Measurement of Material Creep Parameters of Amorphous Selenium by Nanoindentation and the Relationship Between Indentation Creep and Uniaxial Creep

University of Tennessee, Knoxville

Trace: Tennessee Research and Creative

Exchange

Masters Theses Graduate School

8-2004

Measurement of Material Creep Parameters of

Amorphous Selenium by Nanoindentation and the

Relationship Between Indentation Creep and

Uniaxial Creep

James Anthony LaManna Jr. University of Tennessee - Knoxville

This Thesis is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contacttrace@utk.edu.

Recommended Citation

LaManna, James Anthony Jr., "Measurement of Material Creep Parameters of Amorphous Selenium by Nanoindentation and the Relationship Between Indentation Creep and Uniaxial Creep. " Master's Thesis, University of Tennessee, 2004.

To the Graduate Council:

I am submitting herewith a thesis written by James Anthony LaManna Jr. entitled "Measurement of Material Creep Parameters of Amorphous Selenium by Nanoindentation and the Relationship Between Indentation Creep and Uniaxial Creep." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in .

George M. Pharr, Major Professor We have read this thesis and recommend its acceptance:

Kevin M. Kit, Charlie R. Brooks

Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.)

To the Graduate Council:

I am submitting herewith a thesis written by James Anthony LaManna Jr. entitled “Measurement of Material Creep Parameters of Amorphous Selenium by

Nanoindentation and the Relationship Between Indentation Creep and Uniaxial Creep.” I have examined the final electronic copy of this thesis for form and content and

recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Materials Science and Engineering.

George M. Pharr Major Professor

We have read this thesis

and recommend its acceptance:

Kevin M. Kit

Charlie R. Brooks

Accepted for the Council:

Anne Mayhew

Vice Chancellor and Dean of Graduate Studies

Measurement of Material Creep Parameters of Amorphous

Selenium by Nanoindentation and the Relationship Between

Indentation Creep and Uniaxial Creep

A Thesis Presented for the

Master of Science Degree

The University of Tennessee, Knoxville

James Anthony LaManna, Jr.

August 2004

ii

Dedication

This thesis is dedicated to my parents, my wife Anne, our son Timmy, my sisters and the rest of my family. Thanks for all the love and support.

iii

Acknowledgements

First and foremost I would like to thank Dr. George Pharr for everything he has done to help me with this endeavor. He is not only a great advisor, but someone I

consider a friend. He has taught me how to be a materials scientist/engineer, for this I am eternally grateful. I would like to thank Dr. Warren Oliver from MTS Nano Instruments Innovation Center for helping me learn how to operate the Nano Indenter XP and the Testworks software. I would like to thank the rest of the staff at MTS Nano Instruments Innovation Center for their time and helpful ideas. I would like to thank Dr. Kit and Dr. Brooks for serving on my committee. I wish to thank my family and friends for all their support and encouragement. Finally, I would like to thank MTS Systems Corporation Nano Instruments Innovation Center for their financial support of this study.

iv

Abstract

The purpose of this work was to measure the indentation creep parameters of amorphous selenium and to begin to develop a relationship between indentation creep and uniaxial creep. The data for this study was obtained by creating three different indentation creep experiments to test amorphous selenium under multiple loading

conditions at multiple temperatures. The testing temperatures included 40°C, 35°C, 30°C, and 25°C. These specific temperatures were chosen to collect creep data above and below the glass transition temperature of 31°C. This indentation data was used to accomplish several objectives.

The first objective was to prove the loading history independence of amorphous selenium. This was accomplished by showing that at each temperature the data from the different loading conditions all fit on one curve when the indentation strain rate is plotted as a function of the mean pressure applied by the indenter. The second objective was to calculate the creep exponent, n, from the indentation creep equation, n

m I =BP

ε_& , where

I

ε& is the indentation strain rate, Pm is the mean pressure applied by the indenter, and B is

a material constant. The results show that the creep exponent is a strong function of temperature and strain rate and that linear creep is displayed at temperatures above Tg.

The activation energy for creep, Qc, was also calculated from the indentation data. These

results were obtained from the strain rate data at constant values of mean pressure and temperature. The calculated values of Qc ranged from 340 to 380 kJ/mole at 37.5°C and

515 to 530 kJ/mole at 32.5°C. These values were compared to literature data from both uniaxial and indentation experiments.

Another objective of this work was to analyze the indentation load-displacement behavior of amorphous selenium. Below the glass transition temperature it is shown that during the loading portion of a constant loading rate test the load is a function of the displacement squared. This result is compared to a theoretical model suggested in the literature. Experimental results show that above the glass transition temperature the load

v

is a linear function of displacement. A model was derived for the load displacement behavior during linear creep from the indentation creep equation.

The viscosity of amorphous selenium was also calculated from the indentation data. A model found in the literature was used to calculate the viscosity for the linear creep data from the slope of the load -displacement curves. The results show an

estimated viscosity of 5 GPa-s at 35°C and 4 GPa-s at 40°C. These values are compared to literature data from both indentation and uniaxial creep tests.

The final objective of this study was to investigate the relationship between the material constant B in the indentation creep equation and the material constant A in the uniaxial creep equation, n

u Aσ

ε& = , where ε_&u is the uniaxial strain rate, σ is the uniaxial stress and n is the creep exponent. The values for A were taken from the literature and B was calculated from the indentation load-displacement data and the model developed for linear creep. The A/B relationship was calculated to be 0.256 +/- 0.001 at 40°C and 0.266 +/- 0.009 at 35°C. The results were compared to a theoretical model presented in the literature.

vi

Table of Contents

Chapter Page 1.

Introduction

...1 General Purpose………...…....1 State of Stress………...1 Constitutive Equations……….6

Loading History Independence………....7

Transient Behavior………...8

Amorphous Selenium……….14

Specific Objectives...…...14

2.

Literature Review

………...16

Important Selenium Properties...…...16

Shear Viscosity………..16

Glass Transition Temperature………21

Poisson’s Ratio and Young’s Modulus………..23

Indentation Creep Experiments………..27

Load-Displacement Relationship in Elastic Plastic Indentation…………28

A-B Relationship………...30 3.

Experimental Procedures

………..………..35 Sample Preparation………35 Experimental Setup………35 Test Methods………..38 4.

Results

……….…41 Experimental Results……….41

vii

Calculation of the Activation Energy for Creep………... 62

The Elastic-Plastic Behavior of Amorphous Selenium Below Tg……... 65

The Viscous Behavior of Amorphous Selenium Above Tg………. 69

Calculation of Shear Viscosity………. 74

The Experimental Relationship Between A and B……….………. .87

5.

