Tribological and Mechanical Characterization of

Thin Polymer Films

A Dissertation Presented

by

Qian Sheng

to

The Department of Mechanical and Industrial Engineering

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in the field of

Mechanical Engineering

Northeastern University

Boston, Massachusetts

July 2013

i

Abstract

Mechanical and tribological characteristics of polymer films are crucial to their

successful implementation as thin protective coatings. Hot filament chemical vapor

deposition (HFCVD) is a relatively new technique which enables deposition of various

polymers on a variety of surfaces. In this dissertation a toolbox of experimental

techniques was developed to assess the quality of thin polymer films. These include

assessment of friction, durability and interfacial adhesion. While this work is primarily

focused on polytetrafluoroethylene (PTFE), assessments of perfluoroalkoxy (PFA) and

poly(trivinyltrimethylcyclotrisiloxane) (Poly(V3D3)) were also carried out.

Frictional and durability characteristics of thin PTFE films deposited on aluminum

substrates were investigated by using a ball-on-disk and ball-on-plate configurations,

respectively. PFA and Poly(V3D3) were deposited on glass and were likewise tested. The

effects of normal force, sliding speed and surface roughness on the coefficient of friction

(COF) and durability were quantitatively examined by the analysis of their variance

(ANOVA). The results show that native surface roughness of the substrate has the most

significant effects on the COF and durability; and, that the smooth interface is dominated

by adhesion. Experiments indicated that PTFE-thin film durability can be optimized if it

is deposited on a mid-range, (Ra ~ 0.5 μm) surface roughness.

In order to assess the interfacial adhesion properties of thin PTFE films, a micro-

indentation based technique was developed. Experimentally, the technique involved

ii

monitoring the peel-off radius of the PTFE as subjected to different levels of loads by a

conical micro-indenter. The mechanics of PTFE indentation was simulated by the finite

element method (FEM) to obtain quantitative measurements of interfacial fracture

toughness out of the experimental measurements. It was determined that a material

penetration model was necessary to model the indenter-PTFE interaction. A finite

element model utilizing continuum damage mechanics (CDM) was implemented in

ABAQUS/Explicit to simulate the penetration of the bulk PTFE. Effects of different

damage/failure parameters were investigated systematically and compared to

experimental results to determine the adequate bulk fracture energy and damage initiation

strain levels. The bonding in the PTFE-glass interface was modeled by a cohesive zone

mechanics (CZM) model. A normalized relationship between the delamination radius, the

indentation load, coating thickness, and the material properties of the coating was

developed based on the finite element results. Fracture toughness of the PTFE-glass

interface was determined quantitatively. Values reported in this work include a relatively

more comprehensive treatment of the energy spent in penetrating the thin film, and thus

are somewhat lower than those reported in the literature. Both the experimental and the

theoretical aspects of the work also show that it is more difficult to delaminate thinner

coatings.

The results of this work will be useful in assessing the effectiveness of different

deposition techniques, in choosing optimal coating thickness, and in preparing optimal

surface roughness for improved durability.

iii

Curriculum Vitae

The author was born on July 5, 1982 in Nanjing, Jiangsu Province, in the People’s

Republic of China. He studied at Nanjing University of Aeronautics and Astronautics, in

Nanjing, China and received BS and MS degrees from the Department of Mechanical and

Electrical Engineering in July 2004 and April 2007, respectively. He received

scholarships during 2000-2006 and was awarded the honor of excellent graduate student

in 2007. After graduating in 2007, he worked as a mechanical engineer in the Powertrain

Business Unit of Comau, which focuses on production systems in FIAT Corporation, in

Shanghai, China.

He started his Ph.D. studies in the Department of Mechanical and Industrial Engineering

of Northeastern University (Boston, MA, the United States), in the fall of 2008. He

worked as a teaching and research assistant at the Applied (Bio) Mechanics and

Tribology Laboratory and instructed the labs of mechanics of materials and finite element

method for five years. His research topic involved experimental investigation and

numerical modeling for the tribology and mechanics of thin polymer films. His

dissertation advisor was Professor Sinan Müftü.

He is a student member of the American Society of Mechanical Engineers (ASME) and

the Adhesion Society.

iv

Acknowledgements

I thank Professor Sinan Müftü, my dissertation advisor, for his help and guidance on my

research work throughout the last five years. I thank Professor Kai-Tak Wan for fruitful

discussions related to this work and his permission to perform nano-indentation tests in

his laboratory at Northeastern University. I thank Professor Andrew Gouldstone for his

helpful suggestions related to this work and his great help during my teaching assistant

work. I also acknowledge the support from the Department of Mechanical and Industrial

Engineering for providing me the teaching assistantship.

I am grateful for the support and contributions of Aleksandr J. White at GVD

Corporation (Cambridge, MA) to the experimental work of this dissertation. I thank

Michael A. Karnath for starting the initial phase of this work and the journal publication

on Tribology Transactions. I also acknowledge the help from Dr. Guangxu Li, Dr. Jiayi

Shi, and Mr. Michael Robitaille for showing me the use of nano-indenter and helpful

discussions.

I thank all of my colleagues in the Applied (Bio) Mechanics and Tribology Laboratory

for helpful suggestions on my research, and for the friendship throughout my doctoral

studies.

I acknowledge the help and guidance of my teachers at different stages of my education

in China.

v

Finally, I thank my parents for their love, support and education. I remember my

grandparents with love for their perseverance and integrity. I thank my wife, Qingkun Liu,

for her love, support and patience.

vi

Table of Contents

Abstract ................................................................................................................................ i

Curriculum Vitae ............................................................................................................... iii

Acknowledgements ............................................................................................................ iv

Table of Contents ............................................................................................................... vi

List of Figures ..................................................................................................................... x

List of Tables .................................................................................................................. xvii

Nomenclature .................................................................................................................... xx

1 Introduction ............................................................................................................... 1

2 Mechanics Review for Thin-Film Polymer Tribology .............................................. 4

2.1 Indentation of an Elastic Half-Space .................................................................... 4

2.1.1 Point Load on an Elastic Half-Space .......................................................... 4

2.1.2 Hertzian Contact ......................................................................................... 7

2.2 Indentation of a Layered Elastic Half-Space........................................................ 9

2.3 Indentation on an Elastic-Plastic Layered Half-Space ....................................... 14

2.4 Indentation on Soft Thin-Film Material ............................................................. 15

2.5 Material Properties Investigation by Indentation ............................................... 20

2.6 Indentation-Induced Interfacial Delamination ................................................... 25

2.6.1 Definitions of Adhesion Energy and Interfacial Fracture Toughness ...... 25

2.6.2 Mathematical Descriptions of Interfacial Delamination ........................... 27

vii

2.6.3 A Relationship between Non-dimensional Delamination Radius and

Indentation Load ..................................................................................................... 37

2.6.4 Film Thickness Effect on Interfacial Delamination .................................. 39

2.7 Bulk Material Damage/Failure Criteria in Finite Element Analysis .................. 41

2.7.1 Introduction of Bulk Material Damage/Failure ........................................ 41

2.7.2 Implementation of Bulk Material Damage/Failure Criteria in Finite

Element Analysis ..................................................................................................... 42

2.8 Cohesive Zone Model in Finite Element Analysis ............................................ 45

2.8.1 Description of Cohesive Zone Model ....................................................... 45

2.8.2 Implementation of Cohesive Zone Model in ABAQUS ........................... 48

3 Review of PTFE Material Properties ...................................................................... 55

3.1 Molecular Structure of PTFE ............................................................................. 55

3.2 Mechanical Properties of PTFE ......................................................................... 59

3.2.1 Young’s Modulus and Yield Stress .......................................................... 59

3.2.2 Viscoelastic and Plastic Properties of PTFE ............................................. 63

3.2.3 Frictional Characteristics .......................................................................... 66

3.2.4 Wear Characteristics ................................................................................. 76

3.3 Summary ............................................................................................................ 78

4 Friction and Durability of Thin PTFE Films on Rough Aluminum Substrates ...... 80

4.1 Introduction ........................................................................................................ 80

4.2 Materials and Methods ....................................................................................... 81

4.3 Results ................................................................................................................ 83

viii

4.3.1 COF ........................................................................................................... 86

4.3.2 Durability .................................................................................................. 92

4.4 Discussion .......................................................................................................... 96

4.5 Summary and Conclusion ................................................................................ 103

5 Frictional Characteristics of Thin PFA and Silicone Films on Glass Substrates .. 105

5.1 Introduction ...................................................................................................... 105

5.2 Materials and Methods ..................................................................................... 108

5.3 Results .............................................................................................................. 108

5.3.1 Friction Characteristics of Thin PFA Films ............................................ 108

5.3.2 Friction Characteristics of Thin Silicone Films ...................................... 112

5.4 Discussion ........................................................................................................ 115

5.5 Summary and Conclusion ................................................................................ 117

6 Simulation of Material Damage during Indentation of a Soft Polymer ................ 119

6.1 Axi-Symmetric Finite Element Analysis of Thin-Film Indentation ................ 119

6.1.1 Materials and Methods ............................................................................ 119

6.1.2 Results and Discussions .......................................................................... 122

6.2 3D Finite Element Model of Indentation of PTFE Thin-Film by Using a

Material Damage Approach ....................................................................................... 125

6.2.1 Materials and Methods ............................................................................ 125

6.2.2 Results and Discussions .......................................................................... 129

6.3 Summary and Conclusions ............................................................................... 135

7 Interfacial Delamination of PTFE Thin Films ...................................................... 137

ix

7.1 Introduction ...................................................................................................... 137

7.2 Materials and Methods ..................................................................................... 138

7.3 Results and Discussion ..................................................................................... 146

7.4 Summary and Conclusion ................................................................................ 155

8 Summary, Conclusions and Future Work ............................................................. 157

Bibliography ................................................................................................................... 162

Appendix 1 ABAQUS/Explicit Verification by Modeling 2D Interface of Indentation 170

Appendix 2 Material Properties of Fused Silica Using Nano-Indentation ..................... 172

Appendix 3 Mechanical Properties of Thin Polymer Films Using Indentation.............. 175

Appendix 4 Material Properties of Glass Substrates ...................................................... 179

Appendix 5 Green’s Function ......................................................................................... 180

x

List of Figures

Figure 2.1. Concentrated point load on a 3D elastic half space. ........................................ 6

Figure 2.2. Hertzian contact on an elastic half-space (a << R). Note that a is the contact

radius at the interface of spherical indenter and flat surface, δH is the indentation

displacement, R is the indenter radius, p0 is the maximum contact pressure. ..................... 9

Figure 2.3. Spherical indenter on a layered elastic half-space. Note that 2b is the contact

diameter at the interface of indenter and thin film; t is film thickness; ur(r, z) and uz(r, z)

are the horizontal and vertical displacement, respectively. .............................................. 11

Figure 2.4. Stresses on an element of thin-film coating................................................... 11

Figure 2.5. Schematic of indentation on PTFE thin-film coating. ................................... 15

Figure 2.6. Schematic of delamination for three types of indentation cases (adapted from

Ritter et al. [13]). Note that type I indentation is elastic deformation under the indenter;

type II is plastic deformation under the indenter; type III is penetration of coating by

indenter. ............................................................................................................................ 19

Figure 2.7. A schematic representation of load, W, as a function of displacement, δ, for

an indentation test [16]. Note that Wmax is the peak load; δmax is the displacement

corresponded at the peak load; δf is the final depth of the contact impression after

unloading; S is the initial unloading stiffness. .................................................................. 22

Figure 2.8. A section view of an indentation showing parameters in the analysis

(Adapted from [16]). Note that δc is the vertical distance along which contact is made; δs

xi

is the displacement between the initial surface and the surface at the perimeter of the

contact; δf is the final depth of the residual hardness impression. .................................... 23

Figure 2.9. Schematic of driving force/resistance curves (Adapted from Anderson [30]).

........................................................................................................................................... 27

Figure 2.10. The schematic representation of interfacial delamination by conical indenter.

Note that 2b is the contact diameter at the interface of indenter and coating; 2c is the

delamination diameter at the interface of coating and substrate; δ is indentation

displacement; Ψ is half conical angle; W is indentation load; t is film thickness. ............ 28

Figure 2.11. Schematic representation of a delaminated, residually stressed film (adapted

from Marshall and Evans [31]). Note that 2c is delamination diameter, σR is the residual

stress, t is film thickness, ΔR is film expansion radius. ..................................................... 29

Figure 2.12. Schematic representation of a stress-free film with indentation-induced

delamination (adapted from Marshall and Evans [31]). Note that σ0 is the indentation

stress, Δ0 is film expansion radius related to indentation volume, V0. .............................. 33

Figure 2.13. Schematic of annular-plate model for delamination (adapted from Rosenfeld

et al. [32]). ......................................................................................................................... 35

Figure 2.14. Material damage initiation and failure [39]. Note that (a) typical ductile

material; (b) elastic and perfectly plastic material. ........................................................... 42

Figure 2.15. Examples of the traction-separation relationship for the cohesive zone

model, for (a) constant traction; (b) trapezoidal traction; (c) bilinear traction. ................ 47

Figure 2.16. Schematic representation of the bilinear traction-separation law

implemented in ABAQUS. ............................................................................................... 51

xii

Figure 3.1. Schematic representation of PTFE and PE molecular structures [54]. Note

that (a) The zigzag backbone of a PTFE molecular segment; (b) A PTFE molecular chain;

(c) A PE molecular chain. ................................................................................................. 56

Figure 3.2. Crystalline structure of bulk PTFE (Makinson and Tabor [55]). Note that (a)

crystalline block or ‘band’; (b) crystalline slices or ‘striae’ after sliding; (c) hexagonal

array of chains within the slices. ....................................................................................... 59

Figure 3.3. Comparison between experimental data and predicted behavior in uniaxial

tension at different strain-rate (T = 20° C, strain-rates, : 1.2×10-3

/s and 2.3×10-4

/s) [69].

........................................................................................................................................... 65

Figure 3.4. Comparison between predicted and experimental stress relaxation results [69].

........................................................................................................................................... 66

Figure 3.5. Schematic of interfacial and bulk regimes of friction [71]. Note that W is the

normal force, Ffric is the friction force, t1 is the thickness of bulk region, t2 is the thickness

of interfacial region. .......................................................................................................... 68

Figure 3.6. The bulk deformations due to plastic flow and viscoelastic losses [71]. ...... 68

Figure 3.7. The rolling friction of a rigid sphere on bulk PTFE and the quantity, E-1/3

tan(δ), as a function of temperature [71]. .......................................................................... 69

Figure 3.8. Relations between the cohesive energy density (CED) of the polymers and

friction coefficient, wear rate for similar polymer-polymer combinations [77]. .............. 74

Figure 3.9. Frictional and wear characteristics of PTFE with respect to cohesive energy

density (CED). (a) Relations between friction coefficient and the difference in CED for

xiii

dissimilar polymer-PTFE combinations; (b) Relations between the wear rate of polymer

pin and the difference in CED for dissimilar polymer-PTFE combinations [77]. ............ 75

Figure 4.1. Configurations of PTFE frictional and durability tests. ................................ 82

Figure 4.2. Surface roughness, Ra as a function of grit size, g and mean particle diameter,

dp, respectively. ................................................................................................................. 85

Figure 4.3. The COF of aluminum substrate without PTFE thin films. .......................... 86

Figure 4.4. The COF of PTFE on aluminum substrates with different roughness. (a) Ra =

1.28 μm and 2.34 μm; (b) Ra = 0.01 μm and 0.57 μm. ....................................................... 89

Figure 4.5. Durability characteristics of 1 μm PTFE on roughened and polished

aluminum substrates. (a) sliding distance to failure; (b) The COF. .................................. 94

Figure 4.6. The COF comparisons between the prediction by Equation (4.6) and

experiment measurement. ................................................................................................. 99

Figure 4.7. Worn surfaces of PTFE thin films on aluminum substrate with different

roughness Ra (normal force is 5 N, sliding speed is 0.42 mm/s). .................................... 101

Figure 4.8. COF histories on aluminum substrates with different surface roughness. Note

that (a) - (d) were tested on 1 µm PTFE films deposited on aluminum substrate; (e) was

tested on aluminum substrate without PTFE films. ........................................................ 102

Figure 5.1. Schematic representation of PTFE and PFA molecule formulae. ............... 107

Figure 5.2. Schematic representation of poly(V3D3) molecular structure. Note that the

hexagonal units show the intact siloxane rings, acting as cross-linking moieties for

backbone chains [4]. ....................................................................................................... 107

Figure 5.3. The COF as a function of PFA film thickness............................................. 111

xiv

Figure 5.4. The COF as a function of silicone film thickness. ...................................... 114

Figure 5.5. Microscopic observations of 1 μm PFA worn surfaces (sliding speed: 1 mm/s).

......................................................................................................................................... 116

Figure 5.6. Microscopic observations of 1 μm silicone worn surfaces (sliding speed: 1

mm/s). .............................................................................................................................. 117

Figure 6.1. The geometry and mesh configuration of 2D axi-symmetric finite element

model............................................................................................................................... 121

Figure 6.2. Convergence studies of element number in thickness direction of PTFE films

(Hp = 30 MPa). ................................................................................................................ 122

Figure 6.3. Comparison of the experimental and the calculated data to FEA results. ... 124

Figure 6.4. von-Mises stress distribution of indentation at different normal force of (a)

1.5 N and (b) 15 N calculated by finite element analysis................................................ 124

Figure 6.5. The geometry and mesh configuration of 3D finite element model. ........... 128

Figure 6.6. Comparison of the measured and calculated contact width, 2b, to FEA

prediction for 10 μm PTFE films using shear damage model. ........................................ 130

Figure 6.7. Comparison of the measured and calculated contact width, 2b, to FEA result

for 10 μm PTFE films by ductile damage model. ........................................................... 131

Figure 6.8. Effects of equivalent plastic strain, 0pl , (0.1, 0.5, 5% are not shown on the

graph for clarity of illustration) on the contact width, 2b, for Γf = 20 J/m2. ................... 132

Figure 6.9. Effects of bulk fracture toughness, Γf, (40, 500 J/m2 are not shown on the

graph for clarity of illustration) on the contact width, 2b, for0

pl = 1%.......................... 133

xv

Figure 6.10. von-Mises stress distribution of PTFE thin films at steady state

configurations. Note that a is the contact radius at the interface of ball indenter and

substrate; b is the contact radius at the interface of ball indenter and PTFE films. ........ 134

Figure 6.11. von-Mises stress distribution of the glass substrate at steady state

configurations. ................................................................................................................ 135

Figure 7.1. The geometry and mesh configuration of 3D finite element model for the

delamination simulation. ................................................................................................. 142

Figure 7.2. Convergence studies for 1, 5 and 10 μm PTFE delamination simulations (W =

1 N, Γi = 100 mJ/m2, Young’s modulus E = 3 GPa, yield stress σY = 35 MPa for PTFE

material properties). ........................................................................................................ 144

Figure 7.3. 10 μm PTFE delamination contours predicted by finite element simulation

(Young’s modulus, E = 3 GPa, hardness, σY = 35 MPa). ............................................... 145

Figure 7.4. The coordinates of nodes with interfacial delamination. ............................. 146

Figure 7.5. Thickness effects on the interfacial delamination of PTFE thin films (normal

force: 0.5 N). Note that 2c represents the delamination diameter at the interface. ......... 148

Figure 7.6. Load effects on the interfacial delamination of PTFE thin films (film

thickness: 3 μm). ............................................................................................................. 149

Figure 7.7. The predictions of delamination diameter, 2c, as a function of film thickness

and material properties. ................................................................................................... 152

Figure 7.8. The curve fitting of non-dimensional delamination radius, C , and indentation

force, F , from finite element simulation. ...................................................................... 154

xvi

Figure 7.9. The comparison of 10 μm PTFE finite element simulation to Rosenfeld et al’s

formulations with respect to different material properties. ............................................. 155

xvii

List of Tables

Table 2.1. Geometric constant, ε, for different indenters [17, 22]. .................................. 24

Table 3.1. Characteristics of the helical conformation of PTFE [51]. ............................. 58

Table 3.2. Tensile properties of bulk and free-standing films at room temperature [58].

Note that free-standing film is tested by different authors. .............................................. 62

Table 3.3. Mechanical properties of bulk PTFE in the tension test [62] (strain rate, =

5×10-3

s-1

). ......................................................................................................................... 62

Table 3.4. Mechanical properties of bulk PTFE in the compression test (strain rate, =

10-3

s-1

). Note that failure strain and stress were not reported in this compression test [63].

........................................................................................................................................... 63

Table 4.1. Surface roughness parameter of Ra and σ2 for aluminum substrates. Note that

the mean particle diameter dp for each grit size is adapted from Orvis et al. [91]. ........... 84

Table 4.2. Experimental data for frictional characteristics of 1 μm PTFE coating

deposited on glass substrates. ........................................................................................... 90

Table 4.3. ANOVA test for frictional tests of 1 μm PTFE coating on aluminum substrates

(The response is COF). ..................................................................................................... 91

Table 4.4. Two-way ANOVA test for PTFE coatings on Ra = 2.34 μm aluminum

substrates. .......................................................................................................................... 91

Table 4.5. Two-way ANOVA test for PTFE coating on Ra = 1.28 μm aluminum

substrates. .......................................................................................................................... 91

xviii

Table 4.6. Two-way ANOVA test for PTFE coatings on Ra = 0.57 μm aluminum

substrates. .......................................................................................................................... 91

Table 4.7. Two-way ANOVA test for PTFE coatings on Ra = 0.01 μm aluminum

substrates. .......................................................................................................................... 92

Table 4.8. Experimental data for durability characteristics of 1 μm PTFE coating

deposited on aluminum substrates. Note that ave. represents average value and std.

represents standard deviations. ......................................................................................... 95

Table 4.9. ANOVA test for durability tests of 1 μm PTFE coating on aluminum

substrates (The response is sliding distance to failure). .................................................... 95

Table 4.10. ANOVA test for durability tests of 1 μm PTFE coating on aluminum

substrates (The response is COF). .................................................................................... 96

Table 4.11. The determination of parameters in Equation (4.6) by using curve-fitting. .. 98

Table 5.1. Three-way ANOVA for the COF of PFA thin films.................................... 111

Table 5.2. Two-way ANOVA for the COF of 0.3 μm PFA films. ................................. 111

Table 5.3. Two-way ANOVA for the COF of 1 μm PFA films. .................................... 112

Table 5.4. Two-way ANOVA for the COF of 5 μm PFA films. .................................... 112

Table 5.5. Three-way ANOVA analysis for the COF of silicone films. ........................ 115

Table 5.6. Two-way ANOVA analysis for the COF of 0.3 μm silicone films. .............. 115

Table 5.7. Two-way ANOVA analysis for the COF of 1 μm silicone films. ................. 115

Table 6.1. Material properties in 2D axisymmetric finite element analysis [3, 62]. ...... 120

xix

Table 6.2. Material properties used in 3D finite element simulation using material

damage model. Note that 0

pl is the equivalent plastic strain for damage initiation, Γf is the

fracture energy for damage evolution. ............................................................................ 126

Table 7.1. Material properties in 3D FEA delamination model. Note: Diamond and glass

properties were taken from Oliver and Pharr [16]; The PTFE properties were measured

by using nano-indentation, except for Poisson’s ratio, ν, reported in Karnath et al. [3]. 141

Table 7.2. Mesh sizes and aspect ratios for modeling all thick PTFE films. ................. 141

Table 7.3. The number and type of elements implemented in delamination model. ..... 144

Table 7.4. Experimentally measured delamination diameter, 2c, for different film

thickness, t, and normal force, W. Note that Ave. 2c is the average delamination diameter;

