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Determining the Role of Fractal Geometry and Fracture Energy in Brittle Bilayer Materials

Permanent Link: http://ufdc.ufl.edu/UFE0014623/00001

Material Information

Title: Determining the Role of Fractal Geometry and Fracture Energy in Brittle Bilayer Materials
Physical Description: 1 online resource (92 p.)
Language: english
Creator: Smith, Robert
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: atomic, bilayer, ceramic, dimension, energy, force, fractal, fracture, glass, microscopy, strength, toughness, work
Materials Science and Engineering -- Dissertations, Academic -- UF
Genre: Materials Science and Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many applications in the ceramic field require bilayer design to be successful. One example is the application of bilayers in dentistry for all ceramic crowns and bridges. Although fracture in bilayers has been studied, research has not explored if the fracture surface changes as a crack travels from one material to another in a bilayer and if a change in the fracture surface reflects changes in the mechanical properties of the bilayer. The work of fracture, fractal dimension, fracture strength, and apparent fracture toughness were determined for two sets of bilayer bars composed of e.max Ceram veneer bonded to e.max Press core The work of fracture was determined from the actual fracture surface area, using atomic force microscopy, to determine if the work of fracture can be used to estimate the minimum fracture energy of the materials. The location of crack initiation was examined to see if the work of fracture is the same in both sets of bilayers and if the fracture process results in the veneer and core having the same fractal dimension. It was confirmed that the location of crack initiation did change the work of fracture of the bilayers but did not change the fractal dimension of the two materials in the bilayer. The work of fracture was used to estimate the minimum fracture energy of the ceramic monoliths and bilayers. Using the actual fracture surface to determine the work of fracture resulted in a decrease in the work of fracture but did not estimate the theoretical minimum fracture energy of the dental materials. The work of fracture for the bilayer was dependent on the material that the crack initiated in, further indicating that the selection of the materials used in these bilayers is important in designing these bilayer structures. The fractal dimensions of the veneer and core in the bilayers were similar to that of the monolithic specimens. This implies that the fracture process does not change the fractal dimension of these materials and that the fractal dimension is an intrinsic property of these ceramics.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Robert Smith.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Mecholsky, John J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0014623:00001

Permanent Link: http://ufdc.ufl.edu/UFE0014623/00001

Material Information

Title: Determining the Role of Fractal Geometry and Fracture Energy in Brittle Bilayer Materials
Physical Description: 1 online resource (92 p.)
Language: english
Creator: Smith, Robert
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: atomic, bilayer, ceramic, dimension, energy, force, fractal, fracture, glass, microscopy, strength, toughness, work
Materials Science and Engineering -- Dissertations, Academic -- UF
Genre: Materials Science and Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many applications in the ceramic field require bilayer design to be successful. One example is the application of bilayers in dentistry for all ceramic crowns and bridges. Although fracture in bilayers has been studied, research has not explored if the fracture surface changes as a crack travels from one material to another in a bilayer and if a change in the fracture surface reflects changes in the mechanical properties of the bilayer. The work of fracture, fractal dimension, fracture strength, and apparent fracture toughness were determined for two sets of bilayer bars composed of e.max Ceram veneer bonded to e.max Press core The work of fracture was determined from the actual fracture surface area, using atomic force microscopy, to determine if the work of fracture can be used to estimate the minimum fracture energy of the materials. The location of crack initiation was examined to see if the work of fracture is the same in both sets of bilayers and if the fracture process results in the veneer and core having the same fractal dimension. It was confirmed that the location of crack initiation did change the work of fracture of the bilayers but did not change the fractal dimension of the two materials in the bilayer. The work of fracture was used to estimate the minimum fracture energy of the ceramic monoliths and bilayers. Using the actual fracture surface to determine the work of fracture resulted in a decrease in the work of fracture but did not estimate the theoretical minimum fracture energy of the dental materials. The work of fracture for the bilayer was dependent on the material that the crack initiated in, further indicating that the selection of the materials used in these bilayers is important in designing these bilayer structures. The fractal dimensions of the veneer and core in the bilayers were similar to that of the monolithic specimens. This implies that the fracture process does not change the fractal dimension of these materials and that the fractal dimension is an intrinsic property of these ceramics.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Robert Smith.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Mecholsky, John J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0014623:00001


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4ed075298c0d33b27550482e757a0872f208b986







DETERMINING THE ROLE OF FRACTAL GEOMETRY AND FRACTURE ENERGY IN
BRITTLE BILAYER MATERIALS




















By

ROBERT LEE SMITH


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2009





































O 2009 Robert Lee Smith



































To my parents for all of their help and support









ACKNOWLEDGMENTS

I thank Dr. Mecholsky for his guidance and understanding during my time at the

University of Florida. I have learned much about materials science, and life in general, under his

guidance and tutorage.

I thank the other members of my supervisory committee, Dr. Anthony Brennan, Dr.

Wolfgang Sigmund, Dr. Laurie Gower, and Dr. Bhavani Sankar, for their advice and input while

working on my dissertation proj ect. I would especially like to thank Dr. Anthony Brennan for

the discussions that we had that helped me gain a better understanding of my work.

I want to thank my colleague, Dr. Chuchai Anunmana, for making all of the specimens that

l used in my proj ect, for the flexure testing of the specimens, and with helping me with the

microstructural characterization of the specimens. Without his help, I would not have been able

to finish my proj ect and I am grateful for his help.

I thank Robert (Ben) Lee and Allyson Barrett of the Department of Dental Biomaterials for

their assistance in the mechanical testing of my specimens in the dissertation project. I thank Dr.

Anusavice of the Department of Dental Biomaterials for allowing me to use the resources of his

department.

I want to thank Andrew Gerger and Brad Willenberg, respectively, of the MAIC staff for

helping with the use of the atomic force microscope that I used for a maj ority of my research and

for helping me attain the scanning electron microscope micrographs used in this dissertation. I

want to thank Dr. Stephen Freiman with his help in understanding the work of fracture aspects of

my work.

I thank the South East Alliance for Graduate Education and the Professoriate, the Alfred P.

Sloan Foundation, the National Consortium for Graduate Degrees for Minorities in Engineering









and Science, and the National Science Foundation for their support in funding my graduate

study .

I thank my friends and colleagues, especially Dr. Vasana Maneeratana, Ricardo Torres,

and Alma Stephanie Tapia for their help, advice, and support during my time at the University of

Florida. Finally, I thank my mother and father for all the help and support they have given me

that has allowed me to accomplish my goal of receiving my Ph.D.












TABLE OF CONTENTS


page


ACKNOWLEDGMENTS .............. ...............4.....


LIST OF TABLES ................. ...............8.__. .....


LIST OF FIGURES .............. ...............9.....


AB S TRAC T ............._. .......... ..............._ 1 1..


CHAPTER


1 INTRODUCTION ................. ...............13.......... ......


Introducti on ................. ................. 13..............
G oal s ................ ...............13.......... ......

Obj ectives .................. ...............14.......... ......
Structure of Dissertation ................. ...............15........... ...


2 LITERATURE REVIEW ................. ...............16................


Bilayer Composites............... ...............1
Glass Ceramics .............. ...............17....

Diffusion Bonding .............. ...............19....
Work of Fracture ................. ...............20........... ....

Fractal Geometry .............. ...............21....


3 MATERIALS AND METHODS .............. ...............27....


Sample Preparation............... ..................2
Preparation of Glass-Ceramic Core ................. ...............27................
Preparation of Glass Veneer ................. ...............28................
Preparation of Bilayers ................. ...............29................
Microstructural Analysis .............. ...............29....
Phase Identification .............. ...............29....

Aspect Ratio .............. ...............29....
Fracture Strength .............. ...............30....
Fracture T oughness............... ...............3
Strength Indentation .............. ...............32....
Fractography ............ ............ ...............32......
Work of Fracture. ..........._.... ...............33................
Fractal Dimension............... ...............3
Statistical Analysis............... ...............37


4 RESULTS AND DISSCUSION............... ...............4












Microstructural Analysis .............. ...............41....
Phase Identification .............. ...............41....

Aspect Ratio .............. ...............42....
Fracture Strength .............. ...............43....
Fracture Toughness............... ...............4
Work of Fracture ................. ...............45........... ....
Soda-Lime-Silica Glass ................. ...............45.................
Dental M materials .............. ...............48....
Fractal Dimension............... ...............5
Soda-Lime-Silica Glass ................. ...............50.................
Baria Silica Glass .............. ...............52....
Dental M materials .............. ...............53....


5 CONCLUSIONS .............. ...............79....


APPENDIX DETERMINATION OF THE HIURST EXPONENT FOR SILICA GLASS........81


Background on the Hurst Exponent................... ...............8
Measurement Methods for Determining Hurst Exponent .............. ...............82....
Hurst Exponent for Silica Glass .............. ...............84....


BIOGRAPHICAL SKETCH .............. ...............92....










LIST OF TABLES


Table page

3-1 Chemical composition of IPS e.max@ Press ................. ....__....._ ...............39

3-2 Chemical composition of IPS e.max@ Ceram ................. ....__ ............. .......3

3-3 AFM scan areas used to determine optimum scan areas for fractal dimension
measurements of fracture surface of baria silica glass............... ...............40.

4-1 Fracture strength, fracture toughness determined from strength indentation (SI), and
fracture toughness determined from fractography (F) for e.max@ Ceram (EV),
e.max@ Press (EC), bilayer with crack initiation in veneer (BV), and bilayer with a
flaw produced in the core (BC)............... ...............57..

4-2 Work of fracture values calculated for chevron notch specimens using cross-sectional
area (ACS) and actual fracture surface area based on probe tip radius, equilibrium
bond length, and free volume ao. ............. ...............57.....

4-3 Comparison of work of fracture and fracture toughness, based on work of fracture,
for unstable crack growth soda-lime-silica glass specimens .............. .....................5

4-4 Work of fracture TwOF, based on cross-sectional area (Acs) and fracture surface area
(AF) foT Veneer, core, and bilayer bars ................. ...............57........... ..

4-5 Fractal dimensional increment for soda-lime-silica glass fracture surface regions
measured using atomic force microscopy and the Gwyddion and WSxM software.........58

4-6 Fractal dimensional increment for e.max@ Ceram (EV) and the e.max@ Press (EC)
ceramics measured using AFM and Gwyddion software .............. ....................5

4-7 Fractal dimensional increment for bilayer specimens using AFM and Gwyddion ...........58

A-1 Hurst exponent for mirror region of silicate glasses from different experimental
m ethods. .............. ...............87....










LIST OF FIGURES


Figure page

2-1 Diffusion bonding process.. ............ ...............24.....

2-2 Replication technique for creation of samples for slit island method ............... .... ........._..25

2-3 Production of island from polishing .............. ...............25....

2-4 Profie technique using fracture surface replica ................. ...............26........... ..

3-1 Cross-section of bilayer specimens ................. ...............38........... ...

3-2 Diagram of distances used for computation of centroid for bilayer specimens. ................38

3-3 Diagram of distances used for computation of moment of inertia ................. ................39

4-1 X-ray diffraction pattern for e.max@ Press (EC) core ceramic showing that lithium
disilicate are the crystals present in the glass-ceramic............... .............5

4-2 X-ray diffraction pattern of e.max@ Ceram (EV) veneer ceramic showing veneer
material has a amorphous pattern .............. ...............60....

4-3 SEM micrographs of polished and etched surface of e.max Ceram (EV) veneer
ceramic............... ...............61

4-4 SEM image of polished and etched surface of e.max Ceram (EC) core ceramic
showing the presence of needle-like nanofluorapatite crystals ................. ............... ....62

4-5 Energy dispersive X-ray spectroscopy (EDS) of e.max@ Ceram (EV) veneer
ceram ic. .............. ...............63....

4-6 SEM micrograph of polished and etched surface e.max@ Press (EC) core ceramic at
a magnify cation of 10000x. ................ ...............63.......... ....

4-7 SEM micrograph of polished and etched surface e.max@ Press (EC) core ceramic at
a magnify cation of 5000x .............. ...............64....

4-8 SEM micrograph of polished and etched surface e.max@ Press (EC) core ceramic at
a magnify cation of 10000x ................. ...............66.......... ....

4-9 SEM image of fracture surface of chevron-notched soda-lime-silica flexure specimen
showing that fracture surface is all mirror and no mist or hackle is present .....................67

4-10 AFM scan image of soda-lime silica glass fracture surface at a scan area of (750
nm )2............. ...............68...










4-11 Fracture surface area (AF) aS a function of scan area side length (rl) for chevron-
notched sod a-lime-silica bars ................. ...............69........... ....

4-13 Fracture surface area (AF) aS a function of scan area side length (rl).............. ...............71

4-14 Fracture surface area (AF) and adjusted fracture surface area (Adj. AF) aS a function
of scan area side length (rl) for e.max@ Ceram (EV) veneer bars ................. ................. 72

4-15 Fracture surface area (AF) aS a function of scan area side length (rl) for e.max@ Press
(EC) core bars .............. ...............73....

4-16 Fracture surface area (AF) aS a function of scan area side length (rl) for veneer-indent
(BV) and core-indented (BC) bilayers. .............. ...............74....

4-17 Fractal dimension increment (D*) of baria silica glass, plotted against scan area,
determined from Gwyddion for mirror, mist, and hackle region............... .................7

4-18 AFM scan image of e.max@ Ceram (EV) veneer ceramic and e.max@ Press (EC)
core ceramic fracture surface at a scan area of (750 nm)2. ............ .....................7

4-19 SEM images of veneer-indented bilayer (BV) fracture surface ................. ................ ...77

4.20 SEM images of core-indented bilayer (BC) fracture surface............... ................7









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

DETERMINING THE ROLE OF FRACTAL GEOMETRY AND FRACTURE ENERGY IN
BRITTLE BILAYER MATERIALS

By

Robert Lee Smith

August 2009

Chair: John J. Mecholsky Jr.
Major: Materials Science and Engineering

Many applications in the ceramic field require bilayer design to be successful. One

example is the application of bilayers in dentistry for all ceramic crowns and bridges. Although

fracture in bilayers has been studied, research has not explored if the fracture surface changes as

a crack travels from one material to another in a bilayer and if a change in the fracture surface

reflects changes in the mechanical properties of the bilayer. The work of fracture, fractal

dimension, fracture strength, and apparent fracture toughness were determined for two sets of

bilayer bars composed of e.max Ceram@ veneer bonded to e.max Press@ core The work of

fracture was determined from the actual fracture surface area, using atomic force microscopy, to

determine if the work of fracture can be used to estimate the minimum fracture energy of the

materials. The location of crack initiation was examine to see if the work of fracture is the same

in both sets of bilayers and if the fracture process results in the veneer and core having the same

fractal dimension.

It was confirmed that the location of crack initiation did change the work of fracture of the

bilayers but did not change the fractal dimension of the two materials in the bilayer. The work of

fracture was used to estimate the minimum fracture energy of the ceramic monoliths and

bilayers. Using the actual fracture surface to determine the work of fracture resulted in a









decrease in the work of fracture but did not estimate the theoretical minimum fracture energy of

the dental materials. The work of fracture for the bilayer was dependent on the material that the

crack initiated in, further indicating that the selection of the materials used in these bilayers is

important in designing these bilayer structures. The fractal dimensions of the veneer and core in

the bilayers were similar to that of the monolithic specimens. This implies that the fracture

process does not change the fractal dimension of these materials and that the fractal dimension is

an intrinsic property of these ceramics.









CHAPTER 1
INTTRODUCTION

Introduction

The critical element that controls the fracture process has not been fully determined for

bilayer composite materials such as dental prosthetics or ceramic armor. Selection of the proper

material is critical to performance if the material in which fracture initiates controls the fracture

properties of the entire composite. There are several properties that are important in determining

the effect of the location of crack initiation on the fracture process of a material: fracture

toughness, fractal dimension, and work of fracture. Fracture toughness is the resistance to crack

propagation in a material. The fractal dimension is a quantitative measure of the tortuosity of a

surface at multiple length scales. The work of fracture is defined as the total energy consumed to

produce a unit area of fracture surface during the fracture process. These properties can be used

together in multilayer composites to study the effect of the initiation of fracture on the fracture

process in bilayers.

Goals

The goal of the proposed research is to determine if the fracture surface area and the fractal

dimension of materials in a bilayer change as a crack travels from one material into another and

if a change in the fracture surfaces determines the mechanical properties of the bilayer structure.

This goal will be accomplished by testing the fracture toughness, work of fracture, and fractal

dimension of a glassy veneer bonded to a glass-ceramic dental core. Knowledge of the energy

contributed to create the fracture surface in ceramics will help engineers better understand the

total energy to generate fracture. This will help to better design brittle materials and bilayer

structures that are used in several disciplines like dentistry and armor protection, where fracture

resistance and materials selection is critical. Knowledge of changes in the fractal dimension of










fracture surfaces for bonded ceramic materials will give a better understanding of the role that

the fracture process has in determining the fractal dimension of the materials. This will aid in the

further use of the fractal dimension as a tool in characterizing the fracture of ceramic materials.

Obj ectives

To accomplish the goal of this research, the following hypotheses are proposed:


1. Test the hypothesis that the work of fracture based on the actual fracture surface area can

be used to estimate the fracture energy of ceramic materials.

2. Test the hypothesis that the work of fracture of a ceramic bilayer will be the same

regardless of the location of crack initiation.

3. Test the hypothesis that the fractal dimension of the two materials in a bilayer will be the

same regardless of the location of crack initiation.

A veneer and a core dental ceramic were chosen as the materials used to test the stated

hypotheses. The veneer chosen was IPS e.max@ Ceram, a fluorapatite glass-ceramic, and IPS

e.max Press@, a lithium disilicate glass-ceramic. These materials were chosen because bilayers

of the materials are used in dentistry, so the mechanical properties (elastic modulus, fracture

strength, fracture toughness, etc.) of the materials have been studied [1-5]. This allows for

comparison of the mechanical properties studied in this dissertation to what other researchers

have measured. The mechanical properties of the two materials are different enough in order to

determine if location of crack initiation changes the mechanical properties of these materials.

The veneer and core have been shown to bond well [3, 4, 6], producing a strong bond at the

interface between the two materials. This means a crack that propagates in one layer should

continue to propagate into the second layer without delaminating at the interface.

A series of experiments have been determined to test the hypotheses for this proj ect:










1. The microstructure was analyzed and the fracture strength and apparent fracture

toughness were determined for the veneer, the core and for two sets of bilayers.

2. The work of fracture, using the cross-sectional area and the fracture surface area, was

determined for a control group of soda-lime-silica (SLS) glass under stable and unstable

crack growth.

3. The work of fracture, using the cross-sectional area and the fracture surface area, was

determined for the veneer, the core, and for the two sets of bilayers

4. The fractal dimension was measured for a control group of SLS glass and baria silica

glass.

5. The fractal dimension was measured for the veneer, the core, and two set of bilayers.

Structure of Dissertation

Chapter 2 is a compilation of selected literature explaining the concepts of the work of

fracture and fractal dimension, and defining bilayer materials, glass-ceramics, and diffusion

bonding. Chapter 3 describes the materials selected for the study and the process used to prepare

these materials. The experimental procedures used to answer the hypotheses discussed in the

obj ectives are also explained in this chapter. Chapter 4 details the results from the experiments

described in Chapter 3 and interprets the meaning of the results. Chapter 5 summarizes the

dissertation and discusses whether the hypotheses were answered from the experiments. The

dissertation concludes by addressing the concept of the Hurst exponent and its relationship to the

fractal dimension for soda-lime-silica glass in the Appendix.









CHAPTER 2
LITERATURE REVIEW

In this chapter, bilayer composites, diffusion bonding, and glass-ceramics are discussed to

give background on the dental materials and fabrication methods used in the dissertation. The

two parameters discussed in the obj ective, the work of fracture and the fractal dimension, are

defined and background is given on the previous methods used to determine these properties.

Bilayer Composites

A bilayer material is a composite material in which two different materials are bonded

together. Bilayer materials are joined by either bonding the two materials together using a

welding or diffusion bonding technique or by using an interlayer like an adhesive to bond the

two materials together. As in all composites, bilayer materials are designed to combine the

advantages of both materials to produce an optimum structure. Bilayers are used in a variety of

applications from protective coatings on metal substrates that are used in the aerospace industry

to titanium alloys bonded with a bioactive glass coating for bone implants [7].

Ceramic bilayers that are used to produce dental prostheses have had an increase in use

over the years. The use of a bilayer composite allows for the aesthetic and mechanical properties

needed for dental prostheses. All-ceramic dental bilayers for prostheses are made of two

materials: a glass veneer that is matched to the appearance of the surrounding teeth and the glass-

ceramic core that provides the stiffness and fracture resistance that is needed for dental

prostheses. The veneer must also be matched in thermal expansion and contraction coefficient to

minimize the risk for transient crack formation and residue tensile stress.

The mechanical properties of these bilayer materials are dependent on the mechanical

properties of the two materials and the interface between these materials. If a crack initiates and

propagates from the tensile surface of one the less tough layer, e.g., the veneer layer in a veneer-









core all-ceramic dental restoration, the propagation of the crack is dependent on the fracture

toughness of the interface and the second material [5, 8]. If the fracture toughness of the

interface is greater than the fracture toughness of the two materials in the bilayer, the crack will

continue to propagate into the second material. However, if the interface is less resistant to

fracture than the second material, then the crack will propagate along the interface of the two

materials and perhaps nullify the benefits of using a more fracture resistant material in the

bilayer. Taskonak et al. [9] showed that for lithia-disilicate based Eixed partial dentures, cracks

that initiated in the veneer propagated across the interface and into the core. Even though the

core had a greater fracture toughness than the veneer, the crack was not impeded by the core at

the interface. This implies that the interface is tougher than the core. For yttria-stabilized

tetragonal zirconia polycrystals Eixed partial dentures, cracks that initiated in the veneer

propagated through the veneer and stopped at the interface of the two materials. Delamination

occurred at the interface before a secondary crack initiated in the core. This implies that the core

was tougher that the interface and that delamination controlled the fracture process of the bilayer.

Although the mechanical properties of ceramic bilayers and multilayers have been

explored [4, 5, 10-15], research has not explored if the fracture surface changes as a crack travels

from one material to another in a bilayer and if a change in the fracture surface determines the

mechanical properties of the bilayer structure. Analysis of the work of fracture and the fractal

dimension will result in a better understanding of fracture surface of each material in the bilayer

affects the mechanical properties of the composite and if the location of crack initiation changes

the fracture surface and mechanical properties of the bilayer structure.

Glass Ceramics

Glass-ceramics are polycrystalline ceramic materials formed from a base glass through the

controlled nucleation and crystallization of crystals from the glass [16]. The crystals are









contained in a matrix of the base glass. Glass-ceramics are different from traditional ceramics

where the crystalline material is introduced when the ceramic composition is prepared. It has

been known for a long time that glasses can be crystallized to form polycrystalline ceramics.

Reaumur, a French chemist, produced polycrystalline material from glass by heating glass

bottles, packed in sand and gypsum, to a high temperature for several days [17]. Reaumur

experiments resulted in the creation of opaque, porcelain-like obj ects that had low mechanical

strength and a distorted shape compared to the original shape of the bottles. This was due to the

inability to control the crystallization process.

The invention of true glass-ceramics did not occur until the mid-1950s [16]. S. D.

Stookey, a glass chemist and inventor at Corning Glass Works, was planning to precipitate silver

particles in lithia silicate (Li20-SiO2) to create a permanent photographic image. He accidentally

overheated the glass overnight, producing a white ceramic with no net change in shape and an

unusual strength for a glass. Upon further analysis, Stookey determined that lithium disilicate

(Li2Si20s) crystals, nucleating from the silver particles, had been created in the glass. Stookey's

experiment produced a new type of material had been discovered that had applications in several

fields.

Glass-ceramics have advantages over glass when it comes to mechanical properties and

advantages over metals and organic polymers in the areas of thermal, chemical, biological, and

dielectric properties. The presence of grain boundaries and cleavage planes impede fracture

propagation while the crystals cause crack deflection, resulting in a greater fracture toughness

compared to glass [16]. Glass-ceramics exhibit little to no expansion on heating and have a high

temperature stability. This allows glass-ceramic to be used in high temperature application,

where metals cannot be used, e.g., cookware. The high chemical durability and translucency or










opacity of glass-ceramics makes glass-ceramic a useful material in places where the corrosion of

metal is a problem. Another advantage of glass-ceramics is that a glass-working process can be

used to create obj ects resulting in the use of high-speed automatic machining and a faster

processing time when compared with the techniques used in conventional methods.

The most important features of glass-ceramics are that they are applicable over a wide

range of compositions and variations can be controlled through the heat treatment process [17].

Various crystal types can be developed in controlled proportions resulting in the ability to vary

the physical characteristics of glass-ceramics in a controlled manner. This feature allows glass-

ceramics to be designed and created for a multitude of applications.