Conclusions

... .93 Objectives………..93 Future Work………...95

References

...96

Appendix

………99

Vita

……….130

viii

List of Figures

Figures Page

1. Uniaxial Compression Instrument……….………2

2. Schematic of Uniaxial Compression Test……….……….3

3. MTS Nano Indenter XP……….……....4

4. Schematic of Indentation test……….………5

5. Schematic of Loading History Independent Uniaxial Creep Experiment…….……9

6. Schematic of Uniaxial Creep with Transient Behavior………...10

7. Schematic of Uniaxial Creep Experiments at Multiple Stresses……….……11

8. Schematic of Uniaxial Creep Experiment with Abrupt Change in Stress Depicting Loading History Independence….………...12

9. Schematic of Uniaxial Creep Experiment with Abrupt change in Stress Depicting Transient Behavior………..……….13

10. Schematic of a Berkovich Indenter, Illustrating the Half Included Angle α and the Inclined Face Angle β.………19

11. Shear Viscosity Literature Data……….…….22

12. Shear Modulus as a Function of Temperature……….…...25

13. Young’s Modulus as a Function of Temperature……….……..26

14. MTS Nano Indenter XP Setup………..……….…...36

15. Typical Berkovich Indent in Amorphous Selenium at 25°C ……….37

16. Schematic of High Temperature Experimental Setup……….……...39

17. Load and Hold Data 25°C: Load on Sample vs. Displacement into the Surface...42

18. Load and Hold Data 25°C: Mean Pressure vs. Displacement into the Surface………....…43

19. Load and Hold Data 25°C: Indentation Strain Rate vs. Mean Pressure……..…..44

20. Constant Loading Rate Data 25°C: Load on Sample vs. Displacement into Surface……….………...45

ix

21. Constant Loading Rate Data 25°C: Load on Sample vs. Time…….………46 22. Constant Loading Rate Data 25°C: Mean Pressure vs. Displacement into the Surface……….………...48 23. Constant Loading Rate Data 25°C: Indentation Strain Rate vs. Mean Pressure...49

24. Constant

L L&

Data 25°C: Load on Sample vs. Displacement into Surface……..…50

25. Constant

L L&

Data 25°C: Load on Sample vs. Time…….…………...…………..51

26. Constant

L L&

Data 25°C: Indentation Strain Rate vs. Displacement into Surface..52

27. Constant

L L&

Data 25°C: Mean Pressure vs. Displacement into Surface...……....53

28. Constant

L L&

Data 25°C: Indentation Strain Rate vs. Mean Pressure…….…..…..55

29. Comparison of all Data 25°C: Indentation Strain Rate vs. Mean Pressure……...56 30. Comparison of all Data 30°C: Indentation Strain Rate vs. Mean Pressure.…...57 31. Comparison of all Data 35°C: Indentation Strain Rate vs. Mean Pressure…...58 32. Comparison of all Data 40°C: Indentation Strain Rate vs. Mean Pressure.…...59 33. Comparison of all Data: Indentation Strain Rate vs. Mean Pressure…….…...…60 34. Creep Exponent vs. Mean Pressure……….………..….61 35. Constant Values of Mean Pressure for the Calculation of the Activation Energy for Creep……….………..………63 36. The Experimental Calculation of the Activation Energy for Creep, Qc……….…64 37. Stephen’s Viscosity Data: Log(Viscosity) vs. Inverse of Temperature ………....66 38. Comparison of Activation Energy for Creep, Qc……….………...67

39. Experimental Load Displacement curve: 4mN/s at 25°C…….………...68 40. Comparison of Elastic-Plastic Model and Experimental Data…….………..……70 41. Experimental Load-Displacement Curve: 0.001mN/s at 35°C. (Proof of Purely Plastic Experiment)……….………....71 42. Experimental Load-Displacement Curve: 0.001mN/s at 35°C (Proof of Linear

x

Load-Displacement Relationship)………..……...72

43. Optical Image of Amorphous Selenium Indented with a Constant Loading Rate of 0.001mN/s at 35°C……….………...76

44. Optical Image of Amorphous Selenium Indented with a Constant Loading Rate of 0.02mN/s at 40°C……….………...77

45. Interference Microscope Image of Amorphous Selenium Indented at a Constant Loading Rate of 0.001mN/s at 35C (Image 1)………...79

46. Interference Microscope Image of Amorphous Selenium Indented at a Constant Loading Rate of 0.001mN/s at 35C (Image 2)…………..………...………..80

47. Constant Loading Rate Tests 40°C: Displacement into the Surface vs. Time.….82 48. Constant Loading Rate Tests 40°C: Load on the Sample vs. Displacement into the Surface....………...83

49. Constant Loading Rate Tests 40°C: Viscosity vs. Loading Rate….……….84

50. Comparison of Viscosity Data….………...85

51. Comparison of Viscosity Data: Close up of the Area of Interest….……….86

52. Material Constant A as a Function of Inverse Temperature……….………..88

A-1. 30°C Load and Hold Data: Load vs. Displacement into Surface……….100

A-2. 30°C Hold Data: Mean Pressure vs. Displacement into Surface………..101

A-3. 30°C Hold Data: Indentation Strain Rate vs. Mean Pressure………….………..102

A-4. 30°C Constant Loading Rate Data: Load vs. Displacement into Surface………103

A-5. 30°C Constant Loading Rate Data: Mean Pressure vs. Displacement into Surface……….……….104

A-6. 30°C Constant Loading Rate Data: Indentation Strain Rate vs. Mean Pressure..105

A-7. 30°C Constant L L& Data: Load vs. Displacement into Surface……….106

A-8. 30°C Constant L L& Data: Indentation Strain Rate vs. Displacement into Surface.107 A-9. 30°C Constant L L& Data: Mean Pressure vs. Displacement into Surface………..108

xi A-10. 30°C Constant

L L&

Data: Indentation Strain Rate vs. Mean Pressure……….….109

A-11. 35°C Load and Hold Data: Load vs. Displacement into Surface………110

A-12. 35°C Hold Data: Mean Pressure vs. Displacement into Surface……….111

A-13. 35°C Hold Data: Indentation Strain Rate vs. Mean Pressure……….….112

A-14. 35°C Constant Loading Rate Data: Load vs. Displacement into Surface……...113

A-15. 35°C Constant Loading Rate Data: Mean Pressure vs. Displacement into Surface……….114

A-16. 35°C Constant Loading Rate Data: Indentation Strain Rate vs. Mean Pressure.115 A-17. 35°C Constant L L& Data: Load vs. Displacement into Surface………116

A-18. 35°C Constant L L& Data: Indentation Strain Rate vs. Displacement into Surface……….117

A-19. 35°C Constant L L& Data: Mean Pressure vs. Displacement into Surface……….118

A-20. 35°C Constant L L& Data: Indentation Strain Rate vs. Mean Pressure………...119

A-21. 40°C Load and Hold Data: Load vs. Displacement into Surface……….120

A-22. 40°C Hold Data: Mean Pressure vs. Displacement into Surface………..121

A-23. 40°C Hold Data: Indentation Strain Rate vs. Mean Pressure………...122

A-24. 40°C Constant Loading Rate Data: Load vs. Displacement into Surface………123

A-25. 40°C Constant Loading Rate Data: Mean Pressure vs. Displacement………….124

A-26. 40°C Constant Loading Rate Data: Indentation Strain Rate vs. Mean Pressure..125