Std. Dev. 2c is its standard deviation. ............................................................................. 150

xx

Nomenclature

Chapter 2

W Normal force

Wmax Peak normal force

σrr, σθθ, σzz Normal stress components

rr , , zz Averaged normal stresses in the r, θ and z- direction

r , , z Averaged strains in the r, θ and z-direction

ρ Distance in cylindrical coordinates

γrz Shear strain

τrz, τrθ, τθz Shear stress components

Fr, Fz Body forces in the r- and z-direction

ur(r, z), uz(r, z) Displacements in r-, z-direction

r, θ, z Cylindrical coordinate axes

G Shear modulus

E* The composite Young’s modulus

E, Ei Young’s modulus of the materials in contact

ν, νi Poisson’s ratio of the materials in contact

Er Reduced Young’s modulus

a, 2a Contact radius, diameter between the indenter and substrate

b, 2b Contact radius, diameter between the indenter and thin-film layer

R, Ri Radius of the contact bodies

δH Approach distance in the substrate

δ Total indentation approach

p0 Maximum contact pressure

τi (i = 1, 2) Interfacial shear stress

t Film thickness

xxi

f(r) A function describing the indenter profile

Hp Hardness of PTFE

H Hardness of materials

Hc, Hs Hardness of coating, substrate

FH Hertzian contact force

σY Yield stress

k, λ, ω, ς Coefficients in Ritter’s equations

K1, I1, K1’, I1’ Modified Bessel functions of the second kind and their derivatives

a0, b0 Half the indenter diagonal in the substrate and coating, respectively

S Initial unloading stiffness

β A correction parameter for indenter geometry

Ap Projected area of contact

δmax Maximum indentation displacement at peak load

δf Final depth of contact impression

δs Deflection of the surface

δc Contact depth

ε Geometric constant for indenter

F(δc) Area/shape function with different indenter geometry

Ci (i =1, …, 8) Constants in the shape/area function

Wa Adhesion energy

Γi Interfacial fracture toughness, fracture energy release rate

Γc Experimental measurement of interfacial fracture toughness

γf, γs, γfs Surface energies of film, substrate and interface

Γp Plastic dissipation energy

Γfric Energy loss due to friction

ΓR Fracture resistance

U, UR, UR’ Total strain energy of the system

Af Crack area

σ0, σb Stress induced by indentation

xxii

σR Residual stress

σc Critical stress of buckling

Uc, Us Energy stored in the film and remaining system, respectively

UB Energy difference between buckled and unbuckled plates

c, 2c Delamination radius

Ψ Half conical angle

ΔR, Δ0 Expansion radius of the plate

B A constant in the derivation of interfacial fracture toughness

α The slope of buckling load versus displacement

V0 Indentation volume

κ Constant related to material properties

ξ Constant related to indenter geometry

W , C Non-dimensional force and radius

res Non-dimensional residual stress

b Bergers vector

tc Critical film thickness for delamination

k a constant used in the plate theory

D Damage variable

0

pl Equivalent plastic strain for damage initiation

pl

f Equivalent plastic strain for material failure

0

pl Equivalent plastic strain rate

εpl

Equivalent plastic strain variable

pl

fu Equivalent plastic displacement for material failure

upl

Equivalent plastic displacement variable

Γf Bulk fracture energy

L Characteristic length of finite element mesh

η Stress state parameter

xxiii

σi (i = 1, 2, 3) Principal stress

σeqv von-Mises stress

ξs Shear stress ratio

τmax Maximum shear stress

ks Material parameter

p Pressure stress

K0 Initial elastic stiffness

σc0, τc0 Maximum cohesive strengths in normal and shear directions

σm Magnitude of effective traction vector

σm0 Critical traction magnitude for damage initiation

δ0, δc Critical separation for damage initiation and failure, respectively

δmax Maximum separation

δm Magnitude of effective displacement

δn, δs Displacements in normal and shear directions

δmc, δm0 Critical effective displacement for failure and damage initiation

ΓI, ΓII Interfacial fracture toughness of mode-I and –II, respectively

WI, WII Work done mode-I and –II tractions, respectively

φ Phase angle

Γm Interfacial fracture toughness in mixed-mode

α0 Power coefficient in energy-based damage evolution criterion

Chapter 3, 4 and 5

Strain rate

T Temperature

μ The coefficient of friction

μr The rolling coefficient of friction

t1 Thickness of bulk region

t2 Thickness of interfacial region

xxiv

W Normal force

Ffric Friction force

R Indenter radius

Ar Real contact area

p Flow stress

τ Interfacial Shear strength

τ0 A constant of adhesion friction

α Pressure coefficient

tan(δ) Loss tangent of material

Tg Glass-transition temperature

E, E*, Ei Young’s modulus

ν, νi Poisson’s ratio

Ra Average surface roughness

σ2 Variance of surface roughness

g Grit size

dp Mean particle diameter

v sliding speed

c1, c2, c3, c4 Constants for speed effects on COF

p-value null-hypothesis parameter of ANOVA

Chapter 6, 7 and 8

tr Residual thickness of film

a, 2a Contact radius, diameter at the interface of ball indenter and

substrate

b, 2b Contact radius, diameter at the interface of ball indenter and PTFE

coating

c, 2c Delamination radius, diameter at the interface

xxv

E, ν Young’s modulus and Poisson’s ratio

σY Yield stress of PTFE coating

Hp, H Film hardness

W Normal force

Γf Bulk fracture energy

Γi Evaluation of interfacial fracture toughness

Γc Interfacial fracture toughness determined by experiment

K0 Initial elastic stiffness

t Film thickness

α Coefficient of linear interpolation

0

pl Equivalent plastic strain for damage initiation

L0, L1, L2, h Dimensions of finite element mesh

NL0, NL1, Nh, Nt Number of elements in finite element mesh

r Average radius of delamination in simulation

ri Delamination radius in simulation

s Standard deviation of delamination radius

F , C Non-dimensional force and delamination radius, respectively

p, m, n, q Coefficients for non-dimensional force and radius

Appendices

R Indenter radius

E, ν Young’s modulus, Poisson’s ratio

G Shear modulus

H Hardness of materials

b Contact radius of ball indenter and substrate

τ0 Interfacial shear stress

S Initial unloading stiffness

δs Deflection of surface

xxvi

δc Contact depth

ε Correction factor for indenter geometry

δmax Maximum indentation displacement

ΔW, Δδ Variations of indentation load and displacement, respectively

Ap Projected contact area

g(x, s) Green’s function

v(x), p(x) Displacement and distributed force as a function of coordinates

1

1 Introduction

Polymers are a large class of naturally occurring or artificially synthesized, covalently

bonded materials ranging from natural biopolymers, such as DNA, RNA and proteins, to

plastics such as synthetic rubber and silicone [1]. Recently, as more functional polymeric

materials are widely applied in many areas, efforts have been made to investigate

material properties and characteristics of polymers by using theoretical and experimental

methodologies.

Gleason’s group in MIT developed two thin-film polymer coating technologies, including

PTFE (Polytetrafluoroethylene) by using Hot Filament-Chemical Vapor Deposition

(HFCVD) and Poly(V3D3) (poly(trivinyltrimethylcyclotri-siloxane)) by using Initiated

Chemical Vapor Deposition (iCVD) [2, 3]. These polymers have shown extraordinary

characteristics in terms of material properties. In particular, low coefficient of friction

(COF), chemical inertness, and low dielectric constant of PTFE allow its application in

micromechanical devices and integrated circuits [3]. Poly (V3D3) can be applied to

encapsulate implantable microelectronics and be used to protect circuit boards from

environmental effects due to its biocompatibility as well as dielectric and insulating

properties [4, 5].

However, tribological and mechanical properties of these two thin-film coatings are not

clearly understood, which would otherwise significantly influence their performance in

practice. This motivates this dissertation. The general goal of this work is to study the

2

mechanism of friction and wear of polymer coatings and their adhesion properties from

the perspective of contact and fracture mechanics. In particular, this work started by

conducting friction and durability (wear) tests of thin-film PTFE coatings on smooth and

roughened aluminum substrates. Frictional characteristics of thin PFA and Poly (V3D3)

films on glass substrates are also studied. The analysis of variance (ANOVA) is

introduced to quantitatively examine the relative contributions of the effects, including

normal force, sliding speed, and surface roughness on COF and durability.

Another major concern related to successful performance of thin polymer coatings is their

adhesion properties. A micro-indentation based technique was used to characterize

adhesion of thin polymer films. In this method interfacial delamination is induced and the

delamination diameter is compared to a model of interfacial fracture. A 3D finite element

model was developed to investigate the delamination radius as a function of indentation

force, film thickness, material properties and interfacial fracture toughness. In practice,

the interfacial fracture toughness evaluated by indentation tests, is a system parameter,

which consists of contributions from the adhesion energy at the film-substrate interface,

and the plastic dissipation underneath the indenter during the indentation.

Chapter 2 introduces a literature survey related to this work, including indentation

mechanics of a half-space and a layered half-space with elastic or elastic-plastic

properties, mathematical description of interfacial delamination, bulk material

damage/failure criteria and cohesive zone model (CZM) implemented in finite element

analysis. In Chapter 3 we present an overview of PTFE material properties mostly in bulk

3

form and thin-film form. In Chapter 4 the friction and durability characteristics of thin-

film PTFE coatings on rough aluminum substrates are discussed. Effects of the multiple

variables are analyzed based on the statistical variation of the measurements. In Chapter 5

we present the friction characteristics of thin silicone and PFA films deposited on glass

substrates. In Chapter 6 finite element simulation for soft polymer coatings using bulk

damage/failure criteria is presented and compared to the previously developed close-form

formulations. In Chapter 7 experimental investigation and numerical simulation of

interfacial delamination are presented and a relation between non-dimensional

indentation force and delamination radius is developed to evaluate the interfacial fracture

toughness. Chapter 8 gives a summary and conclusions, and recommendations for future

research. Appendices 1-5 contain verification of ABAQUS/Explicit for the indentation

simulation, material properties of thin-film polymer coatings and glass substrates using

nano-indentation and Green’s function.

4

2 Mechanics Review for Thin-Film Polymer

Tribology

In this chapter, mechanics literature related to this dissertation is reviewed. This includes

the mechanics of indentation of a half-space and that of a layered half-space with elastic

or elastic-plastic material properties. The indentation methodology and its mathematical

descriptions are reviewed to help evaluate material properties. The load-interfacial

delamination radius relationships in thin-film indentation are reviewed. The bulk material

damage and cohesive zone models, used in finite element simulations, are presented.

2.1 Indentation of an Elastic Half-Space

2.1.1 Point Load on an Elastic Half-Space

Mechanics of an elastic half-space subjected to a concentrated load, W, as shown in

Figure 2.1, can be modeled by using the equations of equilibrium expressed in the

cylindrical coordinate system. For an axis-symmetric case, the equations of equilibrium

are given as follows,

10r rrrr rz

rFr r z r

(2.1)

1 10zrz zz

rz zFr r z r

(2.2)

5

where σrr, σθθ, σzz are the normal stress components and τrz, τrθ, τθz are the shear stress

components, Fr, Fz are the body forces in unit of load per unit volume. The coordinate

axes r, θ and z are defined in Figure 2.1. The normal and shear stress components σzz and

τrz on the surface of the elastic half-space (z = 0), away the concentrated load are

negligible, and expressed as follows,

,0 0zz r , r (2.3)

0 0,rz r , r (2.4)

The deformation of an elastic half-space due to a concentrated force, W, located at the

origin (Figure 2.1) can be expressed by the following displacement components in the r-

and z-directions [6],

3, 1 2

4r

W rz zu r z

G r

(2.5)

2

3

2 1,

4z

W zu r z

G

(2.6)

where 2 2r z , G is the shear modulus, ν is the Poisson’s ratio. The corresponding

normal and shear stress variations in the half-space are expressed as follows [7],

2

2 2 5

1 31 2

2rr

W z zr

r r

(2.7)

6

2 2 3

11 2

2

W z z

r r

(2.8)

3

5

3

2zz

W z

(2.9)

2

5

3

2rz

W rz

(2.10)

Figure 2.1. Concentrated point load on a 3D elastic half space.

7

2.1.2 Hertzian Contact

Hertz contact theory deals with the contact of two elastic spheres. In what follows friction

and adhesion at the interface are neglected. When two spheres are pressed against each

other a circular contact region with radius, a, develops. It is assumed that a is much

smaller than the radius of curvature of the two contacting bodies. Figure 2.2

schematically depicts the contact of a rigid sphere of radius, R, with an elastic half-space.

In this contact scenario, the normal stress component, σzz, and the shear stress

components, τrz and τθz, are zero outside of the contact region. Therefore, the boundary

conditions are given as follows,

0 0( , )z r and 0 0,rzτ r , r a (2.11)

,0 0zz r , r a (2.12)

The z-direction displacement at the contact interface (r ≤ a) is expressed as follows,

2 2

2 2 1,0 1

2 2H H Hz

r ru r R R r R R

R R

(2.13)

where δH is the approach distance of the spherical indenter.

Using Equation (2.6) and the Green’s function, shown in Appendix 5, the deflection of

the elastic half space under the spherical indenter is found as follows,

8

2 21 1

,0 ( , )z

S

Wu r p s dsd

E r E

2

2 20

*

12

4

πpνa r

E a

for r ≤ a (2.14)

where p0 is the maximum contact pressure, E* is the composite Young’s modulus,

defined as follows,

2 2

1 2

*

1 2

1 11

E E E

(2.15)

where Ei, νi (i = 1, 2) are Young’s modulus and Poisson’s ratio of the materials.

By combining Equations (2.13), (2.14) and (2.15), the contact radius, a, is found as

follows,

1/3

*

3

4

WRa

E

(2.16)

Similarly the contact approach δH is found as,

1/3

2/3

* *2

3 9

4 16H

WW

E a E R

(2.17)

For two contacting elastic spheres, R can be expressed as follows,

1 2

1 1 1

R R R (2.18)

9

where Ri (i = 1, 2) are the radii of contacting bodies.

Figure 2.2. Hertzian contact on an elastic half-space (a << R). Note that a is the contact

radius at the interface of spherical indenter and flat surface, δH is the indentation

displacement, R is the indenter radius, p0 is the maximum contact pressure.

2.2 Indentation of a Layered Elastic Half-Space

Figure 2.3 shows a schematic of indentation of a thin elastic film on a rigid substrate.

Matthewson presents the governing equilibrium equation from which analytical

expressions of stress and strain in the film are obtained [8]. He assumed that the stress

can be averaged in the thickness direction. The stress acting on a small volume of the

10

thin-film coating is illustrated in Figure 2.4. The interface between the film and the

substrate is assumed to be frictionless, where the shear stress τ0 is zero. The equilibrium

equation in r-direction is shown as follows,

1 0rrrrd

dr r t

(2.19)

where rr , are the averaged radial and circumferential stresses, τ1 is the shear stress

acting at the interface of coating-indenter interface, and t is the film thickness.

In the contact region (r < b), the shear stress at z = t is zero and the average strain in the z-

direction, z , is determined by the indenter geometry. Thus, the boundary conditions are

expressed as follows,

0 , , 0rzr t G r t (2.20)

z

f r

t (2.21)

where f(r) is a function describing the indenter profile.

11

Figure 2.3. Spherical indenter on a layered elastic half-space. Note that 2b is the contact

diameter at the interface of indenter and thin film; t is film thickness; ur(r, z) and uz(r, z)

are the horizontal and vertical displacement, respectively.

Figure 2.4. Stresses on an element of thin-film coating.

Outside the contact region (r > b), the average stress in the z-direction, zz , is zero,

therefore, the boundary condition is shown as follows,

12

2

2 01 2

zz r z z

GG

(2.22)

The interfacial shear and normal stress as a function of contact radius were presented for

both spherical and conical indenters. The results indicate the stress distribution is quite

sensitive to the Poisson’s ratio of the coating and the interfacial shear stress for nearly

incompressible materials is largest. The normalized interfacial shear stress distribution

given by Matthewson [8] was reproduced by using our finite element method, and

presented in the Appendix 1.

Chadwick [9] discussed axisymmetric indentation of frictionless spherical indenter on an

incompressible elastic layer by using the Wiener-Hopf integral equations. The governing

equilibrium equations are the same as Equations (2.1) and (2.2). The well-bonded and

slippery interfaces between a thin film and a substrate were considered. In particular, for

both cases, the boundary conditions described in Figure 2.3 are expressed as follows,

2

,0 ,2

z

ru r r b

R (2.23)

,0 0,zz r r b (2.24)

,0 0,rz r r b (2.25)

Note that the contact radius at the indenter-thin film interface is represented by b while

the contact radius at the indenter-substrate (elastic half-space) is represented by a for ease

13

of identification. Additionally, along a well-bonded interface the radial and normal

displacement components are zero,

, 0ru r t , , 0zu r t (2.26)

On the other hand, along a slippery interface the shear stress is zero,

, 0rz r t (2.27)

For these conditions the indentation force, W, as a function of the film thickness t is

shown to be,

2 3

3

2

3

ERW

t

for bonded interface (2.28)

t

ERW

3

2 2 for slippery interface (2.29)

where E is the Young’s modulus of the film, t is the film thickness, and R is the indenter

radius. Equations (2.28) and (2.29) show that the indentation force, W, on bonded layers

is inversely proportional to the cubic of film thickness while only to the film thickness on

slippery layers. This also indicates a larger force is required to equally indent a bonded

layer compared to a slipping layer.

Chen and Engel [10] proposed a general numerical method to analyze the contact stress

of one or two parallel elastic layers bonded to a homogeneous half-space. In particular,

the boundary value problems for both flat and parabolic punches were solved using

14

integral least square approach and Reissner energy method. The results showed that the

normal stress between a layer and flat punch is tensile when the layers are thinner.

However, the normal stress at the interface becomes compressive when no separation is

between the layer and punch.

2.3 Indentation on an Elastic-Plastic Layered Half-Space

Kral et al. analyzed repeated indentations of an elastic-plastic, layered medium by a rigid

sphere using finite element analysis [11]. They found that with full plasticity the contact

pressure develops a high pressure peak at the contact edge, instead of a relatively uniform

contact pressure on a homogeneous half-space. A significant tensile radial stress develops

near the contact edge under the maximum load and increases for thinner, stiffer, and

harder layers and by increasing strain hardening of the layer as well as the substrate. A

tensile hoop stress, formed at the surface near the contact edge, decreases with increasing

strain hardening. Both tensile radial and hoop stresses were thought to be responsible for

ring and radial cracks of the layer interface.

Using the same finite element model, Kral et al. reported the mechanics of the substrate

[12]. In particular, they showed that large tensile, radial and hoop stresses develop in the

film-substrate interface under the maximum load and increases with layer stiffness,

hardness and the strain-hardening exponents. The interfacial shear stress and maximum

von-Mises stress developed in the substrate depend only on strain-hardening exponent in

fully plastic deformation. In addition, the stresses at the interface remain predominantly

15

compressive for both loading and unloading cycles as the tensile hoop stress arises as a

band surrounding the plastic zone in the substrate and prevents the expansion of the

plastic zone.

2.4 Indentation on Soft Thin-Film Material

Karnath et al. developed closed-form formulae to model the indentation of plastically

flowing PTFE layer, schematically shown in Figure 2.5, deposited on the glass substrate

by using Hertz contact theory [3].

Figure 2.5. Schematic of indentation on PTFE thin-film coating.

16

In the case of purely elastic contact, the contact of a spherical indenter and the flat

surface is described by Hertzian contact equations, where the contact radius, a is

evaluated by Equation (2.16). The approach, δH, of the two surfaces is given by using

Equation (2.17).

The indentation of a thin PTFE film of thickness, t, deposited over a substrate by a

spherical indenter can be split into two processes. When the indentation load is below a

critical value, the ball makes contact with the PTFE film only; and the external force, W,

is balanced by the restoring force from the plastically deforming PTFE layer. As the load

is increased, the ball and the glass substrate eventually make contact; and the external

force, W, is balanced by the combined effects of the deforming PTFE film and glass.

These relationships can be expressed as follows,

2

2 2

,

,

p

H p

H πb δ tW

δ tF H π b a

(2.30)

where Hp is the hardness of the PTFE and FH is the Hertzian contact force at the ball

substrate (glass) interface. In addition, the relation of FH to δH is shown as follows,

*2/12/3

3

4ERF HH (2.31)

where R is the radius of ball indenter, E* is the composite Young’s modulus of two

contacting bodies, given in Equation (2.15).

The geometric relationship,

17

RRRb 2222 (2.32)

can be used to describe the width of the indentation on the top of the PTFE film as a

function of the indenter approach distance, δ. Note that when δ is greater than t, the glass

substrate deforms by δH and the following relationship prevails δ = δH + t. By combining

Equations (2.30) – (2.32), a relationship between the external force, W, and the approach

distance of the indenter, δ, is obtained as follows,

1/2 3/2

2 , ( )

( )4 / 3 2 ,

p

H p

H π R δ δ δ tW

δ tR E δ H π R δ δ R δ t

(2.33)

where 3p YH σ and σY is the yield stress of the PTFE film.

Ritter et al. classified the indentation on thin polymer coatings as three different modes

when the coating underneath the indenter has elastic or plastic deformation and the

polymer coating was entirely penetrated, which are schematically shown in Figure 2.6

[13]. In the first mode which the coating was elastically deformed, the equilibrium

equations, Equation (2.19), in terms of normal stresses and strains averaged through the

coating thickness was used to derive the indentation load, W, as a function of contact

radius, b. The mathematical expression was shown as follows,

4 2 42 2

1

6 142

1 2 6 1 2 3 8 2 2

tG tb kb b b GbW b I G b

R t Rt Rt

(2.34)

The coefficients, k, λ, ω, ς, are determined as follows:

18

1/2

3 1 2

2 1k

1 1

1 1 1 1

/ / / 1 6 / 2

/ / / / 1 2

b t K b t K b t t R

kK b t I kb t I kb t K b t

1

1

1 64

4 / 2 1 2

t kbkI

K b t R t

1/2

6 1

4

where 1 /K b t , 1 /I kb t , 1 /K b t , 1 /I kb t are the modified Bessel functions of the

second kind and their derivatives respectively; t is the film thickness, G is the shear

modulus, R is the indenter radius, b is the contact radius at the indenter-coating interface.

In the second mode, which the coating beneath the spherical indenter is plastically

deformed, the indentation force, W, is balanced with the coating hardness. The load, W,

as a function of contact radius, b, is shown as follows [14],

2

cW H b (2.35)

where Hc is the hardness of coating.

19

Figure 2.6. Schematic of delamination for three types of indentation cases (adapted from

Ritter et al. [13]). Note that type I indentation is elastic deformation under the indenter;

type II is plastic deformation under the indenter; type III is penetration of coating by

indenter.

When the coating was completely penetrated, typically by a Vickers indenter (pyramidal

geometry), the indentation force, W, is balanced with the hardness from the substrate as

well as coating, which can be expressed as follows [13, 15],

20

2 2 2

0 0 02 2c sW H b a H a (2.36)

where Hs is the hardness of substrate, a0, b0 are half the indenter diagonal in the substrate

and the coating, respectively.

2.5 Material Properties Investigation by Indentation

In general, micro and nano indentation can be used to determine mechanical properties

(Young’s modulus, E, and the hardness, H) of materials which are difficult to test by

other methods. Therefore, indentation is especially useful for evaluating material

properties of thin films in micro- or nano-scale [16, 17]. ASTM E2546 standard gives the

specifications of instrumented indentation testing. In particular, according to this standard

the test sample thickness should be at least ten times greater than the indentation depth

and six times greater than the indentation radius to avoid the substrate effect or residual

stress concentration [18]. The reason behind the rule of 10% indentation approach with

respect to film thickness is that the plastic zone associated with the indentation is entirely

contained within the film and negligible elastic deformation of the substrate contributes

to the evaluations of material properties [19].

It is found that displacements recovered during first unloading are not elastic in a range

of materials, for example, polymers, which lead to inaccurate evaluation of elastic

properties. It is reported that a nose phenomenon at the loading-unloading peak is

detected as the creeping effect in a wide range of polymers [20]. The ways to minimize

creep effects are to include hold periods of peak load in loading sequence or to use

21

unloading curves obtained after several cycles of loading [16]. Additionally, when a very

thin film (less than 1 μm) is indented, thermal drift, which is brought by changes in the

dimension of the contact (indenter, specimen) from thermal expansion or contraction due

to temperature changes, becomes a significant error source to affect the determination of

material properties [19]. In order to minimize the effects of thermal drift, the indentation

test is typically performed in an insulated enclosure, or in thermal equilibrium,

established by waiting a sufficient period.

Figure 2.7 shows a typical loading-unloading graph of indentation. The material

properties are determined based on the unloading curve. In particular, the initial

unloading stiffness, S, is evaluated as follows,

2r p

WS E A

(2.37)

where β is a correction parameter, Er is the reduced modulus, Ap is the projected area of

contact. The value of correction parameter, β, depends on the shape of the indenter,

which varies from 1.012, 1.034 to 1.067 for Vicker’s, Berkovich and conical indenters,

respectively [21]. The reduced modulus, Er, is defined as follows,

2 21 11 s i

r s iE E E

(2.38)

where Es, Ei are Young’s moduli of the specimen and indenter, respectively, and νs, νi are

the Poisson’s ratios of the specimen and the indenter, respectively. Thus, the Young’s

22

modulus of the specimen, Es, shown in Equation (2.38), is expressed as follows with

respect to Er and Ei,

12

211

1i

s s

r i

EE E

(2.39)

Figure 2.7. A schematic representation of load, W, as a function of displacement, δ, for

an indentation test [16]. Note that Wmax is the peak load; δmax is the displacement

corresponded at the peak load; δf is the final depth of the contact impression after

unloading; S is the initial unloading stiffness.

23

The hardness of specimen, H, is evaluated as follows,

max

p

WH

A (2.40)

A relationship between the projected area of indentation as a function of the tip

displacement was obtained for a variety of indenters. Figure 2.8 schematically shows a

section view of indentation, where the parameters in the analysis are specified. The total

indentation displacement, δ, is expressed in terms of the contact depth, δc, as follows:

s c (2.41)

Figure 2.8. A section view of an indentation showing parameters in the analysis

(Adapted from [16]). Note that δc is the vertical distance along which contact is made; δs

is the displacement between the initial surface and the surface at the perimeter of the

contact; δf is the final depth of the residual hardness impression.

24

Table 2.1. Geometric constant, ε, for different indenters [17, 22].

indenter geometry conical Spherical Berkovich Flat

geometric constant, ε 0.72 0.75 0.75 1.00

The deflection of the surface at the contact perimeter, δs, for different indenters, was

investigated by Sneddon [22], Doerner and Nix [23], and Oliver and Pharr [17]. It is

expressed in terms of the maximum indentation load, Wmax, and the unloading stiffness, S,

as follows,

maxs

W

S

(2.42)

where ε is a geometric constant, given in Table 2.1 for different indenter types.