Diffusion Bonding

Diffusion bonding involves the interdiffusion of atoms across the interface of two

materials, held in contact with each other, resulting in the bonding of the two materials [18]. The

bonding of these materials involves holding a pre-machined component under pressure at an

elevated temperature in a protective environment. The pressure applied to the joining materials

must cause the edges to move within range of the atomic forces but below a pressure that would

cause macrodeformation in either material (Figure 2-1). The temperature range for diffusion

bonding is between 0.5 and 0.8 of the lower melting temperature of either of the two materials

[19]. Diffusion bonding usually requires a minimum of 60 minutes to successful bond the

materials. The main advantages of diffusion bonding are 1) the material at the joint has the same

strength and plasticity as the bulk of the parent materials, 2) the bonding temperature is not as

high as other j oining processes, and 3) more types of joints can be formed compared with other

joining processes [18].

Although diffusion bonding is used mostly to bond metals to metals or metals to non-

metals, it has been used to bond ceramics to ceramics [20]. Ceramic to ceramic bonding is









mainly carried out in the solid phase with heat being applied by radiant, induction, direct or

indirect resistance heating. Some ceramics do not incur the problem of oxidation hindering

bonding since the ceramics themselves are oxides. The bonding of some ceramics, like the

bonding of a glass-ceramic to a glass, has a similar advantage of bonding similar metal alloys:

the ability to increase the mechanical properties of an obj ect but keeping the favorable properties

of the weaker material. This is observed in dental ceramics, where a transparent or translucent

veneer is diffusion bonded to a stronger and more fracture resistant glass-ceramic core. This

makes the dental ceramic stronger than the veneer and allows it to maintain its aesthetic

properties.

Work of Fracture

The fracture energy (y) of a material is defined as the energy per unit area required to

fracture an infinite body into two separate halves [21]. Defined in another way, y is the energy

required to break the bond between two atoms per unit area. When a specimen is tested in a

manner that allows for the stable growth of a crack, the average fracture energy can be

determined for the specimen. This average fracture energy is called the work of fracture (YwOF).

The work of fracture is defined as the total energy consumed to produce a unit area of fracture

surface during the fracture process [21] and is calculated from the equation:


TWOF (2-1)
ACS

where W is the work and is equal to the area under the load-displacement curve and Acs is the

cross-sectional area of the fracture surface. In theory, if it is assumed that all of the work from

loading is dissipated by the breaking of the bonds, then the average y = TwOF [29].

The work of fracture has been used as a method to determine the fracture toughness (Kc)

of materials [22-24] based on Irwin's equations:










Kc =ZEl (2-2)

where E' is the elastic modulus (E) for plane stress or E/(1-v2) [V is Poisson' s ratio] for plain

strain and y = YWOF. The work of fracture has also been used as a separate parameter to describe

the fracture properties of various materials, including biological [25-28], ceramics [23, 29-34],

polymer [35, 36], and metal materials [22, 37]. The advantages of using the work of fracture are

that it does not require knowledge of the stress intensity, the geometry of the notch, the notch tip

acuity, or the linearity of the material.

In most cases, Ke values calculated using YWOF have led to an overestimation of Kc when

compared with values calculated from methods that are more conventional [22, 23]. This is due

to the assumption that y = YWOF based on the equation used to calculate TwOF (Equation 2-1). This

assumption is based on the idea that all of the energy of the fracture process results in the

creation of two atomically smooth surfaces, resulting in the cross-sectional area being used as the

actual fracture area being created by the fracture process [29]. However, it is apparent that

fracture surfaces are not absolutely smooth, even in the fracture mirror region [38]. Since the

energy used to fracture materials results in the creation of a rough and tortuous fracture surface,

then using the actual fracture surface area of the material, which is greater than the cross-

sectional area, to calculated YWOF WOuld result in a better approximation of the theoretical

fracture energy of the material.

Fractal Geometry

Fractal geometry, a non-Euclidean geometry, can be used to quantitatively describe

irregular shapes and surfaces [39], e.g., a maple leaf or a head of cauliflower. Fractal obj ects are

defined as objects that are self-similar, scale invariant, and exhibit fractional dimensions. A self-









similar obj ect is an obj ect that has multiple parts that are similar to the entire obj ect. A scale

invariant obj ect is an obj ect that looks the same at different magnification scales.

From Mandelbrot' s early work in fractal geometry, especially in evaluating the fracture

surface of steels [40], fractal geometry has been shown to be a useful tool in the fields of science

and engineering. The basic quantitative component of fractal geometry is the fractal dimension

(D). The fractal dimension is a non-integer number with a Euclidean and non-Euclidean

component. The Euclidean component describes the topological dimension of the obj ect, e.g., 1

for a line and 2 for a square or circle. The non-Euclidean component, called the fractal

dimensional increment (D*), describes the level of irregularity of the obj ect from its Euclidean

geometry. The fractal dimensional increment ranges from 0 to 1, with the level of irregularity

increasing as the increment increases. For example, broccoli has a fractal dimension of 2.66

[41], where 2 denotes that it is a surface and 0.66 denotes the fractal component and the level of

irregularity in the surface of broccoli.

In materials science, fractal geometry has been used as an analytical tool to quantify the

fracture surfaces of brittle and ductile materials [42-48]. Fracture surfaces have a fractal

dimension of 2.D* with a fractal dimension of 2.1 denoting a fracture surface that is relatively

flat and a fractal dimension of 2.9 denoting a fracture surface that is volume filling.

Several methods have been used to measure the fractal dimension of a fracture surface.

The measurement of the contours of the fracture surface is one of the most frequently used

methods. There are two main boundary contour methods: the slit-island method [40], which

examines the contour parallel to the plane of the fracture surface, and the profile method, which

examines the contour perpendicular to the plane of the fracture surface [39].









The slit island technique uses the fracture surface to determine the fractal dimension. The

fracture surface of a specimen, or a replica of the fracture surface, is plated in a visible coating

such as a gold-palladium sputter coating to make the surface more visible (Figure 2-2). The

specimen is mounted in epoxy or another mountable material and polished parallel to the fracture

surface until parts of the fracture surface, called islands, are visible (Figure 2-3). From the

coastlines of the islands, the fractal dimension is measured using one of two methods:

measurement of the perimeter and area of the islands or measuring the length of the "coastline"

of the islands (the Richardson method) [49]. A straight line drawn through the data points of a

log-log graph of the area versus the perimeter results in a line that has a slope equal to D* (1-

D). For the Richardson method, the perimeter is measured at different length scales using the

following equation:

L~q) Lo91-D)(2-3)

where L(rl) is the measured perimeter length as a function of the step size, Lo is the proj ected

perimeter, and rl is the step size. D* can be determined from the log-log plot of the perimeter

versus the step size.

The profie method measures the profie of the fracture surface (Figure 2-4). The profiles

are obtained using microscopy, e.g., optical, secondary electron, and scanning tunneling. From

these profies, the fractal dimension can be determined using Equation 2-3 where Lo is the

proj ected length of the profie and L(rl) is the measured length of the profie.





_ I_ __ I _


Figure 2-1. Diffusion bonding process. A) Initial contact. B) Plastic deformation. C) Diffusion
processes fill microvoids. D) Bonding completed. [Adapted from Lancaster, J. F.
(1999) Metallurgy of Welding, 6th edn. Woodhead Publishing, Cambridge, UK].





Figure 2-2. Replication technique for creation of samples for slit island method. [Source: Hill,
TJ, Della Bona, A, Mecholsky, JJ (2001) J Mater Sci 36:2651]












Figure 2-3. Production of island from polishing. [Source: Hill, TJ, Della Bona, A, Mecholsky,
JJ (2001) J Mater Sci 36:2651]


I I


I i


~r



















Figure 2-4. Profile technique using fracture surface replica. [Source: Hill, TJ, Della Bona, A,
Mecholsky, JJ (2001) J Mater Sci 36:2651]









CHAPTER 3
MATERIALS AND METHODS

In this chapter, the preparation of the materials used is described for this study. The

mechanical and microstructural procedures used to characterize the materials are discussed and

the measurement of work of fracture and fractal dimension methods for the dental materials is

describe ed.

Sample Preparation

Preparation of Glass-Ceramic Core

The lithium disilicate dental glass-ceramic cores were prepared using a lost-wax method

[4]. A self-curing acrylic resin (Pattemn Resin, GC Corp, Tokyo, Japan) was used to prepare the

master molds for the IPS e.max Press@ (EC) glass-ceramic (Ivoclar AG, Schaan, Liechtenstein).

The EC is a lithium disilicate glass-ceramic (Table 3-1). Impressions were made from the master

mold using a vinyl polysiloxane impression material (Extrude, Kerr Corp., Romulus, MI). The

acrylic resin was poured into the molds and cured to make rectangular bars (25 mm x 4.6 mm x

2.0 mm) using a mill (PGF 100, Cendres & Metaux Sa., Biel-Bienne, Switzerland).

Following the preparation of the resin beams, four specimens were sprued and invested.

The resin beams were placed in a preheating furnace (Radiance, Jelrus Int., Hicksville, NY) and

a two-stage burnout process was used to remove the resin: 1) the furnace is heated up to 2500C at

SoC/min. and held for 30 min. and 2) the furnace was heated up to 850oC at SoC/min. and held

for 1 hr. After burnout, the EC ingot was placed in an investment cylinder and it was transferred

to an isostatic pressing furnace (EP500, Ivoclar AG, Schaan, Liechtenstein). The EC was

pressed at a pressure of 5 bar (0.5 MPa) and a temperature of 9200C for 25 min. After pressing,

the investment cylinders were removed and cooled for 2 hr. in air. The cooled specimens were

divested (removal of investment material) by grit blasting with 80 Clm glass beads (Williams









Glass Beads, Ivoclar North America, Amherst, NY) at an air pressure of 0.25 MPa. The sprues

were cut away using a diamond disc and excess sprue segments were ground from 120 to 1200

grit using silicon carbide (SiC) paper and polished from 8 to 1 micron using diamond paste

(Buehler, Lake Bluff, IL) on polishing cloth (UltrapolThi, Buehler, Lake Bluff, IL) and 1 micron

alumina (Micropolish@ II, Buehler, Lake Bluff, IL) on polishing cloth (Microcloth@, Buehler,

Lake Bluff, IL). All polishing and grinding were preformed on a metallographic polishing wheel

(Model 41-1512, Buehler, Lake Bluff, 1L).

After polishing, the reaction layer from investing was removed by placing the bars in a

plastic bottle containing 20 mL of a 1% hydrofluoric solution (Invex Liquid, Ivoclar AG,

Schaan, Liechtenstein) and ultrasonically cleaned for 10-30 min. After cleaning, the bars were

cleaned again using tap water for 10 s and blown dry using compressed air. After drying, the

bars were grit-blasted with 100 Cpm Al203 particles (Blasting Compound, Williams-Ivoclar North

America Inc., Amherst, NY) at an air pressure of 0.1 MPa. Then, the bars were cleaned using a

pressurized steam spray.

Preparation of Glass Veneer

The glassly veneer, IPS e.max@ Ceram (EV), is described as a fluorapatite glass-ceramic

(Table 3-2). Glass veneers were prepared using a sintering process. A veneer powder was

mixed with a liquid (Ivoclar AG, Schaan, Liechtenstein) to obtain a slurry solution. This

solution was then poured into a silicone mold, condensed by vibration, dried, and sintered in a

furnace (P80, Ivoclar AG, Schaan, Liechtenstein) by heating to 800oC at 600C/min., held for 2

min., and cooled to 180oC. A vacuum was applied between 4500C to 7590C. Following the

sintering process, any excess veneer was ground of with a 75 grit circular diamond disk and

polished down to a 1 Clm finish using a metallographic polishing wheel. All veneer bars have the

same dimensions (25 mm x 4.6 mm x 2.0 mm) as the core ceramic.










Preparation of Bilayers

The veneer and core ceramics were bonded together using a sintering process. The

prepared EC bars were veneered with the EV. The veneer powders were mixed with a liquid

(Ivoclar AG, Schaan, Liechtenstein) to obtain a slurry solution. This solution was brushed onto

the EC. The bilayers were then sintered in a furnace (P80, Ivoclar AG, Schaan, Liechtenstein)

by heating to 800oC at 600C/min., held for 2 min., and cooled to 180oC. A vacuum was applied

between 4500C to 7590C. Three layers of veneer were sintered on each EC bar. Following the

sintering process, any excess veneer was ground of with a 75 grit circular diamond disk and

polished down to a 1 Clm finish using a metallographic polishing wheel. The dimensions of the

bilayers were set to be similar to the monolithic veneer and core (25 mm x 4.6 mm x 2.1 mm)

with the veneer having a thickness of 1.1 mm and the core having a thickness of 1.0 mm.

Microstructural Analysis

Phase Identification

X-ray diffraction (XRD) was performed on the EC and EV to determine if both materials

were glass-ceramics. Using a mortar and pestle, a sample of the glass ceramic was ground into a

fine powder. The powder was placed on a glass slide using double-sided tape. The specimen

was scanned at a rate of 3 0/min. over a 26 range of 10-600.

Aspect Ratio

The aspect ratio of the crystals in the EC glass ceramic specimens was determined using

a scanning electron microscope (SEM). Samples of the glass ceramic were mounted in epoxy

and polished down to a 0.05 Cpm finish. After polishing, the specimens were etched in 2.5 %

hydrofluoric (HF) for 10 s, cleaned in tap water, ultrasonically cleaned for 5 min. in isopropanol,

and air-dried using compressed air. The top and sides of the epoxy mount was painted with

carbon paint before sputter coating with carbon. Three micrographs each were taken from two










specimens. The maj or and minor axes of five crystals from each micrograph were measured to

calculate the aspect ratio of the glass-ceramics.

Fracture Strength

As a control group, the strength of the glass veneer and core glass-ceramic were

measured. Five bars of EC and EV each were prepared for testing. The fracture strengths of the

bilayer bars was tested using the strength indentation technique [50]. A Vickers indent was

placed in the center of the tensile surface of the bars using a microhardness machine (Model,

Buehler, Lake Bluff, IL) with an indent load of 1 kgf for EV and 2 kgf for EC. After indentation,

the bars were tested in four-point flexure using a universal testing machine (Model 4465, Instron

Corp, Norwood, MA) at a crosshead speed of 0.2 mm/min. with a support span of 20 mm and a

load span one-third of the support span. The flexure strength of the bars was calculated using the

equation:

FL
crF =(3-1)
bd 2

where OF is the flexural strength of the specimen, F is the load at failure, L is the support span, b

is the specimen width, and d is the specimen thickness.

For the bilayer bars, two sets of five specimens were fabricated for flexure testing. The

first set was indented on the tensile surface of the EV layer of the bilayer with an indent load of 1

kgf. The second set was indented on the tensile surface of the EC layer of the bilayer with an

indent load of 2 kgf. Because of the difference in the elastic moduli of the two layers, composite

beam theory was used to calculate the fracture strength of the bilayers. We followed the method

outline in Beer and Johnston [51] and applied by Thompson [5] to dental ceramics. The

composite beam was transformed into a uniform beam of EC. This was done by calculating the









transformation factor (n), the centroid (y), and the moment of inertia (I) for the transformed

beam.

The transformation factor is the ratio of the elastic modulus of EC [96 GPa] [52] and EV

[64 GPa] [52]. This ratio is 3/2, therefore n = 1.5 and the width of EV is transformed from 4.6

mm to 3.1 mm (Fig. 3-1). With this transformation, the centroid can be determined for the

composite beam. Using a reference axis set at the tensile surface of the transformed beam

(Fig.3-2), y can be determined using

[ABC +D)F]
y = (3-2)
[BC +EF]

where A, B, C, D, E, and F are the distances for the transformed beam (Fig. 3-2). Once the

centroid has been determined, the moment of inertia for the beam can be determined based on

the distances in Fig. 3-3 and from the equation


I 1 ab3 + ab(c ) 2]i 3 + f (g h) (3-3)

With the centroid and moment of inertia known for the composite beam, the flexure

stress o- can be determined for the composite beam. The flexure stress is calculated from the

equation

M\~c
cr= (3-4)


where M is the maximum moment and c is the distance from the bottom of the beam to the

centroid. For four-point flexure with a load span 1/3 of the support span, M is expressed as

PL
M = (3-5)









Fracture Toughness

The fracture toughness of the bilayer and monolithic bars was determined using two

methods: 1) the strength indentation technique and 2) fractography.

Strength Indentation

The strength indentation technique uses an indent, in this case a Vickers indent, to

introduce a critical crack in the tensile surface of a flexure bar. The advantage of using this

method is that the crack length does not have to be known to calculate the fracture toughness.

The fracture toughness is calculated using the following equation [50]:


Kc = 0.59(E/H)"aFP'1/3 3/4 (3-6)

where Kc is the fracture toughness, E is the elastic modulus of the specimen, H is the hardness of

the specimen, OF is the fracture strength of the specimen, and P is the indent load. For the bilayer

composites, E was determined from the upper bound of the Rule of Mixtures, i.e.,

EB = EEVAEV +EECAEC (3 -7)

where EB is the elastic modulus of the composite, AEV is the area fraction of the veneer in the

bilayer, EEV is the elastic modulus of the core, and AEC is the area fraction of the core in the

bilayer.

Fractography

The fracture toughness of the bars was calculated using fractography. An optical

microscope (Olympus BHMJ, Olympus, Tokyo, Japan) was used to measure the critical flaw

size. The fracture toughness was calculated using the equation [53]:

Kc = Y"F (3-8)









where Y is a geometrical constant for the shape of the crack and loading geometry that has a

value of 1.65 for indent-induced surface cracks, with a local residual stress [54], and c is the flaw

size. The flaw size was calculated as

c = & (3-9)

where a is the crack depth and b is the half of the crack width.

Work of Fracture

The work of fracture was measured for soda-lime-silica (SLS) glass as a control group and

for the bilayer and monolithic dental ceramic bars. Two equations were used to determine TwOF

associated with the following methods: 1) the cross-sectional area approach (Equation 2-1) and

2) the actual fracture surface area (AF) inStead of the cross-sectional area method, i.e.:


TWOF (3-10)
AF

Since the crosshead displacement of the testing machine does not accurately measure the

midpoint deflection (6) of the flexure specimens, 6 was calculated using the equation:

S=~ (3-1 1)

where E is the strain to failure and is determined from the equation


E = F(3-12)


Unstable fracture was accomplished using strength indentation [50] in flexure while stable

fracture was accomplished using a chevron-notch (Figure 4-9) in flexure [55]. Three sets of bars

were used for the unstable fracture test: 1) 45 mm x 4 mm x 3 mm bars were Vickers indented

with a load of 2000 gf load and tested in four-point flexure, 2) 55 mm x 10 mm x 5.7 mm bars

were Vickers indented with a load of 1000 gf and tested in three-point flexure, and 3) 55 mm x

10 mm x 5.7 mm bars were Vickers indented with a load of 2000 gf and three-point flexure.









The chevron-notched bars, with dimensions of 45 mm by 4 mm by 3 mm, were fabricated

using a low speed diamond saw (South Bay Model 650, South Bay Technology, San Clemente,

CA). The notches were cut using a diamond blade with a thickness of 0. 15 mm. The specimens

were tilted at an angle of 250 using a specially made fixture for the diamond saw.

Both the unstable and stable SLS glass specimens' edges were polished using 240-grit

silicon carbide paper to remove or reduce the size of any flaws that were present along the edges.

All specimens were tested using a universal testing machine (Model 4465, Instron Corp,

Norwood, MA). For the unstable test, the crosshead speed was 0.5 mm/min while the stable tests

were conducted at a crosshead rate of 0.03 mm/min.

The fracture surface area was measured using an atomic force microscope (AFM)

(Dimension 3100, Veeco Instruments Inc., Woodbury, NY). Using the software that is provided

with the AFM (Nanoscope 3D, Veeco Instruments Inc., Santa Barbara, CA), the surface area was

determined from the sum of the area of a grid of triangles formed by three adj acent data points in

the scans. Eight square scan areas ((750 nm)2, (625 nm)2, (500 nm)2, (375 nm)2, (250 nm)2, (100

nm)2, (50 nm)2, and (25 nm)2) were used to measure the fracture surface area. Five scans were

completed for each scan area per specimen. All scans were conducted in the tapping mode,

where the AFM tip is oscillated at the cantilever' s resonant frequency causing the tip to

intermittently contact the surface. A tapping tip was used to decrease the wear of the AFM tip

over time on the hard surface of the materials. Previous AFM work using a contact tip showed a

decrease in the surface area measured, over time, due to wear increase the tip radius. All images

were scanned at a scan rate of 2-3 Hz and a resolution of 512 pixels per line. All fracture

surfaces were cleaned with ethanol to remove the dust and other material from the surface and

dried using compressed air.










To determine the fracture surface area, a box counting method was used for which the

average surface area of a portion of a selected mirror, mist, and hackle region was measured

using the AFM. Then, the surface area value for that representative region was multiplied times

the entire cross-sectional area of the region. From the AFM data obtained, AF WAS calculated

using the following equation:

S4 S4 S4
SS AHRR SS z, SS HCL

where A is the area of a given region (mirror, mist or hackle) of the fracture surface, SA is the

surface area value for that region and SS is the scan area used to determine SA. This method

was used to calculate AF for all of the different scan areas.

Fractal Dimension

The fractal dimensions of SLS glass bars and baria silica glass bars, used as a control

group, were determined along with the bilayer and monolithic dental ceramics bars using an

AFM. All scans were conducted using a tapping tip at a resolution of 512 pixels per line and a

scan rate of 2-3 Hz. All fracture surfaces were wiped clean with ethanol to remove dust and

other materials from the surface and dried with compressed air.

To calculate the fractal dimension (D) from the AFM data, two software packages were

used: 1) Gwyddion and 2) WSxM. Gwyddion is a free and open source software program for

scanning probe microscopy data visualization and analysis [56]. Gwyddion uses four methods to

calculate the fractal dimension from AFM data: 1) cube counting, 2) triangulation, 3) variance,

and 4) power spectrum.

Cube counting uses box counting to determine D [57, 58]. A grid of cubes with a cube

edge length m is placed over the surface of the AFM image. N(m) is the number of cubes that









contain at least one pixel. The value of m is decreased by a factor of two until m is equal to the

distance between two pixels. The slope of the log of N(m) versus the log of (1/m) is equal to D.

For the triangulation method [57], a grid of triangles of side length (L) is placed over the

image. The areas of all of the triangles is measured and summed to calculate the surface area

S(1) of the image. The grid size is decreased by a factor of 2 until L is equal to the distance

between two pixels. The slope of the log of S(L) versus the log of 1/L is D-2.

The variance method is based on the scale dependence of the variance of fractional

Brownian motion [59]. An image is divided into cubes and the variance, the power of the RMS

value of the heights, is calculated for each cube. The slope of the log-log plot of the variance is

equal to P and D is determined from the equation:

D = 3 (3-14)

For the power spectrum method [57, 58], every line profile that makes up the image is

Fourier transformed, the power spectrum is analyzed, and power spectra are averaged for all the

lines. The slope of the log-log graph of the power spectra is equal to P and D is determined from

the equation:


D = (3-15)

WSxM is another software program used for data acquisition and processing in scanning

probe microscopy [60]. WSxM uses a modified slit-island method, called flooding, to determine

the fractal dimension for AFM images. A plane parallel to the fracture surface is used to section

the fracture surface image into two halves, producing "islands" of the fracture surface. The log-

log graph of the perimeter versus the area for all islands is calculated and the slope of the graph

is 2/D. The number of islands measured by WSxM can be changed by increasing or decreasing

the size of the islands recognized by the program and the height of the sectioning plane.









To determine the optimal scan areas for determining the fractal dimension, a test was

performed measuring the change in fractal dimension as a function of scan area. The mirror,

mist, and hackle regions of the fracture surface of baria silica glass bars (30mm x 4mm x 3mm)

were Vickers indented at a 500 gf load, fractured in three-point flexure, and scanned at various

scan areas (Table 3-3). Baria silica glass was used because it has a similar D value to that of

SLS glass [D = 0. 10 + 0.01] [61] [3 9]. The values of D were then compared with the fractal

dimension of baria silica glass calculated using the slit island method [61]. Using the optimal

scan areas, the fractal dimension of the veneer, core, and the two sets of veneer/core bilayer bars

was measured and compared.

Statistical Analysis

All statistical analyses were conducted using KaleidaGraph, a graphing and data analysis

software package. ANOVA was used to determine if there was a statistical difference or

similarity between the means of the variables (aspect ratio, fracture strength, fracture toughness,

and fractal dimension) in the experiments. A significance level (a) of 0.05 was used in all

ANOVA tests. A p-value less than a (p < 0.05) denotes a statistically significant difference in

the means. A p-value greater than a (p > 0.05) indicates that the means are statistically the same.

All reported error values are a the standard deviation for those values.





