities. Fuller and Tabor [2.31] use Gaussian distribution of
asperity heights and the JKR model [2.18] to investigate effects of adhesion on contact.
Persson [2.32, 2.33], in his more recent model, starts with a probability density function
of the contact stress. Initially the surfaces are assumed to be smooth, and the contact region
fragments into smaller patches as higher frequency roughness is added. Persson, Bucher and
Chiaia [2.34] apply the model for randomly rough surfaces and find that the contact area varies
linearly with the applied load for most cases, which is a different result than that obtained by the
classical statistical methods which use the probability density function of asperity heights. While
this approach is found useful by some researchers, the discussion about the merits of this method
compared to other models continues [2.35, 2.36].
16
2.4 Multiscale Properties of Surfaces and Fractal Models
2.4.1 Multiscale Properties
Examination of rough surfaces often reveals multiscale features. Smaller scales of
roughness are found when a section of a rough surface is magnified [2.37, 2.38]. The roughness
at smaller scales has been shown to be similar to that at larger scales, following the property
known as self-affinity. The length and height might have different scaling behavior [2.39, 2.40].
The self-affinity of a shape at different length scales is a property employed by fractal models for
surface topography.
The classic example of fractals is given by Mandelbrot [2.40] in his measurements of the
coastline of Britain. As more features and “wiggliness” are observed as the coastline is
magnified, the length of the coastline grows and is dependent on the unit of the measurement.
When the perimeter length of a Euclidean geometrical object, say a circle, is measured, as the
measurement length units (rulers) get smaller and smaller, the value converges to the perimeter
of the object, 2πr for a circle. For natural objects, for example a coastline, the length does not
converge but follows a power law behavior in relation to the length units. This description can be
extended to three dimensions when measuring surface areas: When the surface area is being
measured for a fractal surface, the value increases as the unit area of measurement decreases.
When a surface has detail on arbitrarily small length scales, and when it has a structure
that repeats itself throughout all length scales, it is called a fractal surface [2.41]. There are no
true fractals in nature, however for most natural surfaces the multiscale geometrical
characteristics are seen over a very large range of length scales.
One way of measuring the multiscale nature of the surfaces is using the power spectral
density (PSD). PSD describes the frequency content (in this case, spatial frequencies) of a set of
17
data. It is defined as the Fourier transform of the autocorrelation function of a profile [2.41]. An
equivalent definition of PSD is the squared modulus of the Fourier transform of the data itself,
scaled by a proper constant term
(2.12)
where L is the length of the profile, thus the largest possible wavelength. The fractal dimension
of a surface can be extracted from its PSD, as seen in Fig. 2.4. The PSD of a fractal surface
profile can be related to its fractal dimension by:
)25()(
D
CP
(2.13)
Figure 2.4 Description of a fractal power spectrum on a log-log plot [2.1] Low and high limits
of the measurement frequencies are denoted by ωl and ωh, respectively.
2
0
)exp()(1
)(
L
dxxixzL
P
18
where C is a scaling constant and D is the fractal dimension of a profile vertically cut through the
surface [2.39,2.41,2.42]. For a physically continuous surface, 1<D<2 is obtained for its line
trace.
One other way to extract the fractal dimension from surfaces is using the structure
function. The structure function, given in Eq. (2.14), measures the variance of the surface heights
in accordance with the sampling size τ [2.1]. A surface profile is said to be fractal if the structure
function follows a power law, as in Eq. (2.15).
L
dxxzxzL
S
0
)()(1
)( (2.14)
)2(2)1(2)(
DDGS
(2.15)
2.4.2 Fractal Models
Before the presentation of fractal mathematics by Mandelbrot, Archard [2.37] introduced
a surface model that had multiple roughness scales, as shown in Fig. 2.5. Different roughness
scales were given discrete radii values. Hertz contact formula was used to describe the single
asperity contacts. The relation between the friction force F and normal force N was calculated as
F~N 0.8~1.0
, using varying parameters to model the surface.
Majumdar and Bhushan [2.39, 2.43, 2.44] use the Weierstrass-Mandelbrot (W-M)
function representation to model fractal surface contact. The function is a superposition of a
series of sinusoidal functions scaled at a given ratio γ. The surface height z at a given x location
on a profile is given by:
19
Figure 2.5. Archard’s model of multiscale contact [2.37].
1
)2(
)1( 2cos)(
nn
nD
n
D xGxz
1 < D < 2 γ > 1 (2.16)
where G is a scaling constant, n is the counter for the sinusoidal shapes to be added, n1 is the low
cut off frequency obtained by log γ (1/L), and D is the fractal dimension. The function is
continuous but non differentiable at all points.
To be able to define a contact model at individual asperities, the geometry is obtained at a
length scale defined by a contact spot diameter l. Figure 2.6 shows the asperity geometry, where
R is the curvature, δ is the deflection of the surface, G and D are the fractal roughness and
dimension, respectively. A critical contact spot area ac is defined using an elastic-perfectly
plastic approach to mark the change from the elastic regime to plastic regime, with H as the
hardness and E as the modulus of elasticity:
20
)1/(1
2
)2/(
Dc
EH
Ga
(2.17)
The real contact area is then found to be related to the largest contact spot area aL given in the
following equation, when aL is larger than ac.
(2.18)
Figure 2.6. Geometry of an asperity tip from a fractal profile [2.1].
The three dimensional Weierstrass-Mandelbrot function was developed by Ausloos and
Berman [2.45] and later used by Yan and Komvopoulos [2.46] in their model that uses similar
principles to that of Majumdar and Bhushan [2.43]. The three dimensional (W-M) function is
discussed in section 3.2.
Borodich and Onishchenko [2.47], Warren and Krajcinovic [2.48] and Warren et al.
[2.49] use Cantor set fractal representations in their contact models. In a self affine Cantor set,
the surface is laterally and vertically divided to form protrusions as shown in Fig. 2.7.
Lra
D
DA
2
21
Figure 2.7. A Cantor set illustration [2.48]
In the Cantor set illustration given in Fig. 2.7 [2.48], first the bottom surface length Lo is
divided into three segments and the middle segment is removed, leaving a gap with Lo/3 length
and height ho, leaving two protrusions on the two sides. Then these protrusions are divided into
three segments each, and the middle segment is removed. This recursive algorithm is continued
to obtain a fractal surface. The main parameter that defines how the length after a division is
related to the previous length is a ratio fx. The heights are scaled with a similar ratio fz. If s is the
number of protrusions left on a section after a division (for example, s=2 in Fig.2.7) the fractal
dimension D for a surface obtained by this method is given by
))ln(
)ln(
ln(
)ln(1
x
z
xsf
f
sf
sD (2.19)
The second term in the right-hand-side of the above expression, ln(s)/ln(sfx), is called the Cantor
dimension Dc. For a rigid-perfectly plastic surface, the load and the real contact area is found to
be proportional to δ α, where δ is the deflection, and α is given by:
22
DD
D
c
c
1
1 (2.20)
The authors compare their results with a series of experimental data, and show good correlation
for the pressure vs. deformation relations. Applicability of the method to other surfaces needs to
be investigated [2.1].
Borodich and Onishchenko [2.47] show that it is important to take into consideration the
specific form of the function generating the surface. Their work invalidates the earlier
assumption that the value of fractal dimension is the only crucial factor governing contact
interaction.
2.5 Numerical Models for Multi-Point Contact
When the contact pressure increases over a given macroscopic surface area, more and
more asperities come into contact and it becomes inevitable to account for interaction between
the micro-contacts. Thus the contact models that depend on single asperity contact behavior
become deficient. To model the response of the whole half space, different methods may be
used. For example, Komvopoulos and Choi [2.50] investigate the multi-asperity interaction using
a finite element model, where they study the effects of spacing and radius of the asperities to
deviation of the behavior from that for a single asperity.
Tworzydlo et al. [2.51] conduct FEM simulations of single asperity contact to develop
macroscopic models for friction. They use small strain and include viscous effects and adhesion.
They get a load vs. displacement curve and then use this in a single roughness scale contact
problem with Gaussian height distribution.
23
There are several numerical contact models in the literature that employ the analytical
solution of the deformation of the surface under vertical loading, known as the Boussinesq
solution [2.52, 2.53]. Webster and Sayles [2.54] present a semi-analytical solution for a 2D
profile contact problem. Poon and Sayles [2.55] extend the solution to a simplified 3D
application. They include plasticity, such that the contact pressure is allowed to increase only
until it reaches the hardness of the softer material. Ren and Lee [2.56] develop a moving grid
method to limit the large matrix problem. These models usually start with a prescribed amount of
normal approach between the surfaces, and use a prediction-correction algorithm to converge to
equilibrium. Polonsky and Keer [2.57] use a fast numerical integration technique to calculate the
surface deflections and they employ a conjugate gradient method iteration scheme to reach
contact distribution convergence. Following this work, Liu, Wang and Liu [2.58] develop a 3D
model for thermo-mechanical contact between two rough surfaces.
Hyun et al. [2.59] use a classical 3D finite element approach, wherein a rough self-affine
fractal surface is elastically pressed against a smooth counter surface. They generate the fractal
surface by the successive random midpoint algorithm. They use 10 node tetrahedral elements,
and adaptive meshing to use large elements when sufficiently away from the contact surface. The
mesh for a 512x512 surface grid contains about 911,000 nodes and 568,000 elements. Periodic
boundary conditions are applied. Pei et al. [2.60] consider the same problem but with elastic-
plastic deformations. Molinari et al. [2.61] consider wear of a pin sliding on a rotating shaft
including coupled mechanical and thermal effects. Adaptive re-meshing is used and an iterative
predictor/corrector mechanism is implemented for contact detection and calculation of the
element displacements. An Archard wear rate is used which depends on the sliding velocity,
24
hardness and contact pressure. All three of these studies use dynamic relaxation wherein inertia
is included so that explicit time integration may be used.
In [2.55], Sellgren and Olofsson develop a contact element that models microslip due to
asperity deformations in the commercial FEA software, ANSYS. In [2.63] Sellgren et al. develop
a model for non-linear elastic stiffness of a randomly rough isotropic surface layer in ANSYS.
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25
[2.12] R.L. Jackson, I. Chusoipin, I. Green, “A finite element study of the residual stress and
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contacts,” Nature 435 (2005) 929–932.
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nanotribology,” Journal of Physics D: Applied Physics 41 (2008) 123001.
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[2.16] Y. Mo and I. Szlufarska, “Roughness picture of friction in dry nanoscale contacts,” Phys.
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area and friction vs. load measurements,” J. Colloid Interface Sci. 211 (1999) 395–400.
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Adhesion of Particles,” J. Colloid Interface Sci. 53 (1975) 314-326.
[2.20] Maugis, D., “Adhesion of Spheres: The JKR-DMT Transition Using a Dugdale Model,”
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[2.21] M.P. De Boer, J.A. Knapp, P.J. Clews, “Effect of nanotexturing on interfacial adhesion in
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Roy. Soc. London A295 (1966) 300-319.
[2.23] V.A. Zhuravlev, “On Question of Theoretical Justification of the Amontons-Coulomb
Law for Friction of Unlubricated Surfaces,” Zh. Tekh. Fiz. (J. Technical Phys.– translated
from Russian by F.M. Borodich) 10 (1940) 1447-1452.
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Technology 93 (1971) 398–407.
[2.25] M.S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Phil.
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[2.26] M.S. Longuet-Higgins, “Statistical properties of an isotropic random surface,” Phil.
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[2.27] A.W. Bush, R.D. Gibson and G.P. Keogh, “Strongly anisotropic rough surfaces,” Journal
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26
[2.28] J.I. McCool, “Comparison of Models for the Contact of Rough Surfaces,” Wear 107
(1986) 37-60.
[2.29] W. R. Chang, I. Etsion, D.B. Bogy, “An Elastic-Plastic Model for the Contact of Rough
Surfaces,” Journal of Tribology 109 (1987) 257-263.
[2.30] W. R. Chang, I. Etsion, D.B. Bogy, “Static Friction Coefficient Model for Metallic
Rough Surfaces,” Journal of Tribology 110 (1988) 50-56.
[2.31] K. N. G. Fuller and D. Tabor, “The Effect of Surface Roughness on the Adhesion of
Elastic Solids,” Proc. Roy. Soc. London A345 (1975) 327–342.
[2.32] B. N. J. Persson, “Elastoplastic Contact between Randomly Rough Surfaces” Physical
Review Letters 87 (2001) 116101.
[2.33] B. N. J. Persson, “Theory of rubber friction and contact mechanics” The Journal of
Chemical Physics 115 (2001) 3840-3861.
[2.34] B. N. J. Persson, F. Bucher and B. Chiaia, “Elastic contact between randomly rough
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[2.35] W. Manners and J. A. Greenwood, “Some observations on Persson’s diffusion theory of
elastic contact,” Wear 261 (2006) 600–610.
[2.36] G. Carbone and F. Bottiglione, “Asperity contact theories: Do they predict linearity
between contact area and load?,” Journal of the Mechanics and Physics of Solids 56
(2008) 2555–2572.
[2.37] J.F. Archard, “Elastic Deformation and the Laws of Friction,” Proc. Roy. Soc. London
A243 (1957) 190-205.
[2.38] J.A. Greenwood and J. J. Wu, “Surface roughness and contact: an apology,” Meccanica
36 (2001) 617-630.
[2.39] A. Majumdar and B. Bhushan, “Role of Fractal Geometry in Roughness Characterization
and Contact Mechanics of Surfaces,” Journal of Tribology 112 (1990) 205–216.
[2.40] B.B. Mandelbrot, The Fractal Geometry of Nature (W H Freeman, New York, 1982).
[2.41] K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley,
New York, 1990).
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Roy. Soc. London A370 (1980) 459-484.
27
[2.43] A. Majumdar and B. Bhushan, “Fractal Model of Elastic-Plastic Contact between Rough
Surfaces,” Journal of Tribology 113 (1991) 1–11.
[2.44] A. Majumdar, B. Bhushan, C.L. Tien, “Role of Fractal Geometry in Tribology,” Adv. Inf.
Storage Syst. 1 (1991) 231–266.
[2.45] M. Ausloos and D.H. Berman, “Multivariate Weierstrass-Mandelbrot Function,” Proc.
Roy. Soc. London A400 (1985) 331-350.
[2.46] W. Yan and K. Komvopoulos, “Contact analysis of elastic-plastic fractal surfaces,”
Journal of Applied Physics 84 (1998) 3617-3624.
[2.47] F. M. Borodich and D. A. Onishchenko, “Fractal Roughness in Contact and Friction
Problems (The Simplest Models),” Journal of Friction and Wear 14 (1993) 14-19
[2.48] T. L. Warren, D.Krajcinovic, “Random Cantor Set Models for the Elastic Perfectly
Plastic Contact of Rough Surfaces,” Wear 196 (1996) 1–15.
[2.49] Warren, T. L., Majumdar, A., and Krajcinovic, D., “A Fractal Model for the Rigid-
Perfectly Plastic Contact of Rough Surfaces,” Journal of Applied Mechanics 63 (1996)
47–54.
[2.50] K. Komvopoulos K. and D.H. Choi, “Elastic finite element analysis of multiasperity
contact,” Journal of Tribology 114 (1992) 823-831.
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macroscopic models of contact and friction: formulation, approach and applications,”
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[2.52] J. Boussinesq, “Application des potentiels à l'étude de l'équilibre et du mouvement des
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boundary,” Proc. Roy. Soc. London A228 (1929) 377-420
[2.54] M.N. Webster, R.S. Sayles, “A numerical model for the elastic frictionless contact of real
rough surfaces,” Journal of Tribology 108 (1986) 314-320.
