modification of the mechanical and surface properties at the

UNIVERSITE LIBRE DE BRUXELLES Facult´e des Sciences Appliqu´ees

Service Mati`eres et Mat´eriaux

Characterization and

modification of the mechanical and surface properties at the

nanoscale

Academic year 2009–2010

PhD Director: Professor Marie-Paule Delplancke Author: Enrico Tam

Dissertation presented in order to obtain the title of Philosophical Doctor

(PhD) in Engineering Sciences

ni `a un int´erˆet, ni `a une id´ee pr´econ¸cue, ni `a quoi que ce soit, si ce n’est aux faits eux-mˆemes, parce que,

pour elle, se soumettre, ce serait cesser d’ˆetre.”

Henri Poincar´e (Fˆetes du LXXVe anniversaire de l’ULB, 21 novembre 1909)

Acknowledgments

The writing of a dissertation is a very challenging experience which is obviously not possible without the personal and practical support of numerous people. Therefore, I would here like to express my gratitude to the people who contributed to the accom- plishment of this thesis.

First and foremost I wish to thank Prof. Marie-Paule Delplancke-Ogletree for giving me the chance to participate in this project and for giving me all the resources I needed to succesfully accomplish this work. Her oral and written comments were always extremely perceptive, helpful, and appropriate.

Then, my sincere gratitude goes to all the members of the MiniMicroNano project:

Prof. Philippe Bouillard, Prof. Alain Delchambre, Prof. Frank Ogletree, Prof. Pierre Lambert, Prof. Thierry Massart, Dr Marion Sausse Lhernould et Dr Peter Berke.

Many people on the faculty and staff of the Chemicals and Materials department assisted and encouraged me in various ways during my works. I especially wish to thank them for all the enjoyable lunches and coffee breaks we had.

During the PhD studies I had the great opportunity to spend 6 months at the Lawrence Berkeley National Laboratory. I would like to express my deepest gratitude to all the Molecular Foundry staff. In particular, I am greatful to Prof Miquel Salmeron, Prof.

Frank Ogletree and Dr Paul Ashby for their critical comments and discussions, which were very important for my personal and scientific development.

During these last years spent working abroad (Belgium and USA) I have been fortunate to come across many exceptional friends, without whom life would be bleak. I would like to thank all of them for always supporting me and for all the nice moments we shared together. They were my real source of energy.

This whole thesis would not look like it is without the precious help of some proof- readers. I more than appreciate their help.

My gratitude also goes to the french community of Belgium since this work would not have been possible without their financial support.

Last but not least, I’m particularly greatful to my family who, through my childhood

and study career, have always encouraged me to follow my heart and my adventurous

mind in any direction this has taken me.

Abstract

In the past two decades much effort has been put in the characterization of the me- chanical and surface properties at the nano-scale in order to conceive reliable N/MEMS (Nano and Micro ElectroMechanical Systems) applications. Techniques like nanoinden- tation, nanoscratching, atomic force microscopy have become widely used to measure the mechanical and surface properties of materials at sub-micro or nano scale. Nev- ertheless, many phenomena such us pile-up and pop-in as well as surface anomalies and roughness play an important role in the accurate determination of the materials properties. The first goal of this report is to study the infulence of these sources of data distortion on the experimental data. The results are discussed in the first experimental chapter.

On the other hand, conceptors would like to adapt/tune the mechanical and surface properties as a function of the required application so as to adapt them to the industrial need. Coatings are usually applied to materials to enhance performances and reliability such as better hardness and elastic modulus, chemical resistance and wear resistance.

In this work, the magnetron sputtering technique is used to deposit biocompatible thin layers of different compositions (titanium carbide, titanium nitride and amorphous carbon) over a titanium substrate. The goal of this second experimental part is the study of the deposition parameters influence on the resulting mechanical and surface properties.

New materials such as nanocrystal superlattices have recently received considerable

attention due to their versatile electronic and optical properties. However, this new

class of material requires robust mechanical properties to be useful for technological

applications. In the third and last experimental chapter, nanoindentation and atomic

force microscopy are used to characterize the mechanical behavior of well ordered lead

sulfide (PbS) nanocrystal superlattices. The goal of this last chapter is the under-

standing of the deformation process in order to conceive more reliable nanocrystal

superlattices.