2 1 mm


m


1.1 mm


2.1 mm


m


1.1 mm \


Figure 3-1. Cross-section of bilayer specimens. A) Bilayer before transformation. B) Bilayer
after transformation.


C > A


Figure 3-2. Diagram of distances used for computation of centroid for bilayer specimens.
































Table 3-1. Chemical composition of IPS e.max@ Press [Adapted from Buihler-Zemp P, Voilkel T
(2005) Scientific Documentation IPS e max@ Press: 1]
Standard Composition (wt. %)
Sio2 57.0 80.0
Li20 11.0 19.0
K20 0.0 13.0
P20s 0.0 11.0
ZrO2 0.0 8.0
ZnO 0.0 8.0
+ other oxides 0.0 10.0
+ coloring oxides 0.0 8.0


Table 3-2. Chemical composition of IPS e.max@ Ceram [Adapted from Bithler-Zemp P, Voilkel
T (2005) Scientific Documentation IPS e max@ Ceram:1]
Standard Composition (wt. %)
Sio2 60.0 -65.0
Al203 8.0 -12.0
Na20 6.0 9.0
K20 6.0 8.0
ZnO 2.0 3.0
+ CaO, P20s, F 2.0 6.0
+ other oxides 2.0 8.5
+ Pigments 0.1 1.5


Figure 3-3. Diagram of distances used for computation of moment of inertia.


< a d









Table 3-3. AFM scan areas used to determine optimum scan areas for fractal dimension
measurements of fracture surface of baria silica glass. X mark AFM scan areas used
to measure fractal dimension for a given region of the fracture surface.
Scan area, (Cm)2 Mirror Mist Hackle
60 X X X
40 X X X
20 X X X
10 X X X
5X X X
2X X X
lX X X
0.75 X X X
0.50 X X
0.30 X X
0.10 X









CHAPTER 4
RESULTS AND DISSCUSION

In this chapter, the microstructure for the dental materials is characterized. The fracture

strength, apparent fracture toughness, work of fracture, and fractal dimension are determined for

the monolithic dental materials and the bilayer composites. From these results, the hypotheses

discussed in Chapter 1 are answered.

Microstructural Analysis

Phase Identification

X-ray diffraction (XRD) was conducted for the IPS e.max@ Press (EC) glass-ceramic core

material and IPS e.max@ Ceram EV glassy veneer. XRD was preformed on the EC to determine

if lithium disilicate is the main crystal in the glass ceramic. XRD was performed on the EV to

determine if any crystal phases existed in the glass veneer.

The XRD data (Figure 4-1) showed that EC contains crystals. Peaks at diffraction angles

of 23.90 and 24.40 correspond with two of the three highest peaks for lithium disilicate (PDF 40-

0376; Li2Si20s). This data, along with the knowledge of the fabrication of EC, confirms that EC

is composed mainly of lithium disilicate crystals and is a glass-ceramics using the definition

stated in Chapter 2 for glass-ceramics.

The XRD data (Figure 4-2) shows the pattern for an amorphous structure. This implies

that EV may be a glass and does not contain crystals. SEM micrographs (Figure 4-3) of the

veneer showed what could be nanoparticles, with a number average diameter of approximately

100 nm, that exist in the veneer. It is possible that volume fraction of the particles in the glass

matrix is less than the detection limit for XRD and that this low volume fraction may be

responsible for the small peaks exhibited in the XRD spectra at 25.90, 32.00, and 34.00. These

particles have a spherical shape when compared to the needle-like crystals reported by Ivoclar










(Figure 4-4) [6]. The difference in shape may be due to a difference in heat treatment, which

would affect the growth of crystals. Work by Hoiland et al. has shown that temperature and time

does affect crystal growth for leucite glass-ceramic, with the length of the crystals being

proportional to time at elevated temperatures [62]. Energy dispersive X-ray spectroscopy (EDS)

was also conducted on the veneer specimens to determine the composition of the particles

(Figure 4-5). There was a smaller amount of fluorine (~ 1 wt. %) and an equivalent amount of

calcium (~ 3 wt. %) present in the spectra compared to the amount reported by the manufacturer

(Table 3-2). Based on the EDS and XRD data, it is inconclusive if the particles present in the

veneer specimens are the nanofluorapitite crystals described by Ivoclar [6]. These findings are in

agreement with the work done by Tsalouchou et al. [63] in which the nanofluorapitite crystals

could not be identified for e.max@ Ceram. More analysis, e.g., determine the lattice planes of

the particles from TEM or determining the enthalpy of melting of the particles from DSC or

TGA, is needed to determine the identity of the particle observed in SEM images.

Aspect Ratio

The aspect ratio for the lithium disilicate crystals in the EC glass-ceramic was measured

from SEM micrographs and compared to the aspect ratio of EC report by Holand [2] and Ivoclar

[6]. Using the original etching procedure of 2.5 % HF for 10 s did not etch the glass-ceramic

well enough to measure the aspect ratio (Figure 4-6). To determine the best method to etch the

glass-ceramic, a 2.5 % HF solution was used to etch one polished specimen for 30 s and a 30 %

sulfuric acid (H2SO4)/4 % HF solution [2] was used to etch another polished specimen for 10 s.

SEM micrographs were taken of both specimens at 5000x (Figure 4-7) and 10000x (Figure

4-7). It was determined that the H2SO4/HF solution provided the best etching to measure the

aspect ratio. The aspect ratio for the lithium disilicate crystals was 7.6 + 1.1. The number









average length for the crystals was 5.2 & 2.0 Clm and the number average width was 0.7 & 0.3

lm. This is in agreement with the crystal lengths of 3 6 Clm reported by Hoiland [2].

Fracture Strength

The flexural strength was determined for EC, EV, the veneer indented bilayer (BV), and

the core indented bilayer (BC) to characterize and compare the dental materials and to calculate

the apparent fracture toughness. In order to calculate the flexural strength for the bilayers,

composite beam theory was used to transform the bilayers because of a difference in elastic

modulus between the two materials. Based on the elastic moduli of the two ceramics (96 GPa

for EC [52] and 64 GPa for EV [52]), the transformation factor was calculated to be 1.5. This

transformation factor results in a virtual change in the width of EV from 4.6 mm to 3.1 mm

(Figure 3-1). Using Equation 3-2 and Figure 3-2, y was determined for the BV (~ 1.14 mm) and

BC (~ 0.94 mm).

Using Equation 3-3 and Figure 3-3, the average moment of inertia was calculated to be

2.7 mm4. Using the calculated moment of inertia, Equation 3-4 and 3-5, a was calculated for

each bilayer (Table 4-1).

The measured flexural strength of EC is greater than that of EV. This is expected for the

EC due to the presence of the needle-like lithium disilicate crystals that increases the strength of

the materials due to crack deflection. It is expected that the fracture strength of both bilayers will

have a flexural strength greater than EV and less than EC based on the Rule of Mixtures [64].

Using the Rule of Mixtures, the fracture strength of the bilayer (oB) WAS determined from the

equation

cr, = crEVAEV + ECAEC (4-1)










where OEV is the fracture strength of the veneer, AEV is the area fraction of the veneer in the

bilayer, oEC is the fracture strength of the core, and AEC is the area fraction of the core. Using

Equation 4-1 implies that the strength of each layer can be superimposed to determine the

strength of the composite beam. For the bilayer, oB = 94 MPa. This shows that the fracture

strength of the bilayers should be between the values for the fracture strength of the veneer and

core. Table 4-1 shows that the fracture strength for BV and BC does lie between the fracture

strength of the veneer and core. Therefore, this implies that the location of the crack determines

the fracture strength of the bilayer.

Fracture Toughness

The apparent fracture toughness (Kc) was determined for EC, EV, and the two sets of

bilayers (BV and BC) to compare to measured values reported in the literature. The apparent

fracture toughness was determined using two methods: strength indentation and fractography.

From Equation 3-6 and 3-7, Kc was calculated for each method and displayed in Table 4-1. For

Equation 3-7, the elastic modulus and hardness used was 64 GPa [3] and 5.4 GPa [6] for EV and

96 GPa [3] and 5.5 GPa [6] for EC. Both sets of Kc values for EC agree with the values reported

by Ivoclar [6] and are statistically similar (p > 0.05). Both sets of the apparent fracture

toughness values for EV are not statistically the same (p < 0.05) but are within the range of the

Kc value reported by Taskonak [4]. The difference in the mean of the apparent fracture

toughness for the bilayers was not statistically significant (p > 0.05). Kc for the bilayers was

greater than Kc for the veneer monolithic specimens and less than that for the core monolithic

specimens. The implication of these results is that the apparent fracture toughness of a bilayer is

dependent on the location of crack initiation in the bilayer. Therefore, if the initiating crack is in

the veneer, then the fracture toughness of the veneer controls the strength and toughness of the

composite.









Work of Fracture


Soda-Lime-Silica Glass

The work of fracture (YwOF) WAS determined for EC, EV, and the two sets of bilayers to

determine if the fracture energy could be estimated for the dental ceramics. The work of fracture

was calculated in two different ways: using the cross-sectional area TwOF(CS) (Equation 2-1) and

using the actual fracture surface area TwOF(F) (Equation 3-10). In order to test the accuracy of

using Equation 3-10 to estimate the fracture energy of a brittle material, TwOF WAS determined for

soda-lime-silica (SLS) glass for unstable and stable crack growth.

An atomic force microscope (Dimension 3100, Veeco Instruments Inc., Woodbury, NY)

was used to determine the actual fracture surface area of the SLS glass beams. Using the AFM

software, the surface area was determined from a grid of triangles of a fixed side length, equal to

the distance between pixels, placed over the scan image. The area of all the triangles was

measured and all of the areas were added together to calculate the surface area for the image.

Five square scan areas ((1 Cpm)2, (750 nm)2, (500 nm)2, (375 nm)2, and (250 nm)2) were used to

measure the fracture surface area for the CN specimens while four [(500 nm)2, (150 nm)2, (50

nm)2, and (25 nm)2] were used to measure the unstable fracture specimens. All scans were

conducted using a tapping tip at a scan rate of 3-4 Hz and at a resolution of 512 lines per scan.

Equation 3-13 was used to determine the fracture surface area. Examples of the AFM scan

images of the mirror, mist, and hackle region are illustrated in Figure 4-10.

The fracture surface area versus the scan area was graphed for the stable crack growth

chevron-notched (CN) (Figure 4-11), and the unstable indented specimens (Figure 4-13), to

determine if a relationship existed between the two variables. A number of curve fits were

applied to all the data. The best fit was to a power law relationship, i.e.,










AF =A 9- (4-2)

where rl is the scan area side length, and A and B are constants, existed between the scan area

and AF. Equation (4-2) is similar to the profile length measurement equation (Equation 2-3) and

the direct area measurement equation (Equation A-8). It is assumed here that fracture surfaces

are fractal in nature, in agreement with many studies in the literature [38-40, 42-47]. Therefore,

Equation 4-3 would be expected to be similar to Equations 2-3 and A-8, since the fracture

surfaces are fractal. Based on this, the fracture surface area should increase as the scan area side

length decreases.

In order to determine the actual fracture surface for a specimen, a maximum AF must be

determined for that specimen. Since there is no mathematical limit for Equation 4-2, a physical

limit is necessary based on some physical restraints. There are three possible solutions for the

size limitation: probe tip radius, equilibrium bond length, and the free volume diameter.

The physical limit for determining AF is based on the AFM probe tip radius. The AFM can

only resolve features equal to or greater than the tip radius. For the tapping tip used in these

scans, the tip radius was 8-10 nm. Using the tip radius of 8 nm as rl results in an AF Of 6.3 f 0.3

mm2 (Figure 4-12).

Theoretically, a minimum limit could be the equilibrium bond length for the materials. For

the soda-lime-silica glass, which is composed mostly of silica, the equilibrium bond length

selected was that of Si-O, which is 0.16 nm [65]. Using the equilibrium bond length as rl

generates an AF Of 7. 1 f 0.5 mm2 (Figure 4-12).

Another limit is based on the concept of free volume. Inorganic glasses have open space

due to the disorder in the glassy structures. These open spaces are known as free volume. Swiler

et al [66] suggested from molecular dynamics calculations that it was these free volume areas









that controlled fracture in silica glass. The estimated size of the region was represented by a

radius, ao. West et al. [44] also estimated this region using a different approach. ao was

determined to be approximately 1.2 nm. The proj section of AF based on AFM measurements at a

rl of ao is 6.7 + 0.4 mm2 (Figure 4-12). Using these three limits, Figure 4-12 shows that the

fracture surface area is does increase as the side length of the scan area decreases and that the

fracture surface area is larger than the cross-sectional area.

Using the three previously mentioned limits, yWOF WAS calculated for the CN specimens.

From Table 4-2, all the values are less than the value for TwOF meaSured using Acs (Equation 2-1)

and for the fracture energy measured from traditional fracture mechanics tests [y = 3.5 J/m2]

[67]. The selection of the limit does not greatly affect the results. There is only a 13 % change

for YwOF between using the tip radius and the equilibrium bond length. Thus, our choice of limits

is not critical to determining YWOF, juSt that there is a cutoff length of atomic dimensions.

Using the equilibrium bond length as the limit for the scan area side length, AF WAS

determined for the unstable crack growth specimens (Table 4-3) by extrapolating to the

equilibrium bond length. The values for TwOF(AF) are Similar to the theoretical fracture energy (y)

for fused silica of 1.75 J/m2 calculated by Charles [65] based on the equilibrium bond length of

silica and less than the fracture energy measured from traditional fracture mechanics test [y = 3.5

J/m2] [67]. The results of both experiments confirmed that the actual fracture surface area

provides a lower limit of the work of fracture and that the fracture energy required to break

bonds can be estimated using the work of fracture.

To obtain an accurate value for the fracture energy, the entire fracture surface generated

during crack growth should be measured or estimated. This further implies that for any fracture

surface generated during fracture, the actual fracture surface area should be measured or









estimated to obtain a more accurate value of the energy required to break atomic bonds. The

experiments also show that because of the lower limit, using traditional fracture mechanics tests

to calculate y do not represent the minimum energy needed to rupture an atomic bond and create

the fracture surface.

Dental Materials

After showing that TWOF(F) can be used to estimate the fracture energy of SLS glass, the

fracture surfaces of EC, EV, and the two sets of bilayers were measured using the AFM to

determine if the fracture energy could be estimated for these materials. The crack introduced in

the veneer and core monolithic specimens, from Vickers indentation, resulted in a fracture

surface with no mist or hackle present on the surface. Therefore, only the mirror portion of

Equation 3-13 was necessary to calculate AF for the specimens and N(SS) is equal to the cross-

sectional area of the bars.

Using the same method to determine TwOF(AF) in the SLS glass specimens, the scan areas

were graphed versus the fracture surface area for EV (Figure 4-14) and EC (Figure 4-15) over

scan areas of (250 nm)2 to (25 nm)2. Since the veneer and the core consisted mainly of silica, the

equilibrium bond length of silica [0. 16 nm] was used in determining the minimum scan area.

Based on the scan areas used, YWOF(F) for EV was 5.2 f 1.4 J/m2 and 33 A 17 J/m2 for EC.

Although these two values are significantly greater than the theoretical value for silica of 1.75

J/m2, the values are less than the values reported for yWOF(CS) (Table 4-4). The values for TwOF(F)

may be due to the small sampling of fracture surface area (AF) determined from the AFM data.

On further analysis, AF for the veneer and core appears to have a different level of roughness on

a scale of hundreds of micrometers. To see if the roughness increases AF Of the veneer and core,

the fracture surface of the two materials was measured using an optical profilometer (WYKO

NT1000, Veeco Metrology Group, Tucson, AZ). The fracture surface was examined at two









different scan areas: 0.270 mm2 (Obj ective lens of 5x and field of view lens of 2x) and 0.503

mm2 (Obj ective lens of 5x and field of view lens of 1.5x). The scan area of 0.270 mm2 TOSulted

in AF being 13 % greater than the cross-sectional area (Acs) measured for the veneer and 28 %

greater for the core. For the scan area of 0.503 mm2, there was a 10 % increase in AF for the

veneer and a 21 % increase in AF for the core. Using the data from the large scan area and

applying this information to the data acquired from the AFM, TwOF(F) decreased to 2.9 & 0.8 J/m2

for the veneer and 26. 1 + 13.4 J/m2 for the core.

Although there was a decrease in TWOF(F), it is still greater than that of the theoretical value

for silica. One likely cause for the larger TwOF(F) ValUeS is the speed of the crack for the unstable

crack growth specimens. The speed of the crack does not allow the crack to follow the path of

lowest energy. This results in more energy being supplied than what is required to fracture a

material when compared to the fracture surface area created during fracture, i.e.,

dU
- > (4-1)
dA

where U is the elastic energy stored in system and A is the fracture surface area [24]. But, since

the crack velocity was not measured for the specimens, it is inconclusive if the crack velocity

affects the fracture surface area.

After TwOF for the monolithic dental ceramics, TWOF for the veneer-indented (BV) and core-

indented bilayer (BC) was calculated using the same method as for the monolithic materials.

The scan area side length versus AF WAS plotted for BV (Figure 4-16) and for BC (Figure 4-13).

For BV, TWOF(F) iS 5.6 f 1.4 J/m2 and for BC it is 25 & 6 J/m2. These values follow the trend of

the values for the fracture strength and apparent fracture toughness, with TWOF for BV being

greater than that for EV and TWOF Of BC being less than that for EC. Although optical

profilometry was not used in determining AF for the bilayers, TwOF for BV and BC would









decrease based on the results from the measurement of AF On the monoliths. Based on the values

for TwOF(F) for the bilayers, it was shown that YWOF(F) iS Similar to that of the material in which the

initial crack that caused fracture of the bilayer is located. For fracture in the bilayer, in which

fracture initiated in the tougher material (core), TwOF(F) iS Similar to that of the tougher monolith

(core). For fracture in the bilayer, in which fracture initiated in the less tough material (veneer),

TwOF(F) iS Similar to that of the less tough monolith (veneer). These results based on YWOF(F)

further indicate that selection of both materials in a bilayer is important when designing ceramic

composites.

Fractal Dimension

Soda-Lime-Silica Glass

The fractal dimension (D) of e.max@ Press (EC), e.max@ Ceram (EV), and the two set of

bilayers of EC/EV were measured using an AFM and the two software packages of Gwyddion

and WSxM to determine if location of crack initiation changes D for the veneer and core in the

bilayers. To determine the accuracy of D measurements using these two programs, D was

measured for soda-lime-silica (SLS) glass bars. Three SLS bars (45mm x 4mm x 3 mm) were

indented with a Vickers indent load of 2 kgf and fractured under four-point flexure at a crosshead

speed of 0.5 mm/min. The mirror, mist, and hackle regions of the fracture surface were scanned

using 6 scan areas [(100 nm)2, (250 nm)2, (375 nm)2, (500 nm)2, (750 nm)2, and (1Cpm)2]

Table 4-5 shows the fractal dimension increment (D*) values calculated from the two

programs. For the Gwyddion software, only the triangulation method and the cube counting

method were used to measure D*. The variance and power spectrum methods were not used

because there was too much scatter in the data points used to determine the slopes, producing

inconsistency in the D* values calculated for these methods. The difference in mean values

between the Gwyddion and WSxM software were not statistically significantly for the hackle










region (p > 0.05). For Gwyddion, the triangulation and cube counting were significantly similar

for the mist and hackle region but not for the mirror region (Table 4-5).

The mist and hackle values agree with a D* of 0. 10 + 0.01 for glasses [61, 68] determined

using the slit island method. Thompson et al. [69]showed that for a lithia-disilicate ceramic,

determining D* from the slit-island method and AFM results in the same value of 0.23. Both the

Gwyddion and WSxM software showed an increase in D* for the mirror region when compared

to D* for the mist and hackle regions. Kulawansa [38] had conducted work on measuring D of

SLS glass near the mirror-mist boundary. His work determined that D* was 0.17 & 0.08 with

some values in the rougher regions as high as 0.40. If the fractal dimensions of the three regions

measured using the Gwyddion and WSxM software are averaged, the values for the Gwyddion

software [Triangulation 0. 18 & 0. 10; cube counting 0. 17 & 0.09] agree with Kulawansa's

measured value for D* while WSxM [0.30 + 0.25] does not.

Although WSxM produces a greater D* value for the mirror region, the key point is that

the fractal dimension of the mirror region is greater than the fractal dimension of the mist or

hackle region. This may be due to the roughness in the mirror region being less than that in the

mist or hackle region. Wiederhorn et al. [70] showed that for the mirror region of SLS glass, the

root-mean-square (RMS) roughness decreased as the crack velocity increased. The roughness

exponent, which is related to the fractal dimension, also increased as the RMS roughness

decreased for SLS glass. These results imply that the increase in the D* for the mirror region is

not due to any algorithm errors in Gwyddion or WSxM, but rather due to a change in the

material itself.

To further examine if the greater D* values for the mirror region measured using the

software program is due to the algorithms used in the programs, five line profiles from five AFM









scans [(1Cpm)2, (750 nm)2, (500 nm)2, (375 nm)2, and (250 nm)2] for the mirror region of SLS

glass were used to determine D*. The Richardson's method was used to measure D* by

measuring the length of each profile as a function of step size (50, 40, 30, 20, and 10 mm). D*

from the profiles was 0.35 & 0.08. This value is similar to the values measured using the

triangulation and cube counting method. This shows that D* determined from the algorithms

used to calculate D* in Gwyddion do not result in a significant change in the D. Based on this, it

was determined that the increase in the D* values for the mirror region is not caused by the

software programs used to calculated D*.

Along with greater value for D* for the mirror region calculated from the WSxM software,

there are several other problems with using WSxM. The number of variables that are used in

calculating D*, e.g., the height of the sectioning plane and the minimum island area, results in an

inconsistency in the calculation of D*. The island area and number of islands that is necessary to

accurately calculate D* has not been determined for WSxM. Based on this, WSxM was not used

as a tool to measure the fractal dimension of the brittle materials in this study. All fractal

dimension values reported from this point on were determined from the Gwyddion software

program.

Baria Silica Glass

Using Gwyddion as the software to measure D*, the effect of AFM scan areas on D* was

tested. Baria silica glass (3BaO-5 SiO2) bars (25mm x 4 mm x 2 mm) were used in this

experiment since baria silica glass has the same fractal dimension as other glasses [0. 10 + 0.01]

[61]. The baria silica bars were fractured in three-point flexure, at a crosshead speed of 0.5

mm/min., using a Vickers indent of 500 gf to initiate a crack. The fractal dimension was

determined for the fracture surface of the mirror, mist, and hackle regions at different scan areas









from (100 nm)2 to (60 Cpm)2 and compared to D* calculated from the slit-island technique [0.10 +

0.01] [61].

Figure 4-17 confirms that D* does change when the scan area change. The fractal

dimension increment decreases as scan area decreases for the mirror, mist, and hackle region. As

the scan area increases, the time is takes for the tip to scan a line increases causing a decrease in

the time the tip has to measure the features of the fracture surface. This results in a decrease in

resolution and a decrease in the measurement of the surface area. The measurement of the

surface area using AFM is also limited by the tip radius of the probe. The probe can only resolve

features that are larger than the tip radius. Based on this data, scan areas less or equal to (750

nm)2 for the mirror region, less than (2Cpm)2 for the mist region, and between (1 Cpm)2 and (20

Cpm)2 for the hackle region would be the ideal scan areas used to measure the fracture surface for

glasses using AFM.

Dental Materials

With the limits of the AFM determined for D* measurements, D* was measured for the

e.max@ Press (EC) and e.max@ Ceram (EV) monolithic bars as a control group in order to

establish if a change in D* exist in the bilayer bars. The fracture surfaces (Figure 4-18) of the

three bars each of EC and EV were measured at 250, 375, 500, 625, and 750 (nm)2 to determine

D*. Table 4-6 shows the average D* for the dental core and veneer. D* values for the veneer

are similar to that of SLS and baria silica glass. This is expected for the veneer since it is mostly

glass, based on the previous XRD analysis, and the fracture surface is all mirror. Therefore, the

veneer should have a fractal dimension similar to that of the mirror region of other silicate-based

glasses. The fractal dimension for the core was less than D* values reported for other lithia-

disilicates [0.25 [71], 0.24 [69]]. This difference in D* may be caused by a difference in the

microstructure of the lithia-disilicates, a difference in the scale at which D* was calculated for









the lithia-disilicates compare to the scale used in this experiment, or that the fracture surface of

the core is mostly mirror for EC and different from the fracture surface of the other glass-

ceramics. Thompson [69] measured D* for fully crystallized lithia-disilicate. The amount of

crystallinity may have resulted in the greater value in D*. Naman [72] measured the fractal

dimension of lithia-disilicate glass and lithia disilicate glass-ceramic at 10, 33, and 95 vol. %

crystallinity using the slit-island method. Naman showed that for the mirror region, D* was 0.03

in the glass, 0.05 in the 10 vol. % glass-ceramic, 0.09 in the 33 vol. % glass-ceramic, and 0.16 in

the 95 vol. % glass-ceramic. The value for the 33 vol. % glass-ceramic agree with the values for

the core (EC). Therefore, the difference in the D* measurements is mostly like due to the

fracture surface of the core (EC), which has a fracture surface that is mostly mirror. Future

investigations should be conducted to determine the cause of the fractal dimension values for the

ceramics in the mirror region. One direction to explore for future research is the effect that the

crack velocity has on the fractal dimension.