A-27. 40°C Constant L L& Data: Load vs. Displacement into Surface……….126

A-28. 40°C Constant L L& Data: Indentation Strain Rate vs. Displacement into Surface.127 A-29. 40°C Constant L L& Data: Mean Pressure vs. Displacement into Surface………..128

xii A-30. 40°C

L L&

1

Chapter 1 – Introduction

General Purpose

The general purpose of this study is to establish a relationship between uniaxial creep and indentation creep. Creep is formally defined as time dependent deformation under a constant load and uniaxial testing is the standard for establishing creep properties. In some cases, indentation creep methods have advantages over uniaxial creep tests. Indentation creep tests can be used to measure the creep properties of individual components of material systems, such as a thin film on a substrate or the individual phases of a complex alloy. Uniaxial testing can only yield the average creep properties of the entire sample. Also, uniaxial test specimens are large relative to specimens required for indentation tests. With indentation methods, it is possible to perform over one hundred creep indentation tests on a one centimeter cubed sample. It would be difficult to complete one uniaxial test on a sample that size. To utilize the advantages of

indentation creep, a relationship between indentation creep and uniaxial creep must be developed. This work will begin to develop that relationship.

State of Stress

It is important to understand the states of stress of both testing methods and how they differ. Uniaxial creep testing will be addressed first. Figure 1 shows a typical uniaxial compression test instrument. Figure 2 depicts a schematic of a uniaxial

compression test. This figure emphasizes the fact that during a uniaxial compression test, if friction is negligible and the load is constant, the sample is under a constant state of stress. The state of stress is constant as a function of time and position. In other words, the stress is the same in every position within the sample for the entire experiment. Figure 3 shows the Nanoindenter XP used in this study. Figure 4 displays a schematic of the state of stress during an indentation test. During an indentation test the state of stress is not constant.

2

Figure 1. Uniaxial Compression Instrument.

3

Figure 2. Schematic of Uniaxial Compression Test.

L

Sample

L

Constant State of Stress

Not a Function of Time or Position

Uniaxial Compression

4

Figure 3. MTS Nano Indenter XP.

5

Figure 4. Schematic of Indentation test.

State of Stress is a Function of Time and Position Indentation

Indenter

Sample iso-stress

6

The state of stress is a function of time and position. Time affects the state of stress in the following way. As the indenter penetrates the surface and plastically deforms the sample, the state of stress in the sample will continuously change. If the indenter is then held at a constant depth, the state of stress will still continue to change as the material relaxes and stresses are redistributed. While the state of stress is continually changing as a function of time, it does not change uniformly. Therefore, the state of stress is also a function of position. The stress near the tip of the indenter is higher then the stress in the material farther away from the tip. So regardless of the type of indentation creep test, the state of stress in the sample is influenced by both time and position. After considering the differences in these two methods, it becomes obvious that indentation data can not be directly compared to uniaxial creep results.

Constitutive Equations

Constitutive equations have been established for both uniaxial creep and indentation creep. The uniaxial creep equation used in this study is:

n u Aσ

ε_& = (1)

where ε&_u is the uniaxial strain rate, A is a material constant, σ is the uniaxial stress, and n is the creep exponent. An indentation creep equation has been established that is

analogous to the uniaxial creep equation. This equation helps compare the creep properties of two very different methods in a similar way. It is:

n m I =BP

ε& (2)

where ε_&I is the indentation strain rate, B is a material constant, and Pm is the mean

pressure applied by the indenter. The indentation strain rate and mean pressure are defined as:

7 h h I & & = ε (3) c m _A L P = (4)

where h& is the displacement rate, h is the displacement of the indenter, L is the load on the indenter and Ac is the area of contact between the indenter and the sample. If the

creep exponent, n, is the same in both equations, as is the case for simple creep behavior, data from indentation creep experiments can be used to estimate uniaxial creep data if a relationship between A and B can be established.

Loading History Independence

When load is applied to a sample and it is plastically deformed, in most solid materials, the strain changes the mechanical properties of the material. This phenomenon causes “loading history effects.” A classic example of loading history effects is strain hardening. Materials that strain harden experience an increase in hardness when the material is plastically strained. Loading history is especially important in indentation because the state of stress is so complex. For example, assume a sample is indented with a predetermined load, the load is held constant and the indenter is allowed to creep. If this experiment is performed multiple times, with different loading velocities, loading history will affect the material and the creep results would differ.

If a material is loading history independent then the stresses and strains in the material are not a function of loading history. During indentation, the stresses and strains will only be a function of the current velocity of the indenter. Therefore, the above mentioned experiments would yield the same creep results. Loading history

independence is not a common behavior displayed by solid materials. Water is a good example of a material that is loading history independent. A loading history independent

8

solid is a solid that has atoms that behave like a liquid. The atoms within the solid can reorganize themselves, under load, without altering the properties of the material.

Transient Behavior

Loading history dependence is referred to as transient behavior during uniaxial creep tests. Figure 5 shows a schematic of a theoretical loading history independent constant load uniaxial creep experiment. In this experiment, strain increases as a linear function of time and the slope of the line is Aσn (steady state creep). Figure 6 depicts a schematic of the actual behavior displayed by most materials. In the beginning of the experiment the strain does not increase as a linear function of time. This is referred to as transient behavior and is followed by the predicted steady state creep. Transients are also

observed in experiments where the applied load is changed abruptly altering the stress on the sample. Figure 7 displays a schematic of two uniaxial creep tests at different stresses. Figure 8 shows a schematic of a creep test with an abrupt change in stress (from stress 1 to stress 2 in Fig. 7) that does not display transient behavior (loading history independent), while Fig. 9 shows the same experiment with transient behavior present. Most materials display the behavior depicted in Fig. 9. Transient behavior complicates the analysis of creep data.

Considering the complexity of the relationship between uniaxial creep experiments and indentation creep experiments, eliminating the transient behavior will simplify the model used to quantify that relationship. Removing the transient allows one to assume that the material in the study is loading history independent. This loading history independent assumption is required to define the terms “A” and “B” as constants in the constitutive equations utilized in this study. The removal of the transient behavior from the model makes the selection of a test material more difficult but also increases the likelihood that this study will accomplish all if its goals.

9

Figure 5. Schematic of Loading History Independent Uniaxial Creep Experiment.

t

Aσn

ε

10

Figure 6. Schematic of Uniaxial Creep with Transient Behavior.

t

ε

Transient

11

Figure 7. Schematic of Uniaxial Creep Experiments at Multiple Stresses.

t

ε

σ1 σ2

12

Figure 8. Schematic of Uniaxial Creep Experiment with Abrupt Change in Stress Depicting Loading History Independence.

t

ε

σ1

σ2

Loading History Independent

13

Figure 9. Schematic of Uniaxial Creep Experiment with Abrupt change in Stress, Depicting Transient Behavior.