Contents

1 Introduction 1

1.1 The MiniMicroNano Project . . . . 1

1.2 Present Work . . . . 3

2 Literature Survey (Section A): Surface Properties Characterization 7 2.1 Introduction . . . . 7

2.2 Mechanical Properties by Nanoindentation . . . . 8

2.2.1 Generalities about Nanoindentation . . . . 8

2.2.2 Young’s Modulus and Hardness by Oliver & Pharr . . . . 9

2.2.3 Young’s Modulus and Hardness by the Energetic Approach . . . 13

2.2.4 Continuous Stiffness Measurement . . . 15

2.2.5 Fracture Toughness Characterization . . . 18

2.2.6 Indentation of Thin Layers . . . 21

2.2.7 Indentation and Grain Size Effect . . . 22

2.2.8 Indenters Geometry . . . 23

2.3 Potential Sources of Data Distortion during Nanoindentation . . . 26

2.3.1 Preliminary Discussion . . . 26

2.3.2 Load Frame Compliance . . . 27

2.3.3 Initial Penetration Depth . . . 27

2.3.4 Tip Shape Anomalies . . . 27

2.3.5 Pile-up and Sink-in Phenomena . . . 28

2.3.6 Roughness Effects . . . 29

2.3.7 Creep Phenomenon . . . 31

2.3.8 Discontinuities during the Load-Displacement Curves . . . 32

2.3.9 Influence of Process Parameters . . . 34

2.3.10 Adhesion . . . 34

2.4 Surfaces Properties by Atomic Force Microscopy . . . 35

2.4.1 Generalities about AFM . . . 35

2.4.2 Adhesion Forces by AFM (Pull-off test) . . . 39

2.4.3 Electrostatic Forces by AFM . . . 41

2.5 Mechanical Properties by NanoScratching . . . 42

2.5.1 Generalities about NanoScratching . . . 42

2.5.2 Nanoscale Wear Characterization . . . 43

2.5.3 Friction Behavior Characterization . . . 46

2.5.4 Interfacial Fracture Test (delamination) . . . 47

2.6 Surface Properties by Contact Angle Measurements . . . 48

2.7 Residual Stresses in Coatings by Profilometry . . . 53

2.8 Summary . . . 55

3 Literature Survey (Section B): Surface Properties Modification 57 3.1 Introduction . . . 57

3.2 Physical Vapor Deposition by Magnetron Sputtering . . . 58

3.2.1 Generalities . . . 58

3.2.2 Key Parameters . . . 62

3.2.3 The Main Advantages of Magnetron Sputtering . . . 63

3.2.4 Applications of Sputtered Films . . . 64

3.3 Summary . . . 64

4 Materials and Experimental Technics 65 4.1 Introduction . . . 65

4.2 Bulk Materials Definition (Ti, Ni, Si) . . . 66

4.3 Coatings Definition (TiOx, TiC, TiC, aC-H) . . . 68

4.3.1 Titanium Dioxide Coatings . . . 68

4.3.2 Titanium Carbide Coatings . . . 69

4.3.3 Titanium Nitride Coatings . . . 69

4.3.4 Carbon Coatings . . . 70

4.3.5 Summary about the Coating Properties . . . 71

4.4 Experimental Technics Definitions . . . 72

4.4.1 Hysitron Triboindenter . . . 72

4.4.2 Atomic Force Microscope . . . 74

4.4.3 In-Situ SEM, Nanoindentation and AFM Measurements . . . 75

4.4.4 Coatings Deposition Tools . . . 76

4.4.5 Grain Structure/Size Characterization . . . 77

4.4.6 Coatings Thickness Characterization . . . 78

4.4.7 Crystalline Structure Characterization . . . 79

4.4.8 Coatings Composition Characterization (RBS) . . . 80

4.5 Summary . . . 81

5 Experimental Results (Sec. A): Characterization of Bulk Materials (Ti & Ni) 83 5.1 Introduction . . . 83

5.2 Specimen Preparation . . . 84

5.3 Mechanical Properties by Nanoindentation . . . 86

5.3.1 Discussion about the Calibration . . . 86

5.3.2 Evolution of E&H of Ti&Ni with the Penetration Depth . . . 91

5.3.3 Holding Time & Loading Rate Influence . . . 97

5.3.4 Discussion about the Contact Stiffness Determination . . . 103

5.3.5 Modulus and Hardness by the Energetic Approach . . . 105

5.3.6 Grain Size Effects . . . 106

5.3.7 Roughness Influence . . . 107

5.3.8 Pop-ins Analysis . . . 109

5.4 Mechanical Properties by Nanoscratching . . . 113

5.5 Surfaces Properties by AFM . . . 116

5.5.1 Motivations of the Study . . . 116

5.5.2 Experimental Set-up Definition . . . 117

5.5.3 Qualitative In-Situ SEM-EFM . . . 120

5.5.4 Electrostatic Forces Coming from the Cantilever . . . 122

5.5.5 Spontaneous Charging of the Tip and Surface . . . 124

5.5.6 Tip Diameter and Voltage Effects on the Electrostatic Forces Measurements . . . 125

5.5.7 Roughness Influence on the Electrostatic Forces . . . 127

5.6 Surface Energy by Contact Angle Measurements . . . 131

5.7 Conclusions . . . 133

6 Experimental Results (Sec. B): Deposition & Characterization of Thin Layers 137 6.1 Introduction . . . 137

6.2 Preliminary Study about Hard Thin Layers . . . 139

6.2.1 Aims of the Work . . . 139

6.2.2 Samples Preparation at MISA - Experimental Details . . . 140

6.2.3 Young’s Modulus and Hardness Comparison . . . 140

6.2.4 Statistical Distribution of Young’s Modulus and Hardness . . . . 141

6.2.5 Discussion about the Contact Area Determination . . . 143

6.2.6 Correlation between Roughness and Spread of Hardness and Young’s Modulus Result . . . 144

6.3 Influence of the Titanium Oxide Layers . . . 145

6.3.1 Deposition and Mechanical Characterization of the TiOx . . . . 145

6.4 Preparation and Characterization of Coatings by Magnetron Sputtering 148 6.4.1 Preliminary Discussion about the Substrates . . . 148

6.4.2 Titanium Carbide Characterization . . . 150

6.4.3 Titanium Nitride Characterization . . . 155

6.4.4 Hydrogenated Carbon Characterization . . . 160

6.5 Evolution of Modulus and Hardness across the Layer . . . 163

6.6 Coefficient of Friction & Interfacial Fracture . . . 165

6.7 Conclusions . . . 170

7 Experimental Results (Sec. C): Characterization of Superlattices (PbS) 173 7.1 Introduction . . . 173

7.2 Calibration of the AFM Nanoindentation Tool . . . 175

7.3 Modulus and Hardness Calculation . . . 176

7.4 Investigation of the Deformation Process . . . 178

7.5 Conclusions . . . 183

8 Conclusions and Perspectives 185

9 Annexes: Main Publications 189

Chapter 1

Introduction

1.1 The MiniMicroNano Project

The development of NEMS and MEMS (Nano and Micro Electro Mechanical Systems) is a very big challenge for today’s society. Silicon technology thanks to its easy integration between mechanical and electronic components gave tremendous advance to the microelectronics business over the last forty years.

The object of the Mini-Micro-Nano project is to focus on complementary approaches to N/MEMS development using more conventional engineering materials (metal alloys, ceramics, polymers). Such an approach brings new opportunities and a much wider array of materials properties becomes available.

With new opportunities come new challenges. Critical components such as microassem- blers and movable microstructures (i.e. for human implanted devices) require charac- teristics like smaller dimensions, lighter weight, and increased resistance to wear and deformation. Thus, a better understanding of phenomena taking place at small scale, new techniques, and multifunctional materials are needed. These new challenges can only be handled by a multidisciplinary engineering approach.