[2.55] C.Y. Poon, R.S. Sayles, “Numerical contact model of a smooth ball on an anisotropic
rough surface,” Journal of Tribology 116 (1994) 194–201.
[2.56] N. Ren, S.C. Lee, “Contact simulation of three-dimensional rough surfaces using moving
grid method,” Journal of Tribology 115 (1993) 597–601.
[2.57] I.A. Polonsky, and L.M. Keer, “A numerical method for solving rough contact problems
based on the multi-level multi-summation and conjugate gradient techniques,” Wear 231
(1999) 206-219.
28
[2.58] G. Liu, Q. Wang, and S. Liu, “A three-dimensional thermal-mechanical asperity contact
model for two nominally flat surfaces in contact,” Journal of Tribology 123 (2001) 595-
602.
[2.59] S. Hyun, L. Pei, J.F. Molinari, and M.O. Robbins, “Finite-element analysis of contact
between elastic self-affine surfaces” Physical Review E (Statistical, Nonlinear, and Soft
Matter Physics) 70 (2004) 026117.
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plastic contact between rough surfaces,” J. of the Mechanics and Physics of Solids 53
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dry sliding wear in metals,” Engineering Computations, 18 (2001) 592-610.
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Finite Element Analysis,” Computer Methods in Applied Mechanics and Engineering 170
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contact between rough surfaces,” Wear 254 (2003) 1180-1188.
29
Chapter 3
Multiscale Roughness and Modeling of MEMS Interfaces
This chapter has been modified from the following citation:
C. K. Bora, E. E. Flater, M. D. Street, J. M. Redmond, M.J. Starr, R. W. Carpick, M. E. Plesha,
“Multiscale Roughness and Modeling of MEMS Interfaces,” Tribology Letters 19 (1) (May
2005) 557-568. (E. Flater and M.Street acquired the AFM images. J. Redmond developed the
basis of the summit search algorithm. Rest of the work was accomplished primarily by the first
author, under the supervision of Prof. Plesha and Prof. Carpick.)
3.1 Introduction
The ability to design reliable MEMS devices with sliding surfaces in contact depends on
knowledge of contact and friction behavior at multiple length scales. While friction at
macroscopic scales can be modeled with Amontons’ law, as the dimensions of a structure
become smaller, the importance of surface roughness and surface forces (e.g., adhesion) are
magnified and frictional behavior can change [3.1].
Surface roughness plays a crucial role in contact and friction between surfaces. For
describing the roughness of a surface, statistical parameters for the surface height distribution
function, i.e., root-mean-square (RMS) height, slope and curvature, have been used in several
studies. These parameters can be directly related to the density of summits, summit curvature,
and standard deviation of the summit height distribution function, which are the key inputs to
models of rough contact based on the Greenwood Williamson approach [3.2-4]. However, these
30
parameters, if determined experimentally, can vary with sample size and instrument resolution
[3.5, 3.6].
Examination of rough surfaces shows that they often have multiscale features. That is,
when a section of a rough surface is magnified, smaller scales of roughness appear. This general
characteristic of surfaces was recognized long ago by Archard who described an engineering
surface as consisting of "protuberances on protuberances on protuberances" [3.6, 3.7]. Further,
roughness at smaller scales has been shown to be similar to that at larger scales, but usually with
a different scaling of length and height [3.8, 3.9], a property known as self-affinity. The self-
affinity of a shape at different length scales is a property displayed by fractal models for surface
topography.
A surface is fractal when it is too irregular to be described in traditional geometric
language, when it has detail on arbitrarily small scales, and when it has a structure that repeats
itself throughout all length scales [3.10]. While there are no true fractals in nature that range over
infinitely small to infinitely large length scales, most natural surfaces show multiscale
geometrical characteristics, having roughness over multiple length scales that frequently span
many orders of magnitude.
The power spectral density (PSD) describes the frequency content (in this case, spatial
frequencies) of a set of data. It is defined as the Fourier transform of the autocorrelation function
of a profile [3.10]. An equivalent definition of PSD is the squared modulus of the Fourier
transform of the data itself, scaled by a proper constant term. Figure 3.1 shows two atomic force
microscope (AFM) topographic images from the same region of a polycrystalline silicon MEMS
surface at two different magnifications, and the respective averaged PSD of both. The PSDs are
seen to correlate well over their shared length scales of measurement.
31
Figure 3.1. (a) 512x512 pixel AFM images of the same region of a polycrystalline silicon
surface with RMS roughness ~ 3 nm, taken at 10 µm and 1 µm (inset) scan sizes. (b) Power
spectral density of the two AFM images.
The fractal dimension of a surface can be extracted from its PSD. The PSD of a fractal
surface profile can be related to its fractal dimension by:
10-21
10-20
10-19
10-18
10-17
10-16
10-15
105
106
107
108
109
10 m x 10 m im age
1 m x 1 m im age
Po
we
r, P
(m
2)
F requency, (1/m )
ω-2.0
ω–3.47
a)
b)
a)
32
)25()(
D
CP (3.1)
where C is a scaling constant and D is the fractal dimension of a profile vertically cut through the
surface [3.8, 3.10, 3.11]. For a physically continuous surface, we will obtain 1<D<2.
The PSDs shown in Fig. 3.1(b) are not linear at low frequencies, and over the full range
of frequencies they do not give a single value for the dimension of its fractal function
representation. This surface could be characterized as a multiple-fractal, where for frequencies
less than 1/100 nm–1
(10–7
m–1
), the PSD has varying slope. For the location shown in Fig. 3.1(b)
(~10–7
m–1
), the slope of –2.0 in the log-log graph gives a fractal dimension of 1.5. In the region
where the slope is –3.47, Eq. (3.1) gives D = 0.77, which is a physically impossible value for any
real continuous surface. Because the PSD in Fig. 3.1(b) is not linear, except for high frequencies,
and because the linear region corresponds to an unobtainable fractal dimension for a real surface
(i.e., D < 1), it is not possible to characterize this particular surface with a fractal function.
Nonetheless, the PSD reveals that the surface has roughness at all length scales sampled, and
furthermore it indicates that the effects of the multiple scales of roughness on mechanical contact
phenomena should be taken into consideration [3.8]. A methodology that goes beyond using the
fractal representation is needed.
In this study, we discuss the scale dependence of the average height, the average radius of
curvature, and the density of summits on an actual polycrystalline silicon MEMS surface. Also,
the relationship between the scale dependences and the fractal dimension of the surfaces is
investigated. In the second part, a straightforward multiscale contact model (comparable to
Archard’s idea [3.6, 3.7]) is developed, where the asperity force distributions and the contact
33
area are determined as a function of the length scale using the elastic Hertz contact model at each
length scale.
3.2 Analysis of Surfaces
Determination of the heights, locations, and curvatures of summits on contact surfaces is
necessary to model contact and friction. While this would seem to be a straightforward task, the
multiscale roughness properties of real surfaces make the concept of a "summit" ambiguous and
imprecise. Consider a surface profile whose height is determined at a finite number of discrete
positions, such as by profilometry. A "peak” or “summit" can be defined as a location where the
height is a local maximum. In the case of a two dimensional surface (i.e., a line trace), a sample
point is a “peak” if its height exceeds that at each of its two neighboring sample points, while for
a three dimensional surface (i.e., an AFM image), a sample point is a “summit” if its height
exceeds that at each of its eight neighboring sample points. Ambiguities arise when more
sampling points are used, i.e. when the definition of a summit is changed to require that a pixel is
higher than a larger region of its surrounding neighborhood. To study this phenomenon, we use
AFM topographic imaging like that shown in Fig. 3.1(a) to obtain a pixelized height distribution
for a Si MEMS surface from Sandia National Laboratories SUMMIT process [3.12]. The AFM
images were acquired in contact mode using a Digital Instruments Multimode AFM with a
Nanoscope IV controller with a silicon nitride cantilever having nominal force constant of ~0.05
N/m. The piezo scanner was calibrated using the manufacturer’s recommended procedure. The
tip shape was tested before and after the measurements using in-situ tip imaging samples (Aurora
Nanodevices, Edmonton, Canada) to ensure that it started and remained a sharp, single
protrusion or radius <30 nm, so as to minimize the effect of convolution of tip shape. Numerous
34
tips with blunt, multiple, or asymmetric terminations were rejected. Low loads (in the adhesive
regime) were used to minimize the contact area and enhance the lateral spatial resolution. The
lateral spatial resolution of a contact mode AFM image is approximately determined by the
contact diameter, which we estimate to be of the order of ~2 nm, comparable to the size of one
pixel in our highest resolution images analyzed.
A MATLAB routine is then used to determine the heights and locations of summits by
examining each sample point (pixel) with coordinates (x, y) and comparing its height z to the
heights of n neighboring pixels, where n is called the neighborhood size. For a given value of n,
the region of neighbors surrounding a particular sample point is a square with size dn by dn,
where the box size dn is given by
(3.2)
where L is the physical width of the square AFM image and N is the number of pixels per each
side of the image. When a summit is found for a given n, the MATLAB routine determines the
least squares best fit elliptic paraboloid to the data around the point, to determine the curvature of
the summit in two dimensions. The major axes of the paraboloid are constrained to fall along the
x and y axes of the image, and the max is constrained to occur at the point (x, y) with height z.
The average height, average radius of curvature and number of summits are calculated as
a function of box size dn for 1 µm x 1 µm and 10 µm x 10 µm (L = 1 and 10 µm, respectively)
AFM images of a polycrystalline silicon surface with an overall RMS roughness of 3 nm (as
measured for a 10 µm x 10 µm region), with the results shown in Fig. 3.2. Note that only a
subset of pixels within a central square region of the image is considered when searching for
summits, such that the same portion of the data is considered for all neighborhood sizes.
dn
(2 n 1)L
N
35
Figure 3.2. a) Number of summits per area, b) average height, and c) average radius of
curvature versus the box neighborhood size d for an AFM image of a polycrystalline silicon
surface. The RMS roughness of the surface was measured to be 3 nm for a 10 x 10 µm AFM
image.
In a log-log plot, the height, radius, and number of summits are seen to be almost
perfectly linear functions of the box size dn over almost three orders of magnitude of size. Most
strikingly, even at the smallest box size, (~10 nm), the distributions have not converged to well-
defined values. Scans over smaller regions show this behavior continues for even the smallest
box size we were able to consider (d=2 nm), which is reaching the lateral resolution limit of the
0.01
0.1
1
10
100
1000
104
1 10 100 1000 104
10 m im age
1 m im age
Nu
mb
er
of
Pe
ak
s /
1m
2
d (nm )
1
10
100
1 10 100 1000 104
10 m im age
1 m im age
Av
era
ge
He
igh
t (n
m)
d (nm )
10
100
1000
104
105
1 10 100 1000 104
10 m im age
1 m im age
Av
era
ge
Rad
ius
of
Cu
rvat
ure
(n
m)
d (nm )
a) b)
c)
36
AFM itself. The power law dependence of the summit geometry and density seen in Fig. 3.2
illustrates the scale dependence of these values.
In contrast to a real MEMS surface, Fig. 3.3 shows the average summit radius of
curvature vs. neighborhood size for a hypothetical surface with sinusoidal shape, which is
created by:
nm )10/cos()10/cos(),( yxyxz (3.3)
This surface obviously has a single scale of roughness. As expected, Fig. 3.3 shows that
the average curvature becomes constant for sufficiently small neighborhood sizes, unlike the
MEMS surfaces. For box sizes larger than the period of Eq. (3.3) (i.e., d ≥ 20π), the slope of the
power law approaches 2.0. Because our method of fitting a paraboloid requires the fit to have the
same z coordinate as the summit in the data, this response for large values of d is simply a
mathematical artifact.
Figure 3.3. Average summit radius of curvature vs. neighborhood size for a surface with
a single scale of roughness (a sinusoidal function).
1000
104
105
106
1 10 100 1000
Av
era
ge
Ra
diu
s o
f C
urv
atu
re (
nm
)
d (nm )
d2
37
In other words, fitting a paraboloid to a point at fixed height while the lateral extent of the
fit is enlarged, will produce a power law of 2 if the surface is nominally flat at large length
scales. This can be shown analytically in three dimensions using a least squares fit of a
paraboloid to the sinusoidal surface,
0)cos()cos(22
1
222b
b
b
b
dydxyxR
y
R
x
dR
d (3.4)
where R is the paraboloid's radius of curvature and b is the sampling length (essentially
equivalent to d). Solving for R, its relation to b is found to be:
2bR
(3.5)
This analysis helps validate the curvature calculation in our MATLAB routine since the
analytical and MATLAB results agree. It also shows that the power-law dependence of the
radius of curvature calculation is not by itself an indication of multiscale surface roughness.
However, the lack of convergence of the radius of curvature to a fixed value at small
neighborhood sizes for the MEMS surfaces is indeed an indication of multiscale character, in this
case down to a length scale of ~2 nm.
To compare the foregoing results to those for a model fractal mathematical surface, our
analysis procedure (i.e., pixelation of a continuous map of heights followed by pixel-by-pixel
analysis using our MATLAB program) was applied to three-dimensional fractal surfaces
including the following equation:
38
2/1)2(
,
1
2/122
,
1 0
)3(
ln
}tancos)(2
cos
{cos),(max
ML
GLC
M
m
x
y
L
yx
Cyxz
Ds
nm
n
nm
M
m
n
n
nDs
(3.6)
which is a form of a multivariate Weierstrass-Mandelbrot (W-M) function developed by Ausloos
and Berman [3.13] and later used in a three dimensional elastic-plastic contact model by Yan and
Komvopoulos [3.14].
Equation (3.6) is constructed by taking a two dimensional fractal profile as a "ridge", and
then superposing a number of these ridges at different angles to achieve randomization. Φ is an
array of random numbers to generate phase and profile angle randomization, M is the number of
ridges, and L is the image size in length units. G is the roughness coefficient which is used to
correctly scale the height of the function to fit the modeled surface Ds is the fractal dimension of
the surface (2<Ds<3, as the relation between the fractal dimension of a surface Ds and that of a
profile D is Ds=D+1 for an isotropic surface). γ is a parameter that governs the frequency and
amplitude ratio of successive cosine shapes (γ>1) and thus represents that relative frequency
separation of successive terms in the W-M function. Lmax is the sample size. Finally, nmax is the
number of cosine shapes added for a profile. Note that Eq. (3.6) is perfectly fractal only if
nmax
. For practical applications, finite values of nmax are used, so that cosine shapes with
periods larger than Lmax, and smaller than Lmin are not needed.
39
Figure 3.4. Two W-M surfaces produced using Ds=2.4, G=1.36x10–2
nm, Lmax =1000 nm,
Lmin= 1 nm, M=10, n1=1. γ =1.5 for (a) and γ =5 for (b). All values other than γ are taken from
Yan and Komvopoulos [3.14]). The height scale is in nm.