The relation of the projected contact area, Ap, and the contact depth, δc, which is

expressed as the area or shape function, F(δc), was established experimentally. Pharr [24]

and Sakharova et al. [25] presented the area functions, F(δc), for a range of indenters with

different geometries. In particular, the projected contact area, Ap, as a function of contact

depth, δc, for a geometrically-perfect Berkovich indenter is shown as follows [16],

224.5p c cA F (2.43)

However, the imperfection of the tip can lead to slightly different relations. Oliver and

Pharr presented the relationship of contact area, Ap, and contact depth, δc, to characterize

other indenters which deviate from the Berkovich indenter as follows [17],

25

2 1 1/2 1/4 1/128

1 2 3 824.5p c c c c c cA F C C C C (2.44)

where C1 - C8 are constants. The projected contact area, Ap, is critical for evaluating the

initial unloading stiffness, S, and specimen hardness, Hs.

2.6 Indentation-Induced Interfacial Delamination

2.6.1 Definitions of Adhesion Energy and Interfacial Fracture

Toughness

Volinsky et al. [26] elaborate on the definitions of adhesion energy and interfacial

fracture toughness. In particular, the adhesion energy, or the true work of adhesion at the

interface, Wa, is the amount of energy required to create new surfaces by breaking the

bonds of materials, which is mathematically expressed as follows,

a f s fsW (2.45)

where γf, γs are the surface energies of the film and substrate, respectively, γfs is the

energy of the interface . The parameter, Wa, is an intrinsic property of the film-substrate

interface, which depends on different types of bonding, and it is a constant for a given

film-substrate interface.

Interfacial fracture toughness, Γi, or the practical work of adhesion, is the amount of

energy to delaminate thin films from the substrate. Ideally, the adhesion energy, Wa, is

assumed to be equal to interfacial fracture toughness, Γi, by Griffith fracture theory.

26

However, the process of thin film delamination from the substrate usually involves

plastic deformation, which makes it difficult to extract the adhesion energy from the total

energy measured. In general, the interfacial fracture toughness, Γi, can be quantified as

follows [26, 27],

i a p fricW (2.46)

where Γp is the energy spent in plastic deformation of the film and the substrate, Γfric is

the energy loss due to friction. In fact, Γp and Γfric are functions of the adhesion energy,

Wa, in many cases.

In fracture mechanics, the strain energy release rate, or the crack driving force is used as

a measure of the interfacial fracture toughness, Γi [28, 29]. In a controlled indentation test,

the interfacial fracture toughness, Γi, is determined by obtaining the equilibrium state

with the interfacial fracture resistance, ΓR, which is mathematically shown as follows,

Ri R

f

U

A

(2.47)

where UR is the total strain energy of the system, Af is the crack area.

In particular, when Γi > ΓR, unstable crack growth takes place; when Γi ≤ ΓR, the crack

growth is stable. In the experiments reported in Chapter 7, we measure the interfacial

fracture toughness, Γc, which aims to differentiate the theoretical evaluation of Γi, at

initiation of crack growth instead of obtaining the shape of the R-curve (resistance curve),

which indicates the material resistance to crack extension [30]. Figure 2.9 shows a rising

27

Figure 2.9. Schematic of driving force/resistance curves (Adapted from Anderson [30]).

R-curve for determining the interfacial fracture toughness, Γc. R-curve can be different

for a variety of materials and structures.

Next section describes a general method to theoretically model indentation-induced

delamination by evaluating the strain energy of the system.

2.6.2 Mathematical Descriptions of Interfacial Delamination

Indentation-induced delamination, schematically shown in Figure 2.10, can be used as a

controlled adhesion test to measure the adhesive property of a thin film which is affected

by its fracture resistance and strength.

Marshall and Evans [31] modeled the delamination process by assuming that the thin film

behaves as a clamped plate. They calculated the energy release rate, Γi, from changes in

28

the strain energy of the system, which includes the effects of indentation and the residual

stresses. First, they investigated the delamination of a thin film with biaxial residual

stress, σR. The total strain energy, UR, includes the energy in the film above the crack, Uc,

as well as in the remaining system, Us.

R c sU U U (2.48)

Figure 2.10. The schematic representation of interfacial delamination by conical

indenter. Note that 2b is the contact diameter at the interface of indenter and coating; 2c

is the delamination diameter at the interface of coating and substrate; δ is indentation

displacement; Ψ is half conical angle; W is indentation load; t is film thickness.

29

Figure 2.11. Schematic representation of a delaminated, residually stressed film (adapted

from Marshall and Evans [31]). Note that 2c is delamination diameter, σR is the residual

stress, t is film thickness, ΔR is film expansion radius.

In order to evaluate the energy, Uc, a section of film around the perimeter of the crack

was assumed to be subjected to the residual stress, σR, as shown in Figure 2.11.

The expansion radius of the plate, ΔR, due to the residual stress, σR, can be evaluated as

follows,

1 f R

R

f

c

E

(2.49)

where Ef, νf are the Young’s modulus and the Poisson’s ratio of the film, c is the

delamination radius. The increase of strain energy in the film, Uc, can be calculated as

follows,

30

2 21 f R

c R R

f

t cU ct

E

(2.50)

The total strain energy, UR, in the system becomes,

2 21 f R

R s

f

t cU U

E

(2.51)

Since UR is independent of crack length for an unbuckled plate, UR, is shown as follows:

1 f

R

f

t BU

E

(2.52)

where B is a constant. Thus, the remaining strain energy, Us, can be expressed by

combining Equations (2.51) and (2.52) as follows,

2 21 f R

s

f

t B cU

E

(2.53)

When the edge stress, σR, is greater than a critical value, σc, the film buckles. The

buckling stress for a clamped circular plate was shown as follows [31],

2

2

14.68

12 1c f

tE

c

(2.54)

The difference in strain energy between buckled and unbuckled plates can be expressed

as follows,

31

2 21 1f R c

B

f

t cU

E

(2.55)

where α represents the slope of buckling load versus edge displacement after buckling. It

can be expressed in terms of the Poisson’s ratio, ν, as follows [27],

1

11 0.902 1

(2.56)

Buckling of plate reduces the total strain energy. Thus the total strain energy of the

system, RU , becomes

R R BU U U (2.57)

Based on the definition, the energy release rate, Γi, for the buckled plate R c is

evaluated as follows,

2 21 11

2

f R cRi

f

tdU

c dc E

(2.58)

The second case by Marshall and Evans considers the delamination that occurs at the

interface of a stress-free film as shown in Figure 2.12. The deformation within the plastic

zone around the contact was found by considering volume conservation and radial

displacement mode. The expansion radius Δ0 in terms of indentation volume V0 can be

shown as follows,

32

0 02V ct (2.59)

The stress induced by the indentation at the edge of the plate, σ0, is evaluated as follows,

0 0

0 21 2 1

f f

f f

E E V

v c c t

(2.60)

The total strain energy, UR, when the film is not buckled ( 0 c ), includes the strain

energy, Uc, in the film caused by the stress of indentation, σ0, and the residual strain

energy, Us, stored in the remaining system, as shown in Figure 2.12. The mathematical

expression is shown as follows,

2

2 22 2

0011

2

ff

R c s

f f

t B ct cU U U

E E

(2.61)

where B is constant. The energy release rate, Γi, is calculated as follows,

2 2

011

2 2

fRi

f

tdU

c dc E

(2.62)

When the film is buckled ( 0 c ), the difference of strain energy due to buckling can

be expressed as follows,

2 2

01 1f c

B

f

t cU

E

(2.63)

33

Figure 2.12. Schematic representation of a stress-free film with indentation-induced

delamination (adapted from Marshall and Evans [31]). Note that σ0 is the indentation

stress, Δ0 is film expansion radius related to indentation volume, V0.

The total strain energy, RU , is shown as follows,

R R BU U U (2.64)

Thus the energy release rate, Γi, for buckled plate is shown as follows,

22

0 0

11 1

2f c

i

f

t

E

34

22

0 0

1 11

2 1 0.902 1f c

f

t

E

(2.65)

In the third case, delamination of the interface is analyzed with the combined effects of

residual stress and indentation stress [31]. The total strain energy, U, can be evaluated

with a combination of Equations (2.53), (2.61) and (2.63), where the edge stress is

changed as the sum of residual stress, σR, and indentation stress, σ0, as follows,

2

20 2

0

2 2 2 2

0

11

2

1

ff

i B s R

f

R c R

B ctU U U U c

E

c B c

(2.66)

The derivations of energy release rate, Γi, presented above are based on the assumption of

mixed-mode, where the mode-II predominates the crack tip [29].

Rosenfeld et al. investigated the energy release rate, Γi, for an epoxy by neglecting the

residual stress and buckling [32]. Figure 2.13 shows the interfacial delamination by

conical indenter, which was schematically shown in Figure 2.10, was modeled as an

annular plate. The outside surface of the plate, which adheres to the remaining portion of

coating, was constrained and a fixed pressure, σb (r = b), was applied on the inside

surface. The radial and circumferential stresses, σrr and σθθ, as a function of radius are

shown as follows,

2 2

2 2

1 /

1 /rr b

c r

c b

(2.67)

35

2 2

2 2

1 /

1 /b

c r

c b

(2.68)

where1

1

f

f

v

, νf is the Poisson’s ratio of the epoxy coating.

The total strain energy, UR, of the plate is evaluated by the following integral,

2 2 2c

R rr f rrb

f

tU rdr

E

(2.69)

where t is the coating thickness, Ef is the Young’s modulus of epoxy. The energy release

rate, Γi, can be evaluated based on the strain energy, U, as follows:

22 2

2

2 11 1

2 1 / 1

f bRi

f f f

tdU

c dc E c b

(2.70)

Figure 2.13. Schematic of annular-plate model for delamination (adapted from Rosenfeld

et al. [32]).

36

In order to derive the energy release rate, Γi, as a function of indentation force, W, and the

contact radius, b, hardness of the material, H, is considered. This gives the following

relationship,

22

WH

b (2.71)

Note that the indenter used is a pyramidal indenter with diagonal 2b. For σb, the Tresca

yield criterion gives the following relationship,

b Y H (2.72)

where σY is the yield stress of the film.

For a polymer, the relationship between H and σY is given as follows [32, 33],

2.25 YH (2.73)

By combining Equations (2.72) and (2.73), the stress, σb, is shown as follows,

1.25 0.556b Y H (2.74)

Thus, Equation (2.70) for evaluation of Γi can be rewritten by combing Equations (2.73-

2.74):

22 2

22 1 0.556 2

1 1f

i f f

f

H t Hc

E W

22 2 20.627 1

1 2 1f

f f

f

H t Hc

E W

(2.75)

Jayachandran et al. [34] simulated the indentation of a PMMA coating by using 2D

axisymmetric finite element model and evaluated adhesive shear stress by using

37

Matthewson formulations [14], where the critical contact radius, b, for delamination was

estimated. However, the interfacial shear stress was underestimated without modeling the

interface damage.

2.6.3 A Relationship between Non-dimensional Delamination Radius

and Indentation Load

In order to characterize the interfacial delamination, Marshall and Evans developed the

non-dimensional parameters for indentation load and delamination radius, which

otherwise depend on film material properties, interface fracture toughness and indenter

geometry [28]. The following is a summary of their work. In particular, the non-

dimensional force W is expressed as follows,

2

0

3

12 11

2 1 14.68

VW

t

(2.76)

where V0 is indentation volume which depends on the indenter geometry.

For a pyramidal indenter, the relationship between the projected contact area, Ap, and the

indentation volume, V0, can be shown to be as follows,

2/3

03 2

cotp

VA

(2.77)

where Ψ is the half apex angle of the indenter.

38

Using Equation (2.40), the film hardness, H can be evaluated in terms of the projected

area, Ap, as follows,

2/3

2/3

0

cot

3 2p

WWH

A V

(2.78)

The non-dimensional force W can be calculated by using Equations (2.76) and (2.78) as

follows,

3/2 3/2

3 3

1cot 1 1

7.342

W WW

H t H t

(2.79)

where

cot 1

0.031cot 17.34 2

is a constant that depends on the indenter

geometry and the plate boundary conditions.

The non-dimensional delamination radius, C , is expressed as follows,

2

51

iC cE t

(2.80)

Non-dimensional force,W , given in Equation (2.79), was related to the delamination

radius, C , as follows [28],

22 1

2

1 / 2 1 1

1 res

W W

C

(2.81)

39

where res is the normalized residual deposition stress, defined as follows,

21 1

res R

i

t

E

(2.82)

For the conical indenter, used in micro-indentation, the indentation volume, V0, with

respect to the ratio of normal load, W, to the hardness, H, can be shown to be as follows,

3/2

0

cot

3

WV

H

(2.83)

For this case, the coefficient, ξ, in Equation (2.79) becomes,

3/2

1cot0.024cot 1

7.34

(2.84)

2.6.4 Film Thickness Effect on Interfacial Delamination

Li et al. [33], Sheng et al. [35] and Ritter et al. [15] found that thinner polymer films were

difficult to delaminate at the interface, showing film thickness dependence on the

delamination. Moody et al. showed that the interfacial fracture toughness of epoxy

decreases with decreasing film thickness [36]. The thickness dependence was attributed

to insufficient amount of elastic strain energy in the thinner films for the initiation and

propagation of delamination [27].

Volinsky et al. [37] discussed the contribution of plastic energy dissipation, Γp, in a

ductile thin film, mostly for metals, to the interfacial fracture toughness, Γi, as described

40

in Equation (2.46). A plastic strip model for estimating plastic energy dissipation at the

interfacial crack tip was shown as follows,

2

ln 1Yp

f

tt

E b

(2.85)

where t is film thickness, σY is film yield stress, b is the magnitude of the Burgers vector.

Equation (2.85) shows the thickness dependence on the plastic energy dissipation, where

this model assumes extension of plastic zone through the entire film thickness.

Evans and Hutchinson [29] derived a relation of critical film thickness for delamination,

where the delamination radius and film thickness are sensitive to variations in the

residual stress and the adherence. The equation is shown as follows,

2

0 12

f

c

f

t ckE

(2.86)

where σ0 is the stress induced by the indentation, c is delamination radius, k is a constant

used in the plate theory.

41

2.7 Bulk Material Damage/Failure Criteria in Finite Element Analysis

2.7.1 Introduction of Bulk Material Damage/Failure

Material damage and failure take place often and have been a critical issue in engineering.

Two of the most known accidents with respect to material failure were brittle fracture of

Liberty ships during the World War II, and the explosion of the Challenger Space Shuttle

due to malfunction of an O-ring seal [30]. Continuum Damage Mechanics (CDM), as a

relatively new branch of solid mechanics, is used to investigate damage and failure

characteristics for different materials using numerical methods and boundary integral

equation [38].

Two major mechanisms are used to describe material damage and failure of brittle and

ductile materials. In particular, brittle damage is characterized in the form of cleavage of

crystallographic planes with negligible inelastic deformation, typically observed in

polycrystalline metals at low temperature. Ductile damage behaves quite differently,

where large plastic deformation occurs around crystalline defects and causes localized

necking regions [38].

42

2.7.2 Implementation of Bulk Material Damage/Failure Criteria in

Finite Element Analysis

In continuum damage mechanics, a damage variable, D, is used to indicate the damage

initiation as well as material failure, where the value of D is in a range of 0 - 1. Figure

2.14 schematically shows mechanical characteristics of ductile materials and materials

with the assumption of elastic-perfectly plastic behavior. In particular, damage initiates at

point c (D = 0), where the equivalent plastic strain, 0pl , denotes initiation point of damage.

When the damage variable, D, equals 1, where the plastic energy dissipation decays to

point d, damage accumulation leads to material failure with a complete loss of load-

carrying capability.

(a) (b)

Figure 2.14. Material damage initiation and failure [39]. Note that (a) typical ductile

material; (b) elastic and perfectly plastic material.

43

A literature review shows a number of macroscopic or micromechanical material

properties, such as density, cross-section area, Young’s modulus, yield stress, have been

chosen as phenomenological parameters to characterize material deterioration. For

example, Rabotnov [38] selected the reduction of cross-section area caused by micro-

cracking as the measure of damage initiation. Lemaitre [40] replaced a true stress in the

material constitutive law by an effective stress as damage variable to describe isotropic

ductile damage in metals.

Two primary damage/failure criteria used in ABAQUS include ductile and shear damage.

In particular, the ductile damage criterion is used to predict the initiation of damage

formed by growth and coalescence of voids at local plastic deformations. This

phenomenological model describes the relationship of the equivalent plastic strain, plD , at

the onset of damage, as a function of stress state parameter, η, as well as equivalent strain

rate, pl . The damage variable, D, for ductile damage, is calculated as follows,

,

pl

plplD

dD

(2.87)

1 2 3

2 2 21 2 3 1 2 2 3 1 3

(2.88)

where σi (i = 1, 2, 3) are the principal stresses [41].

The shear criterion is often used to characterize the damage caused by shear band

localization [41]. This model describes the relationship of the equivalent plastic strain,

44

plS , at the onset of damage, as a function of shear stress ratio, ξs, and plastic strain rate,

pl . The damage variable, D, is expressed as follows,

,

pl

plplsS

dD

(2.89)

max

eqv s

s

k p

(2.90)

2 2 2

1 2 2 3 3 1

1

2eqv (2.91)

where τmax is the maximum shear stress, ks is a material parameter, p is pressure stress and

σeqv is the von-Mises stress, σi (i = 1, 2, 3) are the principal stresses.

Material is assumed to degrade linearly after initiation of damage, for numerical

implementation, schematically shown in Figure 2.14(b). A stress-displacement response

was proposed for damage evolution. In order to prevent mesh dependence on strain

localization plastic displacement was defined. An equivalent plastic displacement, upl

,

independent on the characteristic length, L, of the finite element mesh, was introduced to

evaluate the bulk fracture energy, Γf, during the damage evolution. The bulk fracture

toughness, Γf, schematically represented as the green-color region in Figure 2.14, is

expressed as follows [39],

0 0

pl plf f

pl

upl pl pl pl

f L d du

(2.92)

45

pl plu L (2.93)

where σpl

is the stress with plastic deformation, pl

f ,pl

fu are the plastic strain, and

displacement, respectively, at the point of material failure, L is the characteristic length of

finite element mesh, related to the coordinates of integration points among different types

of elements.

As the fracture energy is dissipated in a linear form, the fracture toughness, Γf, as shown

in Equation (2.92), can be expressed as follows,

1

2

pl

f Y fu (2.94)

where σY is the yield stress at the damage initiation.

2.8 Cohesive Zone Model in Finite Element Analysis

2.8.1 Description of Cohesive Zone Model

Cohesive zone model (CZM), which behaves like a non-linear material described by a

traction-separation law, is developed to simulate nucleation and propagation of interfacial

delamination [42]. This non-linear fracture mechanics approach was developed based on

the analysis of plastic zone of a crack tip. Dugdale presented a plastic strip model to

analyze the stress intensity factor ahead of a crack tip, where a constant closure stress

equal to yield stress was applied, by superimposing elastic plane-stress solutions [43].

Barenblatt generalized the plastic strip model and pointed out the stress in the cohesive

46

zone ahead of the crack is supposed to be a function of the separation instead of a

constant yield stress [44]. Cottrell [45] put forward the concept of crack-bridging as a

unifying theory for fracture at various length scales, from atomic bond breaking to large-

scale bridging in fibers of composite materials. A bridging law which relates the surface

tractions in the bridging zone, or cohesive zone to the relative separation displacements

was established to describe the fracture mechanisms of plastic zone at a crack front. Bao

and Suo reviewed the crack-bridging law and emphasized the implications of strength

and resistance toughness for ceramic matrix composite [46].

The bridging law, or traction-separation relation in the cohesive zone model depends on

the material as well as its associated fracture mechanism [42]. In particular, the fracture

mechanism of metals is characterized as large plastic deformation and void nucleation,

growth and coalescence ahead of a crack tip, while the fracture takes place by breaking

atomic bonds in ideally brittle materials, which yields distinctly different traction-

separation relations. The traction-separation relationship for elastic-plastic fracture is

derived from micromechanics models or determined experimentally, while the relation

for brittle fracture can be obtained from an inter-atomic bond potential.

47

(a) (b) (c)

Figure 2.15. Examples of the traction-separation relationship for the cohesive zone

model, for (a) constant traction; (b) trapezoidal traction; (c) bilinear traction.

Some relatively simple traction-separation relationships are defined in analytical and

numerical analysis, as shown in Figure 2.15. In particular, Figure 2.15(a) shows a

constant traction in the bridging zone, which was assumed in the Dugdale model.

Trapezoidal and bilinear relationships are often used in practice. Camanho et al. [47]

predicted the interfacial delamination of composite materials with mixed-mode loading

using bilinear traction-separation relation and gave comparable results with experimental

testing. Sorensen and Jacobsen [48] determined the cohesive law with trapezoidal

traction-separation relations by using J-integral approach and characterized large-scale

failure of carbon fiber-epoxy composites.

Among traction-separation relations, shown in Figure 2.15, two parameters of the

maximum stress, σc0, and the critical displacement, δc, are critical to characterize the

bridging law. The interfacial fracture toughness, Γi, which represents the area of traction-

separation relation, can be evaluated as [46],

48

00

c

i c cd

(2.95)

The strength, σc0, of an inorganic material, is roughly one tenth of its Young’s modulus (~

1010

N/m2), and the critical separation, δc, is on the order of lattice spacing (~ 10

-10 m)

[46]. An order-of-magnitude estimate for interfacial fracture toughness, Γi, can reach ~ 1

J/m2, which is close to the surface energy per unit area of a solid. The strength, σc0, for an

elastic-plastic fracture in metals can reach the yield stress (~ 108 N/m

2) and δc is in the

order of 10-6

m, which gives rise to a fracture energy up to 10 J/m2 and much higher than

the surface energy [42].

2.8.2 Implementation of Cohesive Zone Model in ABAQUS

The cohesive zone model is appropriate to model adhesion and delamination of interfaces

between two dissimilar materials, where the material properties are elastic or elastic-

plastic [49]. In ABAQUS, surface-based cohesive behavior or conventional cohesive

elements by specifying the traction-separation law can be used to model the interfacial

delamination [39]. However, surface-based interface is suitable for simulating a wide

range of cohesive interaction, where the interface thickness is negligibly small. If the

interface adhesive layer has a finite thickness and available macroscopic properties, such

as stiffness and strength, cohesive elements will be a better option for simulations.

Several traction-separation laws are schematically shown in Figure 2.15. A bilinear

traction-separation law as well as its implementation in ABAQUS is presented next as

49

this relation is adopted to investigate interfacial delamination of thin polymer film

through this dissertation.

Figure 2.16 schematically shows the parameters required to define cohesive interface in

mode-I, including initial elastic stiffness, K0, maximum cohesive strength, σc0, and a

critical separation, δc. The interface initially opens elastically with initial elastic stiffness,

K0, until the interface stress equals the maximum cohesive strength, σc0, which indicates

the initiation of damage. A damage variable, D, which is in the 0-1 range, is used to

quantify the interface status and is mathematically expressed using a damage evolution

rule,

max 0

max 0

c

c

D

(2.96)

00

0

c

K

(2.97)

where δ0 is the critical separation for damage initiation and δmax is the maximum

separation during the entire loading history. When the interface is partially delaminated,

with 0 < D < 1, the opening stress related to displacement can be expressed as follows,

01σ D K δ (2.98)

Equation (2.97) applies for both loading and unloading cases. In particular, during

loading (D = 0), the opening stress, σ, linearly increases with displacement, δ. During

unloading, Equation (2.98) can be rewritten as follows,

50

0

1σ

DK δ

(2.99)

Equation (2.99) indicates the damage variable, D, increases by increasing the opening

displacement, δ.

Equation (2.96) and (2.97) substituted into Equation (2.98), the stress at the cohesive

interface is shown as follows,

max 0 max

0 0

0 max 0 max 0

1c c

c c

c c

(2.100)

Since δmax = δ during loading, Equation (2.100) becomes,

0

0

cc

c

δ δσ σ

δ δ

(2.101)

During loading, the maximum separation, δmax, equals the separation, δ, where the

damage parameter, D, increases with increasing the opening displacement. During the

unloading, the maximum separation, δmax, and damage parameter, D, remain constant.

When c , the damage parameter, D, equals 1 and the interface loses its load-carrying

capability (σ = 0).

51

Figure 2.16. Schematic representation of the bilinear traction-separation law

implemented in ABAQUS.

The traction-separation laws of specifying shear and tear modes (mode-II and mode-III)

can be defined separately, with a set of similar parameters as mode-I discussed above.

Under a mixed-mode condition (mode-I and-II), a few damage initiation criteria are

available in ABAQUS, including the maximum stress separation criterion and the

quadratic stress separation criterion. In this work, we adopted the maximum stress

criterion, given as follows,

0 0

max , 1c c

(2.102)

where τ is the interfacial shear stress in either of two shear directions, in the case

of tension and 0 otherwise.

52

The mixed-mode damage initiation criterion shows the critical magnitude of the traction

vector depends on the ratio of the shear to normal traction, defined by a phase angle φ of

mode mix. In particular, φ is expressed as follows,

tan

(2.103)

Note that the Macauley bracket assumes that compression does not cause damage, which

makes Equation (2.103) be valid mathematically.