The main and long term goal of this project is to build a center of excellence at the ULB (Universit´e Libre de Bruxelles) in the field of micromechanical engineering, combining fundamental and applied research simultaneously. Its development is based on the expertise of three ULB partners:

• Chemicals and Materials Department: Surface characterization and modi- fications

• BATir: Simulation of the mechanical behavior at microscale

• BEAMS: Design and production of micro components

The Chemicals and Materials department was in charge of the experimental task in order to characterize and modify the mechanical and surface properties of materials at micro and nanoscales so as to understand the phenomena taking place at the nanoscale.

This constitutes the subject of this thesis. A more exhaustive introduction about the undertaken works and experiments performed in this context is given in the next section.

The BATir department main task was the development of a simulation tool character- izing the deformation involved during the depth sensing nanoindentation experiments.

This becomes particularly useful to understand the mechanical behavior of materials at a very small scale.

The BEAMS department focused on the understanding and modeling of surface forces (such as electrostatic forces) at the nanoscale. Those forces can be neglected at the macroscale but are of great practical impact at a micro or nanoscale.

The main interactions between the three departments are shown in image 1.1.

Figure 1.1:

The project, financially supported by the ”Communaut´e Francaise de Belgique”, spanned over five years and involved three PhD students attached to the relative departments for four years. Four DEA (Diplome d’´etudes approfondies) [1–4] and 2 PhD thesis [5,6]

were alredy obtained in the frame of this project.

An international collaboration with the Lawrence Berkeley National Laboratory (Cal-

ifornia, USA) was also provided thanks to Prof. D.F.Ogletree. This consisted of a 6

months stage at the Molecular Foundry department of the above mentioned laboratory.

1.2. Present Work

1.2 Present Work

This thesis constitutes a dissertation of the study conducted during the four academic years spanning from 2005 to 2009 at the Chemicals and Materials Department of the Universit´e Libre de Bruxelles in order to obtain the title of Philosophical Doctor in Engineering (PhD).

The long-term aims of this work are to lay the scientific basis leading to the design of a microscale gripper capable of manipulating microscopic objects in various environ- ments. The importance of various phenomena taking place at the micro and nanoscale can be totally different from what we expect at the macroscale. For instance, when two objects are brought into contact and then separated, electrostatic forces (which are usually neglected at the macroscale) can play an important role. As a result, the con- ception of reliable micromanipulation systems able to work in different environments is possible only if those phenomena are understood and controlled.

Various issues have to be taken into account to reach this goal. On the one hand, the mechanical properties of the material used to conceive the manipulator tool are of great importance in order to provide reliability. On the other hand, the manipulation by contact of objects at the microscale (generally between 1µm and 1mm) is often disturbed by perturbations related to adhesive surface forces between the manipulated object and the gripper. For example, high surface energies are unwanted since they induce high pull-off forces and can make it impossible to release the manipulated object.

In this optic, controlling the electrostatic forces (in function for instance of the intrinsic surface roughness resulting from different machining techniques) would provide new important design solutions for the conception of new micromanipulators devices.

To reach the aim of the project, an integrated approach combining simulation and experimental work has been conducted. The major techniques used to characterize the surface properties at the nanoscale (nanoindentation, nanoscratching, atomic force microscopy, etc...) were deeply studied. The simulation of the nanoindentation and of the surface forces (mainly electrostatic forces) were provided by the interaction with the BATir and BEAMS departments respectively.

Understanding the phenomena taking place at the nanoscale is, of course, a major and important task but the real challenge would be the modification of the mechanical and surface properties in order to adapt them to the required application. In this context, big efforts were also put in the modification of the surface properties through the deposition of specific thin layers over a bulk substrate.

As a consequence, the efforts done to complete this work are divided into two directions.

Firstly, it was important to understand the nanoscale phenomena and the resulting

surface properties associated with them. Secondly, the attention was focused on the

modification of the surfaces (for instance by physical vapor deposition) in order to

control properties like surface energy and electrostatic forces which, as already stated,

are primordial in the manipulation of small objects.

Since the use of micromanipulation tools (and more generally of MEMS) for human implanted devices is consistently growing, it was decided to take into account mainly biocompatible materials. Bulk titanium was chosen as ’base’ material for its well known biocompatibility and good mechanical properties. Titanium carbide, titanium nitride, titanium oxide and carbon layers, all known to have biocompatible behaviors (even if the extent should be verified) were chosen to be the main coating materials. Bulk nickel was also studied because of its wide utilization in the conception of micromanipulation tools.

Figure 1.2:

collaboration with the BEAMS and BATir departments of the ULB.

The main steps of the PhD work are illustrated on figure 1.2 while the current disser- tation is divided in the following 9 chapters:

The first Chapter is dedicated to the introduction and to the presentation of the Mini-Micro-Nano project.

The second Chapter introduces the main surface characterization techniques, the

theory behind it, the properties derived from it and the instrumentation used to per-

form the tests. Mechanical property values such as modulus and hardness are cal-

culated from nanoindentation experiments based on the idealized elastic contact the-

ory and load-displacement data. Nanoindentation is slightly different from macroscale

hardness tests and requires a number of significant assumptions. In some circumstances

these assumptions can lead to significant sources of error. In order to understand why

the resulting data may be inaccurate, some of the common sources of data distortion

that need to be corrected (such as pile-up and roughness) are discussed.

1.2. Present Work

Nanoscratching, which allows deriving properties such as the coefficient of friction and the delamination resistance is also presented and discussed. Then, the basic theory behind the atomic force microscopy (AFM) is presented. AFMs were mainly used to characterize the electrostatic forces as a function of the tip/sample separation distance.

Finally, some information about the surface energy characterization technique and the residual stresses determination are presented.

Chapter three introduces the techniques utilized to modify the surface properties.

The Physical Vapor Deposition technique (by magnetron sputtering), which was used in our study to deposit thin layers over bulk substrate (titanium carbide, titanium nitride, and carbon layers), is presented. The process parameters to control, the ad- vantages and the main application of the magnetron sputtered films are extensively discussed.

All experimental procedures and materials taken into account during the experimental part are introduced in Chapter four. Any of the additional instrumentation aspects and procedures involved in the characterization and modification of the surface prop- erties are given in this chapter.

Chapter five deals with the characterization of bulk materials (titanium and nickel).

Elastic modulus and hardness were measured using the nanoindentation technique.