There was no significant difference (p > 0.05) for either the cube counting or triangulation

values at the different scan areas. This means that D* was not affected for the range of scan

areas used. The cube counting and triangulation values for EV and EC were not significantly

similar (p < 0.05) to each other though. As Table 4-4 shows, D* from triangulation is greater

than the cube counting values. This is because there is a better linear fit to the data used to

calculate D* for cube counting than triangulation resulting in a lower but more accurate

calculation of D* from cube counting.

With the D* values determined for the monolithic materials, D* was calculated for the

bilayers to determine if D* changes in the material in which the initial crack is not located. The

fracture surfaces of the three veneer-indented bilayer bars (BV) and three core-indented bilayer









bars (BC) was measured at 250, 375, 500, 625, and 750 (nm)2 to determine D*. Table 4-7 shows

the average D* for BV and BC. The difference in the means of triangulation and cube counting

D* values for the veneer and core materials in the BV and BC bilayers were statistically different

(p < 0.05). As previously mentioned, this is due to how D* is calculated for the triangulation and

cube counting methods. For BV and BC bilayers, D* for the veneer and core were statistically

similar (p > 0.05) to D* for the monolithic veneer and core. As mentioned previously, D* for the

core was different from D* determined from other researchers. Although the D* values were

different, the key point is that D* is the same for the core materials in the bilayers and in the

monolith.

To determine if the crack propagation was continuous or if a new crack initiated and

propagated in the non-indented materials in the bilayer, SEM was used to examine the interface

of the bilayer. Figure 4-19 shows examples of the fracture surface of the veneer-indented bilayer

(BV). Figure 4-19B indicates that the wake hackle marks generated by the pores in the veneer

layer, which are due to processing, travel through the interface into the core. This observation

confirms that the materials in the bilayer were bonded well since there was no indication of

delamination or secondary crack initiation at the interface. Thus, the fracture process was

continuous from the veneer, where the indented induced flaw is located on the tensile surface,

into the core material. Figure 4-20 shows examples of the fracture surface of the core-indented

bilayer (BC). Figure 4-20B shows twist hackle marks present in the veneer at the interface. The

presence and direction of these twist hackle marks indicate that the crack propagated

continuously through the bilayer, starting in the tensile surface of the core-indented layer and

propagating across the interface into the veneer.









Based on the D* information for the bilayers, the location of crack initiation and

propagation does not change the fractal dimension of either of the materials in the bilayer when

compared to their monolithic counterparts. This means that the fracture surface of the materials

did not change based on the location of initial crack initiation. This confirms that for the ceramic

bilayer specimens tested in flexure, the fracture process does not change the fractal dimension of

the material in which the initial indented-induced flaw is not located. The fact that the fracture

surface of the ceramic materials does not change supports the idea that the fractal dimension is

an intrinsic property of the material. With the fractal dimension being shown to be a property of

the material and not determined by the fracture process, the fractal dimension can be used as

another property to describe fracture in the ceramic materials.









Table 4-1. Fracture strength, fracture toughness determined from strength indentation (SI), and
fracture toughness determined from fractography (F) for e.max@ Ceram (EV), e.max@ Press
(EC), bilayer with crack initiation in veneer (BV), and bilayer with a flaw produced in the core
(B C)
Specimen No. of o, MPa Kc (SI), MPa-ml/ Kc, (F),
Specimens MPa-ml/2
EV 5 49 & 3 0.83 & 0.04 0.74 & 0.07
EC 5 155 & 23 2.46 & 0.27 2.25 & 0.34
BV 5 67 & 11 1.14 & 0.07 1.21 & 0.14
BC 4 142 & 15 2.30 + 0.18 2.25 & 0.18


Table 4-2. Work of fracture values calculated for chevron notch specimens using cross-sectional
area (ACS) and actual fracture surface area based on probe tip radius, equilibrium
bond length, and free volume ao. Actual area is projected area using equilibrium
bond length.
Cross-sectional Actual area TwOF (fOr Acs) TWOF (Tip YWOF (Bond yWOF (ao)
area (mm2) (mm2) (/m2) Radius) Length) (J/m2)
(J/m2) (/m2)
5.4 & 0.3 7.1 & 0.5 3.5 & 0.1 3.0 + 0.1 2.7 & 0.2 2.6 & 0.2


Table 4-3. Comparison of work of fracture and fracture toughness, based on work of fracture,
for unstable crack growth soda-lime-silica glass specimens
No. of Actual Area W, mJ YWOF (Acs), TWOF, Kc,
specimens (mm2) J/m2 J/m2 MPa-ml/2
3 mm 7 860 & 76 2.7 & 0.6 113 & 30 1.6 & 0.4 0.46 & 0.06
10 mm 1000 gf 7 4002 & 161 8.0 & 2.0 74 112 0.9 & 0.1 0.35 & 0.02
10 mm 2000 gf 8 4438 & 98 6.5 & 0.8 57 & 7 0.8 & 0.1 0.32 & 0.02


Table 4-4. Work of fracture TwOF, based on cross-sectional area (Acs) and fracture surface area
(AF) foT Veneer, core, and bilayer bars
Specimen No. of W, mJ YWOF (Acs), TWOF (AF), YWOF (Adj. AF),
specimens J/m2 J/m2 J/m2
EC 3 3.9 & 1.2 212 & 63 33 A 17 26.1 + 13.4
BC 3 3.6 & 0.7 190 & 37 25 & 6
EV 3 0.6 & 0.1 31 & 4 5.2 & 1.4 2.9 & 0.8
BV 3 0.9 & 0.3 49 & 16 5.6 & 1.4










Table 4-5. Fractal dimensional increment for soda-lime-silica glass fracture surface regions
measured using atomic force microscopy and the Gwyddion and WSxM software
Fracture Surface Region Gwyddion Gwyddion WSxM
Triangulation Cube counting
Mirror 0.29 & 0.05 0.26 & 0.05 0.60 + 0.19
Mist 0.13 & 0.07 0.12 & 0.06 0. 19 & 0. 12
Hackle 0. 12 & 0.06 0.11 & 0.06 0.10 + 0.07




Table 4-6. Fractal dimensional increment for e.max@ Ceram (EV) and the e.max@ Press (EC)
ceramics measured using AFM and Gwyddion software
Specimen Triangulation Cube Counting
EV 0.27 & 0. 11 0.22 & 0.09
EC 0.12 & 0.04 0.09 & 0.03


Table 4-7. Fractal dimensional increment for bilayer specimens using AFM and Gwyddion
Specimen Triangulation Cube Counting
Veneer Indented (BV)
EV 0.29 & 0.07 0.22 & 0.06
EC 0.12 & 0.05 0. 10 + 0.04
Core Indented (BC)
EV 0.31 & 0.10 0.24 & 0.08
EC 0.12 & 0.03 0.09 & 0.03











40000
23.9
35000 124.4

30000


25000


c 20000


15000






500

10 20 30 40 50 60

28


Figure 4-1. X-ray diffraction pattern for e.max@ Press (EC) core ceramic showing that lithium
disilicate are the crystals present in the glass-ceramic. Peaks at diffraction angles of
23.90 and 24.40 correspond with two of the largest peak intensity values for lithium
disilicate (PDF 40-0376).













25.9


32.0


34.0


4500



4000


3500



3000


2500


2000


1500


20 30 40 50


Figure 4-2. X-ray diffraction pattern of e.max@ Ceram (EV) veneer ceramic showing veneer
material has a amorphous pattern. Peaks in pattern at diffraction angles of 25.90,
32.00, and 34.00 could represent peaks for calcium fluoride phosphate (PDF 15-
0876).
























































Figure 4-3. SEM micrographs of polished and etched surface of e.max Ceram (EV) veneer
ceramic. Surface etched using 2.5 % HF solution for 10s. A) Magnifieation of
surface at 500x. B) Magnifieation of box in Figure 4-3A at 5000x. C) Magnifieation
of box in Figure 4-3B, at 30000x, showing presence of nanoparticles.

































Figure 4-3. Continued


Figure 4-4. SEM image of polished and etched surface of e.max Ceram (EC) core ceramic
showing the presence of needle-like nanofluorapatite crystals. Crystals are
approximately 300 nm in length and 100 nm in diameter. [Image courtesy of Ivoclar
Vivadent.]











Counts


5
Ene rgy (keV


Figure 4-5. Energy dispersive X-ray spectroscopy (EDS) of e.max@ Ceram (EV) veneer
ceramic. Amount of Ca (~3 wt. %) present agrees with composition of veneer.
Amount of F (~1 wt. %) present does not agree with composition of veneer.


Figure 4-6. SEM micrograph of polished and etched surface e.max@ Press (EC) core ceramic at
a magnification of 10000x. Surface etched using 2.5 % HF solution, which did not
provided enough contrast to measure the aspect ratio of the crystals.

































Figure 4-7. SEM micrograph of polished and etched surface e.max@ Press (EC) core ceramic at
a magnification of 5000x. A) Surface was etched using 2.5 % HF solution for 30s at
room temperature and did not provide enough contrast to measure the aspect ration of
the crystals. B) Surface etched using 30 % H2SO4/4 % HF solution for 10 s at room
temperature. Etchant provided enough contrast to measure the aspect ratio of the
lithium disilicate crystals.



































Figure 4-7. Continued

































Figure 4-8. SEM micrograph of polished and etched surface e.max@ Press (EC) core ceramic at
a magnification of 10000x. A) Surface was etched using 2.5 % HF solution for 30s at
room temperature and did not provide enough contrast to measure the aspect ration of
the crystals. B) Surface etched using 30 % H2SO4/4 % HF solution for 10 s at room
tempertaure. Etchant provided enough contrast to measure the aspect ratio of the
lithium disilicate crystals.


































Figure 4-8. Continued


Figure 4-9. SEM image of fracture surface of chevron-notched soda-lime-silica flexure
specimen showing that fracture surface is all mirror and no mist or hackle is present.
[Image courtesy of Dr. Jia Hua Yan].































I~m III II II/ 4 115 III oi IIIm 0:1 0:2 0:3 0:4 0:5 U:I
00 ~ ~~~~~~~~ rll IIII "" ol~,r~ T~~bi is








-13n E F

Fiur 4-0 F egtadapiue cniaeo oalm iic ls rcuesraea

a3 sca are ofI (75 nm2 AF mgsso ta h ognssoh ufc
increases~~~~~~~~ ~ ~~~~~~~ frmtemro eint h ake ein n )Mro eini
rog nannoee cl.C n )Ms ego.EadF Hcl ein















-Ap = 6.6-0.029


5.8


5.5.4 -;i


hi
E
E
LL


5.2


5
200


400


600


800


1000


Scan Area Side Length (r), nmn

Figure 4-11. Fracture surface area (AF) aS a function of scan area side length (rl) for chevron-
notched soda-lime-silica bars. AF is shown to increase confirming that AF is greater
than cross-sectional area. Circles represent average AF frOm five scans at that
particular scan area. Error bars represent standard deviation for AF ValUeS.












EBL FV
7T


LI.


0.1


100


1000


Scan Area Side Length (r), nm


Figure 4-12. Projection of the average fracture surface area (AF) fTOm the average AF ValUeS foT
the chevron-notched (CN) specimens based on AFM data. Using the limits of the equilibrium
bond length (EBL), free volume (FV), and tip radius (TR) result in an increase in the AF, with
EBL producing the largest AF Value. CiTCleS represent average AF Of all the CN specimens.
Error bars represent standard deviation for average AF ValUeS.











250
-e- 3 mm
v-- 10 mm- 1000 gf
200 -o -10 mm 2000 gf


--A = 16001-06

-----AF =14001-0.62

-AF = 3001-.1


150



100


20 40 60 80 100 120 140 160

Scan Area Side Length, nm


Figure 4-13. Fracture surface area (AF) aS a function of scan area side length (l). AF is shown to
increase confirming that AF is greater than cross-sectional area. Data points represent
average AF each specimen in at a particular scan area. Error bars represent standard
deviation for AF ValUeS.














22 AF,~ 4 mm' AF= 3611-0.25
Adj. A mm2
20 1-0.25

18

E 16

14

12 (

10


0 50 100 150 200 250 300
Scan Area Side Length (r), nm

Figure 4-14. Fracture surface area (AF) and adjusted fracture surface area (Adj. AF) aS a function
of scan area side length (rl) for e.max@ Ceram (EV) veneer bars. Adjusted surface are
determined from optical profilometry. Data points represent average AF each
specimen in at a particular scan area. Error bars represent standard deviation for AF
values.













-AF = 41-0.26


LI.


O 50


100 150 200 250 300


Scan Area Side Length, nm


Figure 4-15. Fracture surface area (AF) aS a function of scan area side length (rl) for e.max~
Press (EC) core bars. Data points represent average AF each specimen in at a
particular scan area. Error bars represent standard deviation for AF ValUeS.













F


LI.


O 50 100 150 200 250 300
Scan Area Side Length (r), nm

Figure 4-16. Fracture surface area (AF) aS a function of scan area side length (rl) for veneer-
indent (BV) and core-indented (BC) bilayers. Data points represent average AF each
specimen in at a particular scan area. Error bars represent standard deviation for AF
values.


10 B













0.4 I I






0.2 -'1


0.1




0.001 0.01 0.1 1 10 100 1000 104

Scan Area, p~m2
Figure 4-17. Fractal dimension increment (D*) of baria silica glass, plotted against scan area,
determined from Gwyddion for mirror, mist, and hackle region. The horizontal line
represents D* of 0. 1 determined from slit island method.













II 01 Od


9 0nm


21 0V


02 03 0.


32 4 VB












_-167VD


07
-7~ 5 nm A












-163nm C


10pm 01 02 03 0


05 06 07


301-m 0


15 06


Figure 4-18. AFM scan image of e.max@ Ceram (EV) veneer ceramic and e.max@ Press (EC)
core ceramic fracture surface at a scan area of (750 nm)2. COlor scale bar represents

minimum and maximum heights in images. A and C are height images. B and D are

amplitude images. Veneer has a fracture surface similar to soda-lime-silica. Core has

a greater roughness than veneer at same scan area.



























































Figure 4-19. SEM images of veneer-indented bilayer (BV) fracture surface. A) Low
magnification. B) Wake hackle marks traveling from pore in veneer into core show
that fracture of bilayer was a continuous process. White arrow indicates crack
direction.











































B













Figure 4.20. SEM images of core-indented bilayer (BC) fracture surface. A) Low
magnification. B) Twist hackle marks indicate that fracture process was continuous.
White arrows indicate crack direction.









CHAPTER 5
CONCLUSIONS

The obj ectives of this proj ect were to 1) test the hypothesis that the work of fracture

method, using the actual fracture surface area, can be used to estimate the fracture energy of

ceramic materials; 2) test the hypothesis that the work of fracture of a ceramic bilayer will be

the same regardless of the location of crack initiation; and 3) test the hypothesis that the fractal

dimension of the two materials in a bilayer will be the same regardless of the location of crack

initiation.

The first hypothesis was shown to be true. For soda-lime-silica glass, the work of

fracture method using the actual fracture surface area resulted in a decrease in the work of

fracture when compared to using the cross-sectional area. The work of fracture, using the actual

fracture surface area, was shown to be less than the fracture energy determined from traditional

fracture mechanics tests and similar to the theoretical fracture energy for silica. The fracture

energy determined from traditional fracture mechanics tests do not accurately measure the

fracture energy generated during the fracture process because the fracture surface generated

during fracture is not accounted for in these tests. This is the first time it has been

experimentally demonstrated that the actual fracture surface area is critical to determining the

minimum fracture energy. The work of fracture can be used to estimate the minimum fracture

energy required to break the bonds in brittle materials. Therefore, the work of fracture can be

used to determine the energy contributed in the creation of fracture surfaces in ceramics and will

help engineers better understand the contributions to the total energy used to fracture a ceramic

material.

The second hypothesis was shown to be false. The work of fracture values for the bilayer

specimens were similar to those of the monoliths in which the crack was initiated. This










agreement implies that the work of fracture is dependent on the material in which the initial flaw

that causes fracture is located in a bilayer structure. The core, which had a greater measured

work of fracture than the veneer, resulted in a greater work of fracture in the core-indented

bilayer when compared to the veneer-indented bilayer. Based on this information, the work of

fracture should be used as another method in the mechanical characterization when determining

the proper material for selecting and designing bilayer structures.

The third hypothesis was shown to be false. The fractal dimension of the veneer in the

core-indented bilayer was not different from monolithic veneer. The fractal dimension of the

core in the veneer-indented bilayer was not different from the monolithic core. This indicates

that the fractal dimension of the materials in the bilayer did not change based on the site of crack

initiation and propagation and that the fracture surface of the materials did not change when

compared to monoliths of the same materials. Based on this information, the fractal dimension

in each layer of the bilayer structure does not change due to the fracture process. Therefore, the

fractal dimension is identified as being an intrinsic property of the fracture surface of the ceramic

materials.









APPENDIX
DETERMINATION OF THE HURST EXPONENT FOR SILICA GLASS

In this appendix, the Hurst exponent, a parameter that is used to define the fractal

dimension for self-affine objects, i.e., ceramic fracture surfaces, is explained. Background on the

Hurst exponent value for silica glasses is explored and the Hurst exponent is determined for the

mirror region of soda-lime-silica glass.

Background on the Hurst Exponent

Mandelbrot and Passoja determined that fracture surfaces are fractal [40]. A fractal

object can be either self-similar or self-affine. A self-similar surface is isotropic in

magnification, i.e., in a three-dimensional Cartesian coordinate system, x, y, z x, hy, hz

where h is a scalar. A self-affine surface is anisotropic, i.e., x, y, z x, hy, h z where i is

called the Hurst exponent [73]. The Hurst exponent is also known as the Hurst dimension, the

roughness exponent, or the roughness index. Since the fracture surface of ceramics scale

differently out-of-plane compared to in-plane, ceramics are self-affine. For these ceramic

fracture surfaces, there is a characteristic length at which the self-affinity of a surface can be

measured. This length is called the correlation length (5). For lengths greater than 5,

approaches 1. For silica glass, 5 is inversely proportional to crack velocity, i.e., 5 decreases from

80 to 30 nm for crack velocities from 10-10 to 10-4 m/S [74].

The Hurst exponent, like the fractal dimensional increment (D*), is bound by the values 0

and 1, where a i value of I and a D* value of 0 corresponds to a flat surface or line. The Hurst

exponent is related to the fractal dimension (D), but this relationship depends on the way D is

defined [75]. If a curve or profile is covered with boxes (b) of width Ax and height Ay, then the

box dimension (D') is determined from the expression:

N(b; Ax; Ay) ac b-D' (A-1)









where N(b; Ax; Ay) is the number of boxes used to cover the curve. D' is related to i by the

equation:

D'= 2 (A-2)

If the same curve or profile is measured using a ruler length (6) to measure the length of the

curve L, then the divider dimension (Dd) [76] is determined from the expression:

L ac 31-Dd (A-3)

where Dd is related to i by the equation:

Dd = 1/r (A-4).

The Hurst exponent determined by Dd is bound from 0.5 to 1. Because of this difference in the

way the fractal dimension is defined, it is better to use the Hurst exponent when describing self-

affine obj ects to avoid any confusion.

Measurement Methods for Determining Hurst Exponent

There are three methods that are used to measure i of fracture surfaces: perpendicular

sectioning, slit island, and direct surface area [75]. The perpendicular sectioning method

measures i from the profile of the fracture surfaces. The most used form of the perpendicular

sectioning method is the profile length method. The profile length method, measures the length

of a profile using different ruler length and is expressed in the equation

L = Lo31-Dd (A-5)

where Lo is the projected length of the profile. The Hurst exponent is determined using Equation

A-4.

The slit island method, discussed in Chapter 2, measures D from the contour of the

fracture surface. The contours of the fracture surface, or islands, are used to measure the fracture









surface in two ways. The first way uses the perimeter (P) of the islands and determines the

fractal dimension using the expression:

P ac 31-Dd (A-6)

which is similar to the perpendicular section method. The Hurst exponent from this method is

calculated using Equation A-4. The second way the fractal dimension is measured is by

determining the area (A) of the island in comparison to the perimeter. The fractal dimension

measured in this way is expressed as

A ac P D' (A-7)

The Hurst exponent measured using this method is calculated using Equation A-2. The problem

with the slit island method is that since the contour of the fracture surface is used to determine D,

then the self-similarity of the surface is measured instead of its self-affinity. Therefore, there is

some debate as to whether the slit island method can be used to determine the Hurst exponent for

self-similar materials.

The last method is the direct surface area measurement technique [77, 78]. This method

is the profile method conducted in two-dimensions. The fracture surface area of a material is

measured using a form of scanning probe microscopy (SPM) or scanning electron microscopy

(SEM). The fractal dimension is determined from the equation

A =A,32-D (A-8)

where A is the fracture surface area, Ao is the cross-sectional area, and 6 is the side length of the

SPM or SEM scan area. The Hurst exponent is calculated using the equation

(= 1/Dd-1)(A-9)

which is similar to A-4 but Dd has a Euclidean dimension of 2 instead of 1.









Hurst Exponent for Silica Glass

The Hurst exponent has been measured for silicate glasses, using some of the different

methods mentioned previously, to determine if there is universality among all fractured materials

in the Hurst exponent. Atomic force microscopy (AFM) was used in all of the measurements.

The use of AFM has helped researchers achieve an improved view of the fracture surface and the

ability to measure the fractal properties of fracture surfaces at small nanometerr) length scales.

Danguir et al. [79] measured i for soda-lime-silica (SLS) glass. The mirror region was

examined for crack velocities ranging from 10-9 to 10-' m/s. Ten AFM height profiles of lengths

of 1.5 Clm were taken perpendicular and parallel to the crack direction. The Hurst exponent was

calculated based on the expression:

Zmax, (= r' (A-10)

where r is the width of the window used to measure the profiles and Zmax (r) is the difference

between the maximum and minimum heights within the width of the window. It was shown that

there were two different values for i determined by a crossover length (5c). For large length

scales (r > 5c), i = 0.78 and for small length scales (r < 5c) i = 0.5. The direction of the profile

was shown to have no significant effect on i. This worked confirmed that the Hurst exponent is

affected by the length scale used to determine it.

Bonamy et al. [74] measured i for a silica glass. The fracture surface was generated at

crack velocities of 10-"1 to 10-4 m/S. The fracture surface was scanned using an AFM at a scan

area of (1 Clm)2. A one-dimension height-height correlation function (G(x)):


G(x) = lim- [h(x'+) h(x')] dr' (A-11)









where L is the section length, h(x') is the height of the surface at point x', and h(x'+ x) is the

height at a distance x away from point x', was used to determine the fractal dimension based on

the expression:

G(x) ac x4-2D' (-2

Using Equation A-12 and A-2, i = 0.75 for (5 80 nm. This value agrees with the value obtained

by Danguir et al. at large length scales.

Wiederhorn et al. [70] measured i for SLS glass based on the roughness of the fracture

surface. The mirror region of SLS glass was measured at velocities ranging from 10-10 to 10 m/s

for AFM scan areas of (0.5 Cm)2 to (5 Cm)2. The Hurst exponent was determined from the

equation

Rq = aLoc (A-13)

where R, is the root-mean-squared (RMS) roughness, Lo is the edge length of the AFM scan

area, and a is a constant. It was shown that i was inversely proportional to the crack velocity.

The Hurst exponent ranged from 0.18 for a crack velocity of 10 m/s to 0.28 for 10-10 m/s.

The Hurst exponent values reported by Wiederhorn et al. were less than the values

reported by Bonamy et al. and Danguir et al. It is possible that i values measured by Wiederhorn

et al. are the D* values for SLS glass. Based on the D* values in Table 4-5, this assumption

appears to be correct. Based of the lesser i values reported, Wiederhorn et al. showed that this

difference is due to the method used to determine the RMS roughness. The Hurst exponent was

measured for section lengths of 2 and 5 lm. When measuring the line profiles of the fracture

surface from AFM scans, i is determined from the expression

co aoc Loc (A-14)









where co is the RMS roughness for a line and Lo is the length used to measure the line. Using

this method, i = 0.75 for 2 Clm and 0.78 for 5 lm. When the area of the fracture surface is used

to determine roughness, i was 0.92 for 2 Clm and 0.95 for 5 lm. This showed that measuring i

based on one-dimensional (line profiles) and two-dimensional (surface areas) does not result in

the same value and that Equations A-13 and A-14 are not equivalent. Measuring i based on area

roughness is not an accurate method.