ε

σ1 σ2 > σ1

transient

t

14

Amorphous Selenium

Amorphous selenium was selected as the experimental material for this study. There are several reasons for this selection. Most importantly, selenium has shown limited loading history independence over a small range of temperatures. Poisl et al. [1] first proposed the history independence of selenium in 1995. They discovered during limited indentation creep experiments that amorphous selenium is loading history independent at 32.1°C and 34.3°C. It was also important to choose a material that has both uniaxial and indentation data available and amorphous selenium has an abundance of literature data. The literature imperative to this study will be discussed in the next section. For indentation tests, samples must have extremely smooth surfaces. Therefore, sample preparation has to be considered when a material is selected for an indentation study. Luckily, amorphous selenium indentation sample preparation is quite simple. This sample preparation process will be discussed later. Another interesting property of amorphous selenium is the glass transition temperature of 31°C. This temperature is very close to room temperature and allows one to easily test the creep properties of amorphous selenium above and below the glass transition temperature. Finally, previous work has shown that above the glass transition temperature amorphous selenium displays linear viscous behavior [1]. Linear viscous behavior (n=1) can further simplify the constitutive equations used in this study. After reviewing the qualities stated above, it is obvious that amorphous selenium is the ideal material for this work.

Specific Objectives

This study plans to accomplish several specific objectives. The first objective of this study is to prove the loading history independence of amorphous selenium over a wide range of temperatures and loading conditions during indentation creep tests. This is the first objective because the constitutive equations in this study are based on the

assumption of loading history independence and the current information on the loading history independence of amorphous selenium is limited. The indentation data collected while establishing the loading history independence of selenium will be used to

15

accomplish several other objectives. The second objective is to measure the creep exponent, n. This value is an important part of the creep constitutive equation and must be known for the A-B relationship to be established. This study will also calculate the activation energy for creep, Qc. These results will be compared to uniaxial data as well

as data previously collected via indentation. Another objective is to analyze the elastic-plastic behavior of amorphous selenium below the glass transition temperature as well as the viscous behavior of amorphous selenium above the glass transition temperature. These experimental results will be compared to theoretical predictions. The viscosity of a material can also be established from indentation creep data. This is also an objective of this work. The viscosity will be calculated and compared to previous results established by uniaxial creep and indentation creep. The final objective of this work is to begin to establish a relationship between uniaxial creep and indentation creep data by theoretically and empirically calculating the relationship between the uniaxial creep constant, A, and the indentation creep constant, B. The accomplishment of these objectives with further the scientific understanding of indentation creep results and how these results compare to uniaxial data.

16

Chapter 2 – Literature Review

Important Selenium Properties

The properties of amorphous selenium most important to this study include shear viscosity, glass transition temperature, Poisson’s ratio, and Young’s modulus. These properties are well documented in the literature. The majority of the literature data available for amorphous selenium was generated in the 1960’s and 1970’s because amorphous selenium has excellent photoconductive properties and was used in early photocopy machines. Each of the properties stated above will be discussed in detail, including measured values, measurement techniques, and literature sources. Other papers important to the specific objectives of this work will also be discussed.

Shear Viscosity

Shear viscosity is defined as the resistance to flow or the ratio of shear stress to shear strain rate. The relationship is:

γ τ η

&

= (5)

where η is the shear viscosity, τ is the shear stress, and γ_& is the shear strain rate. The measurement of the shear viscosity of amorphous selenium by three different techniques will be reviewed. Cukierman and Uhlmann [2] measured the viscosity of amorphous selenium with a beam bending viscosimeter. In this experiment, a sample was placed on two knife edges and a rod was used to apply force onto the center of the sample. The shear viscosity was calculated from the deflection rate of the sample. They performed this experiment between 30°C and 50°C. Stephens [3] measured the shear viscosity of

17

elongation of a sample was measured and the uniaxial strain rate was calculated. The shear viscosity was calculated from the equation:

u t ε σ η & 3 1 ) ( = (6)

where η(t) is the instantaneous shear viscosity. Stephens conducted tests from 25°C to 40°C. Shimizu et al. [4] used indentation to estimate the shear viscosity of selenium. They began with the relationship:

2 2 2 ₁ 2 h E gk L ν γ − = (7)

where E is Young’s modulus and ν is Poisson’s ratio. The parameters g, k, and γ are all geometrical factors and are defined by:

β π 2 2 = cot = c c h A g (8) β tan = k (9) c h h = γ (10)

where β is the inclined face angle of a conical indenter, and hc is contact depth of the

18 2 2 1 cot 2 h E L ν π β − = , (11)

which is Sneddon’s elastic solution for a cone indenting a semi-infinite solid, when the parameter γ is equal to π/2. This geometrical factor, γ, describes the sink-in /pile-up behavior of the indent. When γ is equal to 1, there is no sink in and no pile up around the indenter. If γ is greater than 1, then material has piled up around the indenter. If γ is less than 1, then the sample has sunk in around the indenter. Shimizu et al. used pyramidal indenters in their analysis. It is common to use pyramidal indenters with Sneddon’s solution for conical indentation by replacing the cone angles with equivalent cone angles for the pyramidal indenters. The equivalent half included angle for the three sided pyramidal Berkovich indenter is 70.3°. The equivalent inclined face angle for the Berkovich indenter is 19.7°; this is illustrated in Fig. 10. The Berkovich indenter is used in Shimizu et al.’s study and also in this work.

The elastic solution is converted to a linear viscoelastic solution by replacing Young’s Modulus, E, with the incremental relaxation modulus, E(t-t’), in Eq. 7 and making it a hereditary integral. The result is:

' ' ) ' ( ) ' ( 1 1 2 ) ( 2 0 2 2 _dt dt t dh t t E gk t L t       − − =

∫

ν γ (12)

where t is time and t’ is an incremental step in time. It is then assumed that the creep test will be performed under a constant rate of penetration, v0. With this assumption, Eq. 12

becomes:

∫

− − = gk v t E t t t dt t L 0 2 2 0 2 ₁ ( ') ' ' ) ( ν γ . (13)

19

Figure 10. Schematic of a Berkovich Indenter, Illustrating the Half Included Angle α and the Inclined Face Angle β.