Figure 3.4 shows two sample W-M surfaces. In Fig. 3.4(a), the popular but otherwise
unremarkable value of γ=1.5 was used, which results in successively added cosine shapes whose
periods and amplitudes are moderately spaced apart, providing a surface with a seemingly
“random” aesthetic character. When γ=5, the difference between the periods and heights of
consecutive cosine shapes is greater. This coarse separation of roughness scales leads to the
a)
b)
40
easily discernible scales of “bumpiness” seen in Fig. 3.4(b). For the W-M function, an
appropriate number of the cosine shapes used for defining the surface (nmax) can be selected
using:
(3.7)
where, Lmin is the period of the smallest cosine shape. Equation (3.7) assures that the cosine
shapes have periods that fully span the length scales from Lmin to Lmax. Thus, the function is
“fractal” for all practical purposes. When Lmin and Lmax are 1 nm and 1 μm, respectively, 18
cosine terms would be needed for γ=1.5, and only 5 cosine shapes would be needed for γ=5. The
frequencies are spaced further apart in the γ=5 case. Several W-M surfaces were then created
with fractal dimensions varying from 2.01 to 2.99, and with the same two values of γ (1.5 and 5)
used to explore the effect of spatial frequency separation. For W-M surfaces with small γ values,
the number of summits per area, average height, and average radius of curvature variation with
neighborhood size give almost perfectly linear distributions on log-log plots, down to the
smallest scales. Examples of the summit density, average height, and radius of curvature as a
function of d for both values of γ are shown for the case of Ds=2.4 are shown in Figs. 5(a), (b),
and (c), respectively. For W-M surfaces with high γ values, these plots show deviations from
linearity, which is due to the large frequency separation between each consecutive cosine term
used in the generation of the surface. This is clearly seen in the plot of the number of peaks vs. d
(Fig. 3.5(a)) and average heights vs d (Fig. 3.5(b)). The average radius of curvature (Fig. 3.5(c))
shows little effect of the frequency separation.
log
/logminmax
max
LLn
41
Figure 3.5. a) Density of summits, b) average height, and c) average radius of curvature vs. box
size d for W-M surfaces crated using γ=1.5 and γ=5, using Ds =2.4 (thus D = Ds – 1 = 1.4)
The results of our “summit search” method match certain analytical predictions.
Majumdar and Bhushan [3.15] derived the radius of curvature R at the tip of an asperity for the
W-M surface as:
(3.8)
10
100
1000
104
105
1 10 100 1000
= 1 .5
= 5
Nu
mb
er
of
Pe
ak
s /
m
d (nm )
0 .1
1
10
1 10 100 1000
= 1 .5 = 5
Av
era
ge
He
igh
t (n
m)
d (nm )
100
1000
104
105
106
1 10 100 1000
= 1 .5
= 5
Av
era
ge
Ra
diu
s o
f C
urv
atu
re (
nm
)
d (nm )
12 D
D
G
dR
a) b)
d1.41
d1.39
c)
42
Using this expression with the box size d as the contact length and with a constant value of G,
the radius of curvature R changes as dD, which matches the slopes found in Fig. 3.5(c).
Wu [3.16] argues that the W-M function given in Eq. (3.6), developed by Ausloss-
Berman, is not exactly a 3D extension of the fractal 2D W-M function, since a vertical cut of the
surface is not necessarily a W-M function, and thus the surface is not isotropic. He later shows
that, despite this observation, surfaces generated by this function share very similar properties,
such as summit curvature, with other fractal functions such as the successive random addition
method [3.16]. Thus, our comparison to the W-M function can be considered as a reasonable
way to illustrate the fractal character of the actual MEMS surfaces.
Figure 3.6 shows the exponents of power law fits (i.e. the slopes on the log-log plots from
Fig. 3.5) for the number of summits, average height, and average radius of curvature
distributions of the W-M surfaces with varying fractal dimensions for the two values of γ. A
relation between the fractal dimension and the power law exponents of these fits is seen. At high
fractal dimensions, the average radius of curvature changes more rapidly with varying
neighborhood size (higher slope value). This effect is reversed for the corresponding behavior of
the density and average height of the summits. In other words, a high fractal dimension means
smaller variance in the density and average height with changing neighborhood size. The change
in γ affects the results somewhat, but further analysis is required to fully understand this effect.
For fractal dimensions D between 2 and 2.5, the relation between R and d shown in Fig. 3.6(c)
follows the behavior described in Eq. (3.8) extremely well, while for higher fractal dimensions
the slopes are slightly lower than expected.
43
Figure 3.6. Exponents of the power law fits for the number of summits, average summit height,
and average summit radius of curvature vs. neighborhood size for W-M surfaces, plotted for a
range of fractal dimensions D. The shaded band running across each figure shows the range of
exponents obtained from the analysis of actual AFM images.
In Fig. 3.6, the range of power law exponents obtained from the “summit search” analysis
of several AFM images of rough polycrystalline silicon MEMS surfaces are shown as shaded
bands running across the graphs. We see that these values are consistent with the W-M fractal
surface properties if we associate the AFM images with low fractal dimensions (Ds=2.1-2.3). The
-2 .1
-2
-1 .9
-1 .8
-1 .7
-1 .6
-1 .5
1.8 2 2.2 2.4 2.6 2.8 3 3.2
= 1.5
= 5
po
we
r la
w e
xp
on
en
t fo
r n
um
be
r o
f p
ea
ks/a
rea
F ractal D im ension Ds
0.3
0.4
0.5
0.6
0.7
0.8
1.8 2 2.2 2.4 2.6 2.8 3 3.2
= 1.5
= 5
po
we
r la
w e
xp
on
en
t fo
r a
ve
rag
e h
eig
hts
F ractal D im ension Ds
0.8
1
1.2
1.4
1.6
1.8
2
1.8 2 2.2 2 .4 2.6 2.8 3 3 .2
= 1.5
= 5
po
we
r la
w e
xp
on
en
t fo
r a
ve
rag
e r
ad
ii
F ractal D im ension Ds
a)
c)
Fit of Eq. 8
b)
44
fractal dimension of these AFM surfaces obtained using PSD analysis varied for different length
scales as discussed earlier, with values ranging from non-fractal values (e.g. Ds=1.77 as in Fig.
3.1) up to approximately Ds=2.5.
3.3 Contact Model
Figure 3.7. Hierarchy of roughness and load distribution among asperities at
different length scales for the contact model we have developed.
The multiscale nature of the MEMS surfaces revealed by our analysis suggests that any
useful contact model must embody this multiscale character. Thus, in our model, surfaces in
contact are modeled with roughness at multiple length scales. The asperities at the largest length
scale have the largest radii and height variation, and upon these lies a second set of asperities
with radii and height variations smaller by a factor s>1 which we call the scaling constant, and so
45
on. Contact between two rough surfaces was approximated by contact between a smooth rigid
surface and a single rough elastic surface. This can be adjusted to represent the behavior of two
rough surfaces as desired [3.4].
Figure 3.7 shows how the total force on the first set of asperities is divided into forces on
asperities at smaller length scales. If only the first scale were considered, then the surface could
be thought of as a Greenwood-Williamson [3.2, 3.3] surface with a given height distribution. In
any event, the total force is proportioned among each contacting asperity at that length scale
using an appropriate single asperity contact model.
In this algorithm, the rough surface is incrementally advanced into the rigid flat counter
surface to determine the asperities at the coarsest length scale that make contact. Then the exact
approach distance is found by interpolation via Newton’s method. This distance is used to
proportion the total force among all of the contacting asperities, using an appropriate single
asperity contact model as discussed below. Thus, the total load F is equal to the sum of the forces
supported by the contacting asperities at that length scale according to:
F Fi
i 1
n1
(3.9)
where n1 is the number of asperities at the first length scale that are in contact. The actual
contact area at this length scale is given by:
1
1
1
n
i
iAA (3.10)
At the next length scale, each of the loads Fi is proportioned among the second
generation asperities that make contact such that the forces and areas are given as:
46
in
j
jiiFF
,2
1
, (3.11)
1 ,2
1 1
,
n
i
n
j
jii
i
AA
(3.12)
where n2,i is the number of second generation asperities in contact that are located on first
generation asperity i.
For example, considering this second scale of roughness, the total force supported is
obtained by combining Eq. (3.9) and Eq. (3.11), whereas the total contact area is given by Eq.
(3.12). The forces and contact areas at subsequent length scales are calculated in the same
fashion. This procedure can be carried out for any number of desired scales of roughness.
When no adhesion is included in the model, the Hertz contact model [3.17] is
appropriate. The single asperity contact area-load relation obtained from the Hertz model is:
3/2
3/2
*4
3L
E
RA (3.13)
In this equation, R is the asperity radius, E* is the composite elastic modulus of the contact, and
L is the total load on the asperity.
There are several significant assumptions in our hierarchical modeling approach. The first
is that the asperities are elastic. The second is that, within one length scale, the mechanical
response of an asperity is not affected by its neighbors. The third is that the roughness of finer
length scales is small enough so that the response of asperities at coarser length scales is
unaffected. This imposes limitations on the applicability of our model, but allows us to capture
47
and interpret the effects of multiscale contact within these limitations. In fact, by using
sufficiently well-spaced generations of asperities, the use of the Hertz model is justified, whereas
such an assumption is questionable in the case of extremely finely-spaced scales of roughness,
such as in the W-M function.
Figure 3.8. (a) True contact area as a function of the number of roughness scales for
non-adhesive (Hertzian) asperities, computed using different scale constants.
(E*=200 GPa, L=10 μN) (b) True contact area plotted vs. the common dimensionless
distance associated with the scales.
20 00
40 00
60 00
80 00
1 104
1.2 104
0 1 2 3 4 5 6 7 8
s = 2.5
s=
s = 5
s = 10
Tru
e C
on
tac
t A
rea
(n
m2)
N umber of Scales
2000
3000
4000
5000
6000
7000
8000
9000
1 104
0.0010.010.11
s = 2 .5
s=
s = 5
s = 10
Tru
e C
on
tac
t A
rea
(n
m2)
Comm on D imensionless D istance
48
Figure 3.9. True contact area as a function of total load, using the model with five
scales of roughness. The circles are the individual results of the model; the solid line
indicates the power law fit.
A surface that appears to satisfy the third assumption of our model is the Weierstrass-
Mandelbrot function, in the form given in Eq. (3.6), with sufficiently high γ value (> ~ 4) to
obtain adequate frequency (asperity size scale) separation. The W-M function with G and D
values given in Fig. 3.4 and with γ=10 was used to create such a surface. Using an appropriate
neighborhood size, the locations, heights, and curvatures of the asperities produced by the largest
cosine function were found by using the “summit search” algorithm described above.
To investigate the effect of changing the scaling constant, summits obtained from the W-
M surface described in the previous section were again used as the first scale of roughness, and
different scaling constants s>1 were used to define the roughness at smaller length scales. For
example, for s=10, the asperity heights at the second order of roughness are one-tenth of the
height of those at the first order, and have radii of curvature that are one-tenth of those at the first
order. In the calculation L=10 μN and E*=200 GPa was used. This E
* value is in the upper range
of reported values for polycrystalline silicon. Figure 3.8(a) shows the computed total contact area
0
2 104
4 104
6 104
8 104
1 105
0 1 105
2 105
3 105
4 105
5 105
6 105
He
rtz
Ca
se
Co
nta
ct
Are
a (
nm
2)
To tal Load (nN )
49
as a function of the number of roughness scales, using different scale constants s. For a given
value of n, the lateral length scale of the asperity size depends on s. For illustration purposes, we
replot the data in Fig. 3.8(a) using a common dimensionless distance axis defined as 1/sn–1
. Thus
the largest scale has an asperity dimension of 1.
Figure 3.9 shows the total area of contact versus the total load applied to the surface,
calculated for five roughness scales using a scaling constant of s=5. The behavior is very close to
linear, and a power law fit shows that the area is related to the total load by a power of 0.94. This
is reminiscent of the well-known result from the G-W model whereby the contact area scales
nearly linearly with load for a collection of equally-sized asperities randomly distributed about a
mean height. The difference in the two models is that G-W model uses a single scale of
roughness [3.2]. In 1940, Zhuravlev [3.3], making assumptions similar to G-W model, found a
similar dependence, namely A α L0.91
.
For all scaling factors tested, the total contact area shown in Fig. 3.8 decreases with
increasing scales of roughness, and appears to converge to well-defined values, but those values
are highly dependent on the particular scaling factor chosen. In Appendix A, we further illustrate
this effect using a simple calculation for a set of Hertzian contacts at the same height. This effect
will be further discussed in the next section.
3.3.1 Constraints on Smaller Scale Roughness Features
A constraint that occurs with real surfaces is that the number of smaller asperities that can
be present on a contacting asperity (i.e., a host asperity) at the larger length scale is limited. In
the example presented in Fig. 3.8, the finer scale contacts were assumed to be hexagonally close-
packed on the contact area of the host asperity, and a limit was introduced according to the ratio
50
(m) of the large asperity contact area to the average of the small asperity contact areas. The
number of small asperity contact spots (Nmax) that would fit on the larger contact area can be
estimated by Nmax α m2, when m is greater than 3. We call this the close-packed constraint, and it
imposes an upper bound to the possible numbers of asperities at successive scales.
Another limitation of the example presented in Fig. 3.8 is that the same scaling constants
are used for both the heights of the asperities and radii of curvature for the asperities. In reality,
the heights and radii scale differently, as seen in the surface analysis results shown in Fig. 3.2.
The data of Fig. 3.2 suggests a better, more realistic way to model multiscale roughness driven
by the experimental observations of the surface properties, as follows. The trends seen in Fig. 3.2
give scaling constants for the heights and radii for the AFM image analyzed. The corresponding
power relations are h α d0.6
, N α d–1.87
, and R α d1.25
. With s used as the scaling constant for
length, the neighborhood size in Fig. 3.2 is scaled by d2=d1/s. The asperity heights for the
subsequent smaller scale are obtained from the heights of the previous scale by sh = s0.6
so that
h2=h1/sh. Similarly, the radii are scaled by sR = s1.25
, and the number of asperities per unit area
scales by sN = s–1.87
. We call this the asperity-density constraint, which originates from real
experimental analysis.
The information regarding number of asperities per unit area provides a more realistic
value for the number of contacts that will be present on the tip –or contact area– of the larger
scale host asperity. If we know the contact area at a particular length scale, then we can multiply
this with the asperity density at the smaller length scale to obtain a limiting value for the number
of asperities that can be present on the host asperity.
We select from the AFM image a set of heights and radii by selecting a neighborhood
size which yields a reasonable number of asperities (e.g., N α 100), and we call this the
51
roughness template. Then the scaling constants for h, R, N can be used to calculate the geometry
and density of asperities at other scales. In other words, the roughness template is scaled up to
provide coarser details of roughness, and is scaled down to obtain finer details of roughness.
The silicon MEMS surface analyzed in Figs. 3.1 and 3.2 was used as an example. The
template scale was found using a summit search box size of ~450 nm, which yielded 103
summits. The summits obtained show a distribution close to an exponential, which is the same
distribution used in previous models in literature such as the G-W model [3.2].
The power law relations cited earlier were used to calculate scaling factors for radii,
height and asperity density for different s values. A load of 1 mN and a modulus of E* = 200
GPa was used. Figure 3.10 shows the prediction of the true contact area as a function of the
smallest length scale used in the computation. In Fig. 3.10, filled marks represent the results
where the number of sub-scale contacts is constrained by a close-packed distribution assumption,
and empty marks represent the results where the experimentally obtained asperity density
constraint method is used.
The calculations shown in Fig. 3.10 are not continued to scales lower than about 1 nm, as
this length approaches atomic spacing. Ten times the equilibrium atomic spacing is a reasonable
estimate for the limiting value of the elasticity (Hertz) solution [3.14], and this provides an
approximate value where to terminate the calculations (i.e., ~4 nm).