As mentioned in Chapter 3, the fracture surface area of chevron-notched SLS glass was

measured using the AFM. The fracture surface of the chevron-notched specimen was all mirror.

Using the direct surface area measurement method, the fracture surface was measured at scan

areas from (1 Clm)2 to (250 nm)2. Using Equations A-8 and A-9, i = 0.97. This value is similar

to that reported by Wiederhorn et al. when using the scan area to determine i but greater than

the values reported by Danguir et al. and Bonamy et al. This difference, as reported by

Wiederhorn et al., is due to the larger length scale, 100 to 1000 nanometers as opposed to tens of

nanometers, used in determining i and determining i based on two-dimensional measurements.

From Chapter 4, the fractal dimension of SLS glass was measured using AFM and the

Gwyddion software. From Table 4-5, D for the mirror region was 2.26 using cube counting and

2.29 using triangulation. For cube counting, the slope of the number of cubes versus the box

length is D'. For triangulation, the slope of the surface area versus the side length of the triangle

is two minus Dd. Therefore, i = 0.74 for cube counting and 0.78 for triangulation. These values

are similar to the values reported by Danguir et al. and Bonamy et al. This showed that data

acquired from two-dimensional measurements could be used to measure i.

As Table A-1 shows, i for SLS and silica glass is dependent on the measurement

technique used and the length scale used to determine i. For length scales greater than the









crossover or critical length for the material, i 0.75. For lengths less than this critical length,

= 0.5. Measurement of i at larger length scales, on the order of hundreds or thousands of

nanometers, results in i approaching 1. Therefore, the length scale used and the measurement

method must be taken into account when measuring i for glasses.



Table A-1. Hurst exponent for mirror region of silicate glasses from different experimental
methods.
Experiment Hurst exponent
Danguir et al. 0.5; 0.78
Bonamy et al. 0.75
Wiederhorn et al.
Initial area roughness 0.18-0.28
Line roughness 0.75 (2 Cpm); 0.78 (5 Cpm)
Area roughness 0.92 (2 Cpm); 0.95 (5 Cpm)
Direct surface area (Smith) 0.97
Triangulation (Smith) 0.78
Cube counting (Smith) 0.74









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BIOGRAPHICAL SKETCH

Robert Lee Smith was born in the city of Chicago to Robert and Loleta Smith. After

graduating from high school, Robert attended Wright State University in Dayton, Ohio for his

undergraduate studies where he earned his bachelor' s degree in materials science and

engineering in 2002. While at Wright State, Robert worked for several years as a co-op in the

Materials and Manufacturing Directorate of the Air Force Research Laboratory at Wright-

Patterson Air Force Base. He worked on several proj ects involving the study of metal matrix

composites and bulk metallic glasses.

Finished with his undergraduate studies, Robert had the opportunity to move to New

Mexico before going to graduate school. He was offered an internship at Los Alamos National

Laboratory, where he worked under Carol Haertling and fellow UF alum Robert Hanrahan Jr.

While at Los Alamos, Robert was elected president of the Los Alamos Student Association.

At the conclusion of his internship, Robert entered the University of Florida in Gainesville

to eamn his PhD in materials science and engineering. Robert worked under the guidance of Dr.

John J. Mecholsky Jr. During his graduate studies at UF, Robert received several fellowships

including the GEM Fellowship and the NSF/SEGEP Fellowship. Taking advantage of the

flexible schedule of graduate school, Robert travelled the globe visiting countries like China,

Brazil, England, and Germany. In between his many trips, Robert earned his master' s degree in

materials science in 2006 and his PhD in 2009.





PAGE 1

DETERMINING THE ROLE OF FRACTAL GEOMETRY AND FRACTURE ENERGY IN BRITTLE BILAYER MATERIALS By ROBERT LEE SMITH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 1

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2009 Robert Lee Smith 2

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To my parents for all of their help and support 3

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ACKNOWLEDGMENTS I thank Dr. Mecholsky for his guidance and understanding during my time at the University of Florida. I have learned much about materials science, and life in general, under his guidance and tutorage. I thank the other members of my superv isory committee, Dr. Anthony Brennan, Dr. Wolfgang Sigmund, Dr. Laurie Gower, and Dr. Bhav ani Sankar, for their advice and input while working on my dissertation project. I would espe cially like to thank Dr. Anthony Brennan for the discussions that we had that helped me gain a better understanding of my work. I want to thank my colleague, Dr. Chuchai A nunmana, for making all of the specimens that I used in my project, for the flexure testing of the specimens, and with helping me with the microstructural characterization of the specimens. Without his hel p, I would not have been able to finish my project and I am grateful for his help. I thank Robert (Ben) Lee and Allyson Barrett of the Department of Dental Biomaterials for their assistance in the mechanical testing of my specimens in the di ssertation project. I thank Dr. Anusavice of the Department of Dental Biomaterials for allowing me to use the resources of his department. I want to thank Andrew Gerger and Brad W illenberg, respectively, of the MAIC staff for helping with the use of the atomic force microscope that I used for a majority of my research and for helping me attain the scanning electron micros cope micrographs used in this dissertation. I want to thank Dr. Stephen Freiman with his help in understanding the work of fracture aspects of my work. I thank the South East Alliance for Graduate E ducation and the Professo riate, the Alfred P. Sloan Foundation, the National Consortium for Gradua te Degrees for Minorities in Engineering 4

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and Science, and the National Science Foundation for their support in funding my graduate study. I thank my friends and colleagues, especially Dr. Vasana Maneeratana, Ricardo Torres, and Alma Stephanie Tapia for their help, advice, and support during my time at the University of Florida. Finally, I thank my mother and father for all the help and support they have given me that has allowed me to accomplis h my goal of receiving my Ph.D. 5

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TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES.........................................................................................................................9 ABSTRACT...................................................................................................................................11 CHAPTER 1 INTRODUCTION................................................................................................................. .13 Introduction................................................................................................................... ..........13 Goals.......................................................................................................................................13 Objectives...............................................................................................................................14 Structure of Dissertation.........................................................................................................15 2 LITERATURE REVIEW.......................................................................................................16 Bilayer Composites.................................................................................................................16 Glass Ceramics.......................................................................................................................17 Diffusion Bonding..................................................................................................................19 Work of Fracture.....................................................................................................................20 Fractal Geometry....................................................................................................................21 3 MATERIALS AND METHODS...........................................................................................27 Sample Preparation.................................................................................................................27 Preparation of Glass-Ceramic Core.................................................................................27 Preparation of Glass Veneer............................................................................................28 Preparation of Bilayers....................................................................................................29 Microstructural Analysis........................................................................................................29 Phase Identification.........................................................................................................29 Aspect Ratio....................................................................................................................29 Fracture Strength....................................................................................................................30 Fracture Toughness.................................................................................................................32 Strength Indentation........................................................................................................32 Fractography................................................................................................................... .32 Work of Fracture.....................................................................................................................33 Fractal Dimension...................................................................................................................35 Statistical Analysis........................................................................................................... .......37 4 RESULTS AND DISSCUSION.............................................................................................41 6

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Microstructural Analysis........................................................................................................41 Phase Identification.........................................................................................................41 Aspect Ratio....................................................................................................................42 Fracture Strength....................................................................................................................43 Fracture Toughness.................................................................................................................44 Work of Fracture.....................................................................................................................45 Soda-Lime-Silica Glass...................................................................................................45 Dental Materials..............................................................................................................48 Fractal Dimension...................................................................................................................50 Soda-Lime-Silica Glass...................................................................................................50 Baria Silica Glass............................................................................................................5 2 Dental Materials..............................................................................................................53 5 CONCLUSIONS.................................................................................................................. ..79 APPENDIX DETERMINIATION OF THE HU RST EXPONENT FOR SILICA GLASS........81 Background on the Hurst Exponent........................................................................................81 Measurement Methods for Determining Hurst Exponent......................................................82 Hurst Exponent for Silica Glass.............................................................................................84 BIOGRAPHICAL SKETCH.........................................................................................................92 7

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LIST OF TABLES Table page 3-1 Chemical composition of IPS e.max Press.....................................................................39 3-2 Chemical composition of IPS e.max Ceram...................................................................39 3-3 AFM scan areas used to determine optimum scan areas for fractal dimension measurements of fracture su rface of baria silica glass.......................................................40 4-1 Fracture strength, fracture toughness determin ed from strength indentation (SI), and fracture toughness determined from fractography (F) for e.max Ceram (EV), e.max Press (EC), bilayer with crack ini tiation in veneer (BV), and bilayer with a flaw produced in the core (BC)..........................................................................................57 4-2 Work of fracture va lues calculated for chevron notch specimens using cross-sectional area (ACS) and actual fracture surface area based on probe tip radius, equilibrium bond length, and free volume a0........................................................................................57 4-3 Comparison of work of fracture and fract ure toughness, based on work of fracture, for unstable crack growth soda-lime-silica glass specimens.............................................57 4-4 Work of fracture WOF, based on crosssectional area (ACS) and fracture surface area (AF) for veneer, core, and bilayer bars...............................................................................57 4-5 Fractal dimensional increment for soda-lime-silica glass fracture surface regions measured using atomic force microsc opy and the Gwyddion and WSxM software.........58 4-6 Fractal dimensional increment for e.max Ceram (EV) and the e.max Press (EC) ceramics measured using AFM and Gwyddion software..................................................58 4-7 Fractal dimensional increment for b ilayer specimens using AFM and Gwyddion...........58 A-1 Hurst exponent for mirror region of sili cate glasses from different experimental methods..............................................................................................................................87 8

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LIST OF FIGURES Figure page 2-1 Diffusion bonding process.................................................................................................2 4 2-2 Replication technique for creati on of samples for slit island method................................25 2-3 Production of island from polishing..................................................................................25 2-4 Profile technique using fracture surface replica.................................................................26 3-1 Cross-section of bilayer specimens....................................................................................38 3-2 Diagram of distances used for comput ation of centroid for bilayer specimens.................38 3-3 Diagram of distances used for co mputation of mome nt of inertia.....................................39 4-1 X-ray diffraction pattern for e.max Press (EC) core ceramic showing that lithium disilicate are the crystals pr esent in the glass-ceramic.......................................................59 4-2 X-ray diffraction pattern of e.max Ceram (EV) ven eer ceramic showing veneer material has a am orphous pattern......................................................................................60 4-3 SEM micrographs of polished and etched surface of e.max Ceram (EV) veneer ceramic........................................................................................................................ .......61 4-4 SEM image of polished and etched su rface of e.max Ceram (EC) core ceramic showing the presence of needle -like nanofluorapatite crystals..........................................62 4-5 Energy dispersive X-ray spectroscopy (EDS) of e.max Ceram (EV) veneer ceramic........................................................................................................................ .......63 4-6 SEM micrograph of polished and etched surface e.max Press (EC) core ceramic at a magnification of 10000x.................................................................................................63 4-7 SEM micrograph of polished and etched surface e.max Press (EC) core ceramic at a magnification of 5000x...................................................................................................64 4-8 SEM micrograph of polished and etched surface e.max Press (EC) core ceramic at a magnification of 10000x.................................................................................................66 4-9 SEM image of fracture surface of chevronnotched soda-lime-silic a flexure specimen showing that fracture surface is all mirror and no mist or hackle is present.....................67 4-10 AFM scan image of soda-lime silica gla ss fracture surface at a scan area of (750 nm)2....................................................................................................................................68 9

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4-11 Fracture surface area (AF) as a function of scan area side length ( ) for chevronnotched soda-lime-silica bars.............................................................................................69 4-13 Fracture surface area (AF) as a function of scan area side length ( )................................71 4-14 Fracture surface area (AF) and adjusted fracture surface area (Adj. AF) as a function of scan area side length ( ) for e.max Ceram (EV) veneer bars.....................................72 4-15 Fracture surface area (AF) as a function of scan area side length ( ) for e.max Press (EC) core bars....................................................................................................................73 4-16 Fracture surface area (AF) as a function of scan area side length ( ) for veneer-indent (BV) and core-indented (BC) bilayers...............................................................................74 4-17 Fractal dimension increment (D*) of baria silica glass, plotted against scan area, determined from Gwyddion for mirror, mist, and hackle region.......................................75 4-18 AFM scan image of e.max Ceram (EV) veneer ceramic and e.max Press (EC) core ceramic fracture surface at a scan area of (750 nm)2.................................................76 4-19 SEM images of veneer-indented bilayer (BV) fracture surface.........................................77 4.20 SEM images of core-indented bilayer (BC) fracture surface.............................................78 10

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Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DETERMINING THE ROLE OF FRACTAL GEOMETRY AND FRACTURE ENERGY IN BRITTLE BILAYER MATERIALS By Robert Lee Smith August 2009 Chair: John J. Mecholsky Jr. Major: Materials Science and Engineering Many applications in the ceramic field requi re bilayer design to be successful. One example is the application of bilayers in dentis try for all ceramic crowns and bridges. Although fracture in bilayers has been stud ied, research has not explored if the fracture surface changes as a crack travels from one material to another in a bilaye r and if a change in the fracture surface reflects changes in the mechanical properties of the bilayer. The work of fracture, fractal dimension, fracture strength, and apparent fracture toughness were determined for two sets of bilayer bars composed of e.max Ceram veneer bonded to e.max Press core The work of fracture was determined from the actual fracture surface area, using atomic force microscopy, to determine if the work of fracture can be used to estimate the minimum fracture energy of the materials. The location of crack initiation was exam ine to see if the work of fracture is the same in both sets of bilayers and if the fracture process results in the veneer and core having the same fractal dimension. It was confirmed that the location of crack init iation did change the work of fracture of the bilayers but did not change the fr actal dimension of the two material s in the bilayer. The work of fracture was used to estimate the minimum fracture energy of the ceramic monoliths and bilayers. Using the actual fracture surface to determine the work of fracture resulted in a 11

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12 decrease in the work of fracture but did not es timate the theoretical minimum fracture energy of the dental materials. The work of fracture for the bilayer was dependent on the material that the crack initiated in, further indicating that the selec tion of the materials used in these bilayers is important in designing these bilayer structures. The fractal dimensions of the veneer and core in the bilayers were similar to that of the monolith ic specimens. This implies that the fracture process does not change the fractal dimension of th ese materials and that the fractal dimension is an intrinsic property of these ceramics.

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CHAPTER 1 INTRODUCTION Introduction The critical element that controls the fractur e process has not been fully determined for bilayer composite materials such as dental prosth etics or ceramic armor. Selection of the proper material is critical to performance if the material in which fracture initiates controls the fracture properties of the entire composite There are several properties th at are important in determining the effect of the location of crack initiation on the fracture process of a material: fracture toughness, fractal dimension, and work of fracture. Fracture toughness is the resistance to crack propagation in a material. The fractal dimension is a quantitative measure of the tortuosity of a surface at multiple length scales. The work of fract ure is defined as the total energy consumed to produce a unit area of fracture surface during the fr acture process. These properties can be used together in multilayer composites to study the eff ect of the initiation of fracture on the fracture process in bilayers. Goals The goal of the proposed research is to determ ine if the fracture surf ace area and the fractal dimension of materials in a bilayer change as a crack travels from one ma terial into another and if a change in the fracture surfaces determines the mechanical properties of the bilayer structure. This goal will be accomplished by testing the fracture toughness, wo rk of fracture, and fractal dimension of a glassy veneer bonded to a glass-ceramic dental core. Knowledge of the energy contributed to create the fract ure surface in ceramics will help engineers better understand the total energy to generate fracture. This will he lp to better design brittle materials and bilayer structures that are used in several disciplines like dentistry and armor protection, where fracture resistance and materials selection is critical. Knowledge of changes in the fractal dimension of 13

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fracture surfaces for bonded ceramic materials will give a better understanding of the role that the fracture process has in determining the fractal di mension of the materials. This will aid in the further use of the fractal dimension as a tool in characterizing the fract ure of ceramic materials. Objectives To accomplish the goal of this research, the following hypotheses are proposed: 1. Test the hypothesis that the work of fracture based on the actual fracture surface area can be used to estimate the fracture energy of ceramic materials. 2. Test the hypothesis that the work of fractur e of a ceramic bilayer will be the same regardless of the location of crack initiation. 3. Test the hypothesis that the fract al dimension of the two materi als in a bilayer will be the same regardless of the loca tion of crack initiation. A veneer and a core dental ceramic were chosen as the materials used to test the stated hypotheses. The veneer chosen was IPS e.max Ceram, a fluorapatite glass-ceramic, and IPS e.max Press, a lithium disilicate glass-ceramic. These materials were chosen because bilayers of the materials are used in dentistry, so the mechanical properties (ela stic modulus, fracture strength, fracture toughness, etc. ) of the materials have been studied [1-5]. This allows for comparison of the mechanical properties studied in this dissertation to what other researchers have measured. The mechanical properties of the two materials are different enough in order to determine if location of crack initiation changes the mechanical properties of these materials. The veneer and core have been shown to bond well [3, 4, 6], producing a strong bond at the interface between the two materials. This mean s a crack that propagates in one layer should continue to propagate into the second laye r without delaminating at the interface. A series of experiments have been determined to test the hypotheses for this project: 14

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15 1. The microstructure was analyzed and the fracture strength and apparent fracture toughness were determined for the veneer, th e core and for two sets of bilayers. 2. The work of fracture, using the cross-sectional area and the fracture surface area, was determined for a control group of soda-limesilica (SLS) glass under stable and unstable crack growth. 3. The work of fracture, using the cross-sectional area and the fracture surface area, was determined for the veneer, the core, and for the two sets of bilayers 4. The fractal dimension was measured for a control group of SLS gl ass and baria silica glass. 5. The fractal dimension was measured for the veneer, the core, and two set of bilayers. Structure of Dissertation Chapter 2 is a compilation of selected literature explaining the concepts of the work of fracture and fractal dimension, a nd defining bilayer materials, glass-ceramics, and diffusion bonding. Chapter 3 describes the materials selected for the study and the pr ocess used to prepare these materials. The experimental procedures used to answer the hypotheses discussed in the objectives are also explained in this chapter. Chapter 4 details the results from the experiments described in Chapter 3 and interprets the mean ing of the results. Chapter 5 summarizes the dissertation and discusses whethe r the hypotheses were answered from the experiments. The dissertation concludes by addressi ng the concept of th e Hurst exponent and its relationship to the fractal dimension for soda-lime-silica glass in the Appendix.

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CHAPTER 2 LITERATURE REVIEW In this chapter, bilayer composites, diffusi on bonding, and glass-ceramics are discussed to give background on the dental mate rials and fabrication methods us ed in the dissertation. The two parameters discussed in the objective, the work of fractur e and the fractal dimension, are defined and background is given on the previous methods used to determine these properties. Bilayer Composites A bilayer material is a composite material in which two different materials are bonded together. Bilayer materials are joined by eith er bonding the two materi als together using a welding or diffusion bonding technique or by using an interlayer like an adhesive to bond the two materials together. As in all composites, bilayer materials are designed to combine the advantages of both materi als to produce an optimum structure. Bilayers are used in a variety of applications from protective coa tings on metal substrates that ar e used in the aerospace industry to titanium alloys bonded with a bioactiv e glass coating for bone implants [7]. Ceramic bilayers that are used to produce dent al prostheses have ha d an increase in use over the years. The use of a bilayer composite al lows for the aesthetic and mechanical properties needed for dental prostheses. All-ceramic dent al bilayers for prostheses are made of two materials: a glass veneer that is matched to the appearance of the surrounding teeth and the glassceramic core that provides the stiffness and fr acture resistance that is needed for dental prostheses. The veneer must also be matched in thermal expansion and contraction coefficient to minimize the risk for transient crack formation and residue tensile stress. The mechanical properties of these bilayer materials are dependent on the mechanical properties of the two materials a nd the interface between these materials. If a crack initiates and propagates from the tensile surface of one the less tough layer, e.g., the veneer layer in a veneer16

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core all-ceramic dental restor ation, the propagation of the crac k is dependent on the fracture toughness of the interface and th e second material [5, 8]. If the fracture toughness of the interface is greater than the fracture toughness of the two materi als in the bilayer, the crack will continue to propagate into the s econd material. However, if the interface is less resistant to fracture than the second material, then the cr ack will propagate along th e interface of the two materials and perhaps nullify the benefits of using a more fracture resistant material in the bilayer. Taskonak et al. [9] showed that for lith ia-disilicate based fixed partial dentures, cracks that initiated in the veneer propagated across the interface and into the core. Even though the core had a greater fracture toughness than the ve neer, the crack was not impeded by the core at the interface. This implies that the interface is tougher than th e core. For yttria-stabilized tetragonal zirconia polycrystals fixed partial de ntures, cracks that initiated in the veneer propagated through the veneer and stopped at the interface of the two materials. Delamination occurred at the interface before a secondary crack initiated in the co re. This implies that the core was tougher that the interface and that delamination c ontrolled the fracture process of the bilayer. Although the mechanical properties of ceram ic bilayers and multilayers have been explored [4, 5, 10-15], research has not explored if the fracture surface cha nges as a crack travels from one material to another in a bilayer and if a change in the fractur e surface determines the mechanical properties of the bilayer structure. Analysis of the work of fracture and the fractal dimension will result in a better understanding of fracture surface of each material in the bilayer affects the mechanical properties of the composite and if the location of crack initiation changes the fracture surface and mechanical prope rties of the bilayer structure. Glass Ceramics Glass-ceramics are polycrystalline ceramic ma terials formed from a base glass through the controlled nucleation and crystall ization of crystals from the glass [16]. The crystals are 17

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contained in a matrix of the base glass. Glass-ceramics are different from traditional ceramics where the crystalline material is introduced wh en the ceramic composition is prepared. It has been known for a long time that glasses can be cr ystallized to form pol ycrystalline ceramics. Raumur, a French chemist, produced polycrystalline material from glass by heating glass bottles, packed in sand and gypsum, to a high temperature for several days [17]. Raumur experiments resulted in the creation of opaque, porcelain-like objec ts that had low mechanical strength and a distorted shape compared to the orig inal shape of the bottles. This was due to the inability to control the cr ystallization process. The invention of true glass-ceramics did not occur until the mid-1950s [16]. S. D. Stookey, a glass chemist and inventor at Corning Glass Works, was planning to precipitate silver particles in lithia silicate (Li2O-SiO2) to create a permanent photographic image. He accidentally overheated the glass overnight, producing a white ceramic with no net change in shape and an unusual strength for a glass. Upon further analys is, Stookey determined that lithium disilicate (Li2Si2O5) crystals, nucleating from the silver particle s, had been created in the glass. Stookeys experiment produced a new type of material had been discovered th at had applications in several fields. Glass-ceramics have advantages over glass when it comes to mechanical properties and advantages over metals and organic polymers in th e areas of thermal, chemical, biological, and dielectric properties. The pr esence of grain boundaries and cl eavage planes impede fracture propagation while the crystals cause crack defl ection, resulting in a greater fracture toughness compared to glass [16]. Glass-ceramics exhibit little to no expansion on heating and have a high temperature stability. This allows glass-cerami c to be used in high temperature application, where metals cannot be used, e.g., cookware. Th e high chemical durability and translucency or 18

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opacity of glass-ceramics makes glass-ceramic a useful material in places where the corrosion of metal is a problem. Another advantage of glass-ceramics is that a glass-working process can be used to create objects resulting in the use of high-speed auto matic machining and a faster processing time when compared with the te chniques used in conventional methods. The most important features of glass-ceramics are that they are applicable over a wide range of compositions and variations can be contro lled through the heat tr eatment process [17]. Various crystal types can be deve loped in controlled proportions re sulting in the ab ility to vary the physical characteristics of glass-ceramics in a controlled manner. This feature allows glassceramics to be designed and created for a multitude of applications. Diffusion Bonding Diffusion bonding involves the interdiffusion of atoms acr oss the interface of two materials, held in contact with each other, resulting in the bonding of the two materials [18]. The bonding of these materials involves holding a pre-machined component under pressure at an elevated temperature in a protective environment. The pressure applied to the joining materials must cause the edges to move within range of th e atomic forces but below a pressure that would cause macrodeformation in either material (Fi gure 2-1). The temperature range for diffusion bonding is between 0.5 and 0.8 of the lower melting te mperature of either of the two materials [19]. Diffusion bonding usually re quires a minimum of 60 minutes to successful bond the materials. The main advantages of diffusion bondi ng are 1) the material at the joint has the same strength and plasticity as the bulk of the parent materials, 2) the bonding temperature is not as high as other joining processes, and 3) more types of joints can be formed compared with other joining processes [18]. Although diffusion bonding is used mostly to bond metals to metals or metals to nonmetals, it has been used to bond ceramics to ceramics [20]. Ceramic to ceramic bonding is 19