β = 19.7°

Berkovich

20

It was then assumed that the viscoelastic material can be modeled by a simple Maxwell spring and dashpot in series. The elastic modulus of the spring is defined as E/(1-ν2) and the viscous dashpot has an elongation viscosity defined as 2(1+ν)η. The relaxation modulus of a viscoelastic material modeled by this Maxwell relationship is:

( )

t_τ E t E( )= exp−

(14) where τ is defined as ) 1 /( ) 1 ( 2 2 ν η ν η τ − + = = E E_m m ₍₁₅₎

where ηm is the Maxwell viscosity and Em is the Maxwell Modulus. Equations 14 and 15

are substituted into Eq. 13 to yield:

[

]

_ _      − − + = _{ν η} τ −τ γ t e t t v gk t L( ) 2(1 ) 2 1 1 0 2 . (16)

If τ=0, the solution in Eq. 16 reduces to the extreme viscous, or perfectly plastic, case:

[

]

v t gk t L 2 0 2 2(1 ) ) ( ν η γ + = (17)

where v0, the constant penetration rate, is equal to h/t. When this substitution is made in

Eq. 17, it yields:

[

]

v h gk t L( ) ₂ 2(1 ν)η ₀ γ + = . (18)

21

For a constant indenter velocity, Eq. 18 reveals a linear relationship between the load on the indenter and the displacement of the indenter into the surface of the material. The shear viscosity is calculated from the slope of this relationship. Shimizu et al. performed indentation tests at a constant penetration, vo, of 0.17um/s at temperatures from 10°C to

42°C. They used Eq. 18 to estimate the shear viscosity of amorphous selenium for each temperature. Experimental results for viscosity measurements from Cukierman and Uhlmann [2], Stephens [3], and Shimizu et al. [4] are plotted together in Fig. 11. This figure shows that the results from all three techniques are in agreement. The results of this study will be compared to this data base for shear viscosity of amorphous selenium.

Glass Transition Temperature

The glass transition temperature, Tg, is a critical temperature used to describe the

behavior of polymers and other amorphous materials. Below Tg a polymer behaves like a

glass; it is brittle and has a relatively high hardness. Above Tg a polymer loses its

strength and becomes ductile. Amorphous selenium has a polymeric type structure. The selenium atoms link together in a chainlike manner, some chains forming eight atom rings [5,7]. Since amorphous selenium has a polymeric structure, it also has a glass transition temperature. Eisenberg and Tobolsky [5] determined the glass transition temperature of amorphous selenium. They calculated Tg by measuring the weight of the

sample in an inert heat transfer liquid with a uniform expansion coefficient as a function of temperature. The heat transfer liquid used was distilled water. When a polymer is below Tg its thermal expansion coefficient is low and nearly constant. Above Tg the

thermal expansion coefficient is high. In Eisenberg and Tobolsky’s experiments, Tg

corresponds to the inflection point on the plot of the weight of the sample in the heat transfer liquid as a function of temperature. The inflection point is due to the change in the thermal expansion coefficient and therefore the density of the sample. Eisenberg and Tobolsky found Tg of amorphous selenium to be 31.0 °C +/- 0.5°C.

22 Log η vs. 1/T 6 7 8 9 10 11 12 13 14 3.05 3.1 3.15 3.2 3.25 3.3 3.35 3.4 (1/T) *103 (K-1) Log η (P a*s)

Cukierman, and Uhlmann Shimizu, Yanagimoto, and Sakai (berkovich) Stephens

experimental temperature range for this work

25°C -40°C

23

Poisson’s Ratio and Young’s Modulus

The Poisson’s ratio and Young’s modulus literature data are discussed together because each of the literature sources described in this study calculate both values by measuring the elastic constants of amorphous selenium. Graham and Chang [6] used the phase comparison method while Vedam et al. [7] and Soga et al. [8] used the pulse superposition method. Both of these methods calculate elastic constants from the

measurement of sound wave velocities within test samples. Since amorphous selenium is an isotropic material, the Poisson’s ratio and Young’s modulus can be calculated from:

) ( 2 1 12 11 C C G= − (19) ) 1 ( 2 +ν = E G (20) ) 2 1 )( 1 ( ) 1 ( 11 _{ν ν} ν − + − = E C (21)

where G is the shear modulus, and C11 and C12 are elastic constants [9]. Table 1 displays

the results for the Poisson’s ratio and Young’s Modulus at 25°C for all three sources. The results are generally in good agreement. Vedam et al. [7] also measured the elastic constants of amorphous selenium as a function of temperature. They displayed these results in a plot of shear modulus. This data is shown in Fig. 12. The results in Fig. 12 were converted to a plot of Young’s modulus as a function of temperature using equation 20 and a Poisson’s ratio of 0.33. This conversion is shown in Fig. 13. Poisson’s ratio and Young’smodulus as a function of temperature will be needed to accomplish some of the specific objectives of this study. It will be argued later that above Tg amorphous

selenium can behave as a Newtonian fluid. The Poisson’s ratio of an incompressible Newtonian fluid is 0.5. When amorphous selenium behaves in this manner, the change in Poisson’s ratio will effect the calculation of Young’s Modulus from Eq. 20.

24

Table 1. Properties Calculated From Elastic Constants for Amorphous Selenium at 25°C.

Source Temperature Young's Shear Poisson's Method

Modulus Modulus Ratio

°C GPa GPa Soga, and pulse superposition Kunugi 25 9.61 3.61 0.331 method

Graham and phase comparison

Chang 25 9.91 3.77 0.324 method

Vedam, Miller,

pulse superposition

and Roy 25 9.8 3.69 0.327 method

25

Shear Modulus vs Temperature (Vedam, Miller, and Roy)

3.2 3.25 3.3 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 20 25 30 35 40 45 Temperature (C) She ar Modulu s ( G Pa )

26

Young's Modulus vs Temperature (Vedam, Miller, and Roy)

8.4 8.6 8.8 9 9.2 9.4 9.6 9.8 10 20 25 30 35 40 45 Temperature (C) Mod u lus ( G P a)

27

Indentation Creep Experiments

The indentation creep methods used in this study were strongly influenced by Lucas [10]. In his study, Lucas describes several indentation creep techniques including a constant loading rate method, a load and hold method, and a new method he developed, the constant strain rate method. Variations of the first two indentation creep methods will be used in this study and explained later. This section will concentrate on explaining the constant strain rate method because it is more complex. Lucas starts with the definition of mean pressure: 2 ch L A L P_m = =

(21)

where Pm is the mean pressure applied by the indenter, A is the contact area between the

indenter and the sample (assuming no sink in or pile up), and c is an indenter geometrical factor. It is noted that A is equal to ch2 in the case of a geometrically similar indenter, such as the Berkovich indenter used in this study (c=24.5). To continue the analysis, Eq. 21 is rewritten as

L P

ch2 _m = _{. (22)}

Equation 22 is then differentiated with respect to time, to give:

L P ch P h

ch& _m + 2 &_m = &

2 . (23)

Equation 23 can be simplified to:

I m m P P L L h h _ε & & & & =       − = 2 1 . (24)

28

Equation 24 shows the relationship between the indentation strain rate,ε&_I , and the mean pressure applied by the indenter, the load on the indenter, and the loading rate. If the mean pressure is held constant, then Eq. 24 leads to:

I L L h h _ε & & & =       = 2 1 . (25)

Equation 25 shows that if the mean pressure can be held constant, then the indentation strain rate can also be held constant if the loading rate divided by the load,

( )

L&_L , is held constant. Under these conditions the indentation strain rate will be equal to one half the constant

( )

L&_L value. This indentation creep method, know as the constant strain rate method, as well as the ones mentioned above, will be instrumental in proving the loading history independence of amorphous selenium.