The magnitude of the effective traction vector, σm, is shown as follows,

2 2

cos sinm

(2.104)

By substituting Equation (2.104) into Equation (2.102), the critical traction magnitude for

damage initiation, σm0, is expressed as follows,

0 00 max ,

cos sin

c cm

(2.105)

In order to characterize damage evolution of the parameter, D, by a combination of

normal and shear deformation across the interface, an effective displacement, δm, is

defined as follows,

2 2

m n s (2.106)

53

where δn and δs are displacements in normal and shear directions, respectively. The

damage parameter, D, in the mixed-mode is evaluated by using the evolution law, shown

in Equation (2.96):

max 0

max 0

mc m

mc m

D

(2.107)

where δmc is the critical effective displacement, depending on the mode mix as well as the

damage evolution criterion. The energy-based damage evolution criterion was adopted

and shown as follows,

0 0

1I II

I II

W W

(2.108)

where α0 is power coefficient, ΓI, ΓII are the interfacial fracture toughness under mode-I, -

II, respectively. WI, WII are the work done by the traction in normal and shear modes,

where were mathematically shown as follows,

0

n

IW d

(2.109)

0

s

IIW d

(2.110)

Also, δm can be expressed in terms of separations, δn, δs as follows,

cos sin

n sm

(2.111)

54

Combining Equation (2.111) and (2.107), the damage parameter, D, is shown as follows,

0 0

0 0

cos sinmc n m mc s m

mc m n mc m s

D

(2.112)

Since the area under bilinear traction-separation law shows the work done by the

tractions, Equations (2.109) and (2.110) can be rewritten as follows,

2

0

1cos

2I m mcW (2.113)

2

0

1sin

2II m mcW (2.114)

The total interfacial fracture toughness in mixed-mode, Γm, can be shown as follows,

0

1

2m I II m mcW W (2.115)

Therefore, the strength of interface with mixed-mode depends on normal and shear

stresses, and the interfacial fracture toughness depends on the toughness in mode-I and -II.

In ABAQUS, five parameters are required to define cohesive interface in mixed-mode,

including the initial stiffness, K0, normal and shear stresses, σc0, τc0, and interfacial

fracture toughness in normal and shear directions, ΓI and ΓII.

55

3 Review of PTFE Material Properties

In this chapter, material properties of PTFE are reviewed. In particular, the literature

related to the molecular structure and mechanical properties, frictional and wear

characteristics in bulk and thin-film forms are surveyed.

3.1 Molecular Structure of PTFE

PTFE (Polytetrafluoroethylene) is an important engineering material with low friction

coefficient and dielectric constant, which is biocompatible, chemically inert and stable

under relatively high temperatures [50, 51]. Because of these characteristics it is widely

used as a solid lubricant, protective film, and an electrical insulator [52].

A number of mechanical properties of PTFE closely depend on its molecular structure,

which consists of 20,000-200,000 repeating units of tetrafluoroethylene (- CF2 - CF2 -)n

[53]. Bunn and Howells [54] investigated the molecular structure of PTFE at different

temperatures by X-ray diffraction. They found that the fluorocarbon molecules of PTFE

have a helical chain that twists at every 13-atom repeating unit. Figure 3.1(a) shows the

schematic of the twisted zigzag backbone in one PTFE segment. Figure 3.1(b) is the

illustration that depicts the fluorocarbon molecules assembled around the backbone of

PTFE segment, while Figure 3.1(c) depicts the molecular structure of PE (polyethylene) –

another polymer composed of hydrocarbon molecules. Compared to the molecular

56

structure of the PE, the smooth profile of the PTFE molecule is one of the main reasons

behind the low coefficient of friction (COF) of PTFE [54].

(a) (b) (c)

Figure 3.1. Schematic representation of PTFE and PE molecular structures [54]. Note

that (a) The zigzag backbone of a PTFE molecular segment; (b) A PTFE molecular chain;

(c) A PE molecular chain.

57

PTFE undergoes three types of thermally activated structural transformations at 19, 30,

and 150° C respectively. Below 19° C, the molecular chain is a 136 helix, i.e. every 13 –

CF2- units presented in six twists form an 180° rotation [51]. This is considered to be the

most stable phase [54]. At 19° C, a first-order crystal transformation occurs where a 157

helical conformation in a hexagonal cell is formed. Above 30° C, the 157 helical

configuration remains, but the conformation disorder increases. Above 150° C, the

conformation disorder of the helix chain significantly increases, which is detrimental to

maintaining the smooth profile of PTFE molecular chain.

It has been reported that the COF of PTFE varies with temperature, and this variation has

been attributed to the different helical conformations of PTFE at different phases [51]. In

particular, the COF values for PTFE below 19° C are somewhat greater than that between

19° 30° C [52]. As mentioned above, a 136 helix conformation exists below 19° C,

while another helix conformation, 157, is found above 19° C. Table 3.1 shows the

residues per turn. This table shows that the density of helical conformation of 157, is

slightly lower than that of 136, which has a very small effect on the PTFE chain

conformation [51]. However, the major difference of inter-chain distance between helical

conformations has significant impact on the conformational disorder where large inter-

chain distance allows the helix to have more room available to untwist or develop

disorders [51]. Bunn and Howells found that the spacing between molecular chains for

different helix conformations and found them to be 5.62 Å below 20° C, and 5.66 Å

above 20° C [54].

58

Makinson and Tabor [55] proposed a model of crystalline structure of bulk PTFE shown

in Figure 3.2. Bulk PTFE is composed of a large number of individual units called

“crystalline blocks” or “bands” with the dimensions of 10 to 100 μm long and 0.2 to 1 μm

wide. Figure 3.2(a) and (b) illustrate that one individual crystalline block is formed by

many small crystalline slices or “striae” which are typically about 200 Å thick. Figure

3.2(c) depicts each individual slice separated by disordered regions consisting of highly

oriented sheets of molecules. This crystalline model indicates that shear takes place more

easily within the amorphous regions between individual slices which makes slip occur

more frequently along the backbone of molecular chain. Recently, Sawyer et al. [56, 57]

investigated the influence of orientation of PTFE molecular structure on the COF values.

They found low friction forces and low barriers to interfacial slip, as the sliding direction

was parallel to PTFE molecular chain’s backbone. In contrast, when the sliding direction

was perpendicular to the PTFE chain backbone, they found high friction forces and high

wear, as molecular reorientation and chain scission happened.

Table 3.1. Characteristics of the helical conformation of PTFE [51].

136 157

Pitch (Å) 2.813 2.786

Unit twist (°) 166.2 168

Rise per residue (Å) 1.298 1.30

Residues per turn 2.167 2.143

59

Figure 3.2. Crystalline structure of bulk PTFE (Makinson and Tabor [55]). Note that (a)

crystalline block or ‘band’; (b) crystalline slices or ‘striae’ after sliding; (c) hexagonal

array of chains within the slices.

3.2 Mechanical Properties of PTFE

In order to thoroughly understand mechanical properties of PTFE, we review the

literature and summarize the findings of the previous investigations. This review includes

the mechanical properties of bulk and thin-film PTFE (modulus of elasticity, hardness,

viscoelasticity and plasticity) and frictional and wear characteristics. This knowledge will

shed little light on our research about PTFE thin-film coatings.

3.2.1 Young’s Modulus and Yield Stress

Wang et al. [58] investigated the thickness dependence of Young’s modulus and hardness

of on-wafer, ultrathin PTFE by using the Dynamic Contact Module (DCM) technique.

Experimental results showed that both Young’s modulus and hardness of PTFE thin film

strongly depend on the film thickness for films thinner than 500 nm. On the other hand

for thicker PTFE films, the Young’s modulus and hardness of the coatings were found to

60

be independent of the thickness, with the reported values of 2.3 GPa and 58 MPa,

respectively. The authors reported that there were significant differences in

thermophysical and mechanical properties of the thin-film polymer and the bulk polymer,

because the molecular structure of ultrathin (less than 100 nm thick) and thin (100 – 1000

nm thick) polymer films reorganize and give rise to a significantly different structure

compared to that of the bulk polymer [58]. Additionally, the Young’s modulus of bulk

and free-standing film measured by other methods is found to be 1/6 of the values

obtained by using nano-indentation, indicating the strength of polymers has a high

pressure dependence.

Table 3.2 gives the tensile properties of the bulk sample and free-standing films at room

temperature [58]. Lucas et al. [59] also measured Young’s modulus and hardness of

PTFE thin-films (500 – 1500 nm thick) on silicon substrate and reported 1 GPa for

Young’s modulus and 30 – 55 MPa for hardness. They found that the material properties

measured were independent of film thickness, which was consistent with the findings by

Wang et al [58]. However, the difference of Young’s modulus at the same indentation

depth of 500 nm reported by these two experiments was possibly caused by different

coating fabrication procedures and indentation heads applied.

Wang et al. [60] further assumed that the higher Young’s modulus and hardness of the

thinner films could be attributed to the first thin layer crystallite, as this thin layer of

PTFE was very dense and well attached to the substrate surface. This explanation was

built on the finding by Jones et al. [61] that polymer films less than 100 nm thick had

61

almost identical molecular properties to the same materials in bulk volume. Wang et al.

[60] also observed by SEM that the lamellar structure of the PTFE films develops

differently at various depth levels; the lamellae were nucleated directly on the surface of

the first ultrathin layer of PTFE crystals and grew into three dimensions. This thickness

dependence of morphology was thought to be critical to the mechanical properties of

PTFE. Since there are always a few of amorphous regions left between the PTFE

lamellae, the thin-film PTFE coating exhibits a lower modulus and hardness at its

amorphous state [60].

Rae and Brown [62] explored mechanical properties of bulk PTFE by using a Hopkinson

bar in tension tests. The tensile tests for samples of Dupont 7A and 7C Teflon (PTFE)

were conducted at a certain range of strain-rates 2 × 10-4

– 0.1 s-1

and temperatures -50 to

23° C. It was found that the tensile mechanical properties of PTFE, such as Young’s

modulus and yield stress1, are affected by strain-rate and temperature, but only to a

limited extent by crystallinity. From the stress-strain diagrams tested at different

temperatures and strain-rates, it was found that bulk PTFE behaves like elastic-plastic

materials until the strain reaches over 50%.

1 Rae and Brown reported that the Young’s modulus was determined from the initial tangent modulus and

the yield stress was calculated with a 2% offset [62].

62

Table 3.2. Tensile properties of bulk and free-standing films at room temperature [58].

Note that free-standing film is tested by different authors.

Materials Young’s Modulus (GPa) Yield Strength (MPa)

Bulk sample 0.41 9.0

15 mm free-standing film 0.4 10.3

15 mm free-standing film 0.4 17

Rae and Dattelbaum [63] also investigated mechanical properties of bulk PTFE by using

compression tests. Same as the tension tests, samples of Dupont 7A and 7C Teflon (PTFE)

were tested in compression at strain-rates in range of 10-4

– 1 s-1

and temperatures in

range of -198 to 200° C. Also, mechanical properties of PTFE were investigated at both

large and small strains in terms of its ductile properties. Experimental results showed the

mechanical properties are significantly affected by strain-rate and temperature.

Additionally, it was found that the Poisson’s ratio at small strains in tension was roughly

0.36, which was different from 0.46 obtained in compression tests.

Table 3.3. Mechanical properties of bulk PTFE in the tension test [62] (strain rate, =

5×10-3

s-1

).

Temperature

(˚ C)

Young’s Modulus

(GPa)

Yield Stress

(MPa)

Failure

Strain

Failure Stress

(MPa)

0 1.29 16.43 1.04 114.29

23 0.79 11.43 1.43 132.14

50 0.51 5.72 1.64 139.29

100 0.29 3.57 1.68 103.57

63

Table 3.4. Mechanical properties of bulk PTFE in the compression test (strain rate, =

10-3

s-1

). Note that failure strain and stress were not reported in this compression test [63].

Temperature

(˚ C)

Young’s Modulus

(GPa)

Yield Stress

(MPa)

0 0.82 16.38

26 0.43 8.57

50 0.38 7.53

100 0.21 4.29

3.2.2 Viscoelastic and Plastic Properties of PTFE

PTFE exhibits viscoelastic properties since it was found that its COF decreases with

slower sliding speeds, higher normal force, and high temperatures, which was believed to

be associated with the occurrence of relaxation between molecular chains in the

amorphous region [52]. Also, its viscoelastic characteristics can be explained by the

peculiar molecular structure of PTFE. In particular, its smooth molecular profile gives

rise to low COF and such low value is obtained once a transfer film with an oriented

molecular structure is created during the sliding. It is worth noting that remarkable

increase in the COF was found when the PTFE was irradiated to produce cross-linked

chains [52].

PTFE also exhibits temperature-dependent creep behavior. PTFE can have either

amorphous or semi-crystalline structure, where the degree of crystallinity and the size

and distribution of the crystallites have a large effect on the mechanical properties [64].

At temperatures well below the glass-transition, long molecular chains are rigid, resulting

64

in brittle behavior; however, at high temperature, backbone bonds rotate and allow

molecules to partially disentangle and move relative to one another [64].

Steijn [65, 66] examined the effects of viscoelasticity on the frictional behavior of PTFE,

and explored the effects of time, temperature and environment on PTFE in pure-sliding

tests. He concluded that the sliding behavior was influenced by the time lapse between

sliding experiments, sliding speed and thermal history of the sliding components.

Khan and Zhang [67] proposed a finite deformation viscoelasto-plastic constitutive

relation, which combined standard linear viscoelastic model and Khan, Huang, and Liang

(KHL)’s viscoplastic model [68] to characterize the strain-rate hardening, creep and

relaxation behavior of PTFE. They used the standard spring-dashpot model to represent

viscoelastic behavior and KHL model to describe viscoplastic behavior. Additionally, the

relaxation of PTFE only depends on initial strain with no effect on succeeding material

behavior; but, creep is influenced by both viscoelastic and plastic deformation, and has an

effect on subsequent material response.

Bergström and Hilbert [69] developed a new constitutive model, referred to as the dual

network fluoropolymer (DNF) model for predicting the time and temperature-dependent

mechanical behavior of PTFE. This model overcame the deficiency of Khan and Zhang’s

constitutive model [67], which can only predict the characteristics of fluoropolymers at

isothermal conditions. The DNF model incorporates experimental characteristics by

decomposing the material behavior of PTFE into a viscoplastic response and a

viscoelastic response. The viscoelastic response is further decomposed into the response

65

of two molecular networks acting in parallel: the first network holds the equilibrium

(long term) of the viscoelastic response and the second network the time-dependent (short

term) deviation from the viscoelastic equilibrium state. They modeled the mechanical

behavior of PTFE, in uniaxial tension and subsequent relaxation by finite element

analysis and found that numerical prediction matched well with experimental results, as

shown in Figure 3.3 and 3.4.

Figure 3.3. Comparison between experimental data and predicted behavior in uniaxial

tension at different strain-rate (T = 20° C, strain-rates, : 1.2×10-3

/s and 2.3×10-4

/s) [69].

66

Figure 3.4. Comparison between predicted and experimental stress relaxation results

[69].

3.2.3 Frictional Characteristics

The frictional characteristics of bulk PTFE was initially investigated by Makinson and

Tabor [55], who showed that the polymer is transferred to the slider surface. The

experiments on both glass and PTFE surfaces showed two separate friction regimes that

depend on sliding speed and temperature. Microscope observations showed a very thin

film was drawn over the surface, where the molecules inside the film were oriented with

the molecular chains parallel to the sliding direction. The low friction values during the

sliding tests were possibly caused by the fact that the shearing of the slices within the

crystal made the disordered regions slip at very small shear stresses.

Pooley and Tabor [70] investigated the relationship between the frictional behavior of

PTFE and its molecular structure. They observed that the coefficient of friction fell to a

low value (µ < 0.1) at the beginning of sliding, once the slider acquired a very thin

67

transfer film of PTFE with preferred orientation. However, the bulk properties of PTFE

were thought to be responsible for the high static friction, because within the bulk of the

PTFE lumpy transfer of PTFE was observed. This was attributed to strong interfacial

adhesion and shearing. They concluded that the frictional characteristics of the polymer

were closely related to the rigidity and smooth profile of the molecules instead of the

degree of crystallinity or the crystalline texture of the polymers.

Briscoe and Tabor [71] investigated the role of mechanical properties on friction of

polymers. They separated the friction force into two components, the adhesion

component and the ploughing component. Figure 3.5 shows a schematic from their work,

where the thin interfacial region has thickness t2. The energy is dissipated in processes

which were believed to resemble plastic shear or fracture in interfacial layers of the

polymer or at the original interface. However, once a large amount of material has been

removed by the rigid slider, marked by the thickness t1, the ploughing component of

friction force arises due to bulk deformation of the material. Figure 3.6 shows that the

plastic losses or viscoelastic losses could occur due to energy dissipation in the bulk

deformation region. In the case of adhesion component, the frictional force is the product

of shear stress and the real contact area. However, practically, it is difficult to measure

the real contact area, therefore evaluations often have to rely on mathematical modeling,

combined with an understanding of the bulk deformation of the solid.

68

Figure 3.5. Schematic of interfacial and bulk regimes of friction [71]. Note that W is the

normal force, Ffric is the friction force, t1 is the thickness of bulk region, t2 is the thickness

of interfacial region.

Figure 3.6. The bulk deformations due to plastic flow and viscoelastic losses [71].

69

Figure 3.7. The rolling friction of a rigid sphere on bulk PTFE and the quantity, E-1/3

tan(δ), as a function of temperature [71].

For the case of a sphere with radius, R, indenting a flat at low loads, the real contact area,

Ar, can be evaluated based on the elastic deformation at the contact interface as follows:

2/3

*

3

4r

WRA

E

(3.1)

where W is the normal force, E* is the composite modulus, mathematically expressed in

Equation (2.15).

As the normal load is increased permanent plastic deformation takes place in the material.

The real contact area, Ar, for a fully developed subsurface plastic region can be shown as

follows:

r

WA

p (3.2)

70

where p is the flow stress, or mean contact pressure of the bulk polymer.

The shear strength τ of high density thin-film polymers was investigated as a function of

applied mean pressure, p, and reported as follows [71]:

0 p (3.3)

where α is the pressure coefficient, τ0 is a parameter, typically a factor of ten lower than

the shear strength of the bulk polymer.

The COF value, µ, can then be evaluated by combining Equations (3.2) and (3.3) as

follows:

0 0fric r

r

F pA

W p A p p

(3.4)

Briscoe [72] assumed that the bulk contact pressure, p, is approximately equal to the

hardness of the polymer at heavy loads. Thus, the COF, μ, at high loads can be estimated

to α since the constant τ0 is rather small compared to the hardness of the polymer, p. He

also found that the COF, µ, calculated in Equation (3.4) matched well with the

experimental measurement of PTFE sliding test on glass, which verified the frictional

work dissipated within a thin surface layer. However, at low loads the COF, μ,

significantly depends upon the surface topography and Equation (3.4) was not applicable

[71].

71

As for the ploughing component of friction, Figure 3.6 illustrates the two situations of

simple plastic flow and viscoelastic grooving occurring at the interface. These are closely

related to the bulk properties of the polymers. The COF, µ, due to the energy loss of

viscoelastic grooving was given as follows [71]:

1/3 1/31/3 2/3 21 tan

2W R E

(3.5)

where W is the normal load, R is indenter radius, E, ν are Young’s modulus and Poisson’s

ratio, respectively, tan(δ) is the loss tangent of the material. Figure 3.8 shows the loss,

1/3

tanE

, as a function of temperature and that the variation of the rolling COF, μr,

with temperature behaves in the same way. The good agreement between theory and

experiment indicated that mechanical properties of the polymer largely govern the

frictional losses.

More recently, Myshkin et al. [73] investigated the effects of load, sliding speed and

temperature on the friction of thin PTFE films. They found that the COF, µ, decreases

with increasing the load in a range of moderate normal loads 0.02 – 1 N, but remains

constant with the load in the range of 10 – 100 N. They also report that the speed-

independent friction was only within a limited range of speed (0.1-10 mm/s) for PTFE.

However, for higher speed values the friction force depends only slightly on the speed.

The sliding speed has a pronounced effect on friction near the glass-transition

temperature, Tg, while friction hardly depends on the sliding speed at lower temperatures

since the segments of main molecular chain were frozen [74]. The trend that the COF, μ

72

increases by increasing the sliding speed was found in several polymer films, including

Polytetrafluoroethylene (PTFE), Polypropylene (PP), high density polyethylene (HDPE),

low density polyethylene (LDPE), Polystyrene (PS), which were investigated by Briscoe

and Tabor [75, 76]. However, Polymethylmethacrylate (PMMA) shows distinctly

different speed-dependence effect on the COF because viscoelastic retardation in

compression was believed as a significant factor in controlling the shear strength below

the glass transition temperature while the strain rate in shear predominated in other

polymers [76]. As for the effects of temperature, the heat induced during sliding results

from the deformation of material in the actual contact spots and the breakdown of

adhesion bonds. This gives rise to the correlation of COF with hardness and shear

strength. It was believed that adhesion is the basic mechanism of friction of polymers at

the highly elastic state over smooth surfaces or when the polymer was heated around the

glass-transition temperature.

Jia et al. [77] investigated the relations between cohesive energy density2 (CED) and

tribological properties in sliding of two polymers. They found that for similar polymer

combinations the friction coefficient is higher when the CED has a high value, as shown

in Figure 3.8. However, Figure 3.9 indicates that for two dissimilar mated polymers the

2 The ratio of cohesive energy over molar volume for monomer unit or atomic group in unit of (J/cm

3),

which provides a criterion to measure the strength of secondary bonds in a polymer or between two

polymers [75].

73

lower friction coefficient is associated with the absolute value of the difference in the

CED values, i.e. larger difference of CED of the two mated polymers gives rise to smaller

coefficient of friction. Thus, they concluded that the sliding friction properties of a

polymer-polymer pair are significantly influenced by the adhesion between two polymers

in contact. Additionally, they examined the wear behavior of various polymer-polymer

combinations with the relations of cohesive energy density (CED) [77]. They found that

the wear rate for similar polymer-polymer combination is decreased with increasing CED,

as shown in Figure 3.8. In Figure 3.9(b), it was observed that the wear rate of dissimilar

polymer-PTFE combinations appeared not to be closely associated with the CED

difference.

Wieleba [78] studied the effects of a number of steel counterface roughness parameters,

on the COF of PTFE composites. He found that the shape of the asperities has the most

significant impacts on the COF, while the height of asperities has the most significant

effects on wear of the PTFE composites.

The draw direction, which easily forms the orientation of the molecular chains of linear

polymer, with respect to friction-induced direction is one of important factors that affect

the tribological properties of PTFE [79]. Liu et al. investigated the dependence of COF

on the orientation of molecular chains of drawn bulk PTFE by using a pin-on-disc tester

[80]. They performed the sliding tests of the drawn PTFE along three different sliding

directions, including parallel with the draw direction, transverse to the draw direction and

perpendicular to the draw direction. They observed that the COF depends on sliding

74

direction; friction-induced orientation occurs on the worn surfaces; and, the transfer films

with high crystallinity are formed on the counterparts.

Figure 3.8. Relations between the cohesive energy density (CED) of the polymers and

friction coefficient, wear rate for similar polymer-polymer combinations [77].

75

(a)

(b)

Figure 3.9. Frictional and wear characteristics of PTFE with respect to cohesive energy

density (CED). (a) Relations between friction coefficient and the difference in CED for

dissimilar polymer-PTFE combinations; (b) Relations between the wear rate of polymer

pin and the difference in CED for dissimilar polymer-PTFE combinations [77].

76

3.2.4 Wear Characteristics

It is known that PTFE has much higher wear rate on the order of 10-7

cm/cm than other

crystalline polymers such as polyethylene, polypropylene, whose wear rates are on the

order of 10-9

cm/cm [81]. Tanaka et al. experimentally investigated the effects of heat

treatment, sliding speed and temperature on wear properties of PTFE, and found that the

wear rate is affected by the width of the bands 3 in the fine structure instead of the

crystallinity. Also, the effects of speed and temperature on the wear rate reflected the

viscoelastic nature of shear deformation at the amorphous region between crystalline

slices [81]. Hollander and Lancaster [82] investigated the effects of the topography of

metal counterface on the wear properties of the polymers and found that the wear rates of

polymers vary inversely with the average radius of curvature of the asperities, R, of metal

counterfaces. Thus, experimental results indicated that the topography of the counterface

for any polymer-metal combination is the predominant factor in determining the

magnitude of the wear rates of polymers. This is not only true for the initial stage of

sliding between fresh surfaces, but also at the later stages where the topography forms by

the sliding process itself. Additionally, the formation of transfer films was found on a

counterface during repeated sliding and believed to play significant roles in reducing the

3 Band: the structure of PTFE appeared as long bands with striations perpendicular to the length of the

bands. This structure is remarkably contrasted to the spherulitic structure of other crystalline polymers [79].

77

localized stresses and increasing the real contact area, which eventually has effects on

wear and the reduction in wear rate.

Steijn [83] discussed the characteristics of polymer wear by considering phenomena

occurring at the interface, including the formation of transfer films, metal pick-up and

surface melting. He focused on the polymer behavior from the response of plastics when

subjected to rubbing and rolling, and therefore separated the primary wear mechanisms

into adhesive and abrasive wear. The experiment results indicated that the wear

characteristics of polymers depend on the behavior of response to the dynamics of the

wear system, including a transfer film formation, surface melting, and elastic or

viscoelastic deformation at the contact region. Also, the wear properties of polymers were

varied and multifaceted since the material properties of polymers deviated from different

elastic moduli and melting points.