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mainly carried out in the solid phase with heat being applied by radian t, induction, direct or indirect resistance heating. Some ceramics do not incur the problem of oxidation hindering bonding since the ceramics themselves are oxides. The bonding of some ceramics, like the bonding of a glass-ceramic to a glass, has a si milar advantage of bonding similar metal alloys: the ability to increase the mechanical properties of an object but keeping the favorable properties of the weaker material. This is observed in de ntal ceramics, where a transparent or translucent veneer is diffusion bonded to a stronger and more fracture resistant glas s-ceramic core. This makes the dental ceramic stronger than the ven eer and allows it to maintain its aesthetic properties. Work of Fracture The fracture energy ( ) of a material is defined as th e energy per unit area required to fracture an infinite body into two separate halves [21]. Defined in another way, is the energy required to break the bond between two atoms per unit area. When a specimen is tested in a manner that allows for the stable growth of a crack, the average fracture energy can be determined for the specimen. This average fracture energy is called the work of fracture ( WOF). The work of fracture is defined as the total en ergy consumed to produce a unit area of fracture surface during the fracture process [21] and is calculated from the equation: CS WOFA W (2-1) where W is the work and is equal to the area under the load-displacement curve and ACS is the cross-sectional area of the fracture surface. In theory, if it is assu med that all of the work from loading is dissipated by the breaki ng of the bonds, then the average = WOF [29]. The work of fracture has been used as a method to determine the fracture toughness (KC) of materials [22-24] ba sed on Irwins equations: 20

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'2 EKC (2-2) where E is the elastic modulus (E) for plane stress or E/(12) [ is Poissons ratio] for plain strain and = WOF. The work of fracture has also been us ed as a separate parameter to describe the fracture properties of various materials, in cluding biological [25-28], ceramics [23, 29-34], polymer [35, 36], and metal materials [22, 37]. The advantages of using the work of fracture are that it does not require knowledge of the stress in tensity, the geometry of the notch, the notch tip acuity, or the linearity of the material. In most cases, KC values calculated using WOF have led to an overestimation of KC when compared with values calculated from methods th at are more conventional [22, 23]. This is due to the assumption that = WOF based on the equation used to calculate WOF (Equation 2-1). This assumption is based on the idea that all of the energy of the fracture process results in the creation of two atomically smooth surfaces, resulting in the cross-sectional area being used as the actual fracture area being created by the fracture process [29]. However, it is apparent that fracture surfaces are not absolutely smooth, even in the fracture mirror region [38]. Since the energy used to fracture materials results in the creation of a rough and tortuous fracture surface, then using the actual fracture surface area of the material, wh ich is greater than the crosssectional area, to calculated WOF would result in a better approximation of the theoretical fracture energy of the material. Fractal Geometry Fractal geometry, a non-Euclidean geometry, can be used to quantitatively describe irregular shapes and surfaces [39], e.g., a maple leaf or a head of cauliflower. Fractal objects are defined as objects that are self-similar, scale invariant, and exhibit fractional dimensions. A self21

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similar object is an object that has multiple part s that are similar to the entire object. A scale invariant object is an object that looks th e same at different ma gnification scales. From Mandelbrots early work in fractal ge ometry, especially in evaluating the fracture surface of steels [40], fractal geometry has been shown to be a useful tool in the fields of science and engineering. The basic quantitative component of fractal geometry is the fractal dimension (D). The fractal dimension is a non-intege r number with a Euclidean and non-Euclidean component. The Euclidean component describes th e topological dimension of the object, e.g., 1 for a line and 2 for a square or circle. The non-Euclidean component, called the fractal dimensional increment (D*), describes the level of irregularity of the object from its Euclidean geometry. The fractal dimensional increment ranges from 0 to 1, with the level of irregularity increasing as the increment increases. For exam ple, broccoli has a fractal dimension of 2.66 [41], where 2 denotes that it is a surface and 0.66 denotes the fr actal component and the level of irregularity in the surface of broccoli. In materials science, fractal geometry has been used as an analytical tool to quantify the fracture surfaces of brittle and ductile materials [42-48]. Fr acture surfaces have a fractal dimension of 2.D* with a fractal dimension of 2.1 denoting a fracture surf ace that is relatively flat and a fractal dimension of 2.9 denoting a fracture surface that is volume filling. Several methods have been used to measure the fractal dimension of a fracture surface. The measurement of the contours of the fractur e surface is one of the most frequently used methods. There are two main boundary contou r methods: the slit-island method [40], which examines the contour parallel to the plane of the fracture surface, and the profile method, which examines the contour perpendicular to th e plane of the fracture surface [39]. 22

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The slit island technique uses the fracture surf ace to determine the fractal dimension. The fracture surface of a specimen, or a replica of the fracture surface, is plated in a visible coating such as a gold-palladium sputter coating to ma ke the surface more visible (Figure 2-2). The specimen is mounted in epoxy or another mountable material and polished pa rallel to the fracture surface until parts of the fractur e surface, called islands, are visible (Figure 2-3). From the coastlines of the islands, th e fractal dimension is measured using one of two methods: measurement of the perimeter and area of the isla nds or measuring the length of the coastline of the islands (the Richardson meth od) [49]. A straight line drawn through the data points of a log-log graph of the area versus the perimeter results in a line that has a slope equal to D* (1D). For the Richardson method, the perimeter is measured at different length scales using the following equation: D OLL1 (2-3) where L( ) is the measured perimeter length as a function of the step size, LO is the projected perimeter, and is the step size. D* can be determined from the log-log plot of the perimeter versus the step size. The profile method measures the profile of the fracture surface (Figure 2-4). The profiles are obtained using microscopy, e.g., optical, seco ndary electron, and scanning tunneling. From these profiles, the fractal dimension can be determined using Equation 2-3 where LO is the projected length of the profile and L( ) is the measured length of the profile. 23

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A B C D Figure 2-1. Diffusion bondi ng process. A) Initial contact. B) Plastic deformation. C) Diffusion processes fill microvoids. D) Bonding completed. [Adapted from Lancaster, J. F. (1999) Metallurgy of Welding, 6th edn. Woodhead Publishing, Cambridge, UK]. 24

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Figure 2-2. Replication technique for creation of samples for slit island method. [Source: Hill, TJ, Della Bona, A, Mecholsky, JJ (2001) J Mater Sci 36:2651] Figure 2-3. Production of island from polishing. [Source: Hill, TJ, Della Bona, A, Mecholsky, JJ (2001) J Mater Sci 36:2651] 25

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26 Figure 2-4. Profile technique us ing fracture surface replica. [Sou rce: Hill, TJ, Della Bona, A, Mecholsky, JJ (2001) J Mater Sci 36:2651]

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CHAPTER 3 MATERIALS AND METHODS In this chapter, the preparation of the materials used is described for this study. The mechanical and microstructural procedures used to characterize the materials are discussed and the measurement of work of fracture and fractal dimension methods for the dental materials is described. Sample Preparation Preparation of Glass-Ceramic Core The lithium disilicate dental glass-ceramic co res were prepared using a lost-wax method [4]. A self-curing acrylic resin (Pattern Resin, GC Corp, Tokyo, Japan) was used to prepare the master molds for the IPS e.max Press (EC) glas s-ceramic (Ivoclar AG, Schaan, Liechtenstein). The EC is a lithium disilicate glass-ceramic (Table 3-1). Impressions were made from the master mold using a vinyl polysiloxane impression material (Extrude, Kerr Corp., Romulus, MI). The acrylic resin was poured into the molds and cure d to make rectangular bars (25 mm x 4.6 mm x 2.0 mm) using a mill (PGF 100, Cendres & Metaux Sa., Biel-Bienne, Switzerland). Following the preparation of the resin beams, four specimens were sprued and invested. The resin beams were placed in a preheating furn ace (Radiance, Jelrus Int., Hicksville, NY) and a two-stage burnout process was used to remove the resin: 1) the furnace is heated up to 250C at 5C/min. and held for 30 min. and 2) the furnace was heated up to 850C at 5C/min. and held for 1 hr. After burnout, the EC ingot was placed in an investment cylinder and it was transferred to an isostatic pressing furnace (EP500, Ivoclar AG, Schaan, Liechtenstein). The EC was pressed at a pressure of 5 bar (0.5 MPa) and a temperature of 920C for 25 min. After pressing, the investment cylinders were removed and cooled for 2 hr. in air. The cooled specimens were divested (removal of investment material) by grit blasting with 80 m glass beads (Williams 27

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Glass Beads, Ivoclar North America, Amherst, NY) at an air pressure of 0.25 MPa. The sprues were cut away using a diamond disc and excess sprue segments were ground from 120 to 1200 grit using silicon carbide (SiC) paper and po lished from 8 to 1 micron using diamond paste (Buehler, Lake Bluff, IL) on polishing cloth (UltrapolTM, Buehler, Lake Bluff, IL) and 1 micron alumina (Micropolish II, Buehler, Lake Bluff, IL) on polishing cloth (Microcloth, Buehler, Lake Bluff, IL). All polishing and grinding we re preformed on a metall ographic polishing wheel (Model 41-1512, Buehler, Lake Bluff, IL). After polishing, the reaction layer from inve sting was removed by placing the bars in a plastic bottle containing 20 mL of a 1% hydrofluoric soluti on (Invex Liquid, Ivoclar AG, Schaan, Liechtenstein) and ultrasonically cleaned for 10-30 min. After cleaning, the bars were cleaned again using tap water for 10 s and blown dry using compressed air. After drying, the bars were grit-blasted with 100 m Al2O3 particles (Blast ing Compound, Williams-Ivoclar North America Inc., Amherst, NY) at an air pressure of 0.1 MPa. Then, the bars were cleaned using a pressurized steam spray. Preparation of Glass Veneer The glassly veneer, IPS e.max Ceram (EV), is described as a fluorapatite glass-ceramic (Table 3-2). Glass veneers were prepared us ing a sintering process. A veneer powder was mixed with a liquid (Ivoclar AG, Schaan, Liechtens tein) to obtain a slurry solution. This solution was then poured into a silicone mold, co ndensed by vibration, drie d, and sintered in a furnace (P80, Ivoclar AG, Schaan, Liechtenstein) by heating to 800C at 60C/min., held for 2 min., and cooled to 180C. A vacuum was applied between 450C to 759C. Following the sintering process, any excess veneer was ground of with a 75 grit circular diamond disk and polished down to a 1 m finish using a metallograp hic polishing wheel. All veneer bars have the same dimensions (25 mm x 4.6 mm x 2.0 mm) as the core ceramic. 28

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Preparation of Bilayers The veneer and core ceramics were bonded together using a sintering process. The prepared EC bars were veneered with the EV. The veneer powders were mixed with a liquid (Ivoclar AG, Schaan, Liechtenstein) to obtain a sl urry solution. This solution was brushed onto the EC. The bilayers were then sintered in a furnace (P80, Ivoclar AG, Schaan, Liechtenstein) by heating to 800C at 60C/min., held for 2 min., and cooled to 180C. A vacuum was applied between 450C to 759C. Three layers of veneer were sintered on each EC bar. Following the sintering process, any excess ve neer was ground of with a 75 grit circular diamond disk and polished down to a 1 m finish using a metallogr aphic polishing wheel. The dimensions of the bilayers were set to be similar to the monolith ic veneer and core (25 mm x 4.6 mm x 2.1 mm) with the veneer having a thickness of 1.1 mm and the core having a thickness of 1.0 mm. Microstructural Analysis Phase Identification X-ray diffraction (XRD) was performed on the EC and EV to determine if both materials were glass-ceramics. Using a mortar and pestle, a sample of the glass ceramic was ground into a fine powder. The powder was placed on a gla ss slide using double-sided tape. The specimen was scanned at a rate of 3/min. over a 2 range of 10-60. Aspect Ratio The aspect ratio of the crystals in the EC glass ceramic specimens was determined using a scanning electron microscope (SEM). Samples of the glass ceramic were mounted in epoxy and polished down to a 0.05 m finish. After polishing, the specimens were etched in 2.5 % hydrofluoric (HF) for 10 s, cleaned in tap water, ultras onically cleaned for 5 min. in isopropanol, and air-dried using compressed air. The top and sides of the epoxy m ount was painted with carbon paint before sputter coating with carbon. Three micrographs each were taken from two 29

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specimens. The major and minor axes of five crystals from each micrograph were measured to calculate the aspect ratio of the glass-ceramics. Fracture Strength As a control group, the strength of the glass veneer and core glass-ceramic were measured. Five bars of EC and EV each were prep ared for testing. The fracture strengths of the bilayer bars was tested using the strength indentation technique [50]. A Vickers indent was placed in the center of the tens ile surface of the bars using a microhardness machine (Model, Buehler, Lake Bluff, IL) with an indent load of 1 kgf for EV and 2 kgf for EC. After indentation, the bars were tested in four-point flexure usi ng a universal testing machine (Model 4465, Instron Corp, Norwood, MA) at a crosshead speed of 0. 2 mm/min. with a support span of 20 mm and a load span one-third of the support span. The fle xure strength of the bars was calculated using the equation: 2bd FLF (3-1) where F is the flexural strength of the specimen, F is the load at failure, L is the support span, b is the specimen width, and d is the specimen thickness. For the bilayer bars, two sets of five specimens were fabricated for flexure testing. The first set was indented on the tensil e surface of the EV layer of the bi layer with an indent load of 1 kgf. The second set was indented on the tensile surface of the EC layer of the bilayer with an indent load of 2 kgf. Because of the difference in the elastic mo duli of the two layers, composite beam theory was used to calculate the fracture st rength of the bilayers. We followed the method outline in Beer and Johnston [51] and applied by Thompson [5] to dental ceramics. The composite beam was transformed into a uniform beam of EC. This was done by calculating the 30

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transformation factor (n), the centroid (y), and the moment of inertia (I) for the transformed beam. The transformation factor is the ratio of the elastic modulus of EC [96 GPa] [52] and EV [64 GPa] [52]. This ratio is 3/2, therefore n = 1.5 and the width of EV is transformed from 4.6 mm to 3.1 mm (Fig. 3-1). With this transforma tion, the centroid can be determined for the composite beam. Using a reference axis set at the tensile surface of the transformed beam (Fig.3-2), y can be determined using EFBC DEFABC y (3-2) where A, B, C, D, E, and F are the distances for the transformed beam (Fig. 3-2). Once the centroid has been determined, the moment of inertia for the beam can be determined based on the distances in Fig. 3-3 and from the equation 2 3 2 3)( 12 1 12 1hgefef dcababI (3-3) With the centroid and moment of inertia known for the composite beam, the flexure stress can be determined for the composite beam The flexure stress is calculated from the equation I Mc (3-4) where M is the maximum moment and c is the distance from the bottom of the beam to the centroid. For four-point flexur e with a load span 1/3 of the support span, M is expressed as 6 PL M (3-5) 31

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Fracture Toughness The fracture toughness of the bilayer and monolithic bars was determined using two methods: 1) the strength indentation technique and 2) fractography. Strength Indentation The strength indentation tec hnique uses an indent, in this case a Vickers indent, to introduce a critical crack in the tensile surface of a flexure bar. The advantage of using this method is that the crack length does not have to be known to calculate the fracture toughness. The fracture toughness is calculated us ing the following equation [50]: 4/3 3/1 8/1/59.0 PHEKF C (3-6) where KC is the fracture toughness, E is the elastic modulus of the sp ecimen, H is the hardness of the specimen, F is the fracture strength of the specimen, and P is the indent load. For the bilayer composites, E was determined from the uppe r bound of the Rule of Mixtures, i.e., ECEC EVEVBAEAEE (3-7) where EB is the elastic modulus of the composite, AEV is the area fraction of the veneer in the bilayer, EEV is the elastic modulus of the core, and AEC is the area fraction of the core in the bilayer. Fractography The fracture toughness of the bars was cal culated using fractography. An optical microscope (Olympus BHMJ, Olympus, Tokyo, Japa n) was used to measure the critical flaw size. The fracture toughness was cal culated using the equation [53]: cYKF C (3-8) 32

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where Y is a geometrical constant for the shape of the crack and loading geometry that has a value of 1.65 for indent-induced surface cracks, with a local residual stress [ 54], and c is the flaw size. The flaw size was calculated as abc (3-9) where a is the crack depth and b is the half of the crack width. Work of Fracture The work of fracture was measured for sodalime-silica (SLS) glass as a control group and for the bilayer and monolithic dental ceramic bars. Two equations were used to determine WOF associated with the following met hods: 1) the cross-sectional area approach (Equation 2-1) and 2) the actual fracture surface area (AF) instead of the cross-sec tional area method, i.e.: F WOFA W (3-10) Since the crosshead displacemen t of the testing machine does not accurately measure the midpoint deflection ( ) of the flexure specimens, was calculated using the equation: d L 6 (3-11) where is the strain to failure and is determined from the equation EF (3-12) Unstable fracture was accomplished using streng th indentation [50] in flexure while stable fracture was accomplished using a chevron-notch (Figur e 4-9) in flexure [55]. Three sets of bars were used for the unstable fracture test: 1) 45 mm x 4 mm x 3 mm bars we re Vickers indented with a load of 2000 gf load and tested in four-point flexure, 2) 55 mm x 10 mm x 5.7 mm bars were Vickers indented with a load of 1000 gf and tested in threepoint flexure, and 3) 55 mm x 10 mm x 5.7 mm bars were Vicker s indented with a load of 2000 gf and three-point flexure. 33

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The chevron-notched bars, with dimensions of 45 mm by 4 mm by 3 mm, were fabricated using a low speed diamond saw (South Bay Mode l 650, South Bay Technology, San Clemente, CA). The notches were cut using a diamond blade with a thickne ss of 0.15 mm. The specimens were tilted at an angle of 25 using a speci ally made fixture for the diamond saw. Both the unstable and stable SLS glass sp ecimens edges were polished using 240-grit silicon carbide paper to remove or reduce the size of any flaws that were present along the edges. All specimens were tested using a universa l testing machine (Model 4465, Instron Corp, Norwood, MA). For the unstable test, the crosshead speed was 0. 5 mm/min while the stable tests were conducted at a crosshead rate of 0.03 mm/min. The fracture surface area was measured us ing an atomic force microscope (AFM) (Dimension 3100, Veeco Instruments Inc., Woodbury, NY). Using the softwa re that is provided with the AFM (Nanoscope 3D, Veeco Instruments Inc., Santa Barbara, CA), the surface area was determined from the sum of the area of a grid of triangles formed by three adjacent data points in the scans. Eight square scan areas ((750 nm)2, (625 nm)2, (500 nm)2, (375 nm)2, (250 nm)2, (100 nm)2, (50 nm)2, and (25 nm)2) were used to measure the fracture surface area. Five scans were completed for each scan area per specimen. A ll scans were conducted in the tapping mode, where the AFM tip is oscillated at the cant ilevers resonant freque ncy causing the tip to intermittently contact the surface. A tapping tip wa s used to decrease the wear of the AFM tip over time on the hard surface of the materials. Previous AFM work using a contact tip showed a decrease in the surface area measured, over time, due to wear increase the tip radius. All images were scanned at a scan rate of 2-3 Hz and a resolution of 512 pixels per line. All fracture surfaces were cleaned with ethanol to remove the dust and other material from the surface and dried using compressed air. 34

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To determine the fracture surface area, a box counting method was used for which the average surface area of a portion of a selected mirror, mist, and hackle region was measured using the AFM. Then, the surface area value for th at representative region was multiplied times the entire cross-sectional area of the region. From the AFM data obtained, AF was calculated using the following equation: HACKLE MIST MIRROR FSS SA A SS SA A SS SA AA (3-13) where A is the area of a given region (mirror, mi st or hackle) of the fracture surface, SA is the surface area value for that region and SS is the sc an area used to determine SA. This method was used to calculate AF for all of the different scan areas. Fractal Dimension The fractal dimensions of SLS glass bars and baria silica glass bars used as a control group, were determined along with the bilayer a nd monolithic dental ceramics bars using an AFM. All scans were conducted us ing a tapping tip at a resoluti on of 512 pixels per line and a scan rate of 2-3 Hz. All fract ure surfaces were wiped clean with ethanol to remove dust and other materials from the surface and dried with compressed air. To calculate the fractal dimension (D) from the AFM data, two software packages were used: 1) Gwyddion and 2) WSxM. Gwyddion is a free and open source software program for scanning probe microscopy data visualization and an alysis [56]. Gwyddion uses four methods to calculate the fractal dimension from AFM data: 1) cube counting, 2) trian gulation, 3) variance, and 4) power spectrum. Cube counting uses box counting to determine D [57, 58]. A grid of cubes with a cube edge length m is placed over the surface of the AFM image. N(m) is the number of cubes that 35

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contain at least one pixel. The value of m is d ecreased by a factor of two until m is equal to the distance between two pixels. The slope of the lo g of N(m) versus the log of (1/m) is equal to D. For the triangulation method [57], a grid of tr iangles of side length (L) is placed over the image. The areas of all of the triangles is measured and summed to calculate the surface area S(l) of the image. The grid size is decreased by a factor of 2 until L is equal to the distance between two pixels. The slope of the log of S(L) versus the log of 1/L is D-2. The variance method is based on the scale dependence of the variance of fractional Brownian motion [59]. An image is divided into cubes and the variance, the power of the RMS value of the heights, is calculated for each cube. The slope of the log-log plot of the variance is equal to and D is determined from the equation: 2 3 D (3-14) For the power spectrum method [57, 58], ever y line profile that makes up the image is Fourier transformed, the power sp ectrum is analyzed, and power spectra are averaged for all the lines. The slope of the log-log gra ph of the power spectra is equal to and D is determined from the equation: 22 7 D (3-15) WSxM is another software program used for data acquisition and processing in scanning probe microscopy [60]. WSxM uses a modified slit-island method, called flooding, to determine the fractal dimension for AFM images. A plane para llel to the fracture surface is used to section the fracture surface image into two halves, producing islands of the fracture surface. The loglog graph of the perimeter versus the area for all islands is calculated and the slope of the graph is 2/D. The number of islands measured by WSxM can be changed by increasing or decreasing the size of the islands recognized by the program and the height of the sectioning plane. 36

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To determine the optimal scan areas for determining the fractal dimension, a test was performed measuring the change in fractal dimens ion as a function of scan area. The mirror, mist, and hackle regions of the fracture surface of baria silica glass ba rs (30mm x 4mm x 3mm) were Vickers indented at a 500 gf load, fractured in three-point flexure, and scanned at various scan areas (Table 3-3). Baria silica glass was us ed because it has a similar D value to that of SLS glass [D = 0.10 0.01][61][39]. The values of D were then compared with the fractal dimension of baria silica glass calcul ated using the slit island method [61]. Using the optimal scan areas, the fractal dimension of the veneer, core, and the two sets of veneer/core bilayer bars was measured and compared. Statistical Analysis All statistical analyses were conducted using KaleidaGraph, a graphing and data analysis software package. ANOVA was used to determ ine if there was a statistical difference or similarity between the means of the variables (aspect ratio, fracture strength, fracture toughness, and fractal dimension) in the expe riments. A significance level ( ) of 0.05 was used in all ANOVA tests. A p-value less than (p 0.05) denotes a statistically significant difference in the means. A p-value greater than (p > 0.05) indicates that the means are statistically the same. All reported error values are the st andard deviation for those values. 37

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EC EV 4.6 mm 2.1 mm 1.1 mm 2.1 mm 1.1 mm 3.1 mm EC EV A B Figure 3-1. Cross-section of b ilayer specimens. A) Bilayer before transformation. B) Bilayer after transformation. F Figure 3-2. Diagram of distances used for computation of centroid for bilayer specimens. B EC EV C E A D y-axis y z-axis 38

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e b EC EV a y-axis g h c f z-axis d Figure 3-3. Diagram of distances used for computation of moment of inertia. Table 3-1. Chemical composition of IPS e.max Press [Adapted from Bhler-Zemp P, Vlkel T (2005) Scientific Documenta tion IPS e max Press:1] Standard Composition (wt. %) SiO2 57.0 80.0 Li2O 11.0 19.0 K2O 0.0 13.0 P2O5 0.0 11.0 ZrO2 0.0 8.0 ZnO 0.0 8.0 + other oxides 0.0 10.0 + coloring oxides 0.0 8.0 Table 3-2. Chemical composition of IPS e.max Ceram [Adapted from Bhler-Zemp P, Vlkel T (2005) Scientific Document ation IPS e max Ceram:1] Standard Composition (wt. %) SiO2 60.0 65.0 Al2O3 8.0 12.0 Na2O 6.0 9.0 K2O 6.0 8.0 ZnO 2.0 3.0 + CaO, P2O5, F 2.0 6.0 + other oxides 2.0 8.5 + Pigments 0.1 1.5 39

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40 Table 3-3. AFM scan areas used to determ ine optimum scan areas for fractal dimension measurements of fracture surface of baria silica glass. X mark AFM scan areas used to measure fractal dimension for a given region of the fracture surface. Scan area, ( m)2 Mirror Mist Hackle 60 X X X 40 X X X 20 X X X 10 X X X 5 X X X 2 X X X 1 X X X 0.75 X X X 0.50 X X 0.30 X X 0.10 X