Load-Displacement Relationship in Elastic-Plastic Indentation

One of the specific objectives of this work is to investigate the elastic-plastic behavior of amorphous selenium below the glass transition temperature. Malzbender et al. [11] has derived an analytical expression for the relationship between the load on the indenter and the displacement into the surface of the sample. It follows the general form of:

2

Kh

L= (26)

where K is a constant. This analysis utilizes the indentation technique described by Oliver and Pharr [12]. They start with the definition of mean pressure:

2 c c m ch L A L P = = .

(27)

29

This definition is recalled from Eq. 21. Equation 27 shows Eq. 21 where h has been replaced with hc, which is the contact displacement. This is done to more accurately

calculate the contact area in the event the material around the indenter sinks in or piles up. The displacement, h, is equal to hc, the contact depth plus hs, the displacement of the

surface at the perimeter of the contact.

Equation 27 can be rearranged to define hc:

m c P L h 5 . 24 = . (28)

The parameter hs is defined as:

S L

h_s =ε (29)

where ε is a geometrical constant equal to 0.75 for a Berkovich indenter and S is the unloading contact stiffness. The unloading contact stiffness is equal to:

c r A E S π 4 = (30)

where Er is the reduced modulus. The reduced modulus takes into account the properties

of the sample and the indenter and is defined as:

s s i i r E E E 2 2 ₁ 1 1 ₌ −ν ₊ −ν (31)

30

where Ei is the Young’s modulus of the indenter, Es is the Young’s modulus of the

sample, νi is the Poisson’s ratio of the indenter, and νs is the Poisson’s ratio of the sample.

The combination of Eqs. 27, 29, and 30 yields:

r m s E LP h 4 π ε = . (32)

Equations 28 and 32 are combined to form the final relationship:

2 2 4 5 . 24 1 h E P P E E L r m m r r −         + = ε π . (33)

Equation 33 shows that if one knows the reduced modulus of a material and the mean pressure applied by the indenter, then the load versus displacement relationship can be modeled. This model will be compared to experimental data for amorphous selenium.

A-B Relationship

The final objective of this work is to relate the indentation material constant, B, to the uniaxial material constant, A. Bower et al. [13] have developed an analytical solution for the linear viscous case. They start with a uniaxial creep constitutive equation:

n u       = 0 0 _σ σ ε ε& & (34)

31

where ε&₀ is the uniaxial reference strain rate and σ0 is the uniaxial reference stress.

Equation 34 fits the form of Eq. 1, where A is defined as: n A 0 0 σ ε_& = . (35)

For the indentation creep problem, Bower et al. assume a creeping half space governed by the uniaxial creep law in Eq. 34. They then use Hill’s similarity transformation to convert the nonlinear viscous creep law to a nonlinear elastic relationship based on the load on the indenter and the displacement of the indenter into the surface of the sample. The result of the transformation is:

) ( 1 0 0 2 _a F n h a L n       = ε σ π & (36)

where a is the contact radius of the indent (assuming an axisymmetric indenter) and the function, F(n), is solely a function of the creep exponent, n. For the linear viscous case (n=1), Bower et al. use Sneddon’s exact linear elastic solution:

      = 0 0 2 ₃ 8 ε π σ π _& & a h a L . (37)

From this solution, the function F(1) is defined as:

π 3 8 ) 1 ( = F . (38)

This general solution for an axisymmetric indenter can be solved for a conical indenter by substituting for “a” in Eq. 37. For a conical indenter, “a” is:

32

α

tan

ch

a= (39)

where α is equal to the half included angle of the conical indenter (the Berkovich

indenter is equivalent to a cone with an α angle of 70.3°, recall Fig. 10), and c is defined as the sink in/pile up coefficient which is defined as:

h h

c₌ c _{. (40)}

For Sneddon’s linear elastic solution c is expressed as:

π 2

=

c . (41)

When Eq. 39 is combined with Eq. 37, the solution for a conical indenter, derived from the general solution for an axisymmetric indenter, is:

                  = α ε π σ tan 1 3 8 5 . 24 2 2 _{0 h 0c} h c h L & & . (42)

The definition of the indentation strain rate is recalled from Eq. 3:

h h I & & = ε . (3)

The mean pressure applied by the indenter, assuming no sink in or pile up, is defined again as:

33 2 5 . 24 h L A L P_m = = . (43)

Combining Eqs. 1, 3, 42, and 43 and rearranging the solution yields:

m I P c _     = 0 0 8 tan 3 σ α ε π ε_& & . (44)

Equation 44 is an analytical linear viscous relationship between indentation strain rate and the mean pressure under the indenter. It is important to note that Eq. 44 fits the form:

m I =BP

ε_& (2)

which is recalled from Eq. 2. Relating Eq. 2 to Eq. 44 reveals the analytical definition of the indentation material constant, B:

0 0 8 tan 3 σ α ε π c B = & . (45)

Recalling Eq. 35 (Bower’s definition of the uniaxial material constant, A) and assuming the linear creep case (n=1), Eq. 35 yields:

0 0 σ ε_& = A . (46)

Combining Eqs. 45 and 46 and rearranging reveals an analytical linear viscous relationship between uniaxial creep and indentation creep:

34 α πtan 3 8c B A ₌ . (47)

Equation 41 defines c for Sneddon’s linear elastic solution. Combining Eqs. 41 and 47 (remembering α is equal to 70.3° for a Berkovich indenter) yields:

1935 . 0 = B A . (48)

An empirical study of the A/B relationship is conducted in this work and the results will be compared to this analytical solution.

35

Chapter 3 – Experimental Procedures

Sample Preparation

The preparation of the amorphous selenium samples was surprisingly simple. The material was obtained from Alfa Aesar in pellet form. The pellets were 2-4mm in

diameter and 99.999% pure. Amorphous selenium has a melting temperature of 217°C, so the pellets were melted into liquid form on a hot plate under a fume hood. A copper mold was designed to produce round specimens 1.25 inches in diameter and about 5 millimeters tall. The copper mold was chilled in cold water before the liquid selenium was poured into the mold. The chilled copper mold removed the heat from the

amorphous selenium fast enough to suppress crystallization. The as-cast samples have a mirror finish, so polishing isn’t unnecessary. If the sample is not cooled fast enough, and the selenium crystallizes, it is gray in color without a mirror finish. To prepare the samples for indentation they were affixed to aluminum cylinders with an acetone based glue.

Experimental Setup

The indentation testing was performed with a MTS Nano Indenter XP and a Berkovich indenter. The MTS Nano Indenter XP equipment is displayed in Fig. 14. A typical indent in amorphous selenium with a Berkovich indenter (25°C) is shown in Fig. 15. The Nano Indenter XP setup had to be modified for testing at elevated temperatures. Luckily, the housing for the Nano Indenter XP provides the necessary environment for elevated temperature control. The heat source inside the housing was a 250 Watt Lamp that was attached to a support member underneath the vibration isolation table. The temperature was controlled with a proportional temperature controller which regulated the amount of power supplied to the lamp from the feedback gathered by a RTD. The temperature inside the housing was monitored with two thermistors, and a thermometer was inserted inside the sample tray.