Lhymn [84] investigated the tribological failure sequence of PTFE and carbon-fiber-

reinforced PTFE on stainless steel plate by microstructural observation and wear rate

measurement. The wear rates of PTFE indicated that a material removal process by a

decohesional flaking mechanism occurs in adhesive wear, which produced extensive

heating at the contact area.

Bahadur [85] studied the development of transfer layers between polymer-polymer as

well as polymer-metal interfaces during sliding, and their role in mechanism of wear. He

found that the wear rate of polymer-metal pair was strongly influenced by the cohesion of

78

transfer film, adhesion of transfer film to the counterface, and protection of rubbing

polymer surface from metal asperities by the transfer film.

Myshkin et al. [73] discussed the common types of wear on polymers, including abrasion,

adhesion and fatigue. They found that PTFE is susceptible to friction transfer, which the

transfer of material from one surface to another occurrs due to localized bonding between

contacting solid surfaces, when rubbing against both metals and polymers. It was

discovered that PTFE is transferred in the form of flakes of very small size at the initial

stage of friction, which changed the roughness of both surfaces in contact. The roughness

of polymer surface underwent large deviations during unsteady wear until the steady

wear was reached, while metal surface roughness was modified due to transfer of PTFE.

However, he concluded that the effects of transfer film on the wear characteristics may

not be significant when small particles of soft polymer material with micrometer size

were transferred onto the hard mating surface, such as metals.

3.3 Summary

The literature survey presented the material properties of PTFE in bulk and thin film

forms. The effects of testing parameters on the frictional and wear characteristics of

PTFE were also investigated and shown as follows:

The COF depends on molecular structure, normal force, sliding speed,

temperature, surface roughness of substrate, adhesion and ploughing at the contact

region, sliding direction with respect to molecular chain orientation.

79

The wear rate depends on normal force, sliding speed, temperature, topography of

substrate, film transfer formation, elastic or viscoelastic deformation at the contact

region.

80

4 Friction and Durability of Thin PTFE

Films on Rough Aluminum Substrates

In this chapter, an experimental evaluation of the friction and durability characteristics of

thin PTFE films, deposited on aluminum substrates with different surface roughness is

presented.

4.1 Introduction

Polytetrafluoroethylene (PTFE) is a chemically inert engineering polymer with low

coefficient of friction (COF), and low dielectric constant [3]. In general PTFE is used as a

low friction coating in thrust bearings [86], and also as a load bearing surface in

applications that require low frictional resistance [87]. PTFE has also been used in thin

film form in micromechanical devices due to its low friction and low surface energy, and

in medical devices due to its chemical and thermal stability [88]. Recently, PTFE-carbon

nanotube (CNT) composites have attracted attention due to their improved wear

resistance [89]. Methods of thin-film PTFE deposition include spin- and dip-coating,

chemical vapor deposition (CVD), and hot-filament chemical vapor deposition (HFCVD)

[2]. The latter deposition technique allows thin-film PTFE to be deposited on almost any

type of substrate, including temperature sensitive materials [90].

81

A survey of the literature related to the frictional and wear characteristics of bulk PTFE

has been presented in Chapter 3.

4.2 Materials and Methods

Friction and durability characteristics of 1 μm thin, PTFE films deposited on aluminum

substrates by HFCVD technique were investigated. In particular, the effects of the normal

load, the sliding speed and the surface roughness were investigated by using a universal

micro-tribotester (UMT-2; CETR, Campbell, CA). The friction tests were conducted by

using the ball-on-plate configuration, schematically shown in Figure 4.1(a), where the

normal force and sliding speed were the independently controlled variables and the

tangential force was measured. A sliding distance of 25-mm was used in the tests. The

coefficient of friction (COF) was defined as the ratio of the tangential force to the normal

force. The reported COF values were chosen from the steady-state regime of the tests,

and each parameter combination was repeated at least five times. In the durability tests,

the ball-on-disk configuration of the instrument was used to monitor the COF of the

interface. Figure 4.1(b) shows such a configuration, where the substrate is attached to a

rotary table. The tests were stopped when the dynamic COF reached the solid-on-solid

friction value. The radius of the test track on aluminum substrates was set to 4 mm. Each

test was repeated at least three times for each testing parameter. The number of tests was

increased when the standard deviation was large.

82

(a) Ball-on-plate configuration for friction test

(b) Ball-on-disk configuration for durability test

Figure 4.1. Configurations of PTFE frictional and durability tests.

Steel (Rockwell hardness of C60-67), spherical balls with the diameter of 6.35 mm were

used as indenters. All of the balls were cleaned with hand soap before the tests, in order

to eliminate the effects of smeared materials. 5052 aluminum was used as the substrate.

83

All the tests were performed in the laboratory environment, where the temperature was in

a range of 20.1 to 27.3° C and the relative humidity (RH) was between 12% to 38%. The

majority of tests were conducted at 25° C and 30% RH.

Thin PTFE films (1 μm thick) were deposited on roughened aluminum plates by using the

HFCVD technique [3]. The surface roughness of the 5052 aluminum substrates were

modified by polishing and by bead blasting with 80, 120, 150, 180, 220 and 320-grit

silica particles. After the tests were completed, the wear tracks were characterized by

using an optical microscope (Meiji, ML 8500).

4.3 Results

The correlation between the various techniques used in this work to adjust the surface

roughness and the average surface roughness (Ra) and the variance of surface roughness

(σ2) are presented in Table 4.1. Figure 4.2 shows the relation between the Ra of the

surface and the size of the silica particles, quantified by units of grit, g, and by mean

particle diameter, dp. Figure 4.2 shows that the following relationships can be established

between Ra versus g, and Ra versus dp,

0.70745.017aR g (4.1-a)

0.6401exp 0.0067a pR d (4.1-b)

Note that the R2 values for these relationships are 0.9122 and 0.9812, respectively.

84

In order to establish a baseline solid-on-solid COF at the aluminum-steel interface,

friction tests were conducted on uncoated roughened aluminum plates. Figure 4.3 shows

the measured COF values as a function of normal load and surface roughness. It is seen

that the COF at the aluminum-steel interface is greater than 0.6. This value of COF was

set as the determinant of failure of the PTFE coating for friction and durability tests of

PTFE coated on substrates.

Table 4.1. Surface roughness parameter of Ra and σ2 for aluminum substrates. Note that

the mean particle diameter dp for each grit size is adapted from Orvis et al. [91].

Aluminum

substrates

Ra

(μm)

Std. Dev.

(μm)

σ2

(μm2)

Std. Dev.

(μm)

dp

(μm)

80-grit 2.340 0.620 9.820 5.430 190

120-grit 1.285 0.035 1.680 0.042 115

150-grit 1.280 0.230 2.640 0.900 92

180-grit 1.110 0.099 1.405 0.120 82

220-grit 0.995 0.007 1.270 0 68

320-grit 0.820 0.014 1.030 0.014 36

Polished 0.010 0.003 0.0002 0.0001

85

(a)

(b)

Figure 4.2. Surface roughness, Ra as a function of grit size, g and mean particle diameter,

dp, respectively.

86

Figure 4.3. The COF of aluminum substrate without PTFE thin films.

4.3.1 COF

The effects of sliding speed and normal force on the COF between the steel, spherical

slider and the PTFE coated aluminum plates are shown in Figure 4.4 and in Table 4.2.

Statistical analysis of the result is presented in Table 4.3-4.7. In addition to the results

presented in Figure 4.4, the effects of sliding speed and normal force on the COF were

also observed in the durability tests. Those cases are presented later in this chapter. Here,

three sliding speed values of 0.1, 1 and 5 mm/s, and four normal load values of 2.5, 5, 10

and 15 N were used. The Ra values of the surface roughness of the plates were 0.01, 0.57,

1.28 and 2.34 µm. Figure 4.4(a) shows the test results for Ra = 1.28 and 2.34 µm. It is

seen that the COF increases with sliding speed in the speed range of 0.1-5 mm/s.

Similarly a trend of increasing COF with increasing normal load in the load range of 2.5-

87

15 N is observed. It also appears that the COF decreases somewhat when the normal load

becomes 15 N. Figure 4.4(b) gives the COF values for the same parameters, but for the

two smoother surfaces with Ra = 0.57 and 0.01 µm. In the case of Ra = 0.57 µm, the COF

is clearly reduced with increasing normal load. A clear increase in COF is seen when the

sliding speed is increased from 0.1 mm/s to the higher values, but no clear change is seen

between 1 and 5 mm/s. In the case of the smoothest surface it is seen that the COF is

nearly independent of the load but depends on the sliding speed.

Results presented in Figure 4.4 show a non-linear relationship between the COF and the

independent variables; normal force, sliding speed and surface roughness. The average

COF values and its corresponding standard deviations for all the variables in this test are

given in Table 4.2. In order to assess the significance of the functional dependencies

stated above, analysis of variance (ANOVA) of the experimental data was carried out.

The null hypotheses tested were that load/speed/roughness has no significance on the

COF. In addition, the effects of the interactions between the variables were also tested. A

three way ANOVA was conducted with the degrees of freedom as shown in Table 4.3.

For each variable the p-value is smaller than 0.05, which indicates that the null

hypothesis is rejected, and that the normal load, the sliding speed, and the surface

roughness, and all three of their combinations have significant effects on COF. By

normalizing the sum of the square (SS) it is possible to obtain the relative contributions

of each factor when the entire data set (Table 4.2) is analyzed as a whole. This shows that

96.36% of the COF is due to surface roughness effects. Sliding speed contributes 1.97%

88

to COF, whereas the contribution of normal load (0.05%) is lower than the error (0.33%).

In order to investigate the effects of normal load and sliding speed on the COF for surface

roughness value, two-factor ANOVA tests were conducted, as shown in Table 4.4 - 4.7.

This analysis shows that sliding speed and normal load and their combinations have

significant effects (p < 0.05) on the COF, except for the case of Ra = 0.57 µm, where the

interaction effects of load and speed are rejected by the ANOVA. The analysis shows that

for each of the rough substrates (Ra = 0.57, 1.28 and 2.34 µm) the sliding speed has more

significance on the COF as compared to normal load. In particular, for the polished

substrate (Ra = 0.01 µm) the sliding speed contributes 97.6% to COF whereas the

contribution of the normal load is negligible.

89

(a) Ra = 1.28 μm and 2.34 μm

(b) Ra = 0.01 μm and 0.57 μm

Figure 4.4. The COF of PTFE on aluminum substrates with different roughness. (a) Ra =

1.28 μm and 2.34 μm; (b) Ra = 0.01 μm and 0.57 μm.

90

Table 4.2. Experimental data for frictional characteristics of 1 μm PTFE coating

deposited on glass substrates.

Normal force

(N)

Sliding speed

(mm/s)

Ra (μm) of

substrates COF (ave.)

COF

(std.dev.)

2.5 0.1 2.34 0.2927 0.0144

5 0.1 2.34 0.3122 0.0059

10 0.1 2.34 0.3202 0.0128

15 0.1 2.34 0.3253 0.0036

2.5 1 2.34 0.3141 0.0128

5 1 2.34 0.3405 0.0136

10 1 2.34 0.3573 0.0017

15 1 2.34 0.3541 0.0069

2.5 5 2.34 0.3356 0.0182

5 5 2.34 0.3452 0.0084

10 5 2.34 0.3540 0.0077

15 5 2.34 0.3358 0.0088

2.5 0.1 1.28 0.2579 0.0083

5 0.1 1.28 0.2513 0.0090

10 0.1 1.28 0.2654 0.0052

15 0.1 1.28 0.2513 0.0023

2.5 1 1.28 0.2696 0.0124

5 1 1.28 0.2729 0.0060

10 1 1.28 0.2862 0.0059

15 1 1.28 0.2868 0.0030

2.5 5 1.28 0.2815 0.0050

5 5 1.28 0.2818 0.0055

10 5 1.28 0.2994 0.0053

15 5 1.28 0.2871 0.0117

2.5 0.1 0.57 0.1579 0.0025

5 0.1 0.57 0.1469 0.0033

10 0.1 0.57 0.1343 0.0046

15 0.1 0.57 0.1318 0.0074

2.5 1 0.57 0.1810 0.0043

5 1 0.57 0.1682 0.0041

10 1 0.57 0.1608 0.0055

15 1 0.57 0.1630 0.0042

2.5 5 0.57 0.1778 0.0061

5 5 0.57 0.1670 0.0029

10 5 0.57 0.1552 0.0047

15 5 0.57 0.1508 0.0023

2.5 0.1 0.01 0.0275 0.0014

5 0.1 0.01 0.0329 0.0006

10 0.1 0.01 0.0377 0.0005

15 0.1 0.01 0.0422 0.0003

2.5 1 0.01 0.0527 0.0017

5 1 0.01 0.0500 0.0008

10 1 0.01 0.0542 0.0006

15 1 0.01 0.0579 0.0006

2.5 5 0.01 0.0971 0.0009

5 5 0.01 0.0995 0.0062

10 5 0.01 0.0987 0.0019

15 5 0.01 0.0954 0.0008

91

Table 4.3. ANOVA test for frictional tests of 1 μm PTFE coating on aluminum substrates

(The response is COF).

Source SS df MS F p Fcri %TTS

Load 0.00135 3 0.00045 9.42 p < 0.05 2.65 0.05

Speed 0.05339 2 0.0267 559.92 p < 0.05 3.04 1.97

Roughness 2.61615 3 0.87205 18290.25 p < 0.05 2.65 96.36

Load × Speed 0.00164 6 0.00027 5.74 p < 0.05 2.14 0.06

Load × Roughness 0.01406 9 0.00156 32.76 p < 0.05 1.92 0.52

Speed × Roughness 0.01692 6 0.00282 59.14 p < 0.05 2.14 0.62

Load×Speed×Roughness 0.00238 18 0.00013 2.77 p < 0.05 1.62 0.09

Error 0.00915 192 0.00005 0.33

Total 2.71504 239

Table 4.4. Two-way ANOVA test for PTFE coatings on Ra = 2.34 μm aluminum

substrates.

Source SS df MS F p Fcri %TTS

Load 0.00751 3 0.0025 22.1437 p < 0.05 2.80 27.75

Speed 0.01159 2 0.0058 51.2621 p < 0.05 3.19 42.83

Interaction 0.002536 6 0.0004 3.7411 0.00392 2.29 9.38

Error 0.005424 48 0.0001 20.05

Total 0.027052 59

Table 4.5. Two-way ANOVA test for PTFE coating on Ra = 1.28 μm aluminum

substrates.

Source SS df MS F p Fcri %TTS

Load 0.002129 3 0.0007 13.4469 p < 0.05 2.7981 13.58

Speed 0.010219 2 0.0051 96.8102 p < 0.05 3.1907 65.18

Interaction 0.000796 6 0.0001 2.5141 0.03388 2.2946 5.08

Error 0.002533 48 0.0001 16.16

Total 0.015678 59

Table 4.6. Two-way ANOVA test for PTFE coatings on Ra = 0.57 μm aluminum

substrates.

Source SS df MS F p Fcri %TTS

Load 0.00543 3 0.0018 87.3099 p < 0.05 2.7981 39.19

Speed 0.00722 2 0.0036 174.1259 p < 0.05 3.1907 52.11

Interaction 0.00021 6 3.497E-5 1.6868 0.14469 2.2946 1.51

Error 0.000995 48 2.073E-5 7.18

Total 0.013855 59

92

Table 4.7. Two-way ANOVA test for PTFE coatings on Ra = 0.01 μm aluminum

substrates.

Source SS df MS F p Fcri %TTS

Load 0.000337 3 0.000112 26.7897 p < 0.05 2.7981 0.80

Speed 0.041286 2 0.020643 4921.982 p < 0.05 3.1907 97.60

Interaction 0.000477 6 7.953E-5 18.962 p < 0.05 2.2946 1.13

Error 0.000201 48 4.194E-6 0.48

Total 0.042302 59

4.3.2 Durability

Durability of the 1 μm thick PTFE film was tested for a smaller set of variables due to the

long duration of such tests. Results are presented in Figure 4.5 and Table 4.8. Normal

force values of 2.5 and 5 N, and sliding speed values of 0.42 mm/s and 4.2 mm/s were

used on three rough (Ra = 0.57, 1.28, and 2.34 μm) and one smooth (Ra = 0.01 μm)

substrate. The sliding distance to failure was monitored during these tests, where failure

was assumed to take place when COF became greater than 0.6. Figure 4.5(a) shows that

sliding distance to failure has a clear dependence on the surface roughness of the

substrate. The smoothest surface, which has the lowest COF value, fails within 1.5 m of

sliding. The case of Ra = 0.57 μm survived for several hundred meters of sliding, whereas

the case of Ra = 2.34 μm failed within a few meters of sliding. Moreover, the sliding

distance to failure, and hence durability increased with increasing speed and decreasing

normal force. These differences were more apparent for the smoother surface (Ra = 0.57

and 1.28 μm), whereas all cases failed fast on the rough surface.

93

The COF values corresponding to the durability distances are plotted in Figure 4.5(b).

Interestingly, the comparison of the results indicates that low friction value to be related

to shorter durability. Tables 4.9 - 4.10 give analysis of variance (ANOVA) results for

durability and COF. The analysis shows that the normal load, sliding speed, surface

roughness and their combinations have significant effects on the durability and COF (p <

0.05), except for the combined effects of load and surface roughness, and load, speed and

roughness for the case of COF. The analysis also indicates that the roughness of the

substrate has the most significant effect on the durability, as compared to the normal load

and the sliding speed.

94

(a)

(b)

Figure 4.5. Durability characteristics of 1 μm PTFE on roughened and polished

aluminum substrates. (a) sliding distance to failure; (b) The COF.

95

Table 4.8. Experimental data for durability characteristics of 1 μm PTFE coating

deposited on aluminum substrates. Note that ave. represents average value and std.

represents standard deviations.

Normal force

(N)

Sliding

velocity

(mm/s)

Ra (μm) of

aluminum

substrates

Sliding

distance

- ave. (m)

Sliding

distance

- std. (m)

COF

(ave.)

COF

(std.)

2.5 4.19 2.34 0.94 0.15 0.251 0.021

2.5 0.42 2.34 28.24 13.46 0.206 0.024

2.5 4.19 1.28 225.89 55.82 0.220 0.036

2.5 0.42 1.28 117.51 5.22 0.214 0.003

2.5 4.19 0.57 906.16 252.49 0.193 0.014

2.5 0.42 0.57 252.53 151.48 0.142 0.009

2.5 4.19 0.01 0.75 0.06 0.069 0.002

2.5 0.42 0.01 1.14 0.56 0.050 0.002

5 4.19 2.34 0.71 0.07 0.197 0.003

5 0.42 2.34 19.90 32.82 0.142 0.017

5 4.19 1.28 153.26 43.42 0.148 0.019

5 0.42 1.28 89.81 43.58 0.143 0.012

5 4.19 0.57 364.56 89.19 0.137 0.004

5 0.42 0.57 212.22 52.94 0.104 0.007

5 4.19 0.01 0.62 0.17 0.068 0.002

5 0.42 0.01 1.14 0.56 0.051 0.002

Table 4.9. ANOVA test for durability tests of 1 μm PTFE coating on aluminum

substrates (The response is sliding distance to failure).

Source SS df MS F p Fcri %TTS

Load 89380.0 1 89380 13.54 0.0009 4.15 3.39

Speed 162328.2 1 162328.2 24.59 p < 0.05 4.15 6.16

Roughness 1460745.8 3 486915.3 73.75 p < 0.05 2.90 55.44

Load × Speed 54390.5 1 54390.5 8.24 0.0072 4.15 2.06

Load × Roughness 172180.6 3 57393.5 8.69 0.0002 2.90 6.54

Speed × Roughness 348633.0 3 116211 17.6 p < 0.05 2.90 13.23

Load×Speed×Roughness 135644.6 3 45214.9 6.85 0.0011 2.90 5.15

Error 211282.8 32 6602.6 8.02

Total 2634585.5 47

96

Table 4.10. ANOVA test for durability tests of 1 μm PTFE coating on aluminum

substrates (The response is COF).

Source SS df MS F p Fcri %TTS

Load 0.02551 1 0.02551 126.27 p < 0.05 4.15 12.62

Speed 0.01114 1 0.01114 55.13 p < 0.05 4.15 5.51

Roughness 0.14436 3 0.04812 238.22 p < 0.05 2.90 71.40

Load × Speed 0.00014 1 0.00014 0.68 0.4163 4.15 0.07

Load × Roughness 0.00942 3 0.00314 15.55 p < 0.05 2.90 4.66

Speed × Roughness 0.00505 3 0.00168 8.33 0.0003 2.90 2.50

Load×Speed×Roughness 0.00011 3 0.00004 0.18 0.9101 2.90 0.05

Error 0.00646 32 0.0002 3.20

Total 0.20218 47

4.4 Discussion

Makinson and Tabor [55], and Briscoe and Tabor [71] found that the COF of the bulk

PTFE increases by increasing the sliding speed or decreasing the normal force and the

temperature. Karnath et al. reported similar observations for the thin-film PTFE [3].

Experimental evidence also shows film transfer has significant effects on the COF [55].

In this work, similar speed and load dependencies on the COF were observed.

Briscoe and Tabor [71] observed that the COF of polymer films, deposited on smooth

glass substrates, are dominated by their intrinsic shear property. In this adhesion

dominated mode, the COF, μ, for polymers was found to depend on the shear strength, τ,

as follows,

0 0fric r

r

F τ αp ττ Aμ α

W p A p p

(4.2)

97

where Ffric is the friction force, W is the normal force, Ar is the real contact area, τ is the

shear strength which, for polymers, depends on a temperature-dependent constant value

τ0 and a pressure-dependent value αp with α as the pressure coefficient. The average

contact pressure, p, can be evaluated as follows,

r

Wp

A (4.3)

2/3

*

3

4r

WRA

E

(4.4)

where2 21 2

1 2

1 11

E EE

, Ei, νi (i = 1, 2) are the Young’s modulus and Poisson’s ratio of

the two contacting bodies, respectively, and R is the radius of indenter. By combining

Equations (4.2) – (4.4), the following expression is obtained,

2/3

0

1/3

3

4

R

W E

(4.5)

For PTFE films, the constants in Equation (4.5) have been reported as τ0 = 1×106 Pa, α =

0.08 [76]. By considering the effects of sliding speed on the COF, the following

relationship is suggested [75],

2/3

01 2 3 41/3 *

3ln exp

4

Rc v c c v c

W E

(4.6)

98

where v is the sliding speed and c1, c2, c3 and c4 are empirical constants. These constants

are determined by using the experimental results on smooth aluminum substrate (Ra =

0.01 μm), and given in Table 4.11.

The COF as a function of normal force, W, is shown in Figure 4.6. For the thin film PTFE

on smooth substrate the friction is dominated by adhesion effects and depends on sliding

speed, v; as well as the intrinsic constants τ0 and α are given by Equation (4.6).

On the other hand, the COF values on rough aluminum substrates are greater than that on

smooth substrates as shown in Figure 4.4. This difference is attributed to the solid-solid

contacts between the substrate and the indenter.

Table 4.11. The determination of parameters in Equation (4.6) by using curve-fitting.

2.5 N 5 N 10 N 15 N

c1 0.349 0.501 0.320 0.485

c2 1.023 1.002 1.062 1.063

c3 0.158 0.165 0.172 0.137

c4 -0.785 -0.811 -0.749 -0.635

99

Figure 4.6. The COF comparisons between the prediction by Equation (4.6) and

experiment measurement.

The durability tests presented here are different than the traditional wear tests, where the

PTFE films are severely removed from the substrates. Figure 4.7 shows microscopic

observations of the substrates after failure for the case of 5 N normal force and 0.42 mm/s

sliding speed. Figure 4.8 shows typical COF histories for different aluminum substrates

during the durability tests. The plots indicate that the PTFE film stabilizes the interfacial

COF to 0.06, on the mirror-polished substrate (Ra = 0.01 μm) while on the rougher

substrates, much larger variations are seen. Figure 4.8(e) shows the frictional test of the

steel ball on the uncoated aluminum substrate. Comparison of Figure 4.8(a) and 4.8(e)

indicates that the large variations in the COF could be due to solid-solid contact.

Interestingly, low COF on the smooth substrate yields lower durability. However, the

100

rough surfaces with medium roughness (Ra = 0.57 and 1.28 μm) give much longer

durability. It is possible that the rough interfaces act as a good reservoir for the storage of

PTFE debris and significantly reduce the wear rate to allow the interface last longer.

Various degrees of mechanical interlocking effects appear on the rough surfaces, which

gradually yield shorter durability. These results can be used to find an optimal roughness

for the PTFE-aluminum interface for improved durability.

The wear mechanism of PTFE is determined by a combination of adhesive and abrasive

wear modes. Adhesive wear is usually associated with conditions in which the asperities

in contact are sheared and the fragments are adhered and removed repeatedly between

two contacting surfaces [92]. Abrasive wear often takes place when a rough hard surface

slides against a softer surface, and the interface is damaged by ploughing and fracture

[92]. Lhymn showed experimentally that different wear mechanisms have significant

effects on the removal volume of PTFE [84]. In particular, the wear depends on the

frictional heat generated at the contact region, proportional to the normal force. Heat

generation induces a very large amount of material removal due to decohesional flaking

[84].