The estimated true contact area at the smallest length scale for the asperity-density
approach (~104 nm
2) is smaller than the close-packed approach (~3 x 10
4 nm
2), as expected. It is
also seen that the contact area converges to a limit faster in the asperity-density method. It was
observed many times in the asperity density approach that there would be only one small asperity
present on the tip of the larger host asperity.
52
Figure 3.10. True contact area as a function of the smallest length scale used in the
simulation, computed using different scale constants, with E*=200 GPa, L=1 mN. Filled
marks represent the results of the close-packed constraint, and empty marks represent the
results from experimental density constraint for the number of sub-asperities.
When two consecutive scales are considered, the contact behavior of the intermediate
size peaks between those two scales is being neglected. The case of the asperity-density
constraint is affected more by this, as the number of small asperities that can be added to
compensate for the lack of intermediate size asperities is limited. Thus the calculated area ends
up being a lower bound estimate, which is imposed by the experimental data. The close-packed
distribution constraint results in a theoretical upper-bound for the contact area that can be
0
5 104
1 105
1 .5 105
2 105
2 .5 105
3 105
3 .5 105
1101001000104
s = 2 .5
s = 2 .5
s =
s =
s = 5
s = 5
s = 10
s = 10
Tru
e C
on
tac
t A
rea
(n
m2)
Sm allest Length Scale (nm )
Close-packed
Constraint
Experim ental
D ensity
C onstraint
Limit of
elasticity
solution
53
calculated in the model, as geometrically there can be no more asperities present in contact. So
the results of our model can be taken as an estimated range for the real contact area.
In the calculation shown in Fig. 3.10, where the scaling constants for radii and heights are
more realistic, we see that using different scaling constants (s = 2.5–5) does not affect the true
contact area as strongly as in Fig. 3.8, where the scaling constants for radii and heights were the
same. Independence of the contact area from the scale constant shows that we can get away with
a “crude” model where large scaling constants and only a few scales are used to represent the
contact behavior. The model surface need not be carrying all the roughness frequencies, thus, for
example, a Weierstrass-Mandelbrot surface with a high γ value can be used for simplicity.
The same AFM image is analyzed with the procedure described by McCool [3.4, 3.18]
which uses a Greenwood-Williamson approach, i.e. has one scale of roughness, with E*=200
GPa, L=1 mN. The analysis gives a nominal asperity radius of 280 nm, summit density of 4.6 x
10–5
nm–2
and a summit height standard deviation of 2.55 nm. The true contact area estimate
from this method is on the order of 105 nm
2 when the apparent contact is 10 μm by 10 μm. This
analysis is described in [3.18] for the same surface.
The nominal asperity radius of 280 nm corresponds to a neighborhood size of ~ 70 nm in
our “asperity search” analysis shown in Fig. 3.2. The contact area estimate of 105 nm
2 is close to
the close-packed area constraint result for this length scale, but exceeds the total contact area we
determine (by considering all length scales down to the atomic limit) by a factor of ~5. Another
key difference between McCool analysis and our model is that we allow the force to be
supported at smaller contact areas, resulting in higher stresses at contact points.
54
3.4 Conclusions
Roughness of polycrystalline silicon MEMS surfaces is strongly scale-dependent.
Analysis of summits for AFM scans of actual MEMS surfaces shows that the height, density, and
geometry of the summits as determined by a “search and fit” routine have a power law
relationship with neighborhood search size.
The analysis of a test surface that has roughness limited to a small range of spatial
frequencies, and analysis of a surface with a single scale of roughness, shows that the summit
search procedure captures the geometry of the smallest summit features when a surface has a
well-defined length scale below which there are no additional details of roughness. When the
same analysis is performed on AFM data of a polycrystalline silicon MEMS surface, additional
details of roughness emerge for even at the smallest neighborhood sizes considered. In other
words, no convergence to a uniform value for the height, density, and geometry of summits is
observed even at the smallest experimentally accessible lateral length scale (~ 2 nm).
The power law behavior obtained from AFM images is similar to fractal W-M surface
results. However, we find that these MEMS surfaces exhibit a range of spectral frequencies over
which the surface is not fractal (the slope of the PSD is less than –3), yet the surface is still
multiscale in nature (no well-defined summit radius, for example). Although the validity of the
method still needs to be explicitly proven, the results indicate that as an alternative to the
conventional power spectral density method [3.8] for determining the fractal dimension of a
fractal surface representation, an “summit search” methodology, which is intuitively more
straightforward and potentially more versatile, may be used for describing surface geometry. The
appropriate method to select other parameters needed for analytical representations of fractal
55
surfaces (such as the W-M function, where γ and G must be determined) are have not yet been
addressed here.
A contact model is developed using multiple length scales for roughness. The smaller
roughness scales are successively modeled as asperities that are superposed on the asperities of
the next larger scale. The total contact area predicted with elastic Hertz behavior approaches a
limit with increasing number of roughness scales.
The calculated area of contact is dependent on the scale constant s that is used when the
radii and the heights of the asperities are scaled with the same constant. When the correct scale
dependence of the heights and radii are used, as obtained from the analysis of summits from
AFM images, the contact area calculated does not depend on the scaling constant s. This is
important as it shows that a simpler surface representation with large scale constants and fewer
scales is still valid. This suggests as well that a large γ value in Weierstrass-Mandelbrot function
can be used to generate a fractal surface model for simplicity. Separation of length scales renders
the use of Hertzian contact mechanics across these length scales more readily believable.
The number of small scale contacts within a large contact area is constrained using two
different methods: The experimental asperity density constraint and the close-packed distribution
assumption. The latter gives a higher area estimate from the former. Together these two values
can be thought as the upper and lower bound estimates for the contact area.
The next step in this modeling approach would be to include adhesion, i.e. JKR [3.19]
and DMT [3.20] single asperity contact models. Preliminary work regarding these models shows
that a technique must be used that accounts for the adhesion associated with the large asperities
when considering a small scale of roughness. The adhesion models mentioned account for
adhesion only at the contact area [3.19] or around it [3.20]. When looking at a roughness scale,
56
the surface forces on the material between two asperities are disregarded. If this is not considered
then a force imbalance occurs.
Plasticity also needs to be added to the model. First of all, the small asperities will likely
experience yielding, and secondly the material properties, specifically the yield strength, may
vary with different length scales.
The “summit search” method and the contact model presented constitute an intuitive
approach to understand the multiscale nature of surfaces, making use of real images of MEMS
surfaces, and numerical computation.
Acknowledgments
We acknowledge the staff at the Microelectronics Development Laboratory at Sandia
National Laboratories for fabricating the MEMS samples, and Maarten P. de Boer and Alex D.
Corwin for useful discussions and feedback. This work was supported by the US Department of
Energy, BES-Materials Sciences, under Contract DE-FG02-02ER46016 and by Sandia National
Laboratories. Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed
Martin Company, for the US Department of Energy under contract DE-AC04-94AL85000.
References:
[3.1] R.W. Carpick and M. Salmeron, “Scratching the surface: Fundamental investigations of
tribology with atomic force microscopy,” Chem. Rev. 97 (1997) 1163-1194.
[3.2] J. A. Greenwood and J. B. P. Williamson, “Contact of Nominally Flat Surfaces,” Proc.
Roy. Soc. London A295 (1966) 300-319.
[3.3] V.A. Zhuravlev, “On Question of Theoretical Justification of the Amontons-Coulomb
Law for Friction of Unlubricated Surfaces,” Zh. Tekh. Fiz. (J. Technical Phys.– translated
from Russian by F.M. Borodich) 10 (1940) 1447-1452.
[3.4] J.I. McCool, “Comparison of Models for the Contact of Rough Surfaces,” Wear 107
(1986) 37-60.
57
[3.5] R.S. Sayles and T.R. Thomas, “Surface topography as a nonstationary random process,”
Nature 271 (1978) 431-434.
[3.6] J.F. Archard, “Elastic Deformation and the Laws of Friction,” Proc. Roy. Soc. London
A243 (1957) 190-205.
[3.7] J.A. Greenwood and J. J. Wu, “Surface roughness and contact: an apology,” Meccanica
36 (2001) 617-630.
[3.8] A. Majumdar and B. Bhushan, “Role of Fractal Geometry in Roughness Characterization
and Contact Mechanics of Surfaces,” ASME J. Tribol. 112 (1990) 205–216.
[3.9] B.B. Mandelbrot, The Fractal Geometry of Nature (W H Freeman, New York, 1982).
[3.10] K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley,
New York, 1990).
[3.11] M.V. Berry and Z.V. Lewis, “On the Weierstrass-Mandelbrot fractal function,” Proc.
Roy. Soc. London A370 (1980) 459-484.
[3.12] E. Garcia and J. Sniegowski, “Surface micromachined microengine,” Sens. Actuators A
48 (1995) 203-214.
[3.13] M. Ausloos and D.H. Berman, “Multivariate Weierstrass-Mandelbrot Function,” Proc.
Roy. Soc. London A400 (1985) 331-350.
[3.14] W. Yan and K. Komvopoulos, “Contact analysis of elastic-plastic fractal surfaces,” J.
Appl. Phys. 84 (7) (1998) 3617-3624.
[3.15] A. Majumdar and B. Bhushan, “Fractal Model of Elastic-Plastic Contact between Rough
Surfaces,” ASME J. Tribol. 113 (1991) 1–11.
[3.16] J.J. Wu, “Characterization of fractal surfaces,” Wear 239 (2000) 36-47.
[3.17] H. Hertz, “The Contact of Elastic Solids,” J. Reine Angew. Math. 92 (1881) 156-171.
[3.18] R.W. Carpick, E.E. Flater, J.R. VanLangendon, and M.P. de Boer, “Friction in MEMS:
From single to multiple asperity contact,” Proc. of the SEM VIII International Congress
and Exposition on Experimental and Applied Mechanics (2002) 282-287.
[3.19] K.L. Johnson, K. Kendall and A.D. Roberts, “Contact Mechanics,” Proc. Roy. Soc.
London A324 (1971) 301-313.
[3.20] B.V. Derjaguin, V.M. Muller and Y.P. Toporov, “Effect of Contact Deformations on the
Adhesion of Particles,” J. Colloid Interface Sci. 53 (1975) 314-326.
58
Chapter 4
A Numerical Contact Model Based on Real Surface Topography
This chapter has been modified from the following citation:
C K. Bora, M.E. Plesha, R.W. Carpick, “A Numerical Contact Model Based on Real Surface
Topography”, Tribology Letters (in review).
4.1 Introduction
Most engineering surfaces are rough, regardless of whether the surfaces are naturally
created or are processed. When two surfaces come into contact, this roughness causes multi-
point contacts such that the actual area of contact is only a small fraction of the available contact
area.
Atomic force microscopy (AFM) and other similar imaging techniques enable
measurement of surface roughness at near-atomistic length scales. There is potentially substantial
revenue in utilizing this high detail surface topography to model and investigate the connection
between micro and macro scales of contact, adhesion, and friction.
Most conventional methods of modeling rough surfaces in contact replace the actual
surface roughness with a distribution of non-interacting hemispheres, using statistical
information about surface heights and slopes at a single length scale [4.1, 4.2] or multiple length
scales [4.3, 4.4]. This allows for application of well-known contact models to the individual
59
hemispherical contact points, and allows for investigating the multiscale geometry of surfaces.
However, modeling a surface using a hierarchy of hemispheres implies a loss of information in
that the high detail topography of the original surface is not directly exploited in the analysis.
Furthermore, most conventional methods do not take into consideration the effect of
contacting zones on their surrounding areas. When the contact pressure increases over a given
macroscopic surface area, an increasing number of asperities, at various distances from each
other, come into contact and it becomes crucial to account for interaction between the micro-
contacts. Furthermore, the elastic deformation due to the compression of a local region tends to
persist over significant lateral lengths. Therefore, the deformation of one asperity influences the
deformation of neighboring asperities, and vice versa. Thus, contact models that are based on
single asperity contact behavior become deficient. Polonsky and Keer [4.5] argue that a
numerical solution technique using actual geometry of real surfaces is necessary to accurately
account for the interaction between the micro-contacts.
One method to directly use the actual surface topography and to model inter-asperity
interactions is to use conventional three-dimensional (3D) finite element discretization. Such
models, while potentially highly accurate, require an enormous number of degrees of freedom
(d.o.f.) and, correspondingly, computer power and time. Hyun et al. [4.6] model a 512x512 pixel
surface contacting a flat surface using over 911,000 nodes with 3 d.o.f. each and 568,000
tetrahedral solid elements; a typical finite element model is shown in Fig. 4.1. To model the
approach between the contacting surfaces, a dynamic method is used wherein inertia is included
in the equations of motion, which requires small time increments and the introduction of
artificial damping. All of these make the solution computationally expensive.
60
Figure 4.1. Side view of a 3D finite element mesh of an elastic body (top) with a 512x512 pixel
resolution rough surface that is pressed onto a flat, rigid substrate (Hyun et al. [4.6]).
Another numerical modeling approach, which is also employed in this work, is to make
use of an analytical solution, such as the Boussinesq solution, to characterize the elastic
deformation of a uniform, planar substrate due to normal direction loads [4.7, 4.8]. In this
method, the displacement effects of multiple points of contact are coupled with each other, and
solved in a system of algebraic equations. There are a number of models in the literature
applying this method. Webster and Sayles [4.9] present a semi-analytical contact solution where
they subdivide the contact area into rectangular segments, on which they assume a constant
pressure. They demonstrate their method for a 2D problem. Poon and Sayles [4.10] describe a
similar method for 3D, and demonstrate application of a simplified version of a 3D contact
problem of a directionally structured rough surface. They include plasticity, such that the contact
pressure is allowed to increase only until it reaches the hardness of the softer material. Ren and
Lee [4.11] develop a moving grid method to avoid large sizes of the matrices that define the
deformation coupling effect between the contact points. These models usually start with a
61
prescribed amount of normal approach between the surfaces, and use a prediction-correction
algorithm to converge to equilibrium. Polonsky and Keer [4.5] use a fast numerical integration
technique to calculate the surface deflections and they employ a conjugate gradient method
iteration scheme to reach contact distribution convergence. Following this work, Liu, Wang and
Liu [4.12] develop a 3D model for thermo-mechanical contact between two rough surfaces.
Dickrell et al. [4.13] discuss a simple numerical model that takes into consideration the
pixelated data of real surfaces as obtained by common profilometry techniques. In their model,
the surface asperities are assumed to be rigid-perfectly plastic and supported by a rigid substrate.
Thus there are no elastic deformations and the softer of the two surfaces is assumed to yield
wherever there is contact, and the material that is displaced by plastic deformation is allocated to
adjacent pixels. In a more recent version of the model, the pixels are given a simple elastic
stiffness; however, lateral coupling of deformation between the pixels is not modeled. In the
present paper, we enhance Dickrell et al.’s approach by using the Boussinesq displacement
relations to create an elastic foundation. We consider only elastic deformations, but the approach
we describe can be further enhanced to include Dickrell et al.’s method to account for plastic
deformation.
Our finite element approach uses a combination of analytically-calculated surface
behavior of a linear, elastic, homogenous, isotropic material subject to normal loads, i.e., the
Boussinesq displacements, to characterize far field deformations, and a surface layer
discretization that directly utilizes AFM data to account for roughness. In essence, the surface
roughness is a thin layer that overlies an elastic substrate. To investigate the development of
contact, we follow a step-by-step approach, which does not require a convergence consideration.
We discuss methods to minimize the size of the matrices and to speed up the detection of contact
62
points. In the following sections we describe our model, show validations through example cases
compared to analytical and other numerical solutions, and discuss accuracy of the method. We
then apply the method to investigate contact behavior of surfaces from actual MEMS-based
friction experiments.