36

Figure 14. MTS Nano Indenter XP Setup.

37

Figure 15. Typical Berkovich Indent in Amorphous Selenium at 25°C.

Amorphous

Selenium

25°C

38

The thermometer was linked to the Testworks software which is used to operate the Nano Indenter. This setup allowed for the temperature to be recorded during each experiment. Two small fans (similar to computer fans) were placed on the floor inside the housing to circulate the air. Finally, an aluminum shield was place directly below the vibration isolation table to minimize direct heating from the lamp. A schematic of the necessary modifications is shown in Fig. 16.

Test Methods

Again, the first objective of this work was to prove the history independence of

amorphous selenium. To accomplish this objective, experiments were designed to indent amorphous selenium under many different loading conditions and at temperatures above and below the glass transition temperature. It was decided that the amorphous selenium samples would be tested by three fundamentally different methods and many different loading conditions at temperatures of 25°C, 30°C, 35°C, and 40°C. These experiments will effectively test the loading history independence of amorphous selenium.

The first method discussed will be the load and hold method. In this method, the indenter was loaded to a prescribed load in ten seconds and then held for an hour or until a depth limit was reached. This method was repeated for multiple loads. The range of prescribed loads was a function of the testing temperature because of the drastic change in properties of amorphous selenium above the glass transition temperature of 31°C. The prescribed loads were as follows: 25°C – 500mN, 400mN, 300mN, 200mN, and 100mN, 30°C – 500mN, 400mN, 300mN, 200mN, 100mN, and 50mN, 35°C – 300mN, 200mN, 100mN, 50mN, 25mN, 10mN, 5mN, and 2mN, 40°C – 100mN, 25mN, 5mN, 2mN, and 1mN.

The second method is the constant loading rate method. During this type of test, the indenter indents the surface of the sample at a constant loading rate until a depth of

39

Figure 16. Schematic of High Temperature Experimental Setup.

Temperature Controller Fan Concrete Blocks Thermistor 1 250 W Nanoindenter Thermometer Thermistor 2 RTD

40

7000 nm is reached. Multiple constant loading rates were selected to produce a variety of loading conditions. The constant loading rates were as follows: 25°C – 4mN/s, 0.3mN/s, and 0.02mN/s, 30°C – 4mN/s, 0.3mN/s, and 0.02mN/s, 35°C – 4mN/s, 0.3mN/s,

0.02mN/s, and 0.001mN/s, 40°C – 4mN/s, 0.3mN/s, and 0.02mN/s.

The final test method used in this study is the constant loading rated divided by

the load,

L L&

, or constant strain rate, method developed by Lucas. As previously discussed,

this method holds the strain rate and the mean pressure constant throughout the test. During a test performed with this method the indenter penetrates the surface of the

sample at a constant

L L&

rate to a depth of 7000nm. The constant values of

L L&

selected

for this study included 0.405 s-1, 0.135 s-1, 0.045 s-1, and 0.015 s-1. These rates were used at all four test temperatures. As the properties of selenium change with the change in

temperature, the constant

L L&

41

Chapter 4 – Results

Experimental Results

The nanoindentation results from the amorphous selenium samples will be

discussed in this section. The results obtained at 25°C will be presented first. The results of the Load and Hold method are described first. Figure 17 shows the load on the sample as a function of displacement into the surface for each of the prescribed loads. The results in Figure 17 represent the average of 10 individual indents for each loading

condition. The tests performed on the amorphous selenium samples were very repeatable with small scatter bars that are usually covered by the data point. Fig. 18 displays the mean pressure as a function of displacement into the surface for each testing condition. It should be noted that the calculation of mean pressure assumes no sink in or pile up. The results in Fig. 18 show that even though the different prescribed loads caused the hold segments to start at different indentation depths, the hardness at the beginning of the hold segment of each sample was approximately 0.4GPa. The different loads did affect the creep rate of the indenter. Therefore, the rate at which the mean pressure dropped during the hold segment was also affected. Figure 19 shows the indentation strain rate as

function of mean pressure for the hold portion or each sample. The data is plotted in this fashion to fit the indentation creep constitutive equation described in chapter 1. When the data is plotted this way, all of the hold data collected from the samples with different prescribed loads falls on to one curve. This result verifies that creep data collected during load and hold tests on amorphous selenium are not affected by the load or indentation depth from which the hold segment is initiated. In other words the indentation creep results are not affected by the loading history.

The second method discussed in chapter 3 is the constant loading rate method. The load on the sample as a function of displacement into the surface for the constant loading rate experiments is shown in Fig. 20. Figure 21 displays the load on the sample as a function of time. From Figs. 20 and 21 it is shown that the loading rates differ by as much as two orders of magnitude and can take less than a minute or more than 3 hrs to

42

Figure 17. Load and Hold Data 25°C: Load on Sample vs. Displacement into the Surface.

Load and Hold Data 25°C

Load On Sample vs Displacement Into Surface

0 100 200 300 400 500 600 0 2000 4000 6000 8000 10000

Displacement Into Surface (nm)

Loa d On S am p le ( m N ) 100mN_200mN 300mN 400mN 500mN

43

Figure 18. Load and Hold Data 25°C: Mean Pressure vs. Displacement into the Surface.

Comparison of Hold Data at 25°C

Mean Pressure vs. Displacement into the Surface

0.15 0.2 0.25 0.3 0.35 0.4 0.45 3000 5000 7000 9000 11000

Displacement into the Surface (nm)

M ean P res su re ( G P a) 100mN 200mN 300mN 400mN 500mN

44

Figure 19. Load and Hold Data 25°C: Indentation Strain Rate vs. Mean Pressure.

Comparison of Hold Data at 25°C Indentation Strain Rate vs. Mean Pressure

0.00001 0.0001 0.001 0.01 0.1 0.1 1

Mean Pressure (GPa)

Inde nt at ion S tr ain R at e ( s -1 ) 100mN 200mN 300mN 400mN 500mN

45

Figure 20. Constant Loading Rate Data 25°C: Load on Sample vs. Displacement into Surface.