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CHAPTER 4 RESULTS AND DISSCUSION In this chapter, the microstructure for the de ntal materials is charac terized. The fracture strength, apparent fracture toughne ss, work of fracture, and fractal dimension are determined for the monolithic dental materials and the bilayer composites. From these results, the hypotheses discussed in Chapter 1 are answered. Microstructural Analysis Phase Identification X-ray diffraction (XRD) was conducted for the I PS e.max Press (EC) glass-ceramic core material and IPS e.max Ceram EV glassy ven eer. XRD was preformed on the EC to determine if lithium disilicate is the main crystal in the glass ceramic. XRD was performed on the EV to determine if any crystal phases existed in the glass veneer. The XRD data (Figure 4-1) showed that EC c ontains crystals. Peaks at diffraction angles of 23.9 and 24.4 correspond with two of the th ree highest peaks for li thium disilicate (PDF 400376; Li2Si2O5). This data, along with the knowledge of the fabrication of EC, confirms that EC is composed mainly of lithium disilicate crysta ls and is a glass-ceramics using the definition stated in Chapter 2 for glass-ceramics. The XRD data (Figure 4-2) shows the pattern for an amorphous structure. This implies that EV may be a glass and doe s not contain crystals. SEM mi crographs (Figure 4-3) of the veneer showed what could be nanoparticles, wi th a number average diameter of approximately 100 nm, that exist in the veneer. It is possible that volume fracti on of the particles in the glass matrix is less than the detection limit for XRD and that this low volume fraction may be responsible for the small peaks exhibited in the XRD spectra at 25.9, 32.0, and 34.0. These particles have a spherical shape when compared to the needle-like crystals reported by Ivoclar 41

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(Figure 4-4) [6]. The difference in shape may be due to a difference in heat treatment, which would affect the growth of crys tals. Work by Hland et al. has shown that temperature and time does affect crystal growth for leucite glass-cer amic, with the length of the crystals being proportional to time at elevated temperatures [62]. Energy disp ersive X-ray sp ectroscopy (EDS) was also conducted on the veneer specimens to determine the composition of the particles (Figure 4-5). There was a smaller amount of fluorine (~ 1 wt. %) and an equivalent amount of calcium (~ 3 wt. %) present in the spectra co mpared to the amount reported by the manufacturer (Table 3-2). Based on the EDS and XRD data, it is inconclusive if the particles present in the veneer specimens are the nanofluorap itite crystals described by Ivoc lar [6]. These findings are in agreement with the work done by Tsalouchou et al. [63] in which the nanofluorapitite crystals could not be identified for e.max Ceram. More analysis, e.g., determine the lattice planes of the particles from TEM or determining the enth alpy of melting of the pa rticles from DSC or TGA, is needed to determine the identity of the particle observed in SEM images. Aspect Ratio The aspect ratio for the lithium disilicate cr ystals in the EC glass-ceramic was measured from SEM micrographs and compared to the aspect ratio of EC report by Holand [2] and Ivoclar [6]. Using the original etchi ng procedure of 2.5 % HF for 10 s did not etch the glass-ceramic well enough to measure the aspect ratio (Figure 46). To determine the best method to etch the glass-ceramic, a 2.5 % HF solution was used to etch one polished specimen for 30 s and a 30 % sulfuric acid (H2SO4)/4 % HF solution [2] was used to etch another polished specimen for 10 s. SEM micrographs were taken of both speci mens at 5000x (Figure 4-7) and 10000x (Figure 4-7). It was determined that the H2SO4/HF solution provided the best etching to measure the aspect ratio. The aspect ratio for the lithium disilicate crystals was 7.6 1.1. The number 42

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average length for the crystals was 5.2 2.0 m and the number average width was 0.7 0.3 m. This is in agreement with the crysta l lengths of 3 6 m reported by Hland [2]. Fracture Strength The flexural strength was determined for EC EV, the veneer indented bilayer (BV), and the core indented bilayer (BC) to characterize an d compare the dental materials and to calculate the apparent fracture toughness. In order to calculate the flexur al strength for the bilayers, composite beam theory was used to transform th e bilayers because of a difference in elastic modulus between the two materials. Based on the elastic moduli of th e two ceramics (96 GPa for EC [52] and 64 GPa for EV [52]), the transfor mation factor was calculated to be 1.5. This transformation factor results in a virtual change in the width of EV from 4.6 mm to 3.1 mm (Figure 3-1). Using Equation 32 and Figure 3-2, y was determin ed for the BV (~ 1.14 mm) and BC (~ 0.94 mm). Using Equation 3-3 and Figure 33, the average moment of in ertia was calculated to be 2.7 mm4. Using the calculated moment of inertia, Equation 3-4 and 3-5, was calculated for each bilayer (Table 4-1). The measured flexural strength of EC is greate r than that of EV. This is expected for the EC due to the presence of the needle-like lithium disilicate crystals that increases the strength of the materials due to crack deflection. It is expect ed that the fracture streng th of both bilayers will have a flexural strength greater than EV and le ss than EC based on the Ru le of Mixtures [64]. Using the Rule of Mixtures, the fracture strength of the bilayer ( B) was determined from the equation ECEC EVEVBAA (4-1) 43

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where EV is the fracture strength of the veneer, AEV is the area fraction of the veneer in the bilayer, EC is the fracture strength of the core, and AEC is the area fraction of the core. Using Equation 4-1 implies that the strength of each layer can be superimposed to determine the strength of the composite beam. For the bilayer, B = 94 MPa. This shows that the fracture strength of the bilayers should be between the values for the fracture strength of the veneer and core. Table 4-1 shows that the fracture streng th for BV and BC does lie between the fracture strength of the veneer and core. Therefore, this implies that the location of the crack determines the fracture strength of the bilayer. Fracture Toughness The apparent fracture toughness (KC) was determined for EC, EV, and the two sets of bilayers (BV and BC) to compare to measured va lues reported in the literature. The apparent fracture toughness was determined using two met hods: strength indentation and fractography. From Equation 3-6 and 3-7, KC was calculated for each method and displayed in Table 4-1. For Equation 3-7, the elastic modulus and hardness used was 64 GPa [3 ] and 5.4 GPa [6] for EV and 96 GPa [3] and 5.5 GPa [6] for EC. Both sets of KC values for EC agree w ith the values reported by Ivoclar [6] and are statistically similar (p > 0.05). Both sets of the apparent fracture toughness values for EV are not statistically the same (p < 0.05) but are within the range of the KC value reported by Taskonak [4]. The differe nce in the mean of th e apparent fracture toughness for the bilayers was not statis tically significant (p > 0.05). KC for the bilayers was greater than KC for the veneer monolithic specimens and less than that for the core monolithic specimens. The implication of these results is th at the apparent fracture toughness of a bilayer is dependent on the location of crack initiation in the b ilayer. Therefore, if the initiating crack is in the veneer, then the fracture toughness of the ve neer controls the strength and toughness of the composite. 44

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Work of Fracture Soda-Lime-Silica Glass The work of fracture (WOF) was determined for EC, EV, and the two sets of bilayers to determine if the fracture energy could be estimated for the dental ceramics. The work of fracture was calculated in two different ways : using the cross-sectional area WOF(CS) (Equation 2-1) and using the actual fracture surface area WOF(F) (Equation 3-10). In order to test the accuracy of using Equation 3-10 to estimate the fr acture energy of a brittle material, WOF was determined for soda-lime-silica (SLS) glass for uns table and stable crack growth. An atomic force microscope (Dimension 3100, Veeco Instruments Inc., Woodbury, NY) was used to determine the actual fracture surf ace area of the SLS glass beams. Using the AFM software, the surface area was determined from a grid of triangles of a fixe d side length, equal to the distance between pixels, placed over the scan image. The area of all the triangles was measured and all of the areas were added togeth er to calculate the surf ace area for the image. Five square scan areas ((1 m)2, (750 nm)2, (500 nm)2, (375 nm)2, and (250 nm)2) were used to measure the fracture surface area for th e CN specimens while four [(500 nm)2, (150 nm)2, (50 nm)2, and (25 nm)2] were used to measure the unstable fracture specimens. All scans were conducted using a tapping tip at a scan rate of 34 Hz and at a resolution of 512 lines per scan. Equation 3-13 was used to determine the fractu re surface area. Examples of the AFM scan images of the mirror, mist, and hackle region are illustrated in Figure 4-10. The fracture surface area versus the scan area was graphed for the stable crack growth chevron-notched (CN) (Figure 4-11), and the unstable indented specimens (Figure 4-13), to determine if a relationship existed between the two variables. A number of curve fits were applied to all the data. The best fit was to a power law relationship, i.e., 45

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B FAA (4-2) where is the scan area side lengt h, and A and B are constants, existed between the scan area and AF. Equation (4-2) is similar to the profile length measurement equation (Equation 2-3) and the direct area measurement equation (Equation A-8) It is assumed here that fracture surfaces are fractal in nature, in agreement with many studi es in the literature [3840, 42-47]. Therefore, Equation 4-3 would be expected to be similar to Equations 2-3 and A8, since the fracture surfaces are fractal. Based on this the fracture surface area should in crease as the scan area side length decreases. In order to determine the actual fract ure surface for a specimen, a maximum AF must be determined for that specimen. Since there is no mathematical limit for Equation 4-2, a physical limit is necessary based on some physical restrain ts. There are three po ssible solutions for the size limitation: probe tip radius, equilibrium bon d length, and the free volume diameter. The physical limit for determining AF is based on the AFM probe tip radius. The AFM can only resolve features equal to or greater than th e tip radius. For the ta pping tip used in these scans, the tip radius was 8-10 nm. Using the tip radius of 8 nm as results in an AF of 6.3 0.3 mm2 (Figure 4-12). Theoretically, a minimum limit could be the equilibrium bond length for the materials. For the soda-lime-silica glass, which is composed mostly of silica, th e equilibrium bond length selected was that of Si-O, which is 0.16 nm [65]. Using the equilibrium bond length as generates an AF of 7.1 0.5 mm2 (Figure 4-12). Another limit is based on the concept of fr ee volume. Inorganic glasses have open space due to the disorder in the glassy structures. These open spaces are known as free volume. Swiler et al [66] suggested from mo lecular dynamics calculations that it was these free volume areas 46

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that controlled fracture in silica glass. The estimated size of the region was represented by a radius, a0. West et al. [44] also estimated th is region using a different approach. a0 was determined to be approximately 1.2 nm. The projection of AF based on AFM measurements at a of a0 is 6.7 0.4 mm2 (Figure 4-12). Using these three limits, Figure 4-12 shows that the fracture surface area is does increas e as the side length of the s can area decreases and that the fracture surface area is larger th an the cross-sectional area. Using the three previously mentioned limits, WOF was calculated for the CN specimens. From Table 4-2, all the values are less than the value for WOF measured using ACS (Equation 2-1) and for the fracture energy measured from traditional fracture mechanics tests [ = 3.5 J/m2] [67]. The selection of the limit does not greatly affect the results There is only a 13 % change for WOF between using the tip radius and the equilibrium bond length. Thus, our choice of limits is not critical to determining WOF, just that there is a cutoff length of atomic dimensions. Using the equilibrium bond length as the limit for the scan area side length, AF was determined for the unstable crack growth speci mens (Table 4-3) by extrapolating to the equilibrium bond length. The values for WOF(AF) are similar to the theoretical fracture energy ( ) for fused silica of 1.75 J/m2 calculated by Charles [65] base d on the equilibrium bond length of silica and less than the fracture energy measured from traditional fracture mechanics test [ = 3.5 J/m2] [67]. The results of both experiments c onfirmed that the actua l fracture surface area provides a lower limit of the work of fracture an d that the fracture energy required to break bonds can be estimated using the work of fracture. To obtain an accurate value for the fracture energy, the entire fr acture surface generated during crack growth should be meas ured or estimated. This furt her implies that for any fracture surface generated during fracture, the actual fracture surface area should be measured or 47

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estimated to obtain a more accurate value of th e energy required to break atomic bonds. The experiments also show that because of the lowe r limit, using traditional fracture mechanics tests to calculate do not represent the minimum energy needed to rupture an atomic bond and create the fracture surface. Dental Materials After showing that WOF(F) can be used to estimate the fracture energy of SLS glass, the fracture surfaces of EC, EV, and the two sets of bilayers were measured using the AFM to determine if the fracture energy could be estimate d for these materials. The crack introduced in the veneer and core monolithic specimens, from Vickers indentation, resulted in a fracture surface with no mist or hackle present on the surface. Therefore, only the mirror portion of Equation 3-13 was necessary to calculate AF for the specimens and N(SS) is equal to the crosssectional area of the bars. Using the same method to determine WOF(AF) in the SLS glass specimens, the scan areas were graphed versus the fracture surface area for EV (Figure 4-14) and EC (Figure 4-15) over scan areas of (250 nm)2 to (25 nm)2. Since the veneer and the core consisted mainly of silica, the equilibrium bond length of silica [0.16 nm] was us ed in determining the minimum scan area. Based on the scan areas used, WOF(F) for EV was 5.2 1.4 J/m2 and 33 17 J/m2 for EC. Although these two values are signi ficantly greater than the theore tical value for silica of 1.75 J/m2, the values are less than the values reported for WOF(CS) (Table 4-4). The values for WOF(F) may be due to the small samp ling of fracture surface area (AF) determined from the AFM data. On further analysis, AF for the veneer and core appears to have a different level of roughness on a scale of hundreds of micrometers. To see if the roughness increases AF of the veneer and core, the fracture surface of the two materials was m easured using an optical profilometer (WYKO NT1000, Veeco Metrology Group, Tucson, AZ). The fracture surface was examined at two 48

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different scan areas: 0.270 mm2 (objective lens of 5x and field of view lens of 2x) and 0.503 mm2 (objective lens of 5x and field of view lens of 1.5x). The scan area of 0.270 mm2 resulted in AF being 13 % greater than the cross-sectional area (ACS) measured for the veneer and 28 % greater for the core. For the scan area of 0.503 mm2, there was a 10 % increase in AF for the veneer and a 21 % increase in AF for the core. Using the data from the large scan area and applying this information to th e data acquired from the AFM, WOF(F) decreased to 2.9 0.8 J/m2 for the veneer and 26.1 13.4 J/m2 for the core. Although there was a decrease in WOF(F), it is still greater than that of the theoretical value for silica. One likely cause for the larger WOF(F) values is the speed of the crack for the unstable crack growth specimens. The speed of the crack does not allow the crack to follow the path of lowest energy. This results in more energy bei ng supplied than what is required to fracture a material when compared to the fracture surface area created during fracture, i.e., A U (4-1) where U is the elastic energy stored in system and A is the fracture surface area [24]. But, since the crack velocity was not measured for the specim ens, it is inconclusive if the crack velocity affects the fracture surface area. After WOF for the monolithic dental ceramics, WOF for the veneer-indented (BV) and coreindented bilayer (BC) was calculated using the same method as for the monolithic materials. The scan area side length versus AF was plotted for BV (Figure 416) and for BC (Figure 4-13). For BV, WOF(F) is 5.6 1.4 J/m2 and for BC it is 25 6 J/m2. These values follow the trend of the values for the fracture strength and apparent fracture toughness, with WOF for BV being greater than that for EV and WOF of BC being less than that for EC. Although optical profilometry was not used in determining AF for the bilayers, WOF for BV and BC would 49

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decrease based on the results from the measurement of AF on the monoliths. Based on the values for WOF(F) for the bilayers, it was shown that WOF(F) is similar to that of the material in which the initial crack that caused fracture of the bilayer is located. For fracture in the bilayer, in which fracture initiated in the tougher material (core), WOF(F) is similar to that of the tougher monolith (core). For fracture in the bilaye r, in which fracture initiated in the less tough mate rial (veneer), WOF(F) is similar to that of the less tough m onolith (veneer). These results based on WOF(F) further indicate that selection of both materials in a bilayer is important when designing ceramic composites. Fractal Dimension Soda-Lime-Silica Glass The fractal dimension (D) of e.max Press (E C), e.max Ceram (EV), and the two set of bilayers of EC/EV were measured using an AFM and the two software packages of Gwyddion and WSxM to determine if location of crack ini tiation changes D for the veneer and core in the bilayers. To determine the accuracy of D m easurements using these two programs, D was measured for soda-lime-silica (S LS) glass bars. Three SLS ba rs (45mm x 4mm x 3 mm) were indented with a Vickers indent load of 2 kgf and fractured under four-point flexure at a crosshead speed of 0.5 mm/min. The mirror, mist, and hack le regions of the fracture surface were scanned using 6 scan areas [(100 nm)2, (250 nm)2, (375 nm)2, (500 nm)2, (750 nm)2, and (1 m)2]. Table 4-5 shows the fractal dimension increment (D* ) values calculated from the two programs. For the Gwyddion software, only th e triangulation method and the cube counting method were used to measure D*. The varian ce and power spectrum methods were not used because there was too much scatter in the data points used to determine the slopes, producing inconsistency in the D* values calculated for these methods. The difference in mean values between the Gwyddion and WSxM software were not statistically signifi cantly for the hackle 50

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region (p > 0.05). For Gwyddion, the triangulation and cube counti ng were significantly similar for the mist and hackle region but not for the mirror region (Table 4-5). The mist and hackle values agree with a D* of 0.10 0.01 for glasses [61, 68] determined using the slit island method. Thompson et al. [ 69]showed that for a lithia-disilicate ceramic, determining D* from the slit-island method and AFM results in the same value of 0.23. Both the Gwyddion and WSxM software showed an increase in D* for the mirror region when compared to D* for the mist and hackle regions. Kula wansa [38] had conducted work on measuring D of SLS glass near the mirror-mist boundary. His work determined that D* was 0.17 0.08 with some values in the rougher regions as high as 0.40 If the fractal dimensions of the three regions measured using the Gwyddion and WSxM software are averaged, the values for the Gwyddion software [Triangulation 0.18 0.10; cube counting 0.17 0.09] agree with Kulawansas measured value for D* while WSxM [0.30 0.25] does not. Although WSxM produces a greater D* value for the mirror region, the key point is that the fractal dimension of the mirror region is great er than the fractal dimension of the mist or hackle region. This may be due to the roughness in the mirror region being less than that in the mist or hackle region. Wiederhorn et al. [70] s howed that for the mirror region of SLS glass, the root-mean-square (RMS) roughness d ecreased as the crack velocity increased. The roughness exponent, which is related to the fractal dimension, also increased as the RMS roughness decreased for SLS glass. These results imply that the increase in the D* for the mirror region is not due to any algorithm errors in Gwyddion or WSxM, but rather due to a change in the material itself. To further examine if the greater D* valu es for the mirror region measured using the software program is due to the algorithms used in the programs, five line profiles from five AFM 51

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scans [(1 m)2, (750 nm)2, (500 nm)2, (375 nm)2, and (250 nm)2] for the mirror region of SLS glass were used to determine D*. The Ri chardsons method was used to measure D* by measuring the length of each profil e as a function of step size (50, 40, 30, 20, and 10 mm). D* from the profiles was 0.35 0.08. This value is similar to the values measured using the triangulation and cube counting method. This sh ows that D* determined from the algorithms used to calculate D* in Gwyddion do not result in a significant change in the D. Based on this, it was determined that the increase in the D* values for the mirror region is not caused by the software programs used to calculated D*. Along with greater value for D* for the mirror region calculate d from the WSxM software, there are several other problems with using WSxM The number of variables that are used in calculating D*, e.g., the height of the sectioning plane and the minimum island area, results in an inconsistency in the calculation of D*. The island area and number of islands that is necessary to accurately calculate D* has not been determined for WSxM. Based on this, WSxM was not used as a tool to measure the fractal dimension of th e brittle materials in th is study. All fractal dimension values reported from this point on were determined from the Gwyddion software program. Baria Silica Glass Using Gwyddion as the software to measure D* the effect of AFM scan areas on D* was tested. Baria silica glass (3BaO-5SiO2) bars (25mm x 4 mm x 2 mm ) were used in this experiment since baria silica glass has the same fractal dimension as other glasses [0.10 0.01] [61]. The baria silica bars were fractured in three-point flexur e, at a crosshead speed of 0.5 mm/min., using a Vickers indent of 500 gf to initiate a crack. The fractal dimension was determined for the fracture surface of the mirror, mist, and hackle regions at different scan areas 52

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from (100 nm)2 to (60 m)2 and compared to D* calculated fro m the slit-island technique [0.10 0.01] [61]. Figure 4-17 confirms that D* does change wh en the scan area change. The fractal dimension increment decreases as scan area decreases for the mirror, mist, and hackle region. As the scan area increases, the time is takes for the tip to scan a line increa ses causing a decrease in the time the tip has to measure th e features of the fracture surface. This results in a decrease in resolution and a decrease in the measurement of the surface area. Th e measurement of the surface area using AFM is also limited by the tip ra dius of the probe. The probe can only resolve features that are larger than th e tip radius. Based on this data, scan areas less or equal to (750 nm)2 for the mirror region, less than (2 m)2 for the mist region, and between (1 m)2 and (20 m)2 for the hackle region would be the ideal scan areas used to measure the fracture surface for glasses using AFM. Dental Materials With the limits of the AFM determined for D* measurements, D* was measured for the e.max Press (EC) and e.max Ceram (EV) mo nolithic bars as a control group in order to establish if a change in D* exist in the bilaye r bars. The fracture surf aces (Figure 4-18) of the three bars each of EC and EV were measured at 250, 375, 500, 625, and 750 (nm)2 to determine D*. Table 4-6 shows the average D* for the dental core and veneer. D* values for the veneer are similar to that of SLS and baria silica glass. This is expected for the veneer since it is mostly glass, based on the previous XRD analysis, and the fracture surface is all mirror. Therefore, the veneer should have a fractal dimension similar to that of the mirror region of other silicate-based glasses. The fractal dimension for the core was less than D* values reported for other lithiadisilicates [0.25 [71], 0.24 [69]]. This diffe rence in D* may be caused by a difference in the microstructure of the lithia-disilicates, a differe nce in the scale at which D* was calculated for 53

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the lithia-disilicates compare to the scale used in this experiment, or th at the fracture surface of the core is mostly mirror for EC and different from the fracture surf ace of the other glassceramics. Thompson [69] measured D* for fully crystallized lithia-disilicate. The amount of crystallinity may have resulted in the greater value in D*. Naman [72] measured the fractal dimension of lithia-disilicate glass and lithia disilicate glass-ceramic at 10, 33, and 95 vol. % crystallinity using the slit-island method. Naman showed that for the mirror region, D* was 0.03 in the glass, 0.05 in the 10 vol. % glass-ceramic, 0.09 in the 33 vol. % glass-ceramic, and 0.16 in the 95 vol. % glass-ceramic. The value for the 33 vol. % glass-ceramic agr ee with the values for the core (EC). Therefore, the difference in th e D* measurements is mostly like due to the fracture surface of the core (EC), which has a frac ture surface that is mostly mirror. Future investigations should be conducted to determine the cause of the fractal dimension values for the ceramics in the mirror region. One direction to explore for future research is the effect that the crack velocity has on the fractal dimension. There was no significant difference (p > 0.05) for either the cube c ounting or triangulation values at the different scan areas. This means that D* was not affected for the range of scan areas used. The cube counting and triangulation values for EV and EC were not significantly similar (p 0.05) to each other though. As Table 4-4 shows, D* from triangulation is greater than the cube counting values. This is because th ere is a better linear fit to the data used to calculate D* for cube counting than triangula tion resulting in a lower but more accurate calculation of D* from cube counting. With the D* values determined for the mono lithic materials, D* was calculated for the bilayers to determine if D* changes in the material in which the initial crack is not located. The fracture surfaces of the three veneer-indented bila yer bars (BV) and three core-indented bilayer 54

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bars (BC) was measured at 250, 375, 500, 625, and 750 (nm)2 to determine D*. Table 4-7 shows the average D* for BV and BC. The difference in the means of triangulation and cube counting D* values for the veneer and core materials in th e BV and BC bilayers were statistically different (p < 0.05). As previously mentioned, this is due to how D* is calculated for the triangulation and cube counting methods. For BV and BC bilayers, D* for the veneer and core were statistically similar (p > 0.05) to D* for the monolithic veneer and core. As mentioned previously, D* for the core was different from D* determined from ot her researchers. Although the D* values were different, the key point is that D* is the same for the core materi als in the bilayers and in the monolith. To determine if the crack propagation was continuous or if a ne w crack initiated and propagated in the non-indented materials in the bilayer, SEM was used to examine the interface of the bilayer. Figure 4-19 shows examples of the fracture surface of the veneer-indented bilayer (BV). Figure 4-19B indicates that the wake hackle marks generated by the pores in the veneer layer, which are due to processi ng, travel through the interface into the co re. This observation confirms that the materials in the bilayer we re bonded well since there was no indication of delamination or secondary crack initiation at the interface. Thus, the fracture process was continuous from the veneer, where the indented induced flaw is locate d on the tensile surface, into the core material. Figure 4-20 shows examples of the fractu re surface of the core-indented bilayer (BC). Figure 4-20B shows twist hackle marks present in th e veneer at the interface. The presence and direction of these twist hackle marks indicate that the crack propagated continuously through the bilayer, starting in the tensile surface of the core-indented layer and propagating across the interface into the veneer. 55