The influence of parameters like the holding time at maximum load and the loading rate was investigated. The effects of roughness and grain size was also taken into account at this stage. Electrostatic forces determined by AFM are derived and the effect of factors like roughness is studied. Then, the results of the nanoscrathing experiments and the surface energy determination are shown and discussed.

Chapter six introduces and discusses the result of the modification of the surface prop- erties. The deposition procedure of thin titanium oxide, titanium carbide, titanium nitride and carbon layers is given and discussed. At the same time, the mechanical properties (modulus and hardness) and their evolution across the layer are studied.

This is done by many different characterization techniques such as nanoindentation, nanoscrathing, atomic force microscope, XPS, RBS, XRD, contact angle measurement.

Chapter seven completes the experimental part and concerns the study of lead sulfide (PbS) nanocrystal superlattices. This advanced material received considerable interest due to its enhanced electronic and optical properties. Nanocrystal superlattices are characterized by an organized structure and are composed of an inorganic nanoparticles core with organic ligands bound to the surface of each core. This new class of material combines the unique properties of the organic nanocrystal core with the properties arising from the interaction between neighboring nanocrystals in the superlattice. The knowledge acquired during the characterization of bulk material and thin layers was necessary to study this new kind of advanced material.

Chapter eight presents the major and most important conclusions and gives some perspectives for future works.

Finally the last Chapter is dedicated to the annexes in which the major publications

realized in the mini-micro-nano project frame are reported.

Chapter 2

Literature Survey (Section A):

Surface Properties Characterization

2.1 Introduction

The study of the mechanical properties at the micro and nanoscale is of great practicle importance for the development of M-NEMS in order to provide the reliability of the conceived systems. For more than a century, researchers in material sciences have tried to develop techniques and experimental tools able to derive the mechanical properties of materials at small scales. In the past two decades, however, a veritable revolution has occurred thanks to the development of new sensors and actuators. This chapter presents an overview of the main techniques to characterize the surface properties at the nanoscale.

Depth sensing nanoindentation provides a highly powerful method for measuring the localized Young’s modulus and hardness, and there has been considerable progresses in the measurement of other mechanical properties such as the hardening/softening behavior, the creep effect, and the residual stresses. The nanoindentation technique current role and some of the physical phenomena connected with it are presented here, with special emphasis on the post treatment methods.

Thin layers are of great practical importance nowadays, thus, the problems associeted with the nanoindentation of thin layers are also discussed in this chapter. Nano- scratching, which is a powerful technique to study, for instance, the delamination of thin layers, is also presented and discussed.

On the other hand, the interaction and adhesion of surfaces brought into contact

is primordial in the conception of micromanipulator tools. AFM can be utilized to

derive the attractive forces between two approaching objects. Surface energy, which is

connected with the adhesion between two bodies, can also be measured with surface

angle measurements. Those techniques are described and illustrated in the last part

of the chapter.

2.2 Mechanical Properties by Nanoindentation

2.2.1 Generalities about Nanoindentation

Nanoindentation is a novel technique developed in the past two decades to measure the mechanical properties of materials [7–24]. This technique is based on high-resolution instruments that continuously monitor the loads and displacements of an indenter as it is pushed into and withdrawn from a material. The load-displacement data (see figure 2.1) obtained from the indentation process, which is often referred to a load-penetration depth curve of indentation, can be used to derive various mechanical properties of materials, most commonly, hardness and Young’s modulus. In addition, the load-displacement curve may contain other information about properties such as the hardening exponents [25–28], creep parameters [29–31] and residual stresses [32–35].

An obvious advantage of depth-sensing indentation test over conventional hardness test is that the contact area of an imprint can be directly determined from the load- penetration depth curve knowing the geometry of the indenter. This feature makes the depth-sensing indentation test particularly suitable for measuring the mechanical properties of materials at small scales where accurate determination of the contact area would be an extremely difficult task for conventional hardness test.

In general, load-displacement curves provide a lot of information but care should be given to the conditions in which these properties were derived. Several adaptations to the basic nanoindentation set-up could be applied to obtain additional information about the processes that occur during nanoindentation testing. For example, in-situ measurements of acoustic emissions and contact resistance can indicate if a phase transformation, fracture or delamination occurred in the sample [36–46]. Environmen- tal control can also be used to examine the effects of temperature and surface chemistry on the mechanical behavior of nanocontacts.

As can be seen, nanoindentation is in continuing development and the aim of the

following paragraphs is to summarize the state of the art about this useful mechanical

properties characterization technique.

2.2. Mechanical Properties by Nanoindentation

2.2.2 Young’s Modulus and Hardness by Oliver & Pharr

An ideal load-penetration curve

obtained after a nanoindentation experiment consists of two parts, loading and unloading, as shown in figure 2.1. The loading part normally

Figure 2.1:

technique. [8]

includes the elastic-plastic deformation of the material [7, 8, 10] and can be expressed as

P = Ah

(2.1)

where P is the indentation load, h is the penetration depth measured from surface, m and A are two constants that are dependent on the geometry of the indenter and the mechanical properties of the material. The relationship in Eq. 2.1 has been confirmed by the experiments of Hainsworth et al. 1996 [10]. According to Oliver and Pharr [7, 8, 10] the unloading part of the indentation process, which is mainly elastic, can be described by

P = Bh

(2.2)

where h

is the elastic depth of the penetration, B is a constant that is related to the elastic properties of materials and the geometry of indenter, and m is a constant, which equals 1, 1.5, and 2 for a flat cylinder punch, sphere or parabola of rotation, and cone, respectively. Experiments conducted on a variety of materials have revealed that the unloading curve is well-described by equation 2.2. Rewriting h

as h − h

one finds

P = B (h − h

)

(2.3)

where h

is the final penetration depth after complete unloading.

1Also called load-displacement curve

Figure 2.2:

To obtain reliable indentation results accurate knowledge of the area of contact is crucial. In order to determine the projected contact area two factors must be known.

One is the geometry of the indenter, i.e. the area function, A = f (h), that relates the cross-sectional area of the indenter to the distance from its tip. This area function can be determined by direct measurement

or can be derived using the calibration method suggested by Oliver and Pharr [8].

A general form that is often used to describe the area function is

A = C

h

+ C

h + C

h

+ C

h

+ . . . (2.4) where the number of terms is chosen to provide a good fit over the entire range of calibration.