4.2 Description of the Model
The topography of a surface, as obtained by AFM imaging, is a set of height data for a
rectangular region of a surface area, as shown in Fig. 4.2. Our model features a one-to-one
representation of each of the contacting surfaces, using rectangular prisms of material that
protrude from each surface at every pixel, and these prisms of material are called voxels. In other
words, voxels are the smallest box-shaped parts of a three-dimensional scan and the name is
derived by contracting the words “volume” and “pixel”.
Figure 4.2. Surface representation with voxels.
The model discretizes each of the two contacting surfaces using two regions. The first
region, defined as the substrate, is an elastic half-space that is discretized using a set of nodes
63
that lie in a horizontal plane and whose deflections are fully coupled with each other. The second
region, the interface, is a thin surface layer consisting of individual, uncoupled springs that
protrude from the substrate at every pixel, as shown in Fig. 4.3. The surface topography, such as
that obtained from an actual AFM image, is represented in the interface domain. Nonlinear
material properties, including adhesion and plastic deformations can be assigned to the springs
that define the interface. While our model uses an elastic half space for the substrate region, it is
possible to use other substrate domain types, such as a thin or thick plate, etc.
Figure 4.3. Surface representation for the finite element model: The elastic horizontal coupling
of the voxels is achieved in the substrate domain. The interface domain is represented with
individual axial springs protruding from the substrate.
In this finite element method, nodes have only normal-direction displacements as d.o.f.
These displacements are coupled with one another within the substrate using the Boussinesq
solution, which provides displacement response for all the surface nodes for a given vertical
loading problem [4.7, 4.8]. The substrate is thus modeled as a super-element, representing the
δ1
δ2
64
deformable half space. Assuming a perfect alignment of the data points on the two contacting
surfaces, the possible contact locations are quantized as the square pixels corresponding to the
voxels of the two surfaces (i.e., two contacting surfaces imaged with 512x512 pixel resolution
will have (512)2 possible contact points.)
4.2.1 The Substrate
The surface deflection for an elastic half space subjected to a normal-direction point load
is described by Boussinesq [4.7] and Love [4.8] as
222
)1()0,,(
yxG
Pyxu
z (4.1)
where x and y are the coordinates of the surface points relative to the point of load application; P
is the point load applied at the origin, as shown in Fig. 4.4(a); G and υ are the shear modulus and
Poisson’s ratio of the elastic half-space respectively; and uz is the displacement in the z direction
which is normal to the interface. This relation is singular at the coordinate system origin, making
it impractical to use as a force-deformation calculation. When the point load is relocated to the
coordinates (s, t), as shown in Fig. 4.4(b), the surface deflections are
22)()(2
)1()0,,(
tysxG
Pyxu
z (4.2)
65
When the loading consists of a normal pressure distribution p(s,t) over an area A, as
shown in Fig. 4.4(c), the displacement solution can be obtained by using Eq. (4.2) to integrate
the displacement effects due to loading over each infinitesimal area dA, or (ds dt), as
dtds
tysx
tsp
Gyxu
A
z22
)()(
),(
2
)1()0,,( (4.3)
Figure 4.4. Depiction of the Boussinesq problem for: a) a point load at the center of the
coordinate system, b) a point load at coordinates (s,t), (c) a pressurized area A, where the pressure
distribution is defined by a function, p(s,t).
(a) (b)
(c)
66
When the pressure distribution on a square pixel is assumed to be uniform, a surface
displacement field can be obtained using the above integral, with a total load of P over a square
area of dimension d x d, as shown in Fig. 4.5, as
22
22
22
22
22
22
22
22
)2()2(2
1
2log)2(
)2()2(2
1
2log)2(
)2()2(2
1
2log)2(
)2()2(2
1
2log)2(
)2()2(2
1
2log)2(
)2()2(2
1
2log)2(
)2()2(2
1
2log)2(
)2()2(2
1
2log)2(),(
ydxdyd
xd
ydxdxd
yd
ydxdyd
xd
ydxdxd
yd
ydxdyd
xd
ydxdxd
yd
ydxdyd
xd
ydxdxd
ydCyxuz
22
)1(where
Gd
PC
(4.4)
Figure 4.5. Boussinesq problem for a square area of dimension d × d with uniform pressure p.
67
Figure 4.6 shows the displacement field produced by a uniform pressure over a unit
square pixel located at the origin of the coordinate system; the displacements are shown inverted
for better visualization, and the boundaries of the pressure region are marked with thick lines.
Equation (4.4) is defined everywhere except locations where a logarithmic operand is equal to
zero, i.e., on line segments x = ±0.5, for –0.5 < y < ∞, and y = ±0.5, for ∞ < x < 0.5. Thus, the
function is defined at the centers of all pixels, which are shown with dots in Fig. 4.6.
Figure 4.6. Displacement field produced by a uniform pressure over a square region of unit
area (boundaries indicated by the heavy lines) centered at (0,0), when the coefficient C in Eq.
(4.4) is assumed to be 1. The displacements are inverted for better visualization.
When the vertical deflections at the center of each surface pixel are defined as d.o.f, Eq.
(4.4) can be used to obtain a flexibility matrix SB that relates forces to displacements, where an
entry sij in the flexibility matrix represents the displacement value at the ith
d.o.f. due to a unit
load at the jth
d.o.f. For given material properties and the pixel size d, this value of sij depends on
the difference of the x and y coordinates between the two d.o.f.. The flexibility matrix that is
x
y
z
68
obtained is symmetric. By taking the inverse of SB, the stiffness matrix *
BK can be obtained,
where an entry kij represents the force required at the jth
d.o.f to cause a unit deflection at the ith
d.o.f. Letting n denote the number of d.o.f. per surface, the stiffness matrix *
BK has a size of n
2.
This stiffness matrix relates the forces at all d.o.f., f, with the deflections at all d.o.f., uz.
*
BK uz=f
nnnn
n
n
kkk
kkk
kkk
21
22221
11211
1SK
*
B
(4.5)
Aside from the substrate d.o.f. described earlier, an additional d.o.f. is included in Eq. (4.5) for
each surface to allow for a possible non-zero far-field displacement of the substrate, as denoted
in Fig. 4.3 with δ1 and δ2; these are called “handle nodes”, or foundation nodes. The stiffness
terms related to the foundation degrees of freedom are augmented to *
BK at row and column
number (n+1). The terms in the last row and column of the resulting matrix B
K are obtained by
considering rigid body displacement capability. No forces should be generated when all the
surface d.o.f. displace by the same amount as the handle node. Equation (4.5) can be rewritten
with a force vector consisting of zeroes, and a displacement vector consisting of ones, both with
size (n+1)
0
0
0
1
1
1
...1
1
1
)1)(1()1(1)1(
)1(
)1(1
nnnnn
nn
n
kkk
k
k
*
B
B
KK
n
j
ijinnikkk
1
,11,where
(4.6)
69
4.2.2 The Interface
In the interface layer, each voxel element is modeled as an axial spring. To determine the
stiffness k of the voxel, we assume the x and y direction strains to be zero throughout the voxel.
This assumption is warranted because the voxel is supported from its sides by adjacent voxels,
except for possibly a small height difference that might make it extend beyond its surrounding
neighbors. Thus, the stress (σz) -strain (εz) relation for the z direction is
zz
E
)21)(1(
)1( (4.7)
where E and υ are the elastic modulus and Poisson’s ratio of the interface material. All pixels
have the same area d2, and hence the stiffness for a voxel with height h above the substrate has
the force displacement relation
2
1
2
1
11
11
P
P
u
uk (4.8)
h
dEk
2
)21)(1(
)1( (4.9)
where u1 and u2 represent the displacements, and P1 and P2 represent the external forces at the
bottom and top nodes of the voxel, respectively, as shown in Fig. 4.7.
When the interface deformability is represented in this fashion, the stiffness of the voxel
is inversely proportional to the height h, which is a somewhat arbitrary term that needs to be
carefully selected. The substrate elasticity represents the exact solution for a flat surface, and any
additional finite stiffness due to the interface adds to the overall flexibility. Further discussion
about the stiffness of the interface layer can be found in the example problems that are treated
later.
70
Figure 4.7. Geometry of a single voxel with area d2 and height h.
4.2.3 Contact between Surfaces
When a pixel from one surface makes contact with a pixel from the other surface, we
model this contact by using an additional very stiff spring element (i.e., a penalty spring).
Stiffness of this contact element has the same form as Eq. (4.8). The topology of the global
stiffness matrix for the entire discretization is shown in Fig. 4.8, where Kc in this case is the
stiffness matrix contributed by the penalty spring elements.
1
0
0
0
0
0
2
2
2
1
1
1
2
2
2
1
1
1
h
B
i
i
B
h
h
B
i
i
B
h
f
f
f
f
f
f
u
u
u
u
u
u
Figure 4.8. Topology of the global stiffness equation including stiffness of the Boussinesq
elements (KB), interface spring elements (Ki), and contact elements (Kc). The subscripts of
displacements (u) and forces (f) represent the handle (h), substrate (B), and interface (i).
y
z
x
h
d d
u1, P1
u2, P2
≡
u1, P1
u2, P2
k
KB1
KB2
Ki1
Kc
Ki2
71
4.3 Algorithmic Considerations
4.3.1 Memory and Speed Considerations
Our approach substantially decreases the number of degrees of freedom compared to a
full three dimensional finite element analysis. Nonetheless, when all the available pixels are
coupled with each other for the Boussinesq half space, the number of equations to be
simultaneously solved becomes large. A grid of fully coupled Np by Np d.o.f. results in a
Boussinesq super element stiffness matrix size of (Np2+1)x(Np
2+1), where Np is the number of
pixels per side of the surface image. For Np =512, this amounts to 262145 equations. The global
stiffness matrix including both surfaces with their substrate and interface layers would involve
over a million degrees of freedom. This is less than the 2.7 million degrees of freedom in the 3D
FEA example in Hyun et al. [4.6], which discretizes one surface of the contact problem; however
the difference is not satisfactory, necessitating further reduction in our system size.
To reduce the size of matrices, two methods were considered. The first method
investigated was the use of a coarse substrate mesh with a manageable size and treating
intermediary points as “slave” d.o.f. In this case, although the substrate still had the general
deflection shape of an elastic half space, the coarsening of the mesh resulted in reduced precision
in the contact area calculation, pressures and displacements. When a load is applied as a uniform
pressure on a larger square area (i.e., a pixel of a coarser mesh), the area of influence of an
individual contact point becomes larger, while the maximum deflection and pressure are under-
estimated. Furthermore, large portions of the stiffness information, namely the equations
contributing to the d.o.f. for voxels that are not in actual contact, are not utilized.
The second method, which we discuss below, involves reducing the super element d.o.f.
to only those that are associated with the voxels in contact. Usually, only less than a few percent
72
of the apparent area is in contact, thus the required stiffness matrix for this method has
substantially smaller size.
To understand the evolution of the contact area with increased compression, an
incremental algorithm is used. One disadvantage of this method is that the stiffness matrix needs
to be reformed with the addition of each new contact point. As the size of the stiffness matrix
becomes large, this may lead to long calculation times. Several methods were implemented to
reduce the program execution time, including: generating nodes only at contact points and
updating the node list at each step; numbering the nodes with element-by-element ordering to
reduce the populated portion of the stiffness matrix; using the same substrate flexibility matrix
for both surfaces and using triple factorization for inverting the matrix; and carefully minimizing
the portion of the area where the next contact point is searched.
4.3.2 The Algorithm
The coding for the finite element analysis was done as an enhancement to the FEMCOD
program skeleton [4.14]. The FEMCOD program has features such as compact column (skyline)
storage and an active column equation solver, which are useful for banded stiffness matrices
such as that shown in Fig. 4.8.
At the start of the algorithm, the surface heights are entered into the program for each
pixel over a square contact region for both contacting surfaces. These values represent the
average heights of the voxels, and the d.o.f. are defined at the center point of each voxel. The
highest sum of any two of the voxel height pairs is determined as the first contact point.
Positioning the surfaces so that they touch at this point without any load allows the gaps between
73
the upper and lower surfaces to be calculated and sorted, to be used in a simplified contact
detection scheme.
Starting with the initial configuration described above, the stiffness matrix is created for a
single point contact. At this stage, there are six degrees of freedom, consisting of the d.o.f. for
the two handle nodes, the two substrate nodes, and the two surface nodes. A unit load (1 nN) is
applied to the top handle node, while keeping the bottom handle node fixed. These linear
equations are solved to obtain the displacements at the contacts.
To determine the next pair of contacting voxels, the displacements of the non-contacting
voxels are calculated under the unit load for the step. Analysis of the whole surface is
cumbersome, and not necessary for this search, as the surfaces are more likely to contact at
locations with low gap values. On the other hand, the contact sequence does not simply follow
the order of the gap values. To efficiently search for the next contact, a reduced candidate
method was prepared, where a set of candidate locations is selected starting from locations with
the smallest initial gaps. The size of this contact candidate list varies according to the number of
existing contacts. Using the force displacement behavior for the load step, the smallest force
required to form another contact is found and the associated location is marked as the next
contact point. Multiple points that require the same smallest force are all included in the next
load step.
New surface and substrate points, interface elements and contact elements are generated
and the connectivity information for the existing Boussinesq elements is updated. The process of
solving for displacements and finding new contacts is iterated until the initially selected
maximum number of contacts is reached. For each of these steps, a unit load is used to determine
the force-displacement behavior. The force increment for every contact point is calculated at
74
each step and added to the previous force value. A final check algorithm is introduced at the end
of the program to verify that no contacts were missed with the reduced candidate contact
detection method.
4.4 Verification Examples
In this section, simulations carried out with the method developed in this paper are
compared with problems having analytical solutions.
For the voxel dimensions considered in the following examples, when the h value is
chosen such that the whole roughness structure is contained in the interface layer, the layer
becomes too soft. In the test cases we considered, it was found that the elastic behavior of the
contacting bodies can be modeled solely with the Boussinesq layer. For this reason, in the
examples discussed in this paper, the interface elements are given a stiffness value that is five
orders of magnitude higher than the substrate layer, making them essentially rigid. In this form,
the Boussinesq layer defines the elasticity and the interface layer is retained to model the
roughness information and future introduction of the nonlinear behavior of the surfaces.
4.4.1 Rigid Cylindrical Punch Pressed into an Elastic Half-Space
As a test example, a problem of a rigid circular punch is investigated. The lower surface
is modeled as a flat elastic substrate, as shown in Fig. 4.9, with E = 200 GPa and υ = 0.25. The
upper surface is modeled as a rigid cylinder protruding from a rigid flat surface, with an elastic
modulus that is five orders of magnitude higher than that of the lower surface.
75
Figure 4.9. Rigid cylindrical punch pressed into an elastic half space.
Figure 4.10(a) shows the discretization of the circular punch for the first model, which
has a 1 nm pixel size. From the discretized circular contact area, an effective radius was obtained
and used in the analytical solution for comparison. Keeping the area of the circular punch
constant, the mesh was refined twice, generating models at one-third and one-ninth of the initial
mesh size, as shown in Fig. 4.10(b,c).