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Based on the D* information for the bilayers, the location of crack initiation and propagation does not change the frac tal dimension of either of the materials in the bilayer when compared to their monolithic counterparts. This means that the fracture surface of the materials did not change based on the location of initial crack initiation. This confirms that for the ceramic bilayer specimens tested in flexure, the fracture process does not change th e fractal dimension of the material in which the initial indented-induced flaw is not located. Th e fact that the fracture surface of the ceramic materials does not change supports the idea that the fractal dimension is an intrinsic property of the material. With the fractal dimension being shown to be a property of the material and not determined by the fracture pr ocess, the fractal dimension can be used as another property to describe fracture in the ceramic materials. 56

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Table 4-1. Fracture strength, fracture toughness determined from strength indentation (SI), and fracture toughness determined fr om fractography (F) for e.max Ceram (EV), e.max Press (EC), bilayer with crack initiati on in veneer (BV), and bilayer w ith a flaw produced in the core (BC) Specimen No. of Specimens MPa KC (SI), MPa-m1/2 KC, (F), MPa-m1/2 EV 5 49 3 0.83 0.04 0.74 0.07 EC 5 155 23 2.46 0.27 2.25 0.34 BV 5 67 11 1.14 0.07 1.21 0.14 BC 4 142 15 2.30 0.18 2.25 0.18 Table 4-2. Work of fracture values calculated for chevron notch specimen s using cross-sectional area (ACS) and actual fracture surface area based on probe tip radius, equilibrium bond length, and free volume a0. Actual area is projected area using equilibrium bond length. Cross-sectional area (mm2) Actual area (mm2) WOF (for ACS) (J/m2) WOF (Tip Radius) (J/m2) WOF (Bond Length) (J/m2) WOF (ao) (J/m2) 5.4 0.3 7.1 0.5 3.5 0.1 3.0 0.1 2.7 0.2 2.6 0.2 Table 4-3. Comparison of work of fracture and fracture toughne ss, based on work of fracture, for unstable crack growth soda-lime-silica glass specimens No. of specimens Actual Area (mm2) W, mJ WOF (ACS), J/m2 WOF, J/m2 KC, MPam1/2 3 mm 7 860 76 2.7 0.6 113 30 1.6 0.4 0.46 0.06 10 mm 1000 gf 7 4002 161 8.0 2.0 74 0.9 0.1 0.35 0.02 10 mm 2000 gf 8 4438 98 6.5 0.8 57 7 0.8 0.1 0.32 0.02 Table 4-4. Work of fracture WOF, based on crosssectional area (ACS) and fracture surface area (AF) for veneer, core, and bilayer bars Specimen No. of specimens W, mJ WOF (ACS), J/m2 WOF (AF), J/m2 WOF (Adj. AF), J/m2 EC 3 3.9 1.2 212 63 33 17 26.1 13.4 BC 3 3.6 0.7 190 37 25 6 EV 3 0.6 0.1 31 4 5.2 1.4 2.9 0.8 BV 3 0.9 0.3 49 16 5.6 1.4 57

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Table 4-5. Fractal dimensional increment for soda-lime-silica glass fracture surface regions measured using atomic force microscopy and the Gwyddion and WSxM software Fracture Surface Region Gwyddion Gwyddion WSxM Triangulation Cube counting Mirror 0.29 0.05 0.26 0.05 0.60 0.19 Mist 0.13 0.07 0.12 0.06 0.19 0.12 Hackle 0.12 0.06 0.11 0.06 0.10 0.07 Table 4-6. Fractal dimensiona l increment for e.max Ceram (EV) and the e.max Press (EC) ceramics measured using AFM and Gwyddion software Specimen Triangulation Cube Counting EV EC 0.27 0.11 0.12 0.04 0.22 0.09 0.09 0.03 Table 4-7. Fractal dimensional increment for bilayer specimens using AFM and Gwyddion Specimen Triangulation Cube Counting Veneer Indented (BV) EV 0.29 0.07 0.22 0.06 EC 0.12 0.05 0.10 0.04 Core Indented (BC) EV 0.31 0.10 0.24 0.08 EC 0.12 0.03 0.09 0.03 58

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0 5000 10000 15000 20000 25000 30000 35000 40000 10 20 30 40 50 60Intensity2 23.9 24.4 Figure 4-1. X-ray diffraction patte rn for e.max Press (EC) core ceramic showing that lithium disilicate are the crystals present in the gl ass-ceramic. Peaks at diffraction angles of 23.9 and 24.4 correspond with two of the largest peak intensity values for lithium disilicate (PDF 40-0376). 59

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1500 2000 2500 3000 3500 4000 4500 10 20 30 40 50 60Intensity2 25.9 32.0 34.0 Figure 4-2. X-ray diffraction pa ttern of e.max Ceram (EV) ve neer ceramic showing veneer material has a amorphous pattern. Peaks in pattern at diffraction angles of 25.9, 32.0, and 34.0 could represent peaks for calcium fluoride phosphate (PDF 150876). 60

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A B Figure 4-3. SEM micrographs of polished and etched surface of e.max Ceram (EV) veneer ceramic. Surface etched using 2.5 % HF solution for 10s. A) Magnification of surface at 500x. B) Magnification of box in Figure 4-3A at 5000x. C) Magnification of box in Figure 4-3B, at 30000x, showing presence of nanoparticles. 61

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C Figure 4-3. Continued Figure 4-4. SEM image of polished and etched surface of e.max Ceram (EC) core ceramic showing the presence of n eedle-like nanofluorapatite crystals. Crystals are approximately 300 nm in length and 100 nm in diameter. [Image courtesy of Ivoclar Vivadent.] 62

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3.0 wt. % 0.9 wt. % Figure 4-5. Energy dispersive X-ray spectroscopy (EDS) of e.max Ceram (EV) veneer ceramic. Amount of Ca (~3 wt. %) presen t agrees with composition of veneer. Amount of F (~1 wt. %) present does not agree with composition of veneer. Figure 4-6. SEM micrograph of polished and etched surface e.max Press (EC) core ceramic at a magnification of 10000x. Surface etched us ing 2.5 % HF solution, which did not provided enough contrast to measure th e aspect ratio of the crystals. 63

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A Figure 4-7. SEM micrograph of polished and etched surface e.max Press (EC) core ceramic at a magnification of 5000x. A) Surface was etch ed using 2.5 % HF solution for 30s at room temperature and did not provide enough contrast to measure the aspect ration of the crystals. B) Surface etched using 30 % H2SO4/4 % HF solution for 10 s at room temperature. Etchant provided enough contra st to measure the aspect ratio of the lithium disilicate crystals. 64

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B Figure 4-7. Continued 65

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A Figure 4-8. SEM micrograph of polished and etched surface e.max Press (EC) core ceramic at a magnification of 10000x. A) Surface was etch ed using 2.5 % HF solution for 30s at room temperature and did not provide enough contrast to measure the aspect ration of the crystals. B) Surface etched using 30 % H2SO4/4 % HF solution for 10 s at room tempertaure. Etchant provided enough contra st to measure the aspect ratio of the lithium disilicate crystals. 66

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B Figure 4-8. Continued Figure 4-9. SEM image of fracture surface of chevron-notched soda-lime-silica flexure specimen showing that fracture surface is all mi rror and no mist or hackle is present. [Image courtesy of Dr. Jia Hua Yan]. 67

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A B C D E F Figure 4-10. AFM height and amplitude scan imag e of soda-lime silica gl ass fracture surface at a scan area of (750 nm)2. AFM images show that the roughness of the surface increases from the mirror region to the hack le region. A and B) Mirror region is rough on a nanometer scale. C and D) Mi st region. E and F) Hackle region. 68

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5 5.2 5.4 5.6 5.8 6 2004006008001000 AF = 6.6 -0.029AF, mm2Scan Area Side Length ( ), nm Figure 4-11. Fracture surface area (AF) as a function of scan area side length ( ) for chevronnotched soda-lime-silica bars. AF is shown to increase confirming that AF is greater than cross-sectional area. Circles represent average AF from five scans at that particular scan area. Error bars represent standard deviation for AF values. 69

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1 10 0.11101001000AF, mm2Scan Area Side Length ( ), nmEBL FV TR 7.1 6.7 6.3 Figure 4-12. Projection of the average fracture surface area (AF) from the average AF values for the chevron-notched (CN) specimens based on AFM data. Using the limits of the equilibrium bond length (EBL), free volume (FV), and tip radi us (TR) result in an increase in the AF, with EBL producing the largest AF value. Circles represent average AF of all the CN specimens. Error bars represent standa rd deviation for average AF values. 70

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0 50 100 150 200 250 20406080100120140160 3 mm 10 mm 1000 gf 10 mm 2000 gf AF = 300 -0.61 AF = 1400 -0.62 AF = 1600 -0.62AF, mm2Scan Area Side Length, nm Figure 4-13. Fracture surface area (AF) as a function of scan area side length ( ). AF is shown to increase confirming that AF is greater than cross-sectional area. Data points represent average AF each specimen in at a pa rticular scan area. Error bars represent standard deviation for AF values. 71

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8 10 12 14 16 18 20 22 24 050100150200250300 AF, mm2 Adj. AF, mm2 AF= 36 -0.25 AF = 41 -0.25AF, mm2Scan Area Side Length ( ), nm Figure 4-14. Fracture surface area (AF) and adjusted fracture surface area (Adj. AF) as a function of scan area side length ( ) for e.max Ceram (EV) veneer bars. Adjusted surface are determined from optical profilometry. Data points represent average AF each specimen in at a particular scan area. E rror bars represent sta ndard deviation for AF values. 72

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8 10 12 14 16 18 20 050100150200250300 AF = 41 -0.26AF, mm2Scan Area Side Length, nm Figure 4-15. Fracture surface area (AF) as a function of scan area side length ( ) for e.max Press (EC) core bars. Data points represent average AF each specimen in at a particular scan area. Error bars represent standard deviation for AF values. 73

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10 12 14 16 18 20 22 050100150200250300 BV BC AF = 43 -0.26 AF = 41 -0.26AF, mm2Scan Area Side Length ( ), nm Figure 4-16. Fracture surface area (AF) as a function of scan area side length ( ) for veneerindent (BV) and core-inden ted (BC) bilayers. Data points represent average AF each specimen in at a particular scan area. E rror bars represent sta ndard deviation for AF values. 74

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0 0.1 0.2 0.3 0.4 0.5 0.0010.010.11101001000104 Mirror Mist HackleD* Scan Area, m2 Figure 4-17. Fractal dimension increment (D*) of baria silica glass, plotted against scan area, determined from Gwyddion for mirror, mist, and hackle region. The horizontal line represents D* of 0.1 determined from slit island method. 75

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A B C D Figure 4-18. AFM scan image of e.max Ceram (EV) veneer ceramic and e.max Press (EC) core ceramic fracture surface at a scan area of (750 nm)2. Color scale bar represents minimum and maximum heights in images. A and C are height images. B and D are amplitude images. Veneer has a fracture surf ace similar to soda-lime-silica. Core has a greater roughness than veneer at same scan area. 76

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A B Wake hackle Veneer (EV) Core (EC) Figure 4-19. SEM images of veneer-indented bilayer (BV) fracture surface. A) Low magnification. B) Wake hackle marks traveling from pore in veneer into core show that fracture of bilayer wa s a continuous process. White arrow indicates crack direction. 77

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78 A Indentation flaw Core (EC) Veneer (EV) B Twist hackle Wake hackle Figure 4.20. SEM images of core-indente d bilayer (BC) fracture surface. A) Low magnification. B) Twist hackle marks indi cate that fracture pr ocess was continuous. White arrows indicate crack direction.

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CHAPTER 5 CONCLUSIONS The objectives of this project were to 1) test the hypothesis that the work of fracture method, using the actual fracture surface area, can be used to estimate the fracture energy of ceramic materials; 2) test the hypothesis that th e work of fracture of a ceramic bilayer will be the same regardless of the location of crack init iation; and 3) test the hypothesis that the fractal dimension of the two materials in a bilayer will be the same regardless of the location of crack initiation. The first hypothesis was shown to be true. For soda-lime-silica glass, the work of fracture method using the actual fracture surface area resulted in a decrease in the work of fracture when compared to using the cross-sectiona l area. The work of fracture, using the actual fracture surface area, was shown to be less than the fracture energy determined from traditional fracture mechanics tests and similar to the theore tical fracture energy for silica. The fracture energy determined from traditional fracture m echanics tests do not accurately measure the fracture energy generated during the fracture process because the fracture surface generated during fracture is not accounted for in these tests. This is the first time it has been experimentally demonstrated that the actual fr acture surface area is criti cal to determining the minimum fracture energy. The work of fracture can be used to estimate the minimum fracture energy required to break the bonds in brittle materials. Therefore, the work of fracture can be used to determine the energy contributed in the crea tion of fracture surfaces in ceramics and will help engineers better understand the contributions to the total en ergy used to fracture a ceramic material. The second hypothesis was shown to be false. The work of fracture values for the bilayer specimens were similar to those of the monoliths in which the crack was initiated. This 79

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80 agreement implies that the work of fracture is de pendent on the material in which the initial flaw that causes fracture is located in a bilayer stru cture. The core, which had a greater measured work of fracture than the veneer, resulted in a greater work of fracture in the core-indented bilayer when compared to the veneer-indented bila yer. Based on this information, the work of fracture should be used as anot her method in the mechanical ch aracterization when determining the proper material for selecting and designing bilayer structures. The third hypothesis was shown to be false. The fractal dimension of the veneer in the core-indented bilayer was not different from monolithic veneer. The fractal dimension of the core in the veneer-indented bilayer was not different from the monolithic core. This indicates that the fractal dimension of the materials in the bilayer did not change based on the site of crack initiation and propagation and that the fracture surface of the materials did not change when compared to monoliths of the same materials. Based on this information, the fractal dimension in each layer of the bilayer structure does not cha nge due to the fracture process. Therefore, the fractal dimension is identified as being an intrinsic property of the fractur e surface of the ceramic materials.

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APPENDIX DETERMINIATION OF THE HURST EXPONENT FOR SILICA GLASS In this appendix, the Hurst exponent, a parame ter that is used to define the fractal dimension for self-affine objects, i.e., ceramic fracture surfaces, is explained. Background on the Hurst exponent value for silica gla sses is explored and the Hurst exponent is determined for the mirror region of soda-lime-silica glass. Background on the Hurst Exponent Mandelbrot and Passoja determined that fr acture surfaces are fractal [40]. A fractal object can be either self-similar or self-affine. A self-similar su rface is isotropic in magnification, i.e., in a three-dimensional Cartesian coordinate system, x, y, z x, y, z where is a scalar. A self-affine surf ace is anisotropic, i.e., x, y, z x, y, z where is called the Hurst exponent [73]. The Hurst expone nt is also known as the Hurst dimension, the roughness exponent, or the roughness index. Since the fracture surface of ceramics scale differently out-of-plane compared to in-plane ceramics are self-affine. For these ceramic fracture surfaces, there is a charact eristic length at which the self-affinity of a surface can be measured. This length is called the correlation length ( ). For lengths greater than approaches 1. For silica glass, is inversely proportional to crack velocity, i.e., decreases from 80 to 30 nm for crack velocities from 10-10 to 10-4 m/s [74]. The Hurst exponent, like the fractal dimensi onal increment (D*), is bound by the values 0 and 1, where a value of 1 and a D* value of 0 corresponds to a flat surface or line. The Hurst exponent is related to the fractal dimension (D), but this rela tionship depends on the way D is defined [75]. If a curve or profile is covered with boxes (b) of width x and height y, then the box dimension (D) is determined from the expression: ';;DbyxbN (A-1) 81

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where N(b; x; y) is the number of boxes used to c over the curve. D is related to by the equation: 2' D (A-2) If the same curve or profile is measured using a ruler length ( ) to measure the length of the curve L, then the divider dimension (Dd) [76] is determined from the expression: dDL1 (A-3) where Dd is related to by the equation: /1 dD (A-4). The Hurst exponent determined by Dd is bound from 0.5 to 1. Becau se of this difference in the way the fractal dimension is defined, it is better to use the Hurst exponent when describing selfaffine objects to avoid any confusion. Measurement Methods for Determining Hurst Exponent There are three methods that are used to measure of fracture surfaces: perpendicular sectioning, slit island, and direct surface ar ea [75]. The perpendicular sectioning method measures from the profile of the fracture surfaces. The most used form of the perpendicular sectioning method is the profile length method. The profile length method, measures the length of a profile using different ruler leng th and is expressed in the equation dDLL1 0 (A-5) where L0 is the projected length of the profile. The Hurst exponent is determined using Equation A-4. The slit island method, discussed in Chapte r 2, measures D from the contour of the fracture surface. The contours of the fracture surface, or islands, are used to measure the fracture 82

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surface in two ways. The first way uses the pe rimeter (P) of the islands and determines the fractal dimension using the expression: dDP1 (A-6) which is similar to the perpendi cular section method. The Hurs t exponent from this method is calculated using Equation A-4. The second wa y the fractal dimension is measured is by determining the area (A) of the island in compar ison to the perimeter. The fractal dimension measured in this way is expressed as 2 D P A (A-7) The Hurst exponent measured using this method is calculated using Equa tion A-2. The problem with the slit island method is that since the contour of the fracture surface is used to determine D, then the self-similarity of the surface is measured instead of its self-affinity. Therefore, there is some debate as to whether the slit island method can be used to determine the Hurst exponent for self-similar materials. The last method is the direct surface area m easurement technique [77, 78]. This method is the profile method conducted in two-dimensions The fracture surface area of a material is measured using a form of scanning probe microscopy (SPM) or scanning electron microscopy (SEM). The fractal dimension is determined from the equation dDAA2 0 (A-8) where A is the fracture surface area, A0 is the cross-sectional area, and is the side length of the SPM or SEM scan area. The Hurst expone nt is calculated using the equation 11dD (A-9) which is similar to A-4 but Dd has a Euclidean dimension of 2 instead of 1. 83

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Hurst Exponent for Silica Glass The Hurst exponent has been measured for si licate glasses, using so me of the different methods mentioned previously, to determine if there is universali ty among all fractured materials in the Hurst exponent. Atomic force microscopy (AFM) was used in all of the measurements. The use of AFM has helped researchers achieve an improved view of the fracture surface and the ability to measure the fractal properties of fractur e surfaces at small (nanometer) length scales. Danguir et al. [79] measured for soda-lime-silica (SLS) glass. The mirror region was examined for crack velocities ranging from 10-9 to 10-7 m/s. Ten AFM height profiles of lengths of 1.5 m were taken perpendicu lar and parallel to the crack direction. The Hurst exponent was calculated based on the expression: rrZ max (A-10) where r is the width of the window used to measure the profiles and Zmax (r) is the difference between the maximum and minimum heights within the width of the window. It was shown that there were two different values for determined by a crossover length ( C). For large length scales (r > C), = 0.78 and for small length scales (r < C) = 0.5. The direc tion of the profile was shown to have no significant effect on This worked confirmed that the Hurst exponent is affected by the length scale used to determine it. Bonamy et al. [74] measured for a silica glass. The fr acture surface was generated at crack velocities of 10-11 to 10-4 m/s. The fracture surface was scanned using an AFM at a scan area of (1 m)2. A one-dimension height-heigh t correlation function (G(x)): ')]'()'([ 1 lim)(0 2dxxhxxh L xGL L (A-11) 84

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where L is the section length, h(x) is the height of the surface at point x, and h(x+ x) is the height at a distance x away from point x, was used to determine the fractal dimension based on the expression: '24)(DxxG (A-12) Using Equation A-12 and A-2, = 0.75 for 80 nm. This value agrees with the value obtained by Danguir et al. at la rge length scales. Wiederhorn et al. [70] measured for SLS glass based on the roughness of the fracture surface. The mirror region of SLS glass was measured at velocities ranging from 10-10 to 10 m/s for AFM scan areas of (0.5 m)2 to (5 m)2. The Hurst exponent was determined from the equation 0aLRq (A-13) where Rq is the root-mean-squared (RMS) roughness, L0 is the edge length of the AFM scan area, and a is a constant. It was shown that was inversely proportional to the crack velocity. The Hurst exponent ranged from 0.18 for a crack velocity of 10 m/s to 0.28 for 10-10 m/s. The Hurst exponent values reported by Wieder horn et al. were less than the values reported by Bonamy et al. and Dangui r et al. It is possible that values measured by Wiederhorn et al. are the D* values for SLS glass. Based on the D* values in Table 4-5, this assumption appears to be correct. Based of the lesser values reported, Wiederhorn et al. showed that this difference is due to the method used to determ ine the RMS roughness. The Hurst exponent was measured for section lengths of 2 and 5 m. Wh en measuring the line profiles of the fracture surface from AFM scans, is determined from the expression 0L (A-14) 85

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where is the RMS roughness for a line and L0 is the length used to measure the line. Using this method, = 0.75 for 2 m and 0.78 for 5 m. When the area of the fracture surface is used to determine roughness, was 0.92 for 2 m and 0.95 for 5 m. This showed that measuring based on one-dimensional (line profiles) and twodimensional (surface areas) does not result in the same value and that Equations A-13 a nd A-14 are not equi valent. Measuring based on area roughness is not an accurate method. As mentioned in Chapter 3, the fracture surface area of chevron-notched SLS glass was measured using the AFM. The fracture surface of the chevron-notched specimen was all mirror. Using the direct surface area measurement method, the fracture surface was measured at scan areas from (1 m)2 to (250 nm)2. Using Equations A-8 and A-9, = 0.97. This value is similar to that reported by Wiederhorn et al. wh en using the scan area to determine but greater than the values reported by Danguir et al. and Bonamy et al. This difference, as reported by Wiederhorn et al., is due to the larger length scale, 100 to 1000 nanometers as opposed to tens of nanometers, used in determining and determining based on two-dimensional measurements. From Chapter 4, the fractal dimension of SLS glass was measured using AFM and the Gwyddion software. From Table 4-5, D for the mirror region was 2.26 using cube counting and 2.29 using triangulation. For cube counting, th e slope of the number of cubes versus the box length is D. For triangulation, the slope of the su rface area versus the side length of the triangle is two minus Dd. Therefore, = 0.74 for cube counting and 0.78 for triangulation. These values are similar to the values reporte d by Danguir et al. and Bonamy et al. This showed that data acquired from two-dimensional measurements could be used to measure As Table A-1 shows, for SLS and silica glass is dependent on the measurement technique used and the lengt h scale used to determine For length scales greater than the 86

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87 crossover or critical length for the material, 0.75. For lengths less than this critical length, = 0.5. Measurement of at larger length scales, on the order of hundreds or thousands of nanometers, results in approaching 1. Therefore, the le ngth scale used and the measurement method must be taken into account when measuring for glasses. Table A-1. Hurst exponent for mirror region of silicate glasses from different experimental methods. Experiment Hurst exponent Danguir et al. 0.5; 0.78 Bonamy et al. 0.75 Wiederhorn et al. Initial area roughness Line roughness Area roughness 0.18-0.28 0.75 (2 m); 0.78 (5 m) 0.92 (2 m); 0.95 (5 m) Direct surface area (Smith) 0.97 Triangulation (Smith) 0.78 Cube counting (Smith) 0.74

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BIOGRAPHICAL SKETCH Robert Lee Smith was born in the city of Chicago to Robert and Loleta Smith. After graduating from high school, Robert attended Wri ght State University in Dayton, Ohio for his undergraduate studies where he earned his b achelors degree in materials science and engineering in 2002. While at Wright State, Robert worked for several years as a co-op in the Materials and Manufacturing Directorate of the Air Force Research Laboratory at WrightPatterson Air Force Base. He worked on several projects involving the study of metal matrix composites and bulk metallic glasses. Finished with his undergraduate studies, R obert had the opportunity to move to New Mexico before going to graduate school. He was offered an internship at Los Alamos National Laboratory, where he worked under Carol Haertli ng and fellow UF alum Robert Hanrahan Jr. While at Los Alamos, Robert was elected presiden t of the Los Alamos Student Association. At the conclusion of his internsh ip, Robert entered the University of Florida in Gainesville to earn his PhD in materials science and engineer ing. Robert worked under the guidance of Dr. John J. Mecholsky Jr. During his graduate studies at UF, Robert receiv ed several fellowships including the GEM Fellowship and the NSF/SE GEP Fellowship. Taking advantage of the flexible schedule of graduate sc hool, Robert travelle d the globe visiting co untries like China, Brazil, England, and Germany. In between his ma ny trips, Robert earned his masters degree in materials science in 2006 and his PhD in 2009. 92