The other parameter needed for the calculation of the projected contact area is the contact depth at peak load. As shown in Fig. 2.2, the contact depth at peak load is given by

h

= h

− h

(2.5)

where h

is the so called contact depth, h

is the maximum penetration depth, which can be directly determined from the load-penetration curve and h

is the deflection of the surface at the perimeter of the contact. It is given by Sneddon’s equation.

h

= ε P

S (2.6)

where ε is a geometric constant and ε = 0.72 for cone, ε = 0.75 for paraboloid of revo- lution, and ε = 1 for flat punch while S is the contact stiffness at the initial unloading.

2This approach is particularly inconvenient, especially at very low loads where direct imaging is difficult to perform.

2.2. Mechanical Properties by Nanoindentation

The contact stiffness, S, is determined in two steps: firstly, a fraction of the unloading curve (normally the unloading data from 20-95% is considered) is fitted with equation 2.3. Secondly, the unloading curve fit is differentiated analytically to determine the slope at maximum load

S = dP

dh |

(2.7)

The contact depth is then determined by recombining equations 2.5, 2.6 and 2.7 h

= h

− ε P

S (2.8)

and the projected contact area can consequently be calculated from the relation

A

= f(h

) (2.9)

The relation connecting the contact area to the contact depth can be directly derived by imaging the indenter tip by atomic force microscopy (AFM), which is the most reliable way, or can be indirectly derived. This second way implies performing several indentations at different increasing maximum load in a material of known Young’s modulus and the final A

= f(h

) relation is found by deriving the contact area for which the experimental Young’s modulus match the expected one.

Now, assuming that indenter and specimen behave like springs in series [14], the elastic deformation of both can be characterized by a single ’reduced modulus’

1

E

= 1 − ν

E

+ 1 − ν

E

(2.10)

where E

is the so-called reduced modulus due to a non-rigid indenter, E

(known) is the Young’s modulus of indenter, Es is the Young’s modulus of the specimen, ν

(known) and ν

(known) are the Poisson’s ratio of the indenter and the specimen, respectively.

Introducing the equation

E

=

√ π 2

√ S

A

(2.11)

which is derived from the Sneddon’s solution (1965) for the elastic deformation of an isotropic elastic material with a flat-ended cylindrical punch, one finds the unknown specimen Young’s modulus Es which is the only unknown parameter remaining.

In addition to the Young’s modulus of the specimen, the data obtained using this method can be used to determine the hardness, H. We define the hardness as the mean pressure the material will support under load. With this definition, the hardness is computed from

H = P

A

(2.12)

where Ac is the projected area of contact at peak load evaluated from equation 2.9.

Figure 2.3:

for ideal elastic(left), rigid-plastic (middle), and elastoplastic materials(right)[47].

Figure 2.3 shows the stress-strain diagrams, the load-displacement curves and the surface profiles at maximum load for an ideal elastic, rigid plastic and elastoplastic material.

Note 1:

Hardness measured using this definition may be different from that obtained from the more conventional definition in which the area is determined by direct measurement of the size of the residual hardness impression. This is because, in few materials, a portion of the contact area under load may not be plastically deformed, and as a result, the contact area measured by observation of the residual hardness impression may be less than that at peak load.

Note 2:

The Poisson’s coefficient is necessary to derive the Young’s modulus of the indented

sample (see equation 2.10). Thus, in the experimental chapters, Young’s modulus

will be derived when indenting on materials with known Poisson’s coefficient (bulk

titanium and nickel) while the only ’reduced modulus’ will be calculated for materials

with unknown Poisson’s coefficient (thin layers). Nevertheless, Young’s modulus and

reduced modulus values are similar.

2.2. Mechanical Properties by Nanoindentation

2.2.3 Young’s Modulus and Hardness by the Energetic Approach In order to improve the characterization of the Young’s modulus and hardness of a given material many authors proposed a different approach based on energetic considerations [48–54].

Figure 2.4:

The principle of the energetic approach was first introduced by Stilwell and Tabor in 1961. In this work, it was shown that the conventional estimation of hardness (maximum applied load divided by the projected contact area H = P

/A

) is equivalent to the plastic work divided by the plastically deformed volume [50]. This hardness characterization method was nevertheless not useful due to the lack of preci- sion of the measurement techniques.

Starting from the conventional expression of hardness one can write:

H

= P

A

⇒ P dh

A

dh ⇒ dW

dV

(2.13)

where W

and V

are the plastic work and the plastic volume respectively.

Integrating this last expression for penetration h ranging from 0 to h

one find:

H

=

0

P dh

R_hmax

0

Adh =

R _hmax

0

dW

0

dV

= W

V

(2.14)

which is the expression for the energetic hardness. As a consequence, for purely elastic indentation, one obtains H

=

and the hardness value is then undefined.

The plastic work as well as the elastic work involved during an indentation experiment

can easily be derived from the load-displacement curve (see figure 2.4).

The other parameter necessary to the energetic hardness determination is the plastic volume V

. Wolf et. al [55] have proposed the calculation of V

by using the contact area determined by the tip area function (see previous sections for more details). It was shown previously that the area of contact can be expressed as a function of the penetration:

A(h) = C

h

+ C

h + C

h

+ C

h

+ . . . (2.15) and the contact volume can then be derived integrating this latter equation:

V (h) =

A(h)dh = C

h

3 + C

h

2 + 2C

h

3 + 4C

h

5 + . . . (2.16) which correspond to the volume of the tip pressed into the sample under load. During the unloading, a part of the deformation is relaxed. In a first order approximation, it can be assumed that this elastic relaxation occurs mainly in the vertical direction, whereas the lateral relaxation is negligible. The plastic volume of the remaining imprint can then be calculated as follow:

V (pl) = h

h

∗ V (h

) (2.17)

Cheng and Cheng [56] demonstrated that the ratio W

/(W

+ W

) can be linked to the material constitutive behavior such as ratio of hardness to reduced modulus H/E

.

W

W

∼ = 1 − x H

E

(2.18)

where W

= W

+ W

.

The value x can be derived by indenting in a sample of known Young’s modulus, for instance using the standard sample (fused quartz) used during the area function calibration.