The analytic solution for a rigid cylindrical punch of radius R contacting the surface of a
semi-infinite body provides the surface displacement and pressure values as [4.15]
RE
P
2
)1(2
(4.10)
222
)(
rRR
Pr
C (4.11)
10nm
10nm
R=2.5nm Region to be
meshed
y
z
x
76
Figure 4.10. Top view of the circular punch with contact areas modeled with
(a) 21 pixels with 1nm x 1nm size, (b) 189 pixels with 1/3nm x
1/3nm size, and
(c) 1701 pixels with 1/9nm x
1/9 nm size.
where δ is the displacement of the punch, P is the applied load, υ is the Poisson’s ratio, and E is
the Young’s modulus of the elastic half space. The theoretical pressure σc at the edge of the
punch approaches infinity. The analytical calculation for the pressure along a radius of the punch
(
c)
(
a)
(
b)
x
10nm
y
10nm
(a) (b)
(c)
77
is shown with the solid line in Fig. 4.11, compared with the calculated pressure values for the
different mesh sizes. The finest mesh size gives excellent agreement with the analytic solution
for pressures and displacement, with errors less than 1%. Table 4.1 compares the center node
pressure values and the displacements with the analytical solution.
Figure 4.11. Radial pressures for the rigid circular punch problem for FE models with different
mesh sizes, and the analytical solution.
Richardson extrapolation [4.16] can be used to improve the results obtained using
multiple mesh sizes according to the following relation, provided that the meshes have
undergone regular refinements:
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 0.2 0.4 0.6 0.8 1 1.2
stre
ss / a
vera
ge s
tress
radial distance / radius of punch
21 points
189 points
1701 pointsanalytical
78
Table 4.1. Center node stress, displacement and calculated errors for the circular rigid punch on
an elastic substrate, under a total load of 1nN.
Mesh Size
(nm)
Displacement
(10-3
nm)
Displacement
Error (%)
Pressure at the
center
(nN/nm2)
Pressure
Error (%)
1 0.955 5.37 0.0261 9.54
1/3 0.920 1.51 0.0245 2.97
1/9 0.911 0.485 0.0240 0.934
Analytical 0.907 0.0238
rr
rr
pp
pp
hh
hh
12
1221
0 (4.12)
where 0 is the extrapolated value of the solution, 1 and
2 are FE approximate solutions
obtained from different mesh sizes, i.e. h1 and h2, respectively, and pr is the rate of convergence
for the model. For our application of Eq. (4.12) to this example, the exact solution 0 is known,
and with the FE results 1 and
2obtained for two mesh sizes h1 and h2, Eq. (4.12) contains one
unknown, namely the rate of convergence pr. Using the results for the two coarsest meshes,
shown in Table 1, we determine pr=1.15 for the rate of convergence for displacements, and
p=1.06 for the rate of convergence for stress. Using these convergence factors, another
application of the extrapolation between the two finer mesh models provides estimates of the
displacement value with an error of 0.01% and the center point stress with an error of 0.031%,
when compared to the analytical solution.
With the values of pr cited above, the rate of convergence is slightly better than linear,
which is slow. Because of the differences between our model and the classical finite element
79
methods, the exact nature of the rate of convergence is not immediately apparent. For this
example, one factor affecting the rate is that the area that is discretized is not the same for each
refinement, as we are approximating a circular edge using piecewise straight lines. Another
factor is our attempt to converge to the asymptotic singular behavior of the stresses at the punch
edge using square areas with uniform pressure. However, the displacement solution is nodally
exact at the d.o.f. for the pressure distribution represented with the discretization, and the results
show outstanding accuracy, even for the coarsest mesh sizes.
4.4.2 Rigid Square Punch Pressed into an Elastic Half-Space
To investigate the effect of approximating a curved boundary using piecewise straight
line segments, we study a similar problem with a square punch, as shown in Fig. 4.12. Beginning
with a single contact point solution, the mesh is refined three times, each time dividing the size
by three, thus increasing the number of contact points by a factor of 9. Table 2 gives the
calculated values for the vertical stress at the center point of the contact area and the
displacement of the punch.
Table 4.2. Center node stresses and displacements.
Mesh size
(nm)
Number of points
defining contact
Displacement for
1nN load
(10-3
nm)
Normalized Pressure
at the center
10 1 0.526 1
3.333 9 0.451 0.511
1.111 81 0.422 0.519
0.3704 729 0.412 0.498
Analytical estimate 0.400 0.456
80
Figure 4.12. Rigid square punch pressed into an elastic half space, with E =200 GPa and υ=0.25.
The analytical solution for this problem is approximate and thus, an exact error analysis
cannot be performed. Borodachev [4.17] offers an approximate solution which is used for
determining the displacement and stress values given in Table 2. A rate of convergence pr can be
calculated using displacement results of three mesh sizes, assuming the rate is uniform. The first
three mesh sizes yield a rate of convergence of pr=0.874 and a second calculation using the
second, third and fourth mesh sizes give pr=0.973. As the first mesh size contributes only to a
uniform pressure distribution, the second convergence rate value is deemed to be more reliable.
This rate is very close to linear, similar to the rates seen in the circular punch problem. Using
pr=0.973, the displacement estimate can be extrapolated to 0.407x10-3
nm and the normalized
pressure at the center of the punch can be extrapolated to 0.486. The convergence rate for this
30nm
30nm
Region to be
meshed
1
0nm
x
z
y
10nm 10nm
81
problem is about the same as that for the cylindrical punch. Even though the contact region is
easier to mesh for this example, modeling the pressure distribution at the edges becomes more
challenging. Because the convergence rates obtained for the finer mesh for both models are close
to 1, a linear rate of convergence is assumed for future examples. As with the circular punch,
while this rate is low, the results show outstanding accuracy, even for the coarsest meshes.
The calculated stresses along the x axis (passing through the center point of the punch,
parallel to the side of the square) are shown in Fig. 4.13. Three mesh sizes using the method
developed in this paper are shown with the square data points. These are compared with
solutions from Borodochev’s approximate analytical solution [4.17] and 3D FEM analysis using
ANSYS, which are shown with the circular data points. In the two ANSYS models, the contact
region is discretized into 36 and 144 elements, respectively, using 20 node quadratic solid
elements. Symmetry is employed to simplify the model. Richardson extrapolation was performed
using the two mesh sizes.
It is seen in Fig. 4.13 that our method overestimates the stress along the axis considered,
while the 3D ANSYS model underestimates it, and both show better agreement with the
approximate analytical result with mesh refinement. The difference between the extrapolated
central stress values is less than 1%. Overall, there is outstanding accuracy of the proposed
model even with the coarsest mesh sizes.
82
Figure 4.13. Comparison of radial pressures for the rigid square punch model to a conventional
3D models and an approximate analytical solution [4.17].
4.4.3 Rigid Spherical Surface Pressed into an Elastic Half-Space
A spherical contact problem, as shown in Fig. 4.14, was modeled to test the step-by-step
contact detection algorithm. In contrast to the previous examples, this example has varying
surface heights.
For two materials with Young’s moduli of E1 and E2 and Poisson’s ratios of υ1 and υ2, an
effective elastic modulus E*, the force-displacement (F-d) relation and the contact area-force (A-
F) relation are as follows [4.15]
2
2
2
1
2
1
*
111
EEE (4.13)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
stre
ss / a
ver
age
stre
ss
distance x from the center / distance to the edge
ANSYS 6x6
approximate analytical
solution
3x3
9x9
27x27
approximate analytical
solution
ANSYS 3x3
ANSYS,extrapolation
83
Figure 4.14. Rigid spherical surface pressed into an elastic half space.
2/32/1*
3
4dREF (4.14)
3/2
*4
3
E
FRA (4.15)
where R is the radius of the sphere, A is the contact area, and F is the contact load. For the
example E1 = 200 GPa, υ1 = 0.25, R=15 nm and the second material is assumed to be infinitely
rigid.
Pixels sizes of 0.5 nm, 1 nm, 2 nm, and 4 nm are used to investigate the performance of
the algorithm. Figure 4.15 shows the spherical surface modeled with the 1 nm pixel size. The
30nm
30nm
Region to be
meshed
x
z
y
R=15nm
84
force displacement behavior does not change with changing pixel sizes in our model, as seen in
Fig. 4.16(a). The contact area calculation seen in Fig. 4.16(b) shows a step-wise increase with
increased load, caused by the discretized nature of the surface, but the overall trend between the
models with different pixel size is consistent.
In both figures, excellent agreement with the Hertz model is seen until a contact area of
about 100 nm2. The Hertz solution is not considered to be valid past this region, as it assumes the
contact radius to be much smaller than the radius of the spherical surface [4.15]. A power law fit
to our data for the 0.5 nm pixel case in the full range shown in Fig. 4.16 gives an area-load
dependence of A α F0.68
, which is in close agreement with the A α F0.667
relation for the Hertz
solution given in Eq. (4.15).
Figure 4.15. Spherical punch model used in the example of a rigid sphere contacting a flat
elastic surface. The sphere has a radius of 15nm and the pixel dimension for this model is 1nm.
(nm)
85
Figure 4.16. (a) load vs. displacement and (b) area vs. load results for a rigid spherical punch
contacting a flat elastic surface, compared to the Hertz solution.
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
0 1 2 3 4 5 6 7
Lo
ad
(n
N)
Displacement (nm)
4nm pixels
2nm pixels
1nm pixels
0.5nm pixels
Hertz
0
50
100
150
200
250
300
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
Co
nta
ct A
rea (
nm
2)
Load (nN)
4nm pixels
2nm pixels
1nm pixels
0.5nm pixels
Hertz
(a)
(b)
86
4.5 AFM Surface: Experiments with Resolution
The AFM topography image of a polycrystalline silicon surface-micromachined
nanotractor actuator was used as a sample case [4.18]. The AFM surface was placed at the
bottom and its contact with a rigid flat surface was modeled. E = 200 GPa, υ = 0.25 were used
for the AFM surface, while the rigid surface was modeled with an E value that is larger by five
orders of magnitude. The image used is of a 5 µm x 5 µm area measured at 512x512 pixel
resolution (Np=512). To investigate the behavior of elastic contact with varying sampling sizes,
the surface resolution was reduced to obtain images of Np=256, Np=128, and Np=64. Let
i,j=1…Np; to reduce the resolution by half, for example, pixels with odd i and odd j indices can
be selected, ignoring the other pixels. This method is consistent with the way an AFM instrument
measures surface heights for different resolutions. An alternate method is also investigated,
which consisted of averaging the neighboring four pixels to obtain a lower resolution height
value. The displacement and force analysis for the first method with varying sets of odd and even
i, j and the alternate method give similar results, with errors within ±2% for 1000 contacts.
Figure 4.17 shows the load vs. displacement and contact area vs. displacement graphs for
the different resolution AFM images. The resolution of the image does not have any significant
effect on the load vs. displacement behavior, while the contact area for a given displacement is
strongly dependent on the resolution. For a given displacement, the high resolution image gives a
much smaller contact area. According to our model, simply dividing a pixel under uniform
pressure into four smaller pixels of equal height does not change the results. However, in the
higher resolution image the four pixels are generally not at the same height. When the highest of
these pixels come into contact, it delays the contact of the remaining pixels. In the purely elastic
case this effect is exacerbated, whereas in a plastic model, the pixel that comes into contact first
87
would likely yield and the surrounding pixels would more easily come into contact. The
differences between the contact areas for the different resolution models will likely be smaller if
plasticity is included.
The results of a simple area calculation representing no elastic coupling (i.e., the substrate
of the rough surface is rigid, and the elastic voxels deform independently from each other)
between the contact points is shown in Fig. 4.17(a) and (b) with dashed lines. The stiffness of an
uncoupled voxel was obtained from the Boussinesq problem with a single pixel under contact;
i.e., using Eq. (4.4) with (x, y)=(0, 0). For the 512x512 pixel image representing a 5µm x 5µm
surface with E = 200GPa and υ = 0.25, the center point of a single pixel under uniform pressure
would deform with a stiffness value of 5.39 10-4
N/m. The area data is obtained by counting the
voxels in the 512x512 image that are higher than the given displacement. For a given
displacement value, the contact area estimated with no elastic coupling is much higher, and the
contact load is much lower than the results from our model, as expected.
Figure 4.18(a) shows the contact area fraction vs. apparent pressure graphs for the
surfaces using our model, shown in dashed lines, and also by using McCool’s statistical method
[4.2] based on Greenwood and Williamson’s model [4.1], shown by the solid lines. The trends
follow a power law, where the exponent increases as the resolution is increased. Increased
resolution also decreases the contact area estimate for a given pressure value. The importance of
the sampling resolution is further demonstrated with the results from the McCool analysis, which
uses size dependant RMS heights, slopes and curvatures as input.
88
Figure 4.17. The solid lines represent (a) load vs. displacement and (b) area vs. displacement
results for the polycrystalline silicon surface at different resolutions, modeled with the elastic
Boussinesq substrate model, pressed against a rigid flat surface. The dashed line represents the
contact area obtained by an elastic response without any coupling between the contact points
(i.e., with a rigid substrate).
L = 0.0023 d2.494
0
1
2
3
4
5
6
7
8
0 5 10 15 20 25 30
Co
nta
ct L
oad (
mN
)
Displacement (nm)
64 x 64
128 x 128
256 x 256
512 x 512
no coupling
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 5 10 15 20 25 30
Co
nta
ct A
rea (
µm
2)
Displacement (nm)
(a)
(b)
89
Overall trends for the 64x64 and 256x256 pixel images look similar between the two
methods, with the fraction of the area in contact becoming smaller as the resolution increases.
However, the power law predicted by our model has a smaller exponent than the McCool’s
method for every resolution image. Our results for the 128x128 pixel image diverge from the
statistical estimate at around 0.8% real contact area, while the 512x512 pixel image results are
different from the beginning of the contact, with the difference typically larger than 15%.
The 512x512 image results in an area-load dependence A α L0.907
. Figure 4.18(b) shows
two sets of Richardson extrapolation results for the different resolution images, using a linear
convergence rate (pr=1). When the data from 128x128 and 256x256 resolution images were
used, the extrapolation gives a trend similar to that of the 512x512 image. The power law trend
of the extrapolation is A α L0.926
. An extrapolation between the 256x256 and the 512x512 images
gives an area load relation of A α L0.947
, which is similar to the McCool method results from the
highest resolution image. (It must be noted that there are other results in the literature for this
relation. One of these is Zhuravlev’s model [4.19], which predicts an exponent of 0.91.) For this
purely elastic case, the exponent becomes larger with the increased image resolution and
extrapolation. The exponent values are within ranges estimated by the statistical models.
90
Figure 4.18. (a) Contact area fraction vs. apparent pressure results for the polycrystalline silicon
surface pressed against a rigid flat, modeled at different resolutions. The results from our model
are shown by the dashed lines, McCool [4.2] analysis results are shown with solid lines. The
dotted line represents the behavior when the substrate effect is suppressed. (b) Two separate
extrapolation calculations (solid lines) are shown with the model results.
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0 50 100 150 200 250 300 350
Fra
ctio
n o
f th
e ar
ea i
n c
onta
ct
Apparent pressure (MPa)
McCool (64x64) A = 0.0002p0.935
McCool (128x128) A = 0.0001p0.9384
McCool (256x256) A = 0.00007p0.9431
McCool (512x512) A = 0.00005p0.9484
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 50 100 150 200 250 300 350
Fra
ctio
n o
f th
e ar
ea i
n c
onta
ct
Apparent pressure (MPa)
extrapolation 256&512 A= 0.00005L0.9470
extrapolation 128&256A= 0.00006L0.9259
91
Figure 4.19. Pressure maps of the polycrystalline silicon surface placed against a rigid flat
surface at different resolutions, when 1.2% of the area is in contact. Values are given in GPa.