The prediction of COF using the modified Briscoe and Tabor equation, shown in Figure

4.6, and the COF history in Figure 4.8(a) indicate that adhesion governs the sliding and

durability of the PTFE film in the case of smooth aluminum substrate. The durability

tests on the rough substrates (Figure 4.7b-d) show a distinctly different wear mode, which

is attributed to an abrasive process.

101

(a) Ra = 0.01 μm (b) Ra = 0.57 μm

(c) Ra = 1.28 μm (d) Ra = 2.34 μm

Figure 4.7. Worn surfaces of PTFE thin films on aluminum substrate with different

roughness Ra (normal force is 5 N, sliding speed is 0.42 mm/s).

102

(a) Ra = 0.01 µm (b) Ra = 0.57 µm

(c) Ra = 1.28 µm (d) Ra = 2.34 µm

(e) Ra = 0.57 µm

Figure 4.8. COF histories on aluminum substrates with different surface roughness. Note

that (a) - (d) were tested on 1 µm PTFE films deposited on aluminum substrate; (e) was

tested on aluminum substrate without PTFE films.

103

4.5 Summary and Conclusion

The goal of this work was to characterize the friction and durability characteristics of thin

PTFE films on roughened aluminum substrates. The following conclusions are reached

based on this work:

The experiments showed that the COF and durability of PTFE film depend on

normal force, sliding speed and surface roughness of substrate.

In particular, the COF increases by increasing the surface roughness, Ra, and the

sliding speed. The durability improves by increasing sliding speed or decreasing

the normal force, but has a non-linear relationship with surface roughness, Ra.

ANOVA analysis indicated that the surface roughness of substrate has the most

significant effects on the COF and durability. The sliding speed contributes more

to the COF and durability as compared to the normal force.

The COF histories of durability tests on different aluminum substrates showed

that the PTFE film well lubricates the smooth surface and stabilizes the COF to

0.06, while it does not lubricate the rough surfaces well, indicating solid-solid

contacts.

The modified equation based on Briscoe and Tabor model predicted the COF on

smooth substrate well, indicating that the adhesion governs the sliding and

durability process.

The durability on rough surfaces was much likely to be associated with abrasive

wear mode.

104

The test results gave guidelines for designing an interface of PTFE film and rough

substrate for improved durability.

105

5 Frictional Characteristics of Thin PFA

and Silicone Films on Glass Substrates

In this chapter, we report on the characterization of the frictional properties of thin PFA

and poly(V3D3) films as a function of film thickness, normal force and sliding speed. A

brief introduction is given to frictional properties of these two polymers. Analysis of

variance (ANOVA) was used to seek correlations between the variables.

5.1 Introduction

PFA (Perfluoroalkoxy) is a polymer in the fluoropolymer family with a molecular

structure similar to PTFE, as shown in Figure 5.1. Physical properties of this polymer,

including chemical stability, high temperature resistance, low COF and anti-stiction are

also similar to PTFE [93]. These extraordinary properties allow it to be widely used as

plastic hardware in laboratory environment, as well as tubing in applications involving

highly corrosive environment. The dielectric properties of PFA are superior to that of

PTFE; for example, the dielectric strength of 100 μm thick PFA-film is four times higher

than the similar PTFE-film [93]. Processing of bulk PFA is easier than PTFE in typical

methods for thermoplastics, including extrusion, injection and moulding, as PFA does not

demonstrate high viscosity at high temperatures.

106

The second thin polymer film that we examined is poly(trivinyltrimethylcyclotrisiloxane),

poly(V3D3), developed by GVD corporation (Cambridge, MA), by using initial chemical

vapor deposition (iCVD)4, a relatively new technique [4]. The polymerization process

includes trivinyl-trimethyl-cyclotrisiloxane (V3D3) treated as a monomer and it is

initiated with tertbutyl peroxide (TBP). The all-dry deposition process generates a highly

cross-linked matrix material, shown in Figure 5.2. Poly(V3D3) coating exhibits high

adhesive strength to silicon substrates and is insoluble in both polar and nonpolar

solvents. Extraordinary dielectric properties were also observed compared to other

polymer materials, for example, parylene-C, and its non-cytotoxic properties allow

potential applications in medical devices [5].

The Young’s modulus and hardness of PFA are reported as 586 MPa and 60 MPa,

respectively [94]. The material properties of poly(V3D3) is not clear. The friction and

durability characteristics of theses polymers have not been thoroughly reported from the

review of literature, which motivates the experimental investigations in this work.

4 iCVD: this technique uses a radical generating initiator species, combined with the monomer and fed to

the CVD reactor. When both species pass over a resistively heated filament array, the temperature is high

enough for the weak initiator to break bonds while the chemical structure of monomer remains unaffected.

107

(a) PTFE (b) PFA

Figure 5.1. Schematic representation of PTFE and PFA molecule formulae.

Figure 5.2. Schematic representation of poly(V3D3) molecular structure. Note that the

hexagonal units show the intact siloxane rings, acting as cross-linking moieties for

backbone chains [4].

108

5.2 Materials and Methods

Friction characteristics of PFA and silicone films were tested by using the method

presented in Chapter 4. The steady-state COF at the interface of thin polymer films,

deposited on glass substrates were monitored. Three thick PFA films (0.3, 1, 5 μm) and

two thick silicone films (0.3, 1 μm) were tested. The normal forces were in the range of

2.5 - 15 N, and the sliding speeds during the test were between 0.01 - 1 mm/s. Each

parameter was measured at least three times.

5.3 Results

5.3.1 Friction Characteristics of Thin PFA Films

The COF for all three thick PFA films (0.3, 1, 5 μm) are presented in Figure 5.3. In

general, the highest sliding speed gives the highest COF values on 0.3 and 5 μm PFA

films while the sliding speed of 0.01 and 0.1 mm/s give comparably lower COF. For the 1

µm films the COF values are comparable for all speeds. The medium sliding speed, 0.1

mm/s gives lowest COF at a low load range of 2.5 – 5 N while 0.01 and 1 mm/s gives

higher COF at a low load range. In order to assess the significance of the normal load, the

sliding speed and the film thickness on the COF, analysis of the variance (ANOVA) of

the experimental data was performed. The null hypotheses of the analyses were

load/speed/film thickness have no significance on the COF. Additionally, the interaction

effects between the variables for COF were also tested. Table 5.1 shows the results of the

three-way ANOVA analysis with the specified degrees of freedom. The p-values for all

109

variables were found to be less than 0.05, indicating that the load, the sliding speed and

the film thickness and the interactions of these variables having significant effects on the

COF. The relative contribution of each variable was obtained by normalizing the sum of

squares (SS). This showed that the sliding speed and the film thickness contribute 26.67%

and 33.22% to the COF, respectively. These contributions are much greater than the

contribution from the normal force (4.23%). In order to evaluate the effects of normal

force and sliding speed on the COF for each separate film, two-way ANOVA analysis

were carried out, as presented in Tables 5.2-5.4. These results indicate that sliding speed

has the most significant effect on the COF as compared to the normal force and their

interactions for the 0.3 and 5 μm PFA films. For the 1 μm PFA films, the interaction of

sliding speed and normal force has the most significance on the COF, and the sliding

speed contributes more to COF as compared to normal force.

110

(a) 0.3 μm

(b) 1 μm

111

(c) 5 μm

Figure 5.3. The COF as a function of PFA film thickness.

Table 5.1. Three-way ANOVA for the COF of PFA thin films.

Source SS df MS F p Fcri %TTS

Load 0.0078 3 0.0026 33.28 p < 0.05 2.6895 4.23

Speed 0.0490 2 0.0245 314.51 p < 0.05 3.0812 26.67

Thickness 0.0610 2 0.0305 391.76 p < 0.05 3.0812 33.22

Load × Speed 0.0039 6 0.0007 8.36 p < 0.05 2.1845 2.13

Load × Thickness 0.0041 6 0.0007 8.75 p < 0.05 2.1845 2.23

Speed × Thickness 0.0309 4 0.0077 99.00 p < 0.05 2.4566 16.79

Load×Speed×Thickness 0.0215 12 0.0018 22.98 p < 0.05 1.8437 11.69

Error 0.0056 72 0.0001 3.05

Total 0.1838 107

Table 5.2. Two-way ANOVA for the COF of 0.3 μm PFA films.

Source SS df MS F p Fcri %TTS

Load 0.0017 3 0.0006 10.36 0.00015 3.0088 7.54

Speed 0.0129 2 0.0065 118.83 p < 0.05 3.4028 57.67

Interaction 0.0065 6 0.0011 19.89 p < 0.05 2.5082 28.96

Error 0.0013 24 5.429E-5 5.82

Total 0.0224 35

112

Table 5.3. Two-way ANOVA for the COF of 1 μm PFA films.

Source SS df MS F p Fcri %TTS

Load 0.0009 3 0.0003 12.60 p < 0.05 3.0088 9.86

Speed 0.0027 2 0.0014 56.74 p < 0.05 3.4028 29.60

Interaction 0.0050 6 0.0008 34.67 p < 0.05 2.5082 54.27

Error 0.0006 24 2.423E-5 6.26

Total 0.0093 35

Table 5.4. Two-way ANOVA for the COF of 5 μm PFA films.

Source SS df MS F p Fcri %TTS

Load 0.0093 3 0.0031 19.88 p < 0.05 3.0088 10.51

Speed 0.0642 2 0.0321 206.77 p < 0.05 3.4028 70.51

Interaction 0.0139 6 0.0023 14.89 p < 0.05 2.5082 15.23

Error 0.0037 24 0.0002 4.09

Total 0.0911 35

5.3.2 Friction Characteristics of Thin Silicone Films

The COF values for 0.3 and 1 μm thick silicone films are presented in Figure 5.4. The test

conditions, load and sliding speed, were the same as in the friction tests of the PFA films.

Note that 5 µm thick film was not available for these tests. In general, Figure 5.4

indicates that the COF increases by decreasing the sliding speed. This trend is opposite to

the one observed for the PTFE friction tests [3]. It is also seen that the COF decreases by

increasing the normal force, and eventually stabilizes. This trend is similar to that

exhibited by PTFE coatings. A film thickness dependence similar to PTFE is also

observed.

In order to evaluate the functional dependencies of the tested variables, ANOVA was

performed on the experimental data. Results of the three-way ANOVA test is shown in

113

Table 5.5, where the null hypothesis was stated as the load/speed/film thickness as well

as the interaction of these variables have no significance on the COF. The p-values for

the load, speed and thickness indicate the null-hypothesis is rejected, which shows that

load/speed/film thickness have significant effects on the COF. In particular, the film

thickness and sliding speed contribute 39.01% and 16.68% to the COF, respectively.

These are much greater than the other effects. In addition, the interactions of load and

speed, and thickness and speed also have significant effects on COF, whereas the

interactions of load and thickness (p > 0.05) do not.

Two-way ANOVA analysis was also carried out for each separate silicone film, in order

to investigate the effects of normal force and sliding speed on the COF. Tables 5.6 - 5.7

show the ANOVA analysis results, indicating that the sliding speed contributes more to

the COF as compared to normal force.

114

(a) 0.3 μm

(b) 1 μm

Figure 5.4. The COF as a function of silicone film thickness.

115

Table 5.5. Three-way ANOVA analysis for the COF of silicone films.

Source SS df MS F p Fcri %TTS

Load 0.0321 3 0.0107 8.15 p < 0.05 2.734 8.42

Speed 0.0637 2 0.0318 24.23 p < 0.05 3.126 16.68

Thickness 0.1489 1 0.1489 113.32 p < 0.05 3.976 39.01

Load × Speed 0.0280 6 0.0047 3.55 p < 0.05 2.229 7.33

Load × Thickness 0.0042 3 0.0014 1.06 0.3771 2.734 1.09

Speed × Thickness 0.0118 2 0.0059 4.50 0.0162 3.126 3.10

Load×Speed×Thickness 0.0300 6 0.0050 3.80 p < 0.05 2.229 7.86

Error 0.0631 48 0.0013 16.52

Total 0.3817 71

Table 5.6. Two-way ANOVA analysis for the COF of 0.3 μm silicone films.

Source SS df MS F p Fcri %TTS

Load 0.0093 3 0.0031 5.02 p < 0.05 3.009 24.20

Speed 0.0127 2 0.0064 10.30 p < 0.05 3.009 33.13

Interaction 0.0016 6 0.0003 0.42 0.86 2.508 4.08

Error 0.0148 24 0.0006 38.59

Total 0.0384 35

Table 5.7. Two-way ANOVA analysis for the COF of 1 μm silicone films.

Source SS df MS F p Fcri %TTS

Load 0.0270 3 0.009 4.48 p < 0.05 3.009 13.89

Speed 0.0627 2 0.0314 15.61 p < 0.05 3.403 32.28

Interaction 0.0564 6 0.0094 4.68 p < 0.05 2.508 29.02

Error 0.0482 24 0.0020 24.82

Total 0.1944 35

5.4 Discussion

The wear track of the 1 μm PFA and silicone films were observed under an optical

microscope after the sliding tests. Figure 5.5- 5.6 show the microscopic observations for

1 mm/s in the load range of 2.5 - 15 N. Worn PFA surfaces show clean and regular

appearance while the silicone films give a “crystalline” worn track; the piled-up material

is discontinuous at places. The distinct worn appearances of those two polymers are

116

possibly related to interfacial shear strength, which gives rise to different COF values

[76]. However, no experimental data exists in the literature for the τ0 and α values of

these materials to make a comparison. Obtaining this data was beyond the scope of this

dissertation.

(a) 2.5 N (b) 5 N

(c) 10 N (d) 15 N

Figure 5.5. Microscopic observations of 1 μm PFA worn surfaces (sliding speed: 1

mm/s).

117

(a) 2.5 N (b) 5 N

(c) 10 N (d) 15 N

Figure 5.6. Microscopic observations of 1 μm silicone worn surfaces (sliding speed: 1

mm/s).

5.5 Summary and Conclusion

We investigated the frictional characteristics of PFA and silicone thin films with respect

to the normal force, sliding speed and film thickness. The observations and conclusions

are as follows:

The COF of PFA increases by increasing the sliding speed while the COF of

silicone increases by reducing the speed.

118

The ANOVA analysis showed that sliding speed and film thickness have more

significant effects on the COF as compared to normal force. The coupling of the

load and speed is not significant.

119

6 Simulation of Material Damage during

Indentation of a Soft Polymer

The goal of this chapter is to demonstrate the use of material damage mechanics in

simulating indentation of very soft materials. To this end the experiments used to obtain

the contact width, 2b, reported by Karnath et al. [3] in COF measurements were

simulated. 2D axi-symmetric and 3D finite element analyses with material damage are

presented.

6.1 Axi-Symmetric Finite Element Analysis of Thin-Film Indentation

6.1.1 Materials and Methods

Indentation of a 10 μm thick PTFE thin-film deposited on a glass substrate by a spherical

indenter (Figure 2.5) was modeled in the axisymmetric configuration. The PTFE film and

the glass were assumed to behave in elastic/perfectly-plastic manner and the ball indenter

was modeled to be rigid. The indenter diameter was 6.25 mm. The PTFE material

properties were based on the experimental results obtained by Rae and Brown [62], and

Rae and Dattelbaum [63]. Figure 6.1 gives material properties used in this analysis.

120

Table 6.1. Material properties in 2D axisymmetric finite element analysis [3, 62].

Material Young’s modulus, E (GPa) Poisson’s ration, ν Yield stress, σY (MPa)

PTFE 0.496 0.44 5 – 10

Glass 72 0.22 1000

Commercially available finite element analysis programming package,

ABAQUS/Explicit (Simulia, Providence, RI), was used. The indentation problem is

verified in Appendix 1. The coating-substrate interface was assumed to be perfectly

bonded, and the bottom surface of the substrate was fixed in all degrees of freedom. The

symmetry axis was modeled by constraining the x-component of the displacement as

shown in Figure 6.1. The interface between the indenter and the coating was given a COF

value of 0.06. 8 quadrilateral elements were placed in the thickness direction of the

coating, and adaptive remeshing was implemented to prevent excessive element

distortion. A total number of 91272 CAX4R reduced integration elements was assigned

to the PTFE film and the substrate. The surface-based contact pair algorithm was

implemented in contact simulations. This indentation force was applied on the indenter in

a ramped manner with a rise time of 0.1 second, and kept constant thereafter.

Figure 6.1(b) shows the finite element mesh used in this simulation. A convergence study

was carried out by varying the number of elements between 2 and 10 in the thickness

direction. The film hardness in the simulation is set as Hp = 30 MPa. A plot of the contact

width, 2b, with number of elements used through the thickness of the thin-film shows that

121

(a) The geometry of modeling indentation

(b) Fine meshes around the contact region in ABAQUS/Explicit

Figure 6.1. The geometry and mesh configuration of 2D axi-symmetric finite element

model.

122

Figure 6.2. Convergence studies of element number in thickness direction of PTFE films

(Hp = 30 MPa).

convergence is achieved with 4 elements, as shown in Figure 6.2. The mesh size in the

glass substrate was the same down to a depth of 150 μm, and thereafter it was gradually

increased in regions away from the contact region.

6.1.2 Results and Discussions

The indentation process was modeled by the FEM as described above. The contact width,

2b, as schematically defined in Figure 2.5 was predicted and compared to experimental

measurements and a closed-form solution given in Equations (2.32) – (2.33).

Figure 6.3 shows comparison of the finite element results, the predictions of Equations

(2.32) - (2.33), and the experimental measurements for the 10 μm thick PTFE film. The

123

PTFE hardness, Hp, was adjusted as a fitting parameter. Two sets of finite element results

are shown with the PTFE hardness values of Hp = 15 MPa and 30 MPa. The analytical

results are shown for Hp = 0 and 10 MPa. Note that the Hp values used in both of these

approaches are close to physically measured values (40 MPa) in Appendix 2. This figure

shows good agreement between the measured and the analytically calculated values.

These results indicate that the contribution of the Hertzian contact between the steel ball

and the glass substrate dominates the interfacial contact conditions. For most of the load

range, the ball directly contacts the glass substrate. The finite element analysis does not

predict the contact width, 2b, as a function of normal force, W, as well as the analytical

relationship, especially for low load situations.

Figure 6.4 shows the interfacial conditions predicted by the finite element analysis, where

the residual indentation half-width, b, and residual thickness, tr, are shown. In the case of

1.5 N normal force, b and tr are 136.8 μm and 7.95 μm, respectively. When the

indentation force is increased to 15 N, these values become 288 μm and 2 μm. The finite

element analysis predicts a micron-scale thin PTFE film to remain trapped in the ball-

substrate interface. The compliance of the interface modeled by the finite element method

is close, but not identical to the experimental conditions as seen in Figure 6.4. Thus, it is

concluded that the PTFE layer has a smaller contribution than modeled by FEM, and the

model presented through Equations (2.32) – (2.33) describes the physics of the problem

more closely.

124

Figure 6.3. Comparison of the experimental and the calculated data to FEA results.

(a) (b)

Figure 6.4. von-Mises stress distribution of indentation at different normal force of (a)

1.5 N and (b) 15 N calculated by finite element analysis.

125

6.2 3D Finite Element Model of Indentation of PTFE Thin-Film by

Using a Material Damage Approach

Experimental evidence and analytical formulation show that the indenter is making direct

contact with the substrate. This indicates that the PTFE film is penetrated easily. In order

to model penetration of the PTFE film by the indenter, we employed material damage

model available in ABAQUS, introduced in Section 2.7.2, to model the indentation of

this soft PTFE film.

6.2.1 Materials and Methods

The steel ball and the glass substrate were modeled as elastic perfectly-plastic materials

with properties given in Table 6.2. The PTFE thin film was modeled as an elastic/

perfectly-plastic material and the damage of the material was considered. Shear and

ductile material damage models were investigated as possible damage criteria for this soft

polymer film. Additionally, the equivalent plastic strain 0

pl (0.1 - 50%) and fracture

energy Γf (10 – 1000 J/m2) were used for defining the initiation and evolution of the bulk

material damage, respectively. Section 2.7 presents the physical interpretation of these

two damage parameters, 0

pl and Γf and their implementations in ABAQUS. It is assumed

that the facture energy of PTFE film is dissipated in a linear form during damage

evolution. Table 6.2 gives the material properties used in this 3D finite element analysis.

126

Table 6.2. Material properties used in 3D finite element simulation using material

damage model. Note that 0

pl is the equivalent plastic strain for damage initiation, Γf is the

fracture energy for damage evolution.

Material E (GPa) ν σY (MPa) 0

pl

Γf (J/m2)

Steel 210 0.30 380 N/A N/A

PTFE 0.496 0.44 10 0.1 - 50 10 - 1000

Glass 72 0.22 1000 N/A N/A

Figure 6.5(a) shows the configuration used in the analysis. The film was assumed to be

bonded to the substrate. The bottom surface of substrate was constrained in all degrees of

freedom. The surfaces along the quarter-symmetric axis were fixed in x- or z-directions,

respectively. A constant pressure was applied on the top surface of ball indenter, with

which was initially ramped in 0.1 second to the final value. The element-based general

contact algorithm was implemented, which allows the contact between the indenter and

the PTFE film even after elements are removed due to material damage.

The ball indenter was partitioned into two regions, including a fine mesh region-I and

free mesh region-I. The PTFE film was partitioned as fine mesh region-II and free mesh

region-II. The fine mesh region-I subtends 6.5˚, and makes the total length of the fine

mesh region-I be equal to that of the fine mesh region-II. We find the use of comparable

element size in the fine mesh region-I and –II to be essential for convergence of these

contact simulations. In particular, the free mesh region-I in the ball indenter was meshed

with 45,487-C3D4 tetrahedral elements, and the fine mesh region-I was meshed with

48,600-C3D8R hexahedral elements. The fine mesh region-II and free mesh region-II in

127

PTFE film were meshed with 80,000 and 16,000-C3D8R hexahedral elements,

respectively. The substrate was meshed with the number of 120,000 graded C3D8R

hexahedral elements with gradually increasing size away from the contact region. Figure

6.5(b) shows the mesh for 3D finite element model around the contact region. The

thickness direction in both fine mesh region-I and -II was given 4 and 8 elements for

convergence studies. Figure 6.6 indicates that 8-element in thickness direction achieves a

converged solution.

128

(a) The geometry of 3D finite element model

(b) Meshes around the contact region in ABAQUS

Figure 6.5. The geometry and mesh configuration of 3D finite element model.

129

6.2.2 Results and Discussions

Figure 6.6 shows comparison of the calculated and the measured contact width, 2b, to the

finite element results for the 10 μm thick PTFE film by using the shear damage model.

The damage initiation was assumed to start at 1% of equivalent plastic strain, and the

material fails when the fracture energy, Γf, reaches 20 J/m2. The predicted contact width,

2b, agrees with the measured and the calculated (Equations 2.32 – 2.33) results. The pile-

up of PTFE films predicted in finite element analysis possibly increases the calculation of

contact width, 2b as compared to the measured values. Difference in using 4 or 8-

elements through the film-thickness is small. Note that the error bars indicating the

standard deviation of numerical analysis, shown in Figure 6.6, are due to the rectangular

mesh configuration we adopted. The particular algorithm of collecting all data points and

evaluating the standard deviation is presented in section 7.2.

When the contact width, 2b, predicted by using shear damage model and 2D axi-

symmetric model without material failure in Figure 6.3 and 6.6 are compared, it is seen

that the thin-layer of coating that is trapped in the ball-glass interface leads to

overestimation of the material compliance. The improved prediction by using the material

damage model verifies the contribution of Hertzian contact at the ball-glass interface,

which dominates interfacial contact conditions at loads higher than 2.5 N.

Figure 6.7 shows the contact width, 2b, predicted by using the ductile material damage

criterion. Comparison of Figure 6.6 and 6.7 shows that damage models have significant

130

effects on the prediction of contact width, 2b, and the shear damage model is more

suitable for simulating PTFE damage. Piggot [95] found that the interface failure of fiber-

enforced polymers, for example, polyethylene (PE), polypropylene (PP), and poly(methyl

mechacrylate) (PMMA) was caused by high mean shear stress. The fact that shear

fracture dominates the polymer failure is consistent with the damage mechanism used in

this finite element simulation.

Figure 6.6. Comparison of the measured and calculated contact width, 2b, to FEA

prediction for 10 μm PTFE films using shear damage model.

131

Figure 6.7. Comparison of the measured and calculated contact width, 2b, to FEA result

for 10 μm PTFE films by ductile damage model.

Next, the effects of equivalent plastic strain, 0pl , and fracture energy, Γf, on the prediction

of contact width, 2b were investigated. Figure 6.8 shows the effects of equivalent plastic

strain, 0pl ( = 0.1%, 0.5%, 1%, 5%, 10%, 20%, and 50%), for Γf = 20 J/m

2. Results

indicate that the equivalent plastic strain, 0pl , has negligible effects on contact width, 2b,

in the range of 0.1 – 10% strain. Joyce performed fracture experiments on bulk PTFE to

evaluate the fracture toughness, Γf, and reported that the offset yield stress of 1% plastic

strain, at room temperature, is 10 - 15 MPa [96, 97]. In order to be consistent with this

finding of damage initiation value, we used 1% plastic strain in our simulations.

