ANALYTICAL AND NUMERICAL MODELING OF CONTACT USING …

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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deformation and stresses in an elastic-plastic layered medium subjected to indentation

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repeated indentation of a half-space by a rigid sphere,” Journal of Applied Mechanics 60

(1993) 829-841.

[2.9] L. Kogut, and I. Etsion, “Elastic-plastic Contact Analysis of a Sphere and a rigid flat,”

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[2.11] R.L. Jackson, I. Green, “A finite element study of elasto-plastic hemispherical contact

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25

[2.12] R.L. Jackson, I. Chusoipin, I. Green, “A finite element study of the residual stress and

deformation in hemispherical contacts,” Journal of Tribology 127 (2005) 484-493.

[2.13] B. Luan and M. O. Robbins, “The breakdown of continuum models for mechanical

contacts,” Nature 435 (2005) 929–932.

[2.14] I. Szlufarska, M. Chandross, and R. W. Carpick, “Recent advances in single-asperity

nanotribology,” Journal of Physics D: Applied Physics 41 (2008) 123001.

[2.15] Y. Mo, K. T. Turner, and I. Szlufarska, “Friction laws at the nanoscale,” Nature 457

(2009) 1116–1119.

[2.16] Y. Mo and I. Szlufarska, “Roughness picture of friction in dry nanoscale contacts,” Phys.

Rev. B 81 (2010) 035405.

[2.17] 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.

[2.18] K.L. Johnson, K. Kendall and A.D. Roberts, “Contact Mechanics,” Proc. Roy. Soc.

London A324 (1971) 301-313.

[2.19] 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.

[2.20] Maugis, D., “Adhesion of Spheres: The JKR-DMT Transition Using a Dugdale Model,”

J. Colloid Interface Sci. 150 (1992) 243-269.

[2.21] 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, 2001).

[2.22] J.A. Greenwood and J.B.P. Williamson, “Contact of Nominally Flat Surfaces,” Proc.

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.

[2.24] P.R. Nayak, “Random process model of rough surfaces,” Journal of Lubrication

Technology 93 (1971) 398–407.

[2.25] M.S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Phil.

Trans. R. Soc. Lond. A249 (1957) 321–387.

[2.26] M.S. Longuet-Higgins, “Statistical properties of an isotropic random surface,” Phil.

Trans. R. Soc. Lond. A250 (1957) 157–174.

[2.27] A.W. Bush, R.D. Gibson and G.P. Keogh, “Strongly anisotropic rough surfaces,” Journal

of Lubrication Technology 101 (1979) 15–20.

26


(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

surfaces: Comparison of theory with numerical results” Physical Review B 65 (2002)

184106.

[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.


A243 (1957) 190-205.


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).

[2.42] M.V. Berry and Z.V. Lewis, “On the Weierstrass-Mandelbrot fractal function,” Proc.

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.

[2.51] W.W. Tworzydlo, W. Cecot, J.T. Oden, and C.H. Yew, “Computational micro- and

macroscopic models of contact and friction: formulation, approach and applications,”

Wear 220 (1998) 113-40.

[2.52] J. Boussinesq, “Application des potentiels à l'étude de l'équilibre et du mouvement des


elastic solids),” (Gauthier Villars, Paris, 1885).


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.

[2.60] L. Pei, S. Hyun, J.F. Molinari, and M.O. Robbins, “Finite element modeling of elasto-

plastic contact between rough surfaces,” J. of the Mechanics and Physics of Solids 53

(2005) 2385-2409.

[2.61] J.F. Molinari, M. Ortiz, R. Radovitzky, and E.A. Repetto, “Finite-element modeling of

dry sliding wear in metals,” Engineering Computations, 18 (2001) 592-610.

[2.62] U. Sellgren and U. Olofsson, “Application of a Constitutive Model for Micro-Slip in

Finite Element Analysis,” Computer Methods in Applied Mechanics and Engineering 170

(1999) 65-77.

[2.63] U. Sellgren, S. Bjorklund, S. Andersson, “A finite element-based model of normal

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


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


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.





(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.


A243 (1957) 190-205.


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.


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.


London A324 (1971) 301-313.



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.


(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.


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


elastic solids),” (Gauthier Villars, Paris, 1885).



[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.




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.


London A324 (1971) 301-313.



[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).



[5.10] 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%.

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