See Fig. 4.20 for zoomed-in images with finer detail of the regions bordered with dashed lines.
10 20 30 40 50 60
10
20
30
40
50
60
20 40 60 80 100 120
20
40
60
80
100
120
50 100 150 200 250
50
100
150
200
250
50 100 150 200 250 300 350 400 450 500
50
100
150
200
250
300
350
400
450
500
10 20 30 40 50 60
10
20
30
40
50
60
20
40
60
80
100
120
140
64 x 64 128 x 128
256 x 256 512 x 512
92
Figure 4.20. Zoomed-in pressure maps of the polycrystalline silicon surface placed against a
rigid flat surface at different resolutions, when 1.2% of the area is in contact. Values are given
in GPa. The respective images correspond to the areas bordered with dashed lines in Fig. 4.19.
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
10
20
30
40
50
60
5 10 15 20 25 30
5
10
15
20
25
30
10
20
30
40
50
60
10 20 30 40 50 60
10
20
30
40
50
60
10
20
30
40
50
60
20 40 60 80 100 120
20
40
60
80
100
120
10
20
30
40
50
60
20 40 60 80 100 120
20
40
60
80
100
120
10
20
30
40
50
60
32 x 32
subset from
128 x 128 image
16 x 16
subset from
64 x 64 image
64 x 64
subset from
256 x 256 image
128 x 128
subset from
512 x 512 image
93
The contact area vs. load behavior of a simple model with no elastic coupling between
contacts is shown with a dotted line Fig. 4.18(a). For a given contact load, a much smaller
contact area is estimated when substrate coupling effects are neglected. This shows the
importance of including those effects in the calculation.
Figures 4.19 and 4.20 show the actual pressure distribution of the contact spots for the
four different image sizes. At 1.2% of the total area in contact, the calculated maximum elastic
contact pressures are close to 150 GPa. Although this value is well past the hardness of the
material, the solution was extended to these pressure levels to study the behavior and make
comparisons to the statistical models, and also to study the effects of the resolution on the
pressure distribution. While the calculations on the smallest image size give only a crude
estimate of the contact locations, for the image at 128x128 pixels, it is actually possible to
identify the contact shapes, pressure distributions and intensities within individual contact points.
The estimated pressures become higher and the shapes become smoother with further increase in
resolution.
For the AFM surface example, a comparison of the elastic stresses at each pixel to
material hardness (H) shows the number of contact points where the stress exceeds the yield
stress, and the fraction of the contact where stress exceeds the hardness for each step (Fig. 4.21).
A hardness value of 11 GPa was used for the polycrystalline silicon material [4.18]. According
to this comparison, the plastic region occupies 60-70% of the real contact area through all stages
of contact development, even at the smallest loads. With a proper plastic response model, the
contact area estimate would be higher than what is observed in our results.
94
Figure 4.21. The number and percentage of contact points that are estimated to be experiencing
pressure values above the material hardness. For the 512x512 pixel example, when the contact
rea is 1% of the total surface area, 1854 of the 2562 pixels experience pressure values, p > H.
4.6 Conclusion
This model is developed to preserve and fully utilize the high-detail surface topography
data obtained from AFM profilometry. It makes use of analytical solutions to simplify the
treatment of the elastic foundations, using degrees of freedom only in the normal direction, and
suppressing the need to model the substrate material in full three dimensional detail. For the
examples of rigid punches with different geometries pressed into an elastic half space, we show
that our method yields results that are in excellent agreement with analytical and 3D finite
element solutions, even using coarse mesh sizes. The strength of the method is that the solution
of the surface displacement is nodally exact for the employed pressure distribution.
0
500
1000
1500
2000
2500
0
10
20
30
40
50
60
70
80
0 0.2 0.4 0.6 0.8 1 1.2
num
ber
of
conta
ct p
oin
ts w
ith p
> H
perc
en
tage
of
co
nta
ct p
oin
ts w
ith
p>
H
contact area / total surface area (%)
percentage of contact points with p > H
95
Examples of Richardson extrapolation are demonstrated for the trial cases and the real
AFM surface model. Despite the linear convergence rate, the method can be reliably utilized to
obtain high-accuracy estimates of the contact area-pressure relations using lower-resolution
image results.
In the tests using AFM surfaces, the resolution of the image strongly affects the contact
area estimate. The solutions presented in this paper are completely elastic, and the differences in
the responses for the different image size estimates are expected to decrease with the addition of
plasticity. According to our estimate, a large portion (60-70%) of the real contact area will
undergo plastic deformation starting at the smallest loads, and continuing through all stages of
contact development.
The method presented can be used to investigate effects of using different materials,
surface roughening and texturing methods, and differences between unworn and worn surfaces.
Possible future directions, in addition to elastic-plastic material behavior, are inclusion of
adhesion, and modification to include solutions for a surface shear distribution using a
Boussinesq-Cerruti solution [4.8].
References:
[4.1] J.A. Greenwood and J.B.P. Williamson, “Contact of Nominally Flat Surfaces,” Proc.
Roy. Soc. London A295 (1966) 300-319.
[4.2] J.I. McCool, “Comparison of Models for the Contact of Rough Surfaces,” Wear 107
(1986) 37-60.
[4.3] W. Yan and K. Komvopoulos, “Contact analysis of elastic-plastic fractal surfaces,”
Journal of Applied Physics 84 (1998) 3617-3624.
[4.4] A. Majumdar and B. Bhushan, “Fractal Model of Elastic-Plastic Contact between Rough
Surfaces,” Journal of Tribology 113 (1991) 1–11.
[4.5] I.A. Polonsky, and L.M. Keer, “A numerical method for solving rough contact problems
based on the multi-level multi-summation and conjugate gradient techniques,” Wear 231
(1999) 206-219.
96
[4.6] S. Hyun, L. Pei, J.F. Molinari, and M.O. Robbins, “Finite-element analysis of contact
between elastic self-affine surfaces” Physical Review E (Statistical, Nonlinear, and Soft
Matter Physics) 70 (2004) 026117.
[4.7] J. Boussinesq, “Application des potentiels à l'étude de l'équilibre et du mouvement des
solides élastiques (Application of potentials to the study of equilibrium and motion of
elastic solids),” (Gauthier Villars, Paris, 1885).
[4.8] A.E.H. Love, “The stress produced in a semi-infinite solid by pressure on part of the
boundary,” Proc. Roy. Soc. London A228 (1929) 377-420.
[4.9] M.N. Webster, R.S. Sayles, “A numerical model for the elastic frictionless contact of real
rough surfaces,” Journal of Tribology 108 (1986) 314-320.
[4.10] C.Y. Poon, R.S. Sayles, “Numerical contact model of a smooth ball on an anisotropic
rough surface,” Journal of Tribology 116 (1994) 194–201.
[4.11] N. Ren, S.C. Lee, “Contact simulation of three-dimensional rough surfaces using moving
grid method,” Journal of Tribology 115 (1993) 597–601.
[4.12] G. Liu, Q. Wang, and S. Liu, “A three-dimensional thermal-mechanical asperity contact
model for two nominally flat surfaces in contact,” Journal of Tribology 123 (2001) 595-
602.
[4.13] D.J. Dickrell, M.T. Dugger, M.A. Hamilton, W.G. Sawyer, “Direct Contact-Area
Computation for MEMS Using Real Topographic Surface Data,” J. Microelectromech.
Syst. 16 (2007) 1263-1268.
[4.14] M. E. Plesha, R. D. Cook, and D. S. Malkus, FEMCOD - Program Description and User
Guide, (University of Wisconsin – Madison,1998).
[4.15] W.C. Young, Roark’s Formulas for Stress & Strain, 6th
ed., (McGraw-Hill, New York,
1989).
[4.16] R.D. Cook, and D.S. Malkus, M.E. Plesha, R.J. Witt, Concepts and Applications of Finite
Elements Analysis 4th
ed., (Wiley, New York, 2001).
[4.17] N.M. Borodachev, “Impression of a punch with a flat square base into an elastic half-
space”, International Applied Mechanics 35 (1999) 989-994.
[4.18] M.P. de Boer, D.L. Luck, W.R. Ashurst, R. Maboudian, A.D. Corwin, J.A. Walraven,
and J.M. Redmond, “High-performance surface-micromachined inchworm actuator”, J.
Microelectromech. Syst. 13 (2004) 63-74.
[4.19] V.A. Zhuravlev, “On Question of Theoretical Justification of the Amontons-Coulomb
Law for Friction of Unlubricated Surfaces,” Zh. Tekh. Fiz. (J. Technical Phys.– translated
from Russian by F.M. Borodich) 10 (1940) 1447-1452.
97
Chapter 5
Conclusions and Future Directions
5.1 Refining the Multiscale Model and the Surface Analysis Technique
In Chapter 3, a novel method is presented to detect the multiscale properties of the
surface roughness by analyzing actual peaks on the surface. The results of this surface analysis
were then used to estimate the real contact area using a hierarchical roughness scales. It was
shown that the real contact estimate depends on at what scale the roughness is defined.
Furthermore it was shown that when the different scaling behaviors of the geometrical properties
(height, curvature, separation) were correctly defined, the contact area estimate does not depend
on the scaling constant, i.e. the spatial ratio between two consecutive scales.
5.1.1 Introduction of Adhesion and Plasticity to the Model
Adhesion plays an especially important role when the sizes are small and external
mechanical forces of the contact are relatively low. When the contact size is in the order of a few
nanometers, as in the case of an AFM tip, the contact behavior diverts from the Hertzian
behavior [5.1]. Initial attempts to include adhesion in the multiscale contact model using JKR
[5.2] and DMT [5.3] models have led to unstable and diverging results. The behavior shows that
a technique must be used that accounts for the adhesion associated with the larger scale asperities
when considering the smaller scales of roughness. The adhesion models mentioned account for
98
adhesion only over the contact area [5.2] or around it [5.3]. When considering a particular
roughness scale, the surface forces on the material between two asperities are disregarded. If
these forces are not considered, a force imbalance occurs between two consecutive scales.
To overcome this complication, a large area attraction model can be used. [5.4]
According to this model, the attraction between two nominally flat surfaces is related to the
separation between the two surfaces and the surface properties as follows:
2
012/~ dA
H (5.1)
where Г is the adhesion, with units of energy per area, AH is the Hamaker constant, related to the
Van der Waals force between the two surfaces, and d0 is the mean separation between the two
surfaces. Using this understanding, adhesion for each scale can be found by using a mean
separation defined for the scale. Adjusting the total adhesion force between the different scales
might be necessary.
Plasticity can be added to model via methods that extend the elastic Hertzian behavior
[5.2]. Small contact areas undergo the largest pressures, so this analysis is likely to result in a
small scale cut-off for the hierarchy of scales. The change in material properties between scales,
specifically the yield strength, needs be taken into account for accurate analysis [5.5].
5.1.2 Further Investigation of the Multiscale Properties of the Surfaces
The multiscale properties of surfaces are usually attributed to the self affinity of the
surfaces. The polycrystalline surface samples that are used in the surface analysis examples show
a behavior that cannot be fully explained by fractal geometry and this phenomenon has not been
99
investigated in the literature in a significant way. Further investigation of this property using
different fractal analysis techniques (structure function method, two dimensional power spectral
analysis [5.6] and comparison to other fractal surfaces, i.e. random midpoint displacement
method), would yield more accurate identification of the multiscale characteristics of these
surfaces.
Multiscale behavior plots given in Chapter 3 are linear for practical purposes. However a
more detailed investigation would be necessary to explain the slight change in the slopes of the
two images taken from the same sample at two different image resolutions (i.e. in the average
height vs. linear spacing (d) plot, Fig. 3.2). A possible explanation to this change in slope is the
presence of grain boundaries on the surface at distances from each other which are comparable to
the linear spacing value where this behavior is seen. This may be investigated with analysis of
surface data taken from the same sample at different image sizes (from 10μm down to 100nm)
and resolutions. Another possible topic of investigation is the effect of the AFM tip radius and
the noise in the AFM data on the multi-scalar properties.
5.2 Possible Improvements to the Boussinesq Finite Element Analysis Model
The model described in Chapter 4 aims to preserve and fully utilize the high-detail AFM
data while accurately and sufficiently modeling the full interaction between contact points. It
makes use of analytical solutions to simplify the problem, by only requiring the normal direction
d.o.f. to be analyzed, and suppressing the need to model the bulk material. The model is
compared to well-defined analytical problems and it is shown to be accurate within 1% for
practical loads. The effects of varying surface resolution are discussed. Once the elastic solution
100
is acquired and plasticity and adhesion are included in the system, it would be possible to come
up with a close estimate for the true contact area.
5.2.1 Addition of Plasticity
Plasticity can be included in our model in the following way: The voxels for which the
stress exceeds the hardness value can be eliminated from the stiffness calculation in the
Boussinesq (substrate) level, and the plastic behavior can be introduced in the interface elements,
using a method explained by Dickrell et al. [5.7]. This method operates on the idea of
distributing the plastic portion of the volume to the neighboring voxels, thus conserving the
volume.
5.2.2 Possible Changes to the Program to Improve Accuracy and Speed
Figure 5.1 shows the increased time required for each step for the 512x512 pixel surface
size. To obtain a 1/100 real contact area to apparent contact area ratio, which corresponds to
2562 pixels in contact for the 512x512 pixel problem, a total time of 121 hours was required.
The speed of the calculations is dependent on the processor speed. This solution was obtained on
an older work station with a 2.8GHz Pentium 4 processor and 512MB physical RAM. Using a
newer machine, the required time could be reduced.
Appendix B explains some of the features of our algorithm that increase the speed of
execution. Other possible improvements to improve accuracy and speed include:
101
Figure 5.1. Time required for each step for the 512x512 problem. With the increased number of
contacts the number of linear equations, thus the time required for each step increases.
i. Inclusion of plasticity: The stiffness matrix is recalculated after each added contact point
and the time required to process the stiffness matrix is dependent on the number of elastic
contacts. If the plastic portion, which contributes to 60% to 70% of the area, is eliminated
from the stiffness calculation, the time requirement can be greatly decreased for
additional displacements.
ii. Elimination of the elastic interface springs: In the given examples, as the elastic response
was sufficiently modeled within the Boussinesq substrate level, the interface springs were
assigned a large stiffness value (5 orders of magnitudes higher than the bulk stiffness)
and were essentially rigid. The interface springs were kept in the model to be used for the
inclusion of plasticity. The size of the stiffness matrix can be reduced by introducing an
interface spring only at a location when yielding occurs.
0
5
10
15
20
25
30
35
40
45
0 500 1000 1500 2000 2500 3000 3500 4000
tim
e p
er s
tep
(m
in)
Number of elastic contacts
102
iii. Limiting the area of influence of a contact on the substrate: Figure 5.2 shows the
displacement behavior of pixels along an axis passing through a single loaded pixel (the
values are divided by the displacement directly under the load). The displacement effect
diminishes by a ratio of 100 at the 28th
pixel, and by a ratio of 1000 at the 283rd
pixel
from the load. This notion can be used to decrease the size of the Boussinesq flexibility
matrix. The loss in precision that would result by this method needs to be investigated.