132

Figure 6.8. Effects of equivalent plastic strain, 0pl , (0.1, 0.5, 5% are not shown on the

graph for clarity of illustration) on the contact width, 2b, for Γf = 20 J/m2.

Figure 6.9 shows the contact width, 2b, as a function of fracture energy, Γf. The

equivalent plastic strain is set 1% and the fracture toughness, Γf, is varied in the range of

10, 20, 30, 40, 50, 100, 500 and 1000 J/m2. The result indicates that the fracture

toughness, Γf, has significant effects on the prediction of contact width, 2b. Bulk fracture

toughness values, Γf, which are in the range of 20 - 50 J/m2, give results similar to

experiments. Note that this range is only several percent of the toughness values reported

by Joyce [96]. This work shows that PTFE is easily deformable at a high contact pressure,

resulting in low fracture toughness.

133

Figure 6.9. Effects of bulk fracture toughness, Γf, (40, 500 J/m2 are not shown on the

graph for clarity of illustration) on the contact width, 2b, for0

pl = 1%.

Steady state configurations of the thin-film are represented in Figure 6.10 for different

normal loads (1.25, 2.5, 5 and 10 N), and for 0pl = 1% and Γf = 20 J/m

2. The indentation

radius, b, and the contact radius, a, are computed based on such results, as marked on this

figure. Close inspection of the figures reveals material pile-up at the edge of indentation

radius, b, where the maximum von-Mises stress in the PTFE film is also found. The

volume of the damage material increases with load. At low load levels the PTFE film is

partially penetrated and the stresses generated in the film provide the equilibrium with

respect to the indentation force; whereas, at higher loads, the indenter makes direct

contact with the glass substrate, as hypothesized in the analytical results. The von-Mises

stress distribution of the glass substrate at steady state configurations is shown in Figure

134

6.11. Note that the maximum stress at the normal load of 10 N is lower than the specified

yield stress, 1 GPa, which indicates glass substrate elastically deformed.

Figure 6.10. von-Mises stress distribution of PTFE thin films at steady state

configurations. Note that a is the contact radius at the interface of ball indenter and

substrate; b is the contact radius at the interface of ball indenter and PTFE films.

135

Figure 6.11. von-Mises stress distribution of the glass substrate at steady state

configurations.

6.3 Summary and Conclusions

Simulations of the indentation of thin PTFE films on a glass substrate were carried out by

using the FEM with a material damage model. The material damage parameters pertinent

to PTFE were determined by comparison to experimental results.

Shear damage model was found to be more suitable to model the interfacial

contact of soft polymer films as compared to ductile damage model.

The effects of equivalent plastic strain, 0

pl , and bulk fracture toughness, Γf, as

indicators of the damage initiation and evolution, respectively, on the contact

width, 2b, were investigated.

136

The simulation results as compared to experimental measurements showed that

bulk fracture toughness, Γf, has more significant effects on the contact width, 2b

predictions as compared to the equivalent plastic strain, 0

pl .

This work showed that the equivalent plastic strain 0

pl = 1% and fracture

toughness Γf = 20 J/m2 represent the experimentally obtained results well.

137

7 Interfacial Delamination of PTFE Thin

Films

Interfacial adhesion is a major concern with respect to the performance of thin polymer

films in developing new thin film processes, such as hot filament chemical vapor

deposition (HFCVD). In this chapter, we present an experimental investigation and

numerical simulation of the interfacial fracture toughness of PTFE by micro-indentation.

7.1 Introduction

Indentation is one of the common techniques, (Figure 2.10), to evaluate the interfacial

fracture toughness and shear strength of the thin film-substrate interface. Evans and

Hutchinson derived the stress intensity factor of a circular interfacial crack by assuming

mixed-mode fracture (I and II) at the crack tip, and showed that the stress intensity varies

directly with the indentation volume and inversely with the delamination radius and film

thickness [29]. Marshall and Evans gave a generalized relationship between the

normalized load and the delamination diameter to quantitatively predict the interfacial

fracture toughness. They showed that the delamination diameter depends on the normal

load and the film thickness, and verified the results experimentally by using a ZnO film,

deposited on a silicon substrate [28]. Ritter et al. evaluated the interfacial shear strength,

based on Matthewson’s formulations for the cases when the films were elastically or

138

plastically deformed or penetrated. Delamination of a soft polymer coating was found to

be caused by the shear failure at the perimeter of the contact region [13].

In this work, micro-indentation experiments were performed to induce delamination at

the interface of a PTFE film and a glass substrate. A 3D finite element model was

developed to simulate the delamination of the PTFE-film, and the results were compared

to experimental measurements. The interfacial fracture toughness, Γc, for the system was

then determined.

7.2 Materials and Methods

HFCVD was used to deposit 1, 2, 3, 5 and 10 μm thick PTFE films on glass substrates.

Micro-indentation tests of the PTFE films were conducted by using a micro-tribotester

(UMT-2 by CETR, Campbell, CA). A Rockwell C indenter (200 μm radius and 120°

cone angle) was used in the tests. Five indentation loads (W = 0.5, 0.75, 1, 2 and 3 N)

were used. The loading rate during the tests was 0.01 mm/s and the hold period was 30

seconds. Each indentation load was repeated twenty times at different locations. An

optical microscope (MX51 by Olympus, Japan) was used to examine the interface after

the indentation, and measure the delamination diameter, 2c, as shown schematically in

Figure 2.10. Young’s modulus, E, and hardness, Hp, of PTFE films were measured by

using a nanoindenter with a Berkovich tip (Nano Bionix, UTM-150, MTS, Eden Prairie,

MN). The detailed results of nano-indentation are provided in Appendices 3-4. The yield

stress, σY, which is obtained by using the relationship Hp = 3σY, is reported in Table 7.1.

139

As shown in Chapter 6, the PTFE film is penetrated very easily by the indenter during

indentation; and, straightforward finite element simulation of the compression of the

coating under the indenter is not adequate for modeling this very soft material [3]. In

order to simulate the penetration process, 3D finite element analysis that considers

material damage under the indenter was carried out by using ABAQUS/Explicit (Simulia,

Providence, RI). Shear damage criterion was applied to the bulk of the PTFE film by

giving equivalent plastic strain, 0pl (1%), as the damage initiation, and bulk fracture

toughness, Γf (20 J/m2), as the damage evolution criteria.

For the glass-PTFE interface, bilinear traction-separation law was used to specify the

delamination, as shown schematically in Figure 2.16. In particular, the normal stress and

shear stress were given 5 MPa as delamination initiation, and the stiffness, K0 is given as

4×104 N/mm

3. Interfacial fracture toughness, Γi, was given in a range of 50 - 1000 mJ/m

2

for delamination evolution. This approach allows for simulating mode-I and -II fracture

effects in the film-substrate interface. The PTFE film and the substrate were modeled as

elastic-perfectly plastic, while the indenter was modeled as elastic, due to its relatively

high Young’s modulus. Bergström and Boyce [98] presented a modified constitutive

model for time-dependent mechanical behaviors of elastomers and soft biological tissues

under cyclic loading. Bergström and Hilbert [69] developed a Dual Network

Fluoropolymer (DNF) constitutive model to simulate the time and temperature-dependent

mechanical behavior of bulk fluoropolymers including PTFE. In this work, we did not

140

use these models for simplicity and assumed isothermal condition for thin PTFE films.

The material properties used in the finite element analysis are given in Table 7.1.

The finite element mesh density and the aspect ratio of the 3D hexahedral (C3D8R)

elements near the contact region were found to be critical for the convergence of the

predicted delamination radius. Table 7.2 shows dimensions of the fine mesh regions and

mesh sizes for modeling the delaminations of 1, 5 and 10 μm PTFE films. Figure 7.1(a)

shows the geometry of simulation and mesh configurations. In particular, the fine mesh

regions-I and –IV, on the PTFE film, and the indenter, respectively, were given

comparable mesh sizes. Fine mesh region-III, of the substrate was given comparable

mesh size, to the fine mesh region-IV, considering the eventual contact of the indenter

and the substrate. Fine mesh region-II of the PTFE film was given mesh size comparable

to the substrate, in order to predict the delamination. The mesh regions of the PTFE film

and the substrate away from the contact were gradually increased. Figure 7.1(b) shows

the finite element mesh for the 10 μm thick PTFE film. The element-based general

contact algorithm was implemented, which allows the contact between the indenter and

the PTFE film even after elements are removed due to material damage. Table 7.3 shows

the number of elements and element types implemented in the simulations.

141

Table 7.1. Material properties in 3D FEA delamination model. Note: Diamond and glass

properties were taken from Oliver and Pharr [16]; The PTFE properties were measured

by using nano-indentation, except for Poisson’s ratio, ν, reported in Karnath et al. [3].

Material E (GPa) ν σY (MPa) 0pl (%) Γf (J/m

2)

Diamond 1141 0.07 N/A N/A N/A

PTFE 3 0.44 35 1 20

Glass 72 0.22 1000 N/A N/A

Table 7.2. Mesh sizes and aspect ratios for modeling all thick PTFE films.

t (μm) L0 (μm) L1 (μm) L2 (μm) h (μm) Nt NL0 NL1 Nh Ratio

1 100 300 10 10 4 150 175 40 2.67

5 200 600 10 10 8 125 150 8 2.56

10 200 600 10 10 8 75 125 8 2.13

142

(a) The geometry

(b) Elements around the contact region for 10 μm PTFE film

Figure 7.1. The geometry and mesh configuration of 3D finite element model for the

delamination simulation.

143

A constant pressure was applied on the top of the conical indenter. The bottom of

substrate was constrained in all degrees of freedom. The surfaces along the quarter-

symmetric axis were fixed in x- or z-directions, respectively.

Convergence of the finite element model was tested by using different mesh sizes.

Figure 7.2 shows the convergence studies for 1, 5 and 10 μm thick PTFE delamination

diameters for normal force, W = 1 N, and interfacial fracture toughness, Γi = 100 mJ/m2.

The number of elements is in the range of 4 - 10. This plot indicates that use of 8-

elements in the thickness direction achieves the converged results for the 5 and 10 μm

thin films, and 4-elements for the 1 μm thin film. The error bars shown in Figure 7.2 are

due to the use of a rectangular finite element mesh in an axisymmetric problem.

Figure 7.3 shows the delamination contours predicted by the finite element method for W

= 1 N, E = 3 GPa, σY = 35 MPa. The interfacial fracture toughness, Γi, is given in the

range of 50 – 200 mJ/m2. In ABAQUS, the parameter, CSDMG, monitors interfacial

delamination, with CSDMG = 1 indicating the interface is completely delaminated, and

CSDMG = 0 indicating no delamination.

144

Figure 7.2. Convergence studies for 1, 5 and 10 μm PTFE delamination simulations (W =

1 N, Γi = 100 mJ/m2, Young’s modulus E = 3 GPa, yield stress σY = 35 MPa for PTFE

material properties).

Table 7.3. The number and type of elements implemented in delamination model.

Film

thickness

# of elements in film

and substrate

# of elements in

indenter

# of elements in

fine mesh region-I

# of elements in fine

mesh region-IV

1 μm 380,208 (C3D8R) 44,309 (C3D8R, C3D4) 90,000 (C3D8R) 32,000 (C3D8R)

5 μm 397,953 (C3D8R) 1,352 (C3D8R, C3D4) 80,000 (C3D8R) 352 (C3D8R)

10 μm 309,422 (C3D8R) 1,292 (C3D8R, C3D4) 45,000 (C3D8R) 480 (C3D8R)

In order to evaluate the delamination radius, the coordinates of the nodes ,i iC x z with

interfacial delamination as shown in Figure 7.4 are found by using a Python script. Since

the parameter, CSDMG, gives node-based values including the coordinates, the

coordinates, ,i iC x z are regrouped into (n+1) small data sets based on the x-coordinate

in range of (0 - xn). Within each small set, the maximum z-coordinate is searched and

145

each node on the delamination contour is denoted as, max10,C z , 1 max 2,C x z , …,

max,n nC x z . The radius of each node on the delamination boundary was then calculated

as ri (i = 0, 1, …, n) with respect to the origin. Then the averaged radius, r , and its

standard deviation, s, were evaluated as follows,

0

1

1

n

i

i

r rn

(7.1)

2

0

1 n

i

i

s r rn

(7.2)

(a) 1 N, 50 mJ/m

2 (b) 1 N, 100 mJ/m

2

(c) 1 N, 150 mJ/m

2 (d) 1 N, 200 mJ/m

2

Figure 7.3. 10 μm PTFE delamination contours predicted by finite element simulation

(Young’s modulus, E = 3 GPa, hardness, σY = 35 MPa).

146

Figure 7.4. The coordinates of nodes with interfacial delamination.

7.3 Results and Discussion

Figure 7.5 shows optical micrographs of the indented surfaces of five different PTFE

films for W = 0.5 N. No interfacial delamination was observed on the 1 and 2 μm thick

films, in Figure 7.5(a) and (b). On the other hand, typical delaminated interfaces for 5 and

10 μm thick films are seen in Figure 7.5(d) and (e). Figure 7.6 shows the load effects on

the delamination diameter, 2c, of the 3 μm thick film, where the interfacial delamination

was clearly seen at 0.75, 2 and 3 N, as shown in Figure 7.6(b), (d) and (e). These

observations show that delamination takes place for film thickness values of 3 μm and

higher, and indicate the thickness-dependence.

As the delamination takes place more readily on 5 and 10 μm films, we measured the

delamination diameter, 2c, at each indentation force for five repeats. Table 7.4 shows the

147

averaged 2c and its standard deviation. The Student’s T-test was used to test if the film

thickness has significance on the delamination diameter, 2c. The null-hypothesis was that

the delamination diameter, 2c, is not a function of film thickness. The p-value (0.32 >

0.05) indicates that the null hypothesis cannot be rejected and thus there is no significant

difference for 2c between 5 and 10 μm PTFE films.

148

(a) 1 μm (b) 2 μm

(c) 3 μm (d) 5 μm

(e) 10 μm

Figure 7.5. Thickness effects on the interfacial delamination of PTFE thin films (normal

force: 0.5 N). Note that 2c represents the delamination diameter at the interface.

2c

2c

149

(a) 0.5 N (b) 0.75 N

(c) 1 N (d) 2 N

(e) 3 N

Figure 7.6. Load effects on the interfacial delamination of PTFE thin films (film

thickness: 3 μm).

In order to develop a general understanding of the effects of interfacial fracture toughness,

Γi, and the other material properties, interfacial delamination was modeled for 1, 5 and 10

150

μm films by using a range of Γi (50 – 1000 mJ/m2), E (0.5, 3 GPa), Hp (30, 105 MPa) and

t (1, 5, 10 μm) values. Figure 7.7(a) - (c) show the computed variation of the delamination

diameter, 2c, with different PTFE material properties, for 1, 5, and 10 μm thick films,

respectively. The delamination diameter, 2c, increases with increasing normal force, W,

but decreases with increasing interfacial fracture toughness, Γi. For the large Γi values,

the gradient of 2c with respect to Γi is lower. Figure 7.7(a) - (c) also shows that for fixed

Γi and W values, a larger delamination diameter develops under the more compliant

coating.

Table 7.4. Experimentally measured delamination diameter, 2c, for different film

thickness, t, and normal force, W. Note that Ave. 2c is the average delamination diameter;

Std. Dev. 2c is its standard deviation.

t (μm) load, W (N) Ave. 2c (μm) Std. Dev. 2c (μm)

5 0.5 179.41 7.07

5 0.75 177.49 12.11

5 1 187.17 8.93

5 2 201.68 12.30

5 3 181.99 7.02

10 0.5 146.86 13.65

10 0.75 177.00 7.75

10 1 187.27 6.95

10 2 203.27 7.94

10 3 204.49 4.93

151

(a) 1 μm PTFE film

(b) 5 μm PTFE film

152

(c) 10 μm PTFE film

Figure 7.7. The predictions of delamination diameter, 2c, as a function of film thickness

and material properties.

The results presented in Figure 7.8 are normalized by defining the following non-

dimensional force, F , and delamination radius, C ,

2

nW

FHt

(7.3)

i

mp qc E

Ct Et H

(7.4)

where W is the indentation force, E is the Young’s modulus, H is the hardness of the film,

t is the film thickness, c is the delamination radius, Γi is the interfacial fracture toughness.

With these definitions, the relationship between the variables is established as follows,

C F (7.5)

153

where the coefficients α = 1.5 × 10-3

, n = 0.299, p = 0.953, m = 0.381 and q = -0.915 are

found by curve fitting as shown in Figure 7.8. Note that the R2 value for the curve fit is

0.99. The complete relationship between non-dimensional variables is shown as follows,

3

0.3810.953 0.915 0.299

21.5 10ic E W

t Et H Ht

(7.6)

The interfacial fracture toughness, Γc, can be obtained from this equation as follows,

1/0.3810.915 0.299 0.953

20.0015c

E W cEt

H Ht t

(7.7)

By using the experimentally measured values, the interfacial fracture toughness, Γc, for 5

and 10 μm PTFE films are determined by using Equation (7.7). In particular, Γc for the 5

and 10 μm thick coatings are found as 196 ± 112 mJ/m2 and 721 ± 231 mJ/m

2,

respectively. Figure 7.8 shows the comparison of the normalized experimental values and

the numerical simulations, where 1 μm PTFE simulation results are not shown. We also

used Rosenfeld et al.’s method [32], described in Equation (2.74), to calculate the

interfacial fracture toughness, Γc. The values of Γc for 5, 10 μm PTFE films were found as

1190 ± 471 mJ/m2 and 2434 ± 703 mJ/m

2, respectively, which are much higher than the

values evaluated using Equation (7.7). The energy spent in the penetration of PTFE film

was not considered in the Rosenfeld et al.’s model, which results in the overestimation of

the interfacial fracture toughness.

154

The load-dependence of the delamination radius, c, is also investigated. Equation (7.6)

gives the relation of delamination radius, c, and normal force, W, as follows,

0.772 1.35991/0.953 0.3137

0.3998 1.274

i

t Ec W

H

(7.8)

Rosenfeld et al.’s method shown in Equation (2.74) indicates that 0.5c W , which is

reasonably close to Equation (7.8). Figure 7.9 shows the comparison of our finite

element simulation to Rosenfeld et al.’s formulation for 10 μm PTFE delamination.

Figure 7.8. The curve fitting of non-dimensional delamination radius, C , and indentation

force, F , from finite element simulation.

155

Figure 7.9. The comparison of 10 μm PTFE finite element simulation to Rosenfeld et al’s

formulations with respect to different material properties.

7.4 Summary and Conclusion

Adhesion properties of PTFE thin films deposited on glass were investigated. Micro-

indentation tests were used to induce the interfacial delamination of PTFE thin films on

glass substrate. Following observations and conclusions are made:

The experiments showed that the delamination diameter, 2c, depends on the film

thickness and the normal force. The critical thickness for the occurrence of

delamination was found to be 3 μm.

The entire indentation process was simulated by using the finite element method,

with material damage model and bilinear traction-separation law. The numerical

156

results indicated that the delamination diameter, 2c, depends on film thickness,

material properties, and indentation force.

The occurrence of delamination depends on the strain energy stored in the film

during the indentation, indicating thinner films are less likely to delaminate.

Predicted interfacial fracture toughness, Γc, for 5 and 10 μm PTFE films are less

than those predicted by Rosenfeld et al.’s formulation. The difference is due to the

energy spent during penetration to cause PTFE film damage/failure.

157

8 Summary, Conclusions and Future Work

In this dissertation, we investigated the frictional and durability characteristics of thin-

film PTFE coatings deposited on rough and smooth aluminum substrates. We also

studied the frictional characteristics of thin PFA and Poly(V3D3) films on glass substrates.

The analysis of variance (ANOVA) was used to quantitatively analyze the effects of

normal force, sliding speed and surface roughness on the COF and durability.

Material damage/failure approach was introduced to numerical simulation of indentation

on a soft polymer coating because 2D-axisymmetric numerical model without using

continuum damage mechanics underestimated the contact width, compared to

experimental measurement. Different damage criteria were used and the effects of

equivalent plastic strain and fracture toughness on the prediction of contact width were

studied.

Interfacial delamination of thin PTFE films was also investigated. Micro-indentation with

a Rockwell C indenter was performed on films with different thickness and delamination

diameters were measured by using optical microscope. Numerical model using shear

damage/failure criterion and cohesive zone model was developed to predict the

delamination diameter. A relationship of non-dimensional indentation force and

delamination radius was found by curve-fitting and was used to evaluate the interfacial

fracture toughness.

158

The conclusions are summarized as follows,

Frictional and durability characteristics of thin-film PTFE coatings

It was observed that the COF and durability depend on normal force, sliding speed

and surface roughness from the substrate. In particular, the COF increases by

increasing the surface roughness, Ra, and the sliding speed. Load-dependence of COF

is observed on rough surfaces, but is negligible on smooth substrates. The durability

is improved by increasing the sliding speed or decreasing the normal force, but has a

nonlinear relationship with surface roughness, Ra. The ANOVA results indicated that

surface roughness of substrate has the most significant effects on the COF and

durability. The sliding speed contributes more to the COF and durability as compared

to the normal force.

We also observed that the interface of PTFE and smooth aluminum substrates is well

lubricated in friction and durability tests. A modified equation based on Briscoe and

Tabor friction model [75, 76], was developed to predict the COF on the smooth

surface and had good agreements with the experiments. It indicated that the adhesion

governs the sliding and durability process on smooth substrates.

Frictional characteristics of thin-film PFA and Poly(V3D3) coatings

We observed that the COF of PFA increases by increasing the sliding speed while

that of silicone increases by reducing the speed. The ANOVA showed that the sliding

159

speed and the film thickness have greater significance on the COF as compared to the

normal force.

Mechanics simulation of material damage for soft polymer coatings

The simulation results, as compared to the experiment, showed that shear

damage/failure model was suitable to characterize the interfacial contact of soft

polymers on rigid substrates.

We investigated the effects of equivalent plastic strain and bulk fracture toughness as

indicators of the damage initiation and evolution, respectively, on the prediction of

contact width. The results showed that the bulk fracture toughness has more

significant effects on the contact width as compared to the equivalent plastic strain.

We also found that the equivalent plastic strain 0

pl = 1% and fracture toughness Γf =

20 J/m2 predict the experimentally obtained results well. It showed that indentation

simulation can be used to evaluate the bulk fracture toughness for thin-film polymers,

which are difficult to determine using typical experimental methodologies.

We also observed that thin-film polymer coating at a high contact pressure has less

resistance to be deformed, indicated by simulation results.

Interfacial delamination of thin-film PTFE coatings

We performed the micro-indentation on thin PTFE films to induce interfacial

delamination. The experiments showed that the delamination diameter depends on the

160

film thickness and the normal force. In our work, the critical thickness for the

occurrence of delamination was found to be 3 μm. Thinner films were observed to

delaminate less likely because the occurrence of delamination depends on the strain

energy stored in the film during the indentation.

We employed the finite element method, using material damage/failure approach and

cohesive zone model, to simulate the delamination process. The results showed that

the prediction of delamination diameter is a function of film thickness, material

properties, indentation load and interfacial fracture toughness. The relationship of

non-dimensional indentation force and radius was developed to evaluate the

interfacial fracture toughness of 5 and 10 μm PTFE coatings. The evaluations of

interfacial fracture toughness were found to be much smaller as compared to those

values using Rosenfeld et al.’s equations. The difference is due to the energy spent

during penetration to cause PTFE film damage/failure, which is not considered in the

mathematical description of Rosenfeld et al.’s model.

Future Work

In the future, a simulation of scratch on rough surface may be modeled to investigate

the effects of surface roughness on the COF of soft polymer coatings. More refined

nonlinear material properties may be included in the finite element model to study the

effects of creep response on the prediction of contact radius and adhesion properties.

The experiments related to the evaluation of non-linear material properties may be

proposed, including drop-ball and peel tests. The frictional and durability tests on

161

other thin polymer films may be performed by considering different indenter and

substrate materials, which probably expands the overview and understandings to the

mechanisms of polymer tribology in thin-film form.

162

Bibliography

1. Young, R.J. and Lovell, P.A., Introduction to polymers. Second ed. 1991:

Chapman&Hall.

2. O'Shaughnessy, W.S., Functional thin film polymers for biopassivation of

neuroprosthetic implants. 2007, Massachusetts Institute of Technology.

3. Karnath, M.A., Sheng, Q., White, A.J., and Muftu, S., Frictional characteristics

of ultra-thin polytetrafluoroethylene (PTFE) films deposited by hot filament-

chemical vapor deposition (HFCVD). Tribology Transactions, 2011. 54: p. 36-43.

4. O'Shaughnessy, W.S., Gao, M., and Gleason, K.K., Initiated chemical vapor

deposition of trivinyltrimethylcyclotrisiloxane for biomaterial coatings. Langmuir,

2006. 22: p. 7021-7026.

5. O'Shaughnessy, W.S., Murthy, S.K., Edell, D.J., and Gleason, K.K., Stable

biopassive insulation synthesized by initiated chemical vapor deposition of poly

(1,3,5-trivinyltrimethylcyclotrisiloxane). Biomacromolecules, 2007. 8: p. 2564-

2570.

6. Williams, J.A., Engineering Tribology. 1994: Oxford University Press Inc., New

York.

7. Muftu, S., Class notes for plane contact problems. 2009, Northeastern University.

p. 1-32.