Knowing the hardness, it is now possible to derive the reduced modulus associated with the material (and consequently the Young’s modulus).

E

= xH

1 − W

/W

(2.19)

2.2. Mechanical Properties by Nanoindentation

2.2.4 Continuous Stiffness Measurement

Continuous Stiffness Measurement (CSM) is a recent and extremely powerful technique [16] that offers a significant improvement in nanoindentation testing. The CSM allows the measurement of the contact stiffness at any point along the loading curve and not just at the point of unloading as described in the preceding section. This is accomplished by imposing a harmonic force P = P

, which is added to the nominally increasing load on the indenter, as shown in figure 2.5.

Figure 2.5:

By continuously measuring the displacement response h(ω) during the loading of the indenter at the excitation frequency and the phase angle between the two, it is possible to solve the in-phase and out-of-phase portions of the response results. This permits an explicit determination of the contact stiffness, S, as a continuous function of depth.

Considering the masse m of the indenter, the spring constant, K

, of the leaf springs that support the indenter, the stiffness of the indenter frame K

= 1/C

(where C

is the compliance of the load frame) and the damping coefficient, C, due to the air in the gaps of the capacitor plate displacement sensing system, combined with the contact stiffness, S, it is possible to schematize the overall response as shown in figure 2.6.

As a result, knowing the imposed driving force P = P

e

and the displacement response of the indenter h(ω) = h

e

, the schematic of figure 2.6 can be solved for the contact stiffness S passing from the determination of the displacement signal,

¯¯

¯¯

P

h(ω)

¯¯

¯¯

=

(S

+ K

)

+ K

− mω

+ ω

C

(2.20)

or from the phase angle between the force and the displacement signal,

tan(φ) = ωC

(S

+ K

)

+ K

− mω

(2.21) where ω is the frequency of the oscillation.

Solving equations 2.20 and 2.21 for the contact stiffness, S, and for the damping due to the air in the gaps between the capacitor plates ωC one finds (the damping of the contact itself is regarded to be negligible):

S =





1

Pos

h(ω)

cos φ − (K

− mω

) − K





−1

(2.22)

ωC = P

h(ω) sin φ (2.23)

This technique makes possible the measurement of mechanical properties of materials such as hardness and Young’s modulus as a continuous function of depth from a single indentation experiment. It makes also possible to obtain data at very small penetration depths. This makes CSM a powerful tool for measuring mechanical properties of nanometric films.

Figure 2.6:

2.2. Mechanical Properties by Nanoindentation

In addition, treating non uniform materials in which the microstructure and the me- chanical properties change with indentation depth becomes possible. Furthermore, utilizing the continuous stiffness technique, creep measurement on the nanoscale can be performed by monitoring changes in displacement and stress relaxation.

Moreover, CSM is less sensitive to thermal drift because it is carried out at frequencies greater than 40 Hz. This allows the accurate observation of creep in small indents to be carried out over a long time period. Also, the performance of fatigue tests at the nanoscale is permitted through load cycles of a sinusoidal shape at high frequencies.

The fatigue behavior of thin films and microbeams can be studied by monitoring the

change in contact stiffness because it has been demonstrated that the contact stiffness

is sensitive to damage formation.

2.2.5 Fracture Toughness Characterization

Fracture toughness is a term that quantitatively describes the material’s resistance to fracturing in the presence of a crack. Large values of fracture toughness mean that the material will probably undergo ductile fracture while low values of fracture toughness indicate that the material will probably face brittle fracture.

To explain the fracture toughness, one can assume that the stability of the crack is assessed as follow [57–60]. For simplicity, the displacement at a defined load can be considered constant and the problem can be simplified characterizing the crack by its area A. Increasing the load will lead to a larger crack (area A+dA), and the strain energy released can be evaluated with respect to the change in crack area.

If the displacement is constant, the force level is dictated by the stiffness (or com- pliance) of the indented body. Thus, when the crack increases in size, the stiffness decrease and that leads to a decreasing of the force level. The fact that the force level decrease under the same displacement indicates that the elastic strain energy stored in the body is decreasing (is being released). Generally, the higher the load, the higher the cracks and the higher the elastic strain energy released.

When the released strain energy exceeds a critical value, the crack will grow spon- taneously. For brittle material, the crack will grow (in an unstable manner) if the the elastic energy released by the crack is greater than the critical energy required to increase the crack surface. For ductile materials, the energy associated with the plastic deformation must be taken into account. In this case, the energy involved in the propagation of the crack can be much larger than the one observed for brittle ma- terials since the work necessary for the plastic deformation can be much greater than the surface energy.

In practice, if the elastic energy released is higher than the sum of the surface energy and plastic deformation energy, the crack will propagate.

Nanoindentation can be used to evaluate the fracture toughness of materials and inter- faces by measuring the residual crack after complete unloading. This can be done for instance by taking some AFM or high resolution SEM images of the residual imprint.

The type of the created cracks, as well as their formation mechanism, varies according to the nature of the material and the indenter geometry. The two main crack geometry regimes for sharp indenters are described by the Half-Penny and the Palmqvist model depending on the crack formation evolution [61, 62].

Half-Penny models are connected with both median and radial cracks. Median cracks consist of penny shaped cracks which lie perpendicular to the indented surface while radial cracks are cracks propagating radially from the tip center. Palmqvist models are developed for shallow radial cracks occurring on the specimen surface at the edge of the plastic contact impression, usually at the indentation corner.

It was demonstrated by Palmqvist that the length of cracks, which emanate from the

corner of an indent, can be empirically related to the toughness of the investigated

material [62].

2.2. Mechanical Properties by Nanoindentation

Figure 2.7:

Attention is usually given to the length of the radial cracks as measured from the corner of the indentation and then radially outward along the specimen surface as shown in figure 2.8.

Figure 2.8:

The numerical values of the fracture toughness in the case of both Palmqvist and Half-Penny models can then be derived measuring the length of the radial cracks and applying the Laugier’s equations [64, 65]:

K

= x

a

l

¶_1/2µ

E H

¶_2/3

P max

c

(2.24)

K

= x

E

H

¶_2/3

P max

c

(2.25)

Where Kc is the fracture toughness, a,c,l describe the length of radial cracks (see image

2.8), E the Young’s modulus, H the hardness, x

and x

two constants (usually

0.015).