Figure 5.2. Normalized displacements on an axis parallel to the side of the pressure region, due
to a single loaded pixel, as described in Fig. 4.5.
iv. Starting from a specified final load or displacement: While the algorithm is prepared to
model the full step-by-step development of the contact, it is possible to modify it so that
the calculation begins from a prescribed displacement and uses a correction method to
0.0001
0.001
0.01
0.1
1
0 100 200 300 400 500
norm
aliz
ed d
ispla
cem
ent
distance from the load in pixels
103
reach convergence. This could potentially decrease the time required for the program,
depending on the problem solved. The speed of convergence in relation to the roughness
structure, plasticity, etc. needs to be investigated.
v. Using a correction method to change the matrix inverse for additional contacts: Let A
be the global stiffness matrix with size mxm, and M be the modified stiffness matrix
when an additional d.o.f. is introduced into the system. (b is a vector of size m and c is a
scalar stiffness value.)
cT
b
bAM (5.2)
The inverse of the resulting matrix M can be calculated using A-1
as follows [5.8]:
kk
kk11
11
1T
11T11
1
Ab
bAAbbAA
M
where k = c – bTA
-1b
(5.3)
This method of finding inverses of modified matrices is more efficient than forming the
stiffness matrix and solving the equations at each step. Although, for this method to
work, at the beginning, the Boussinesq flexibility matrix would have to be calculated and
inverted for a larger set of candidate points, rather than the actual contacts. The stiffness
of the handle nodes would also need to be modified at each step.
5.2.3 Other Functionalities That Can Be Introduced to the Model
For inclusion of adhesion, similar to the idea discussed in section 5.1.1, a large area
attraction model can be used [5.4]. According to this model, the attraction between two
104
nominally flat surfaces is related to the separation between the two surfaces and the surface
properties.
The pressure distribution obtained from the contact model can be used together with a
pressure dependent surface shear model to estimate the friction coefficient of the contact.
The surface displacement calculation can be extended to include solutions for a surface
shear distribution using a Boussinesq-Cerruti solution [5.9], which is an enhancement to the
Boussinesq solution allowing a horizontal load at the surface. More enhanced models related to
this would require investigation of interlocking mechanisms of voxels when the surfaces are
being loaded relative to each other in the horizontal direction.
5.3 Final Remarks
The two methods discussed in this thesis represent two main ways of using high precision
topography info for modeling of contact. The model described in Chapter 3 assumes spherical
contact geometry and does not take into account the interaction between the contact points. On
the other hand the method in Chapter 4 (Boussinesq substrate model) uses all the information
from a single AFM image, and does not consider the presence of other length scales.
A natural direction that follows is perhaps a third model that would consider the
important aspects of both of these methods. In the numerical model described in Chapter 4, the
effects of sampling size is seen when the results from images with different resolutions are
compared. Fractal parameters may be obtained from these images, and used in a hierarchic
method of multiscale contact as the one described in Chapter 3 to further refine the real contact
area.
105
Addition of plasticity and adhesion are necessary steps before directly comparing the
model results to actual experiments, such as those obtained at the micrometer level with Sandia’s
Nanotractor device [5.10]. Effects of using different materials, surface roughening and texturing,
also behavioral differences between unworn and worn surfaces are possible areas of
investigation.
References:
[5.1] R.W. Carpick and M. Salmeron, “Scratching the surface: Fundamental investigations of
tribology with atomic force microscopy,” Chem. Rev. 97 (1997) 1163-1194.
[5.2] K.L. Johnson, K. Kendall and A.D. Roberts, “Contact Mechanics,” Proc. Roy. Soc.
London A324 (1971) 301-313.
[5.3] B.V. Derjaguin, V.M. Muller and Y.P. Toporov, “Effect of Contact Deformations on the
Adhesion of Particles,” J. Colloid Interface Sci. 53 (1975) 314-326.
[5.4] M.P. De Boer, J.A. Knapp, P.J. Clews, “Effect of nanotexturing on interfacial adhesion in
MEMS,” Proc. 10th Int. Conf. on Fracture (2001), Honolulu, Hawaii.
[5.5] Y.F. Gao, A.F. Bower, "Rough surface plasticity and adhesion across length scales,"
Proceeding of the International Workshop on Nanomechanics, Asilomar (2004) CA.
[5.6] A. Majumdar, B. Bhushan, “Characterization and Modeling of Surface Roughness and
Contact Mechanics,” Handbook of Micro/Nanotribology, Ed. Bharat Bhushan (CRC
Press LLC, Boca Raton, 1995).
[5.7] D.J. Dickrell, M.T. Dugger, M.A. Hamilton, W.G. Sawyer, “Direct Contact-Area
Computation for MEMS Using Real Topographic Surface Data,” J. Microelectromech.
Syst. 16 (2007) 1263-1268.
[5.8] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in
Fortran: The Art of Scientific Computing, 2nd
Edition (Cambridge University Press, New
York, 1996).
[5.9] A.E.H. Love, “The stress produced in a semi-infinite solid by pressure on part of the
boundary,” Proc. Roy. Soc. London A228 (1929) 377-420.
[5.10] M.P. de Boer, D.L. Luck, W.R. Ashurst, R. Maboudian, A.D. Corwin, J.A. Walraven,
and J.M. Redmond, “High-performance surface-micromachined inchworm actuator,” J.
Microelectromech. Syst. 13 (2004) 63-74.
106
APPENDIX A
Contact Area and Length Scales:
Decreasing of contact area with increasing number of length scales
To further investigate the effect reported in Chapter 3, whereby the contact area decreases
with an increasing number of length scales, we use a Hertzian contact model for the following
two simple cases: (1) a single asperity interface consisting of one asperity with radius R
contacting a rigid flat surface under load L, and (2) a multi-asperity contact where n identical
asperities all at the same height and radius r ≤ R contact a rigid flat surface, again under load L.
These are illustrated in Figure A.1. In the second case, the area-load relation in equation (3.13)
becomes
3/23/2
*4
3
n
L
E
rnA
n (A.1)
Figure A.1. Two cases of contact: (a) a single asperity surface under load L, (b) multi-asperity
contact with n identical asperities with same radii under L.
(b)
(a)
107
Figure A.2 shows the area ratio An/A from combining equation (3.13) and equation (A1),
as a function of the radius ratio which is analogous to the scale factor s for the multiscale model.
The number of asperities used in the calculation was n=100. For a high scaling constant, i.e., r
<< R, the ratio for the n-asperity model is small, that is, the total contact area for 100 asperities is
much smaller than for a single asperity at the same load. However, this area increases as r
approaches R. In other words, for a large scale factor, dividing a single asperity into multiple
asperities, with the same total load, decreases the total contact area significantly.
It must be noted that if s (or, for this example, the radius ratio) becomes too small, the
assumption of asperities at one scale not affecting asperities at the larger scale breaks down.
Figure A.2. Area ratio An/A for the Hertz solution for multiple vs. a single asperity
contact, plotted as a function of radius ratio, using n=100 asperities at the same
height.
0
1
2
3
4
5
0 10 20 30 40
Ra
tio
An/A
Ratio R / r
108
APPENDIX B
Structure of the Program Described in Chapter 4
In the initial part of the main program, the label of the input file, the number of pixels
per side of the surface, the pixel size, and the maximum allowed number of contacts are read in
from the first two lines of the input file, and operational constants, array storage limits are
calculated accordingly.
The CREATEINPUT subroutine is called to continue the data input: The material
properties, i.e. Young’s Moduli and Poisson’s ratios of the lower surface, upper surface and the
contact elements are read, respectively. The surface height information of both surfaces is read
and stored in the first part of the real number array.
Surfaces are arranged for the situation when there is only one point of contact. The top
surface is flipped about the horizontal axis as shown in Fig. B.1. The pixel locations are
numbered starting from corner A, in a row-wise fashion. The initial gap at each point is
calculated and stored in the second block of the real value array. The ascending order of the gap
sizes is determined using a “merge-sort” algorithm and the corresponding numbers of the
locations are saved in the initial part of the integer array. Nodes and elements are set for the first
contact point. At this point there are six nodes, each representing a single region of the two
surfaces. There are two Boussinesq elements, two surface springs and a single penalty element.
(When there are multiple points with a zero gap value at the initial contact, the first one of these
is selected as the initial contact point. The rest are added to the stiffness at the second step,
without additional displacement.) The displacement of the lower handle node is prescribed to be
zero, while the top handle node was given a unit (1 nN) load.
109
Figure B.1. The two squares on the left represent the matrices for height information of the
lower and upper surfaces, as seen on the input data file. To setup the contact, the upper surface
is flipped around a “horizontal” axis and placed on top of the lower surface so that the corners
A, B, C, D of the lower surface come against corners G, H, E, F of the surface, respectively.
All the data associated with the problem is saved in two arrays, one for the integers and
the other for the real numbers. In the main program, the memory pointers are calculated. These
are addresses of individual blocks in the integer and real arrays. The sizes of the blocks and the
data stored in them are explained in tables B.1 and B.2. The real array is initialized with zeroes
for necessary ranges. Using the element connectivity data for each element type, the “skyline”
shape of the stiffness matrix, i.e. the column heights are obtained using the COLHT subroutine.
Element subroutines are called to construct the stiffness matrix: In the BOUSSINESQ
subroutine, a flexibility matrix is constructed using the force-displacement relations, coupling all
the d.o.f. in the Boussinesq layer. The handle node is added in a way that satisfies rigid body
displacement requirements. SPRING subroutine is called twice, first to add surface springs, then
to add the penalty springs. (The high stiffness defined for the rigid contact element is used for
both.)
110
TABLE B.1. Structure of the real array
MEMORY
POINTER
Block Size (for Np=512,
Mcon=1000)
Data stored, notes
MPSURF 2 x Np2
(524288) Lower and upper surface height data,
respectively.
MPGAPS Np2 (262144) Gaps at the current contact point
MPCORD 3xMAXNODE (12008) Node coordinates (3 dimensions for nodes in 4
layers and two handle nodes
MPFEXT MAXNODE (4002) External node entries (single entry: block can
be eliminated)
MPDISP MAXNODE (4002) nodal displacement solution for unit load at a
load step
MPDISP2 4x Np2+2 (1048578) nodal displacement at last contact
MPFORCE MAXCONT (1000) Total spring force
MPWORK MAXNODE (4002) Work array – internal dummy matrices
MPSTIF (flexible) (2121244) Stiffness Matrix. Number represents the largest
size
MPEND Marks the end of the array.
MPEND= 3981268 for 1000 contact problem
Note: MAXCONT: Maximum number of contacts,
MAXNODE: Maximum number of nodes (4xMAXCONT+2)
111
TABLE B.2. Structure of the integer array.
MEMORY
POINTER
Block Size (for Np=512,
Mcon=1000)
Data stored, notes
IPSORT Np2
(262144) Sorted addresses of the gaps pointing at
MPGAPS block in the real array, row-wise
numbering
IPKFIX MAXNODE (4002) Gaps at the current contact point
IPMAT 2 (2) Element type 1 material properties
IPMAT2 2 (2) Element type 2 material properties
IPMAT3 2 (2) Element type 3 material properties
IPIADR MAXNODE (4002) Diagonal addresses of the stiffness matrix
IPNOD1 2xMAXCONT+2 (2002) Element type 1 connectivity
IPNOD2 4xMAXCONT (4000) Element type 2 connectivity
IPNOD3 2xMAXCONT (2000) Element type 3 connectivity
IPCONT MAXCONT (1000) Contact location, counting
IPSITU 5xMAXCONT (5000) Situation 0: no contact, 1: contact, designated
for the first 5xMAXCONT gaps
IPEND Marks the end of the array.
IPEND= 284156 for 1000 contact problem
Note: MAXCONT: Maximum number of contacts,
MAXNODE: Maximum number of nodes (4xMAXCONT+2)
112
The prescribed zero displacement of the bottom handle node is introduced by eliminating
the first rows in the stiffness matrix. The linear equations are solved by calling the TRFACT
subroutine twice, first to factorize the stiffness matrix into three parts L, D, U and secondly to
solve the equations using the negative unit external load at the second handle.
The subroutine POSTP is called to post-process the displacement data. The “candidate
array” is selected starting from the smallest gaps and contact is searched within these points. It
was determined that the nth
contact happens within 2n pixels with the smallest gaps, up to 1000
points. While searching for the nth
contact, the candidate array is designated as 2.5n highest
spots. This provides a factor of safety against missing a spot. Fig. B.2 shows the order of the
found contact point according to the ascending gap size at each step.
It was found that depending on the varying surface roughness structure with increasing
contact percentages, this method might lead to missed contacts, as the required candidate array
size goes above the provided 2.5n. If the missed nodes are found later on, when they showed up
in the expected candidate array, the mistake is corrected as the force and displacement are
recalculated as they were supposed to be after that point on. However, there remains a small
error in the stiffness (slope of the F vs. d plot) up to that point, over a range determined by the
size of the displacement mistake. For a 512x512 images, the first missed contact point occurred
at 2285th
pixel (~0.87% true contact area), and only 10 pixels were added out of order up to the
3200th
point. These points can be found slightly above the line which designates candidate array
size, shown in Fig. B.2. The program searches for other missed contacts when the aimed number
of contacts is reached, by going over a much larger candidate array, and this would add 10 more
pixels that were not accounted for. A similar calculation for the 64x64 resolution of the same
113
surface starts missing contacts at 212nd point (~5.2% true contact area). The difference shows
the effect of the roughness structure and image resolution for this routine.
The contact forces are obtained using the displacement data associated with the unit load.
Then using the Boussinesq surface displacement formula, the displacements at the “candidate
nodes” are found. For each candidate node, the necessary multiplier for the load is calculated to
“close the gap,” and the smallest one is selected as the step load to reach next contact. If there are
multiple nodes with the smallest value, all of them are added at once. With the step load,
displacements at all the candidate nodes are calculated and used to modify the gap array.
Figure B.2. “How ‘out of order’ are the contact locations?” The figure shows the order of the
found contact point according to the ascending gap size at each step. The dark line shows how
the candidate array was set up. For example at step 1117, the pixel found to come into contact
had the 2249th
smallest gap, and at step 1787, the pixel with the 241st smallest gap came into
contact. (The two examples are marked with stars.)
114
The nodes and elements are recreated for the increased area of contact. The program is
redirected back to the COLHT to determine the shape of the stiffness matrix, and the element
subroutines are called to construct and solve the stiffness matrix again. This loop continues until
the maximum number of contacts is met, after which the candidate array assumption is verified
by calculating final gaps at a larger group of pixels.
Techniques Used to Enhance the Program Capacity and Speed
Several methods were implemented to keep the program time to a minimum. Some of
these methods are: generating nodes only at contact points, regenerating the node list at each step
instead of initially assigning nodes to all possible positions, numbering the nodes element by
element to reduce the populated part of the stiffness matrix, using the same substrate flexibility
matrix for both surfaces, as both surfaces have coincident nodes at contact points, using triple
factorization for inversion to use the symmetry of the stiffness matrices.
The search for the next contact is a time consuming step in the program. Instead of sifting
through all pixels, limiting the search to a range of pixels which contributes to the smallest
values of gaps decreases the required time significantly. One way to accomplish this is to start
with a static array of candidates; another is to start with a smaller array and extend it at each step.
Instead of setting a static candidate array from the beginning, a scheme was created to increase
the number of candidates according to the number of contacts. In this scheme, the candidate
array starts with 100 smallest gaps. After the 40th
contact is completed, the number of candidates
is increased proportional to the number of existing contacts. The list of gaps was obtained using
a “merge-sort” algorithm (sorting of 512x512=262144 gaps with this method takes a few
115
seconds; a simpler “bubble-sort” method results in times that are three orders of magnitude
longer.)
The initial calculations were conducted on a 1.75GHz CPU speed, 1GB memory
computer (with an additional virtual memory of 1GB). When a 3.35 GHZ computer was used,
the time required decreased by 30%.