8. Matthewson, M.J., Axi-symmetric contact on thin compliant coatings. J. Mech.

Phys. Solids, 1981. 29(2): p. 89-113.

9. Chadwick, R.S., Axisymmetric indentation of a thin incompressible elastic layer.

SIAM J. Appl. Math., 2002. 62(5): p. 1520-1530.

10. Chen, W.T. and Engel, P.A., Impact and contact stress analysis in multilayer

media. Int. J. Solids Structures, 1972. 8: p. 1257-1281.

11. Kral, E.R., Komvopoulos, K., and Bogy, D.B., Finite element analysis of repeated

indentation of an elastic-plastic layered medium by a rigid sphere, part I: surface

results. J. of Applied Mechanics, 1995. 62(1): p. 20-28.

12. Kral, E.R., Komvopoulos, K., and Bogy, D.B., Finite element analysis of repeated

indentation of an elastic-plastic layered medium by a rigid sphere, part II:

subsurface results. Journal of Applied Mechanics, 1995. 62(1): p. 29-42.

163

13. Ritter, J.E., Lardner, T.J., Rosenfeld, L., and Lin, M.R., Measurement of adhesion

of thin polymer coatings by indentation. J. Appl. Phys, 1989. 66(8): p. 3626-3634.

14. Matthewson, M.J., Adhesion measurement of thin films by indentation. Appl. Phys.

Letter, 1986. 49(21): p. 1426-1428.

15. Ritter, J.E., Sioui, D.R., and Lardner, T.J., Indentation behavior of polymer

coatings on glass. Polymer engineering and science, 1992. 32(18): p. 1366-1371.

16. Oliver, W.C. and Pharr, G.M., An improved technique for determining hardness

and elastic modulus using load and displacement sensing indentation experiments.

J. Mater. Res., 1992. 7(6): p. 1564-1583.

17. Oliver, W.C. and Pharr, G.M., Measurement of hardness and elastic modulus by

instrumented indentation: Advances in understanding and refinements to

methodology. J. Mater. Res., 2004. 19(1): p. 3-30.

18. Standard practice for instrumented indentation testing. 2011, ASTM International:

PA, United States. p. 1465-1487.

19. Fischer-Cripps, A.C., Critical review of analysis and interpretation of

nanoindentation test data. Surface and coatings technology, 2006. 200: p. 4153-

4165.

20. Briscoe, B.J., Fiori, L., and Pelillo, E., Nano-indentation of polymeric surfaces. J.

Phys. D.: Appl. Phys., 1998. 31: p. 2395-2405.

21. King, R.B., Elastic analysis of some punch problems for a layered medium. Int. J.

Solids Structures, 1987. 23(12): p. 1657-1664.

22. Sneddon, I.N., The relation between load and penetration in the axisymmetric

boussinesq problem for a punch of arbitrary profile. Int. J. Engng Sci., 1965. 3: p.

47-57.

23. Doerner, M.F. and Nix, W.D., A method for interpreting the data from depth-

sensing indentation instruments. J. Mater. Res., 1986. 1(4): p. 601-609.

24. Pharr, G.M., Measurement of mechanical properties by ultra-low load indentation.

Materials Science and Engineering A, 1998. 253: p. 151-159.

25. Sakharova, N.A., Fernandes, J.V., Antunes, J.M., and Oliveira, M.C., Comparison

between Berkovich, Vickers and conical indentation tests: A three-dimensional

numerical simulation study. Int. J. of Solids and Structures, 2009. 46: p. 1095-

1104.

164

26. Volinsky, A.A., Moody, N.R., and Gerberich, W.W., Interfacial toughness

measurements for thin films on substrates. Acta Materialia, 2002. 50: p. 441-466.

27. Li, M., Carter, C.B., Hillmyer, M.A., and Gerberich, W.W., Adhesion of polymer-

inorganic interfaces by nanoindentation. J. Mater. Res., 2001. 16(2): p. 3378-

3388.

28. Rossington, C., Evans, A.G., Marshall, D.B., and Khuri-Yakub, B.T.,

Measurements of adherence of residually stressed thin films by indentation. II.

Experiments with ZnO/Si. J. Appl. Phys., 1984. 56(10): p. 2639-2644.

29. Evans, A.G. and Hutchinson, J.W., On the mechanics of delamination and

spalling in compressed films. Int. J. Solids Structure, 1984. 20(5): p. 455-466.

30. Anderson, T.L., Fracture mechanics. 3 ed. 2005, United States: Taylor & Francis

Group, LLC.

31. Marshall, D.B. and Evans, A.G., Measurement of adherence of residually stressed

thin films by indentation. I. Mechanics of interface delamination. J. Appl. Phys.,

1984. 56(10): p. 2632-2638.

32. Rosenfeld, L.G., Ritter, J.E., Lardner, T.J., and Lin, M.R., Use of the

microindentation technique for determining interfacial fracture energy. J. Appl.

Phys., 1990. 67(7): p. 3291-3296.

33. Li, M., Palacio, M.L., Carter, C.B., and Gerberich, W.W., Indentation

deformation and fracture of thin polystyrene films. Thin solid films, 2002. 416: p.

174-183.

34. Jayachandran, R., Boyce, M.C., and Argon, A.S., Mechanics of the indentation

test and its use to assess the adhesion of polymeric coatings. J. Adhesion Sci.

Technol., 1993. 7(8): p. 813-836.

35. Sheng, Q., White, A.J., and Muftu, S., Experimental and numerical investigations

on interfacial delamination of thin-film PTFE coatings, in 2013 Adhesion

Conference. Daytona, FL.

36. Moody, N.R., Bahr, D.F., Kent, M.S., Emerson, J.A., and Reedy, E.D., Film

thickness effects on interfacial fracture of epoxy bonds. Mat. Res. Soc. Symp.,

2002. 710: p. 141-146.

37. Volinsky, A.A., Tymiak, N.I., Kriese, M.D., Gerberich, W.W., and Hutchinson,

J.W. Quantative modeling and measurement of copper thin film adhesion. 1999.

38. Neto, E.D., Peric, D., and Owen, D.R., Computational methods for plasticity.

2008, UK: John Wiley & Sons Ltd.

165

39. ABAQUS (ver. 6.10) documentation. Providence, RI: Simulia.

40. Lemaitre, J., A continuous damage mechanics model for ductile fracture. J. of

Engineering Materials and Technology, 1985. 107(1): p. 83-89.

41. Hooputra, H., Gese, H., Dell, H., and Werner, H., A comprehensive failure model

for crashworthiness simulation of aluminum extrusions. Inter. J. Crash, 2004. 9(5):

p. 449-463.

42. Mei, H., Gowrishankar, S., Liechti, K.M., and Huang, R., Initiation and

propagation of interfacial delamination in integrated thin-film structures, in IEEE

Intersociety Conf. Therm. Thermomechanical Phenom. Electron. Syst., ITherm.

2010: Las Vegas, NV.

43. Dugdale, D.S., Yielding in steel sheets containing slits. J. of the Mechanics and

Physics of Solids, 1960. 8: p. 100-104.

44. Barenblatt, G.I., The mathematical theory of equilibrium cracks formed in brittle

fracture. Adv. Appl. Mech., 1962. 7: p. 55-129.

45. Cottrell, A.H. The Bakerian Lecture, 1963. Fracture. in Proceedings of the R. Soc.

of London. Series A, Math. and Phys. Sci. 1963.

46. Bao, G. and Suo, Z., Remarks on crack-bridging concepts. Appl. Mech. Rev.,

1992. 45: p. 355-366.

47. Camanho, P.P., Davila, C.G., and DeMoura, M.F., Numerical simulation of

mixed-mode progressive delamination in composite materials. J. of Composite

Materials, 2003. 37(16): p. 1415-1438.

48. Sorensen, B.F. and Jacobsen, T.K., Determination of cohesive laws by the J

intergral approach. Engineering fracture mechanics, 2003. 70: p. 1841-1858.

49. Hutchinson, J.W. and Evans, A.G., Mechanics of materials: top-down approaches

to fracture. Acta Mater., 2000. 48: p. 125-135.

50. Schadler, L.S., Brinson, L.C., and Sawyer, W.G., Polymer nanocomposites: a

small part of the story. JOM, 2007. 59(3): p. 53-60.

51. Biswas, S.K. and Vijayan, K., Friction and wear of PTFE - a review. Wear, 1992.

158: p. 193-211.

52. Karnath, M.A., Frictional characteristics of thin polytetrafluoroethylene films

deposited on glass by hot filament chemical vapor deposition method, in

Mechanical Engineering. 2008, Northeastern University: Boston.

166

53. Burris, D.L., Santos, K., Lewis, S.L., Liu, X., Perry, S.S., Blanchet, T.A.,

Schadler, L.S., and Sawyer, W.G., Polytetrafluoroethylene matrix

nanocomposites for tribological applications Tribology of Polymeric

Nanocomposites, 2008. 55: p. 403-438.

54. Bunn, C.W. and Howells, E.R., Structures of molecules and crystals of fluoro-

carbons. Nature, 1954. 174: p. 549-551.

55. Makinson, K.R. and Tabor, D. The friction and transfer of polytetrafluoroethylene.

in Proceeding of the royal society of London. 1964.

56. Sawyer, W.G., Perry, S.S., Phillpot, S.R., and Sinnott, S.B., Integrating

experimental and simulation length and time scales in mechanistic studies of

friction. J. Phys.:Condens. Matter, 2008. 20: p. 354012 (14pp).

57. Barry, P.R., Jang, I., Perry, S.S., Sawyer, W.G., Sinnott, S.B., and Phillpot, S.R.,

Effect of simulation conditions on friction in polytetrafluoroethylene (PTFE). J.

Computer-Aided Mater. Des., 2008. 14: p. 239-246.

58. Wang, J., Shi, F.G., Nieh, T.G., Zhao, B., Brongo, M.R., Qu, S., and Rosenmayer,

T., Thickness dependence of elastic modulus and hardness of on-wafer low-k

ultrathin polytetrafluoroethylene films. Scripta mater, 2000. 42: p. 687-694.

59. Lucas, B.N., Rosenmayer, C.T., and Oliver, W.C. Mechanical characterization of

sub-micron polytetrafluoroethylene (PTFE) thin films. in Mat. Res. Soc. Symp.

Proc. 1998.

60. Wang, J., Kim, H.K., Shi, F.G., Zhao, B., and Nieh, T.G., Thickness dependence

of morphology and mechanical properties of on-wafer low-k PTFE dielectric

films. Thin solid films, 2000. 377-378: p. 413-417.

61. Jones, R.L., Kumar, S.K., Ho, D.L., Briber, R.M., and Russell, T.P., Chain

conformation in ultrathin polymer films. Nature, 1999. 400: p. 146-149.

62. Rae, P.J. and Brown, E.N., The properties of poly(tetrafluoroethylene) (PTFE) in

tension. Polymer, 2005. 46: p. 8128-8140.

63. Rae, P.J. and Dattelbaum, D.M., The properties of poly(tetrafluoroethylene)

(PTFE) in compression. Polymer, 2004. 45: p. 7615-7625.

64. Colak, O.U., Modeling deformation behavior of polymers with viscoplasticity

theory based on overstress. Int. J. of Plasticity, 2005. 21: p. 145-160.

65. Steijn, R.P., The effect of time, temperature, and environment on the sliding

behavior of polytetrafluotoethylene. Asle transactions, 1966. 9: p. 149-159.

167

66. Steijn, R.P., Friction and wear of plastics. Metals engineering quarterly, 1967. 10:

p. 9-21.

67. Khan, A. and Zhang, H., Finite deformation of a polymer: experiments and

modeling. Int.J. of Plasticity, 2001. 17: p. 1167-1188.

68. Khan, A.S. and Liang, R., Behaviors of three BCC metal over a wide range of

strain rates and temperatures: experiments and modeling. Int. J. of Plasticity,

1999. 15: p. 1089-1109.

69. Bergstrom, J.S. and Hilbert, L.B., A constitutive model for predicting the large

deformation thermomechanical behavior of fluoropolymers. Mechanics of

Materials, 2005. 37: p. 899-913.

70. Pooley, C.M. and Tabor, D. Friction and molecular structure: the behavior of

some thermoplastics. in Proc. R. Soc. Lond. A. 1972.

71. Briscoe, B.J. and Tabor, D., Friction and wear of polymers: the role of

mechanical properties. The British Polymer Journal, 1978. 10: p. 74-78.

72. Briscoe, B.J. The friction of polymers: a short review. in Friction and traction:

proceedings of the 7th Leeds-Lyon symposium on tribology. 1981.

73. Myshkin, N.K., Petrokovets, M.I., and Kovalev, A.K., Tribology of polymers:

Adhesion, friction, wear, and mass-transfer. Tribology International, 2005. 38: p.

910-921.

74. Vinogradov, G.V., Bartenev, G.M., Elkin, A.I., and Mikhailov, V.K., Effect of

temperature on friction and adhesion of crystalline polymers. Wear, 1970. 16: p.

358-368.

75. Briscoe, B.J. and Tabor, D. Shear properties of thin organic films. in Symposium

on lubricant properties in thin lubricating films presented before the division of

petroleum chemistry, Inc. 1976. New York.

76. Briscoe, B.J. and Tabor, D., Shear properties of thin polymeric films. Journal of

Adhesion, 1978. 9(2): p. 145-155.

77. Jia, B., Liu, X., Cong, P., and Li, T., An investigation on the relationships

between cohesive energy density and tribological properties for polymer-polymer

sliding combinations. Wear, 2008. 264: p. 685-692.

78. Wieleba, W., The statistical correlation of the coefficient of friction and wear rate

of PTFE composites with steel counterface roughness and hardness. Wear, 2002.

252: p. 719-729.

168

79. Liu, X., Li, T., Liu, X., and Lv, R., Study on tribological properties of

polytetrafluoroethylene drawn uniaxially at different temperature. Macromol.

Mater. Eng., 2005. 290: p. 172-178.

80. Liu, X., Li, T., Liu, X., Lv, R., and Cong, P.H., An investigation on the friction of

oriented polytetrafloroethylene (PTFE). Wear, 2007. 262: p. 1414-1418.

81. Tanaka, K., Uchiyama, Y., and Toyooka, S., The mechanism of wear of

polytetrafluoroethylene. Wear, 1973. 23: p. 153-172.

82. Hollander, A.E. and Lancaster, J.K., An application of topographical analysis to

the wear of polymers. Wear, 1973. 25: p. 155-170.

83. Steijn, R.P., Characteristics of polymer wear. Wear tests for plastics: selection

and use, ASTM STP 701, R. G. Bayer, Ed., ASTM, 1979: p. 3-17.

84. Lhymn, C., Microscopy study of the frictional wear of polytetrafluoroethylene.

Wear, 1986. 107: p. 95-105.

85. Bahadur, S., The development of transfer layers and their role in polymer

tribology. Wear, 2000. 245: p. 92-99.

86. Mahieux, C.A., Experimental characterization of the influence of coating

materials on the hydrodynamic behavior of thrust bearings: a comparison of

Babbitt, PTFE, and PFA. J. of Tribology, 2005. 127(3): p. 568-74.

87. Jackson, R.L. and Green, I., Experimental analysis of the wear, life and behavior

of PTFE coated thrust washer bearings under non-axisymmetric loading.

Tribology Transactions, 2003. 46(4): p. 600-607.

88. Lau, K.K., Caulfield, J.A., and Gleason, K.K., Structure and morphology of

fluorocarbon films grown by hot filament chemical vapor deposition. Chem.

Mater., 2000. 12: p. 3032-3037.

89. Chen, W.X., Li, F., Han, G., Xia, J.B., Wang, L.Y., Tu, J.P., and Xu, Z.D.,

Tribological behavior of carbon-nanotube-filled PTFE composites. Tribology

letters, 2003. 15(3): p. 275-278.

90. Limb, S.J., Labelle, C.B., Gleason, K.K., Edell, D.J., and Gleason, E.F., Growth

of fluorocarbon polymer thin films with high CF2 fractions and low dangling

bond concentrations by thermal chemical vapor deposition. Appl. Phys. Letter,

1996. 68(20): p. 2810-2812.

91. Orvis, K.H. and Grissino-Mayer, H.D., Standardizing the reporting of abrasive

papers used to surface tree-ring samples. Tree-ring research, 2002. 58(1/2): p. 47-

50.

169

92. Bhushan, B., Introduction to tribology. 2002, New York: John Wiley&Sons, Inc.

93. Korinek, P.M., Properties and applications of Perfluoroalkoxy copolymers.

Kunststoffe German Plastics, 1987. 77(6): p. 11-12.

94. Fluoroplastic comparison - typical properties DuPont. Available from:

http://www2.dupont.com/Teflon_Industrial/en_US/tech_info/techinfo_compare.ht

ml.

95. Piggott, M.R., A new model for interface failure in fibre-reinforced polymers.

Composites Science and Technology, 1995. 55: p. 269-276.

96. Joyce, J.A., Fracture toughness evaluation of Polytetrafluoethylene. Polymer

engineering and science, 2003. 43(10): p. 1702-1714.

97. Joyce, P.J. and Joyce, J.A., Evaluation of the fracture toughness properties of

Polytetrafluoroethylene. International J. of Fracture, 2004. 127: p. 361-385.

98. Bergstrom, J.S. and Boyce, M.C., Constitutive modeling of the time-dependent

and cyclic loading of elastomers and application to soft biological tissues.

Mechanics of materials, 2001. 33: p. 523-530.

170

Appendix 1 ABAQUS/Explicit Verification by

Modeling 2D Interface of Indentation

In order to verify if ABAQUS/Explicit is applicable to the analysis of indentation

interface, a 2D-axisymmetric finite element model as shown schematically in Figure 6.1

was used. An analysis of this configuration was given by analytical functions by

Matthewson [8]. The same mesh and boundary conditions were employed. The coating

and the substrate were modeled as elastic. In particular, the Young’s modulus and the

Poisson’s ratio of substrate were given as 72 GPa, and 0.22, respectively. A variety of the

Poisson’s ratios and thus shear moduli were assigned to the coating to investigate the

variations of interfacial shear stress as a function of contact radius. Table A1.1 shows

material properties and contact radius, b, evaluated in the finite element analysis.

Figure A1.1 shows the normalized interfacial shear stress variation in the radial direction

as predicted by finite element analysis and Matthewson’s close-form formulation [8].

Results predicted by the finite element method indicate a similar trend to analytical

solution. The deviations are attributed to i) finite element mesh, which affects the

evaluation of contact radius, and ii) the assumption of averaged stress in Matthewson’s

model.

171

Table A1.1 Material properties of glass using nano-indentation.

Indenter radius,

R (mm)

Poisson’s

ratio, ν

Young’s modulus, E

(MPa)

Shear modulus, G

(MPa)

Contact radius, b

(mm)

3.125 0.3 5000 1923 0.131

3.125 0.4 5000 1786 0.112

3.125 0.45 5000 1724 0.095

Figure A1.1 The comparisons of interfacial shear strength between finite element

analysis and Matthewson’s solution [8].

172

Appendix 2 Material Properties of Fused Silica

Using Nano-Indentation

The sample of fused silica is selected as the calibration of nano-indentation. The

experimental method and instrument were presented in Section 7.2. Figure A2.1 shows

the load-displacement graphs of fused silica for two test repeats. In order to compare to

material properties investigated by Oliver and Pharr [16], the maximum indentation load

was set as 120 mN. The material properties are determined as follows. The initial

unloading stiffness, S, is evaluated using Equation (2.37), shown as follows,

36

9

Δ 120.325 100.249 10

Δ 1021.969 540.031 10

WS

δ

(N/m) (A2.1)

The deflection of the surface, δs, is calculated using Equation (2.42) and shown as

follows,

39max

6

120.325 100.72 347.93 10

0.249 10s

Wδ ε

S

(m) (A2.2)

The contact depth, δc, is evaluated using Equation (2.41),

9 9

max 1021.969 347.93 10 674.04 10c sδ δ δ (m) (A2.3)

The contact area, Ap, for Berkovich indenter, was estimated by using Equation (2.43) and

shown as follows,

173

Figure A2.1 Indentation test on fused silica.

2

2 9 1224.5 24.5 674.04 10 11.13 10p cA δ (m2) (A2.4)

Thus, the hardness of fused silica, Hs, is calculated using Equation (2.40),

3

max

12

120.325 1010.81

11.13 10s

p

WH

A

(GPa) (A2.5)

The reduced modulus, Er, is evaluated based on Equation (2.37),

610

12

0.249 106.397 10 63.97

2 2 1.034 11.13 10r

p

S π πE

β A

(GPa) (A2.6)

174

The Young’s modulus of fused silica, Es, is evaluated using Equation (2.39),

1 12 2

2 21 1 0.071 1

1 1 0.18 65.663.97 1141

i

s s

r i

νE ν

E E

(GPa) (A2.7)

where Young’s modulus of indenter, Ei = 1141 GPa, the Poisson’s ratio, νi = 0.07 [16].

The Young’s modulus obtained by Oliver and Pharr is 69.3 GPa with the standard

deviation of 0.39 GPa.

175

Appendix 3 Mechanical Properties of Thin

Polymer Films Using Indentation

Nano-indentation is used to characterize mechanical properties of polymers in thin-film

and bulk forms. The experimental method and background are presented in Section 2.5

and 7.2, respectively. Figures A3.1 - A3.4 show the load-displacement graphs for 5, 10

µm PTFE films, bulk PTFE and 5 µm silicone film, respectively. The Young’s modulus

and the hardness were evaluated by using Oliver and Pharr’s theory [16]. Table A3.1

shows the material properties of polymers, including PTFE and silicone, in thin-film and

bulk forms, respectively. It indicates that thin-film PTFE coatings have greater Young’s

modulus and hardness than bulk PTFE, but lower than 5 µm silicone films. The student’s

T-test was performed to test if the material properties are as a function of film thickness

among 5 and 10 µm PTFE films. The null-hypothesis is that the material properties of

PTFE are not a function of film thickness. The p-values shown in Table A3.2 indicate

that film thickness has no significance on the material properties between 5 and 10 µm

PTFE films.

Lucas et al. [59] reported that the material properties of PTFE film on silicon wafer are

independent of the film thickness (0.5 – 15 µm). Wang et al. [58] examined the thickness

effects on material properties of ultra-thin PTFE films (48.1 – 1141 nm), spin coated on

the silicon wafers. Thickness-dependence effect is seen on thinner films (< 500 nm).

When the thickness is greater than 500 nm, the test results indicate no thickness-

176

dependence on material properties. In particular, Young’s modulus, E = 2.3 GPa, and

hardness, H = 58 MPa. Wang et al. [58] also found that PTFE in thin-film form exhibits

higher modulus compared to bulk sample and free-standing film. It is possible that

polymers have a high pressure dependence of the strength, which gives rise to higher

modulus in compression than in tension. The molecular organizations in ultra-thin and

thin polymer films are different from those in bulk forms, resulting in significance in

thermophysical and mechanical properties [60].

Figure A3.1 The load-displacement graph for 5 µm PTFE film.

177

Figure A3.2 The load-displacement graph for 10 µm PTFE film.

Figure A3.3 The load-displacement graph for bulk PTFE (thickness, t = 1.5 inches).

178

Figure A3.4 The load-displacement graph for 5 µm silicone film.

Table A3.1 Material properties of polymers in thin-film and bulk forms.

5 µm PTFE 10 µm PTFE bulk PTFE 5 µm silicone

Young’s modulus (GPa) 2.71 2.80 0.85 3.55

Std. Dev. (GPa) 0.26 0.17 0.07 0.26

Hardness (GPa) 0.11 0.10 0.04 0.23

Std. Dev. (GPa) 0.01 0.01 0.01 0.02

Table A3.2 The student’s T-test of material properties for 5 and 10 µm PTFE films.

Young’s modulus, E Hardness, H

p-value 0.517 0.095

179

Appendix 4 Material Properties of Glass

Substrates

Material properties of glass substrate, where thin polymer films are supposed to be

deposited by using HFCVD, are also examined. Table A4.1 shows the Young’s modulus

and the hardness of glass with ten test repeats. The evaluations of Young’s modulus and

hardness are given as 62.59 ± 7.83 GPa, and 9.45 ± 1.85 GPa, respectively.

Table A4.1 Material properties of glass using nano-indentation.

Test # W (mN) δmax (nm) E (GPa) H (GPa)

1 4.99 208.23 54.34 9.26

2 4.99 215.34 56.66 7.60

3 4.99 200.77 72.71 7.92

4 4.99 197.91 62.12 9.87

5 4.99 213.71 49.56 9.35

6 4.99 207.84 60.21 8.26

7 4.99 188.11 73.57 10.16

8 4.99 195.38 67.44 9.49

9 4.99 203.91 63.59 8.45

10 4.99 183.28 64.66 14.13

Average 62.59 9.45

Std.Dev. 7.83 1.85

180

Appendix 5 Green’s Function

A Green’s function g(x,s) represents the effect/displacement at a position x due to a force

of unit magnitude acting at s, which is schematically shown in Figure A5.1. Note that a

Green’s function is symmetrical, i.e. g(x,s) = g(s,x). The displacement v(x) for any

distributed force, p(x), applied in the range l1 ≤ x ≤ l2, can be obtained by using an

influence (Green’s) function as follows [7],

2

1

,l

lp xv x g x s dx (2.11)

Figure A5.1 Schematic of the displacement for distributed force, p(x) using the Green’s

function [7].