Lateral cracks can also be present. They are usually generated below the surface regardless of whether the threshold for median crack initiation is exceeded or not. If the applied load is too high for the given specimen then these lateral cracks tend to divert upwards toward the surface, and can result in material removal at the surface.

This process is also called ”chipping” [60].

Table 2.1 summarizes the fracture toughness for the most common materials.

Material Fracture Toughness

[MPa m^1/2]

Metals Aluminium alloy 36

Steel alloy 50

Titanium alloy 44-66

Ceramics Aluminium oxide 3-5

Silicon carbide 0.7-0.8

Concrete 0.2-1.4

Polymers Polymethyl methacrylate 1

Polystyrene 0.8-1.1

Table 2.1:

It is important to mention that the presence of residual stresses can influence the fracture resistance. From the theoretical point of view, the apparent surface crack length is a combination of both the intrinsic fracture toughness of the material and the pre-existing residual stresses. Compressive stresses will lead to a diminution of the surface crack length relative to the equilibrium length in the absence of stresses.

Tensile stresses will do the opposite [66].

2.2. Mechanical Properties by Nanoindentation

2.2.6 Indentation of Thin Layers

Measuring the mechanical properties of bulk materials has been the subject of the preceding paragraphs. This chapter will deal with the use of nanoindentation in the measurement of mechanical properties of thin films. Any measurement performed on the whole sample will inevitably be dominated by the bulk substrate. Nanoindentation, since it allows to perform indentation at a very small depth, offers a possible solution to avoid the substrate influence and to the problem of measuring exclusively the properties of the film.

However, the problems comes in part from the presence of an interface between the film and substrate. The quality of the interface can be affected by many variables, resulting in a range of effects on the apparent elastic and plastic properties of the film.

In particular, when the deformation region around the indent approaches the interface, the indentation curve may exhibit features due to the thin film, the bulk material, the interface, or a combination of all three. As a direct consequence of these complications, models for thin-film behavior must attempt to take into account not only the properties of the film and substrate, but also the interface between them.

If the standard nanoindentation analysis routines are to be used and only the thin film properties has to be measured, it is essential that the plastic zone and the elastic strain field are both confined to the film and do not reach the substrate.

Recently published ISO and ASTM standards (ISO: 14577-4 and ASTM: E 2546-07) state strong rules to conduct indentation tests on thin films:

• the surfaces must be smooth enough compared to the penetration depth which means that the indentation depth should be at least 20 times higher than the Ra average roughness of the tested material;

• the maximum indentation depth should not exceed 10% of the film thickness to avoid the influence of the substrate.

As a result, when indenting films of a micron of thickness, the penetration depth must be less than 100 nm, and the surface roughness must be less than 5 nm. Clearly, this is difficult to achieve in practice.

In addition, these rules are questionable. There are film/substrate combinations for which 10% is very conservative, while for other combinations even 5% may be too deep. Generally, for a very soft film on a hard substrate, nanoindentations of 50% of the film thickness can be al-right, but less than 10% can be necessary when indenting a hard film on a soft substrate. For a very hard film on a soft substrate, the surface film behaves like an elastic membrane or a bending plate.

Due to all of these complications, the use of nanoindentation to study thin film me-

chanical properties should be a danger if the interpretation of the resulting data is not

done correctly. It can be easy to misguidedly take the values of E and H obtained dur-

ing nanoindentation testing to be absolute values and discover later that those values

contained significant errors.

2.2.7 Indentation and Grain Size Effect

The increasing of the yield strength with a decreasing of the grain size is known as the Hall-Petch effect. Yang and Vehoff studied the influence of the grain size on the mechanical properties of nanonickel examined by nanoindentation [67].

They considered different situations in which the ratio between the grain size over the imprint size was progressively modified (see figure 2.9 on the left-hand side). Four ten- dencies were observed for the hardness performing indentations with the same imprint size into samples presenting different grain sizes (see figure 2.9 on the right-hand side).

First, in the region where the indent size is much smaller than the grain size (region 1 in figure 2.9 on the right-hand side), the hardness increases by reducing the grain size.

Then, if the grain size keeps decresing (keeping the same imprint size) the hardness is found to decrease (region 2 in figure 2.9 on the right-hand side). This is due to the fact that the indentation starts affecting the surrounding grains leading to an increasing plastic zone in relation to the grain size. With further decreasing of the grain size (for the same imprint size) the indent becomes much larger than the grain and the microindentation situation is encountered. In this region the hardness increases again by decreasing the grain size (region 3 in figure 2.9 on the right-hand side). At last, an abnormal decrease of hardness was observed with a further reduction of the grain size.

Figure 2.9:

The increasing of hardness with a decreasing grain size was also reported by other authors. Swadener et al. [68] found that hardness increases with a decreasing grain size or with a decreasing tip radius. A dependency of indenter shape on the measured hardness value was also described by Mirshams et al. [69]. Their results indicate that the nanoindentation hardness result is strongly related to the indenter geometry and less sensitive to the nanoindentation depth.

As as result, given the importance of the indenter shape, the main commercially avail-

able indenters for the nanoindentation experiments and their characteristics will be

introduced in the next paragraph.

2.2. Mechanical Properties by Nanoindentation

2.2.8 Indenters Geometry

The probe tip is a very important part of indentation testing. The geometry and the material it is made from can dramatically influence the raw data. The material is important because the stiffness influences the measured Young’s modulus and hardness values (refer to section 2.2.2 for more details) and because of local adhesion that might occur between the sample and the tip. The most common shapes of indenters are spheres, flat punches, cones, and pyramids. Each of these has its own pressure distributions, and therefore different load requirements for deformation.

Tips are usually categorized as sharp and blunt. This is primarily a relative term with no rigorous criteria, very dependent upon context. For example, a tip that is extremely sharp may reach critical stresses at very low loads, acting as a knife cutting through the material creating new surfaces with little elastic deformation. This would mask important information such as initial yield strength. With blunt tips elasticity plays a more important role. The larger contact area means that higher loads must be applied to initiate plasticity. This would also mean that the tip is not as sensitive to surface anomalies. As a result, depending on the properties being studied, there may be an optimal tip radius.