Load Rate Data 25C

Load On Sample vs Displacement Into Surface

0 50 100 150 200 250 300 350 400 0 1000 2000 3000 4000 5000 6000 7000

Displacement Into Surface (nm)

Loa d On S am p le ( m N ) 0.02mN/s 4mN/s 0.3mN/s

46

Loading Rate Tests 25C Load on Sample vs. Time

0 50 100 150 200 250 300 350 400 450 0 2000 4000 6000 8000 10000 12000 Time (s) Lo ad on S am p le ( m N ) 4 mN/s 0.3 mN/s 0.02 mN/s

47

penetrate a sample to a depth of 7000nm. This large difference in loading rates provides a wide range of loading conditions. Figure 22 shows the mean pressure as a function of displacement into the surface. This figure shows that during this type of experiment the mean pressure is both a function of the loading rate and also the displacement into the surface of the sample. Also, like the previous method, the mean pressure decreases as the displacement into the surface increases. Figure 23 shows the indentation strain rate as a function of mean pressure. This data, like the results from the previous method, can also be plotted approximately on one line. This method shows that indentation loading rate does not affect the indentation creep properties of amorphous selenium. The mean pressure is only a function of the indentation strain rate, while the indentation strain rate is only a function of the current velocity of the indenter. This result is further proof of the history independence of amorphous selenium.

The final indentation experimental method is the constant strain rate method. Figure 24 shows the load on the sample as a function of displacement into the surface for each of the constant loading rate divided by the load samples. This figure looks a lot like Fig. 20, the load on the sample as a function of displacement into the surface for the constant loading rate tests. The difference between these methods becomes obvious when comparing Fig. 21, the load as function of time for the constant loading rate method, with Fig. 25. This figure displays the load as a function of time for the constant strain rate method. It shows that the loading rate during this type of experiment is not constant and actually increases rapidly as the indenter penetrates into the surface of the sample. The velocity of the indenter must increase during this method to keep the strain rate and the mean pressure constant as a function of displacement into the surface. Figures 26 and 27 show the strain rate and the mean pressure as a function of displacement into the surface. These figures show that indeed both the indentation strain rate and the mean pressure are constant during the entire experiment. Since this is a load controlled experiment and the indentation strain rate is measured, the concept of the constant strain rate method is proven. Therefore, when the indentation strain rate is plotted as a function of mean pressure, each constant value of loading rate divided by the load (which is equal

48

Figure 22. Constant Loading Rate Data 25°C: Mean Pressure vs. Displacement into the Surface.

Comparison of Loading Rates at 25°C Mean Pressure vs. Displacement into the Surface

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 1000 2000 3000 4000 5000 6000 7000

Displacement into the Surface (nm)

M ean P res su re ( G P a) 4mN/s 0.02mN/s 0.3mN/s

49

Figure 23. Constant Loading Rate Data 25°C: Indentation Strain Rate vs. Mean Pressure.

Comparison of Loading Rates Data at 25°C Indentation Strain Rate vs. Mean Pressure

0.00001 0.0001 0.001 0.01 0.1 0.1 1

Mean Pressure (GPa)

Inde nt at ion S tr ain R at e ( s -1 ) 4mN/s 0.3mN/s 0.02mN/s

50

Ldot/L Data 25°C

Load On Sample vs Displacement Into Surface

0 100 200 300 400 500 0 1000 2000 3000 4000 5000 6000 7000

Displacement Into Surface (nm)

Load O n Sample (mN) 0.015 /s 0.045 /s 0.135 /s 0.405 /s Figure 24. Constant L L&

51

Ldot/L Tests 25C Load on Sample vs. Time

0 100 200 300 400 500 600 0 100 200 300 400 500 600 700 Time (s) Lo ad on S am p le ( m N ) 0.015 s-1 0.045 s-1 0.135 s-1 0.405 s-1 Figure 25. Constant L L&

Data 25°C: Load on Sample vs. Time.

52

Figure 26. Constant L L&

Data 25°C: Indentation Strain Rate vs. Displacement into Surface.

Ldot/L data 25°C

IndentationStrain Rate vs Displacement Into Surface

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 1000 2000 3000 4000 5000 6000 7000

Displacement Into Surface (nm)

Inde nt at ion S tr ain R at e ( s-1) 0.015 s-1 0.045 s-1 0.135 s-1 0.405 s-1

53

Comparison of Ldot/L Data at 25°C Mean Pressure vs. Displacement into the Surface

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 1000 2000 3000 4000 5000 6000 7000

Displacement into the Surface (nm)

M e an P ress ure (G Pa) _0.015 0.045 0.135 0.405 Figure27. Constant L L&

54

to the strain rate times two) yields a different mean pressure. This result is displayed in Fig. 28. Results were collected for each temperature in the same manner as the results described for the 25°C experiments. The results for the experiments performed at 30°C, 35°C and 40°C have been compiled in the appendix of this thesis. The appendix shows that for each temperature and experimental method the plot of the strain rate as a function of mean pressure displays no sign of history dependence. In every case the results

collapse into a singular set of data.

For the mechanical properties of amorphous selenium to truly be considered loading history independent the experimental results must not only agree when one changes the loading conditions within a method, but the results from each method must also be in agreement. To compare the data from each method, all of the results were plotted together in an indentation strain rate as a function of mean pressure plot. This plot was repeated for each temperature. These results are presented in figures 29-32. Figure 33 shows all the data from all four temperatures. These figures show that for each temperature the results from the different methods all come together to form a single data set. These results solidify the fact that amorphous selenium is loading history

independent over a wide range of loading conditions and temperatures.

Creep Exponent Calculation

The creep exponent, n, is calculated from Eq. 2. Therefore, by definition, the creep exponent is the slope of the data in figures 29-33. For most of the data, the creep exponent is not constant as a function of mean pressure. To calculate the change in the creep exponent as a function of the mean pressure, an Excel spreadsheet was used to calculate an average slope for the log of indentation strain rate and the log of mean pressure for every 7 data points. Since the plot of indentation strain rate as a function of mean pressure is the same for every experimental loading condition at each temperature, one or two loading conditions were selected to represent the entire range of mean

pressure for each temperature. The results of these calculations are plotted in Fig. 34. This figure displays several interesting features. The 25°C and 30°C (below Tg) data

55

Figure 28. Constant L L&

Data 25°C: Indentation Strain Rate vs. Mean Pressure. Comparison of Ldot/L Data at 25°C

Indentation Strain Rate vs. Mean Pressure

0.001 0.010 0.100 1.000

0.100 1.000

Mean Pressure (GPa)

Indent at ion S tr a in R a te ( s -1 ) 0.015s-1 0.405s-1 0.135s-1 0.045s-1

56

Figure 29. Comparison of all Data 25°C: Indentation Strain Rate vs. Mean Pressure.

Comparison of all Data at 25°C Indentation Strain Rate vs. Mean Pressure

0.00001 0.0001 0.001 0.01 0.1

0.1 _{Mean Pressure (GPa)} 1

Inde nt at ion S tr ain R at e ( s -1 ) 100mN 200mN 300mN 400mN 500mN 4mN/s 0.3mN/s 0.02mN/s Pdot/P

57

Figure 30. Comparison of all Data 30°C: Indentation Strain Rate vs. Mean Pressure.

Comparison of all Data at 30°C Indentation Strain Rate vs. Mean Pressure

0.00001 0.0001 0.001 0.01 0.1 1 0.01 0.1 1

Mean Pressure (GPa)

In de n ta tion S tr ain R at e ( s -1 ) 50mN 100mN