Berkovich Indenters

The most frequently used indenter in nanoindentation testing is the Berckovich tip (see figure 2.10 on the left), a three-sided pyramid with the same depth-to-area relation as the four-sided Vickers pyramid commonly used in microhardness work.

The total included angle on this tip is 142.3 degrees, with a half angle of 65.35 degrees.

This makes it a relatively flat tip. This tip is used primarily for bulk materials and thin films thicker than 100nm. The average radius of curvature for a Berkovich tip is typically between 100nm and 200nm.

Figure 2.10:

Conical Indenters

The simple cylindrical symmetry makes it attractive from a modelling standpoint.

These tips can be used for indentation and scratch testing. Because of the non- directional geometry at the end of these tips, they are good scratching tips for harder materials, where plastic deformation is desired (e.g. to find friction coefficient).

The cone is also attractive because the complications associated with the stress con- centrations at the sharp edges of the indenter are absent. An image of a typical conical tip used in nanoindentation is shown on figure 2.10.

The problem with these tips is that they are difficult to manufacture. In fact, because of their geometry, it is very difficult or impossible to get the radius of curvature at the end as small as that obtained for the three sided pyramidal tips.

Cube-Corner Indenters

The total included angle of this tip is 90 degrees (figure 2.10). This tip produces much higher stresses and strains in the vicinity of the contact, which is useful, for example, in producing very small, well-defined cracks around hardness impressions in brittle materials. Such cracks can be used to estimate the fracture toughness at relatively small scale.

There are three classes of these tips, which are dependent on the radius of curvature which is less than 40nm, 40-60nm, and 60-100nm respectively.

Spherical Indenters

Another important indenter geometry is the sphere. Spherical contact differs from the sharp contact of the Berckovich indenters in the way the stresses develop during indentation. For spherical indenters, the contact stresses are initially small and produce only elastic deformation.

Specialty Tips

These tips are used for very specialized applications [70].

Vickers: A Vickers indenter is a four sided pyramidal tip. The imprints made by a Vickers indenter will be square in shape. These indenters are good for very large load work where the results are to be compared to macroscale indentations made with a Vickers indenter.

Knoop: A Knoop indenter is a four sided pyramidal tip where the opposite sides are

parallel. The angles between the sides are not 90 degrees, so the indent is shaped like

a parallelogram.

2.2. Mechanical Properties by Nanoindentation

Fluid Cell Tips

Fluid cell tips are similar to the standard tips except that they are mounted on a long, thin shaft which extends the tip away from the transducer. This allows to perform indentation in a fluid. These tips are used for electrochemical experiments and other experiments where the sample is to be submerged in a liquid. Most tip shapes are available as a fluid cell tip.

Remark:

Indenters are usually made from diamond since it is important that the indenter is

harder than the specimen under consideration. If this is not the case, plastic deforma-

tion will occur within the indenter rather than the specimen or in both.

2.3 Potential Sources of Data Distortion during Nanoin- dentation

2.3.1 Preliminary Discussion

In the previous section we described how to calculate idealized Young’s modulus and hardness from the displacement that occurs as a load is applied. There are some aspects associated with tools used during indentations such as the load frame com- pliance, the initial penetration depth and the tip shape anomalies that should be accounted for. In depth-sensing indentation measurement, there are other sources of error that are caused by the material and the testing parameters. Unlike those as- sociated with the apparatus, these distortions are material specific and therefore much harder to identify and correct.

Accurate knowledge of the contact area is crucial in determining a truly representative hardness and Young’s modulus value. Unfortunately, some materials exhibit deforma- tion behaviors that can lead to significant miscalculations in the contact area. Some- times, and under certain loading conditions some materials seem to pile-up around the indenter deviating from the elastic behavior theory and conducing to an increase of the contact area. Other materials experience sink-in, seeming to collapse from the area surrounding the indenter.

Moreover, surface anomalies and roughness, that are not significant issues at the macro-scale, can remarkably influence nanoindentation. In fact, depending upon the scale of the roughness it can mean that only a fraction of the indenter tip is truly in contact with the surface leading to a miscalculation of the contact area.

In some cases and with some material pairs adhesion is also an issue that must be accounted for since it can influence the penetration process as well as the unloading.

Time dependent deformation is also extremely important. Creep is defined as con- tinued plasticity at a constant load. This phenomenon is often observed when a hold period at maximum load is applied.

Materials that demonstrate displacement bursts (pop-ins) at the onset of plasticity have shown this property to be time dependent. In the material systems that are subject to time dependent plasticity, the measured mechanical properties may vary depending upon the testing parameters.

In addition, other sources of data distortion are directly connected with the process parameters. Thus, distortion connected with loading rate, holding time and impact velocity could rise from the technique itself and this can also have a significant influence on the indentation results.

All these sources of data distortion will be discussed in details in the following sections.

To conclude, it is worth pointing out that the identification and the comprehension of

these source of errors help achieve more repeatable and reliable results.

2.3. Potential Sources of Data Distortion during Nanoindentation

2.3.2 Load Frame Compliance

An important consideration is how well the load frame compliance is known. In fact, the measured displacements are the sum of the displacement in the specimen and in the load frame. As a result, to accurately determine the specimen displacement, the frame compliance must be known with high precision. Considering large indentations made in materials with high Young’s modulus this becomes extremely important since the load frame displacement can be a significant fraction of the total displacement.

2.3.3 Initial Penetration Depth

In practice, when indentations are performed, it is necessary to establish a contact with the specimen surface. This initial contact is usually set using the smallest obtainable force of the instrument (normally in the order of 2µN). As a result, all subsequent depth measurement taken from that point will be affected by a small penetration depth error (initial penetration depth) resulting from the initial contact force. This initial penetration depth should be added to all depth measurements in order to correct for this initial displacement.

2.3.4 Tip Shape Anomalies

As a practical matter, if there is suspicion the tip may be damaged, its geometry should be imaged, for example using Scanning Electron Microscopy. Figure 2.11 shows two spherical indenters which present an improper geometry and that should not be used during nanoindentation experiments.

Figure 2.11: