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Experimental and molecular dynamics determination of fractal fracture in single crystal silicon

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Experimental and molecular dynamics determination of fractal fracture in single crystal silicon
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Thesis (Ph. D.)--University of Florida, 1993.
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EXPERIMENTAL AND MOLECULAR DYNAMICS DETERMINATION OF
FRACTAL FRACTURE IN SINGLE CRYSTAL SILICON












BY

YUEH-LONG TSAI


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


1993














ACKNOWLEDGMENTS


I would like to express my sincerest gratitude to Dr. J. J. Mecholsky, my

supervisory committee chairman, for his invaluable guidance, support and encouragement

throughout the research portion of this project, and for his numerous suggestions and

corrections during the preparation of this dissertation. I am also thankful to the rest of my

committee members, Dr. R. T. DeHoff, Dr. F. Ebrahimi, Dr. P. A Mataga, and Dr. J. H.

Simmons, for their encouragement and advice throughout the research of this project.

I would like to express my special thanks to T. P. Swiler, from whom I learned a

lot about the MD technique and UNIX computer operation system. He also allows me to

use his computer codes to operate the MD simulation and MD movies.

I would also like to extend my thanks to Dr. T. J. Mackin, Mr. Z. Chen, Mr. L.

Hehn, Mr. J. Niaouris and A. Naman, with whom I had many precious discussions on

various aspects of this research. Special thanks must also be given to Ms. J. Y. Chan for

her help in obtaining some of the experimental data.

Finally, I am deeply indebted to my parents for their love and encouragement

throughout my entire life and to my wife, Yu, for her patience, understanding, and

support during the course of this work. I am especially grateful at this time because my

daughter, Gina, arrived in the world and she makes my school life more fruitful.














TABLE OF CONTENTS

Pages

ACKNOWLEDGMENTS......................................................................... ii

A B STR A C T ................................... ...... ...... ................ ..... .................... v

CHAPTER

1. IN TR O D U C TIO N .............................................................................. 1

2. FUNDAMENTAL.............................................................................. 7

Structure of Single Crystal Silicon ..................................................... 7
Failure A analysis ............ ....................... ........................ .. .............. 9
Fracture M echanics .............................................................. 9
Fracture Surface Analysis....................................................... 11
Indentation Fracture Mechanics......................................................... 15
Fractography-Indentation Analysis............................................ 15
Strength-Indentation Technique................................................ 18
Fractal G eom etry .......................................................................... 19
Fractal D im ension ................................................................ 19
Fracture Surface Analysis by Fractal Geometry............................ 27
Molecular Dynamics Simulation........................................................ 29

3. MOLECULAR DYNAMICS TECHNIQUES............................................. 33

O overview .................................................................................... 33
Potential Determination .................................................................. 34
Fundamentals About the MD Simulation.............................................. 36
Initial Conditions ................................................................. 36
Interactions Between Atoms.................................................... 39
MD Simulation Procedures .............................................................. 41
The Concept of Time Step...................................................... 42
The Method to Update Particle Positions.................................... 42
Periodic Boundary Condition .................................................. 43
Interactions Due to Two-Body Potential ..................................... 44
Interactions Due to Three-Body Potential.................................... 44
Updating of the Atom Positions ............................................... 45
Verlet's Algorithm ...................................................... 45
Gear's Algorithm........................................................ 46
Temperature Calculation........................................................ 48
M .D M ovie .................................. ....... ..... .... .......... ........... 48









Potential Energy of the System ................................................ 49
System Pressure Calculation ................................................... 49
Pair Correlation Function....................................................... 50
Bond Angle Distribution ........................................................ 51
Introduction of Strain............................................................ 52

4. EXPERIMENTAL AND SIMULATION PROCEDURE .............................. 53

Experimental Procedure.................................................................. 53
Sample Preparation............................................................... 53
Fracture Surface Analysis....................................................... 55
Toughness Measurement........................................................ 55
Fractal Dimension Determination............................................. 57
M.D. Simulation Procedure ............................................................. 61
Determination of Input Data.................................................... 61
Simulation Procedure ............................................................ 63
Fractal Analysis Using Simulation Results .................................. 67

5. RESULTS AND DISCUSSION.............................................................. 71

Experimental Results ..................................................................... 71
Fracture Surface Analysis....................................................... 71
Fracture Toughness Measurement............................................. 90
Fractal A analysis .................................................................. 100
Relationship Between Toughness and Fractal Dimension................. 104
Results From Molecular Dynamics Simulation ...................................... 111
Validity of the Applied Potential .............................................. 111
Fracture Using MD Simulation................................................ 120
Toughness in Different Orientations.......................................... 128
Strength Dependence on the Strain Rate ..................................... 130
Strength Dependence on the Crack Size...................................... 134
Fractal Analysis Using MD Results........................................... 142
Comparison Between Measured And Simulated Results ........................... 146
Comparison of Toughness ..................................................... 148
Comparison Between Experimental and Simulation Results for
Fractal Dimension............................................................... 148

6. C O N C LU SIO N S ............................................................................... 151

APPENDIX A. SUMMARY OF DATA FROM DIFFERENT INVESTIGATORS 154

APPENDIX B. STATISTICAL ANALYSIS................................................. 155

APPENDIX C. YOUNG'S MODULUS CALCULATION............................... 158

R E FERE N C E S .................................................................................... 160

BIOGRAPHICAL SKETCH .................................................................... 164












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

EXPERIMENTAL AND MOLECULAR DYNAMICS DETERMINATION OF
FRACTAL FRACTURE IN SINGLE CRYSTAL SILICON

By

Yueh-Long Tsai

December 1993


Chairman: Dr. J. J. Mecholsky
Major Department: Materials Science and Engineering


Single crystal silicon was selected as a model material in which to study the

correlation of the fracture surface features as characterized by their fractal dimension for

different orientations of fracture with the fracture toughness of the material as measured

using the strength-indentation, and fracture surface analysis techniques. Single crystal Si

was selected for several reasons: Si is a brittle, monoatomic material that will obviate the

complication of microstructure, i.e., grains, pores, etc. And Si has been well studied so

that many of its properties are well characterized. Flexure bars were indented with a

Vickers indent at various loads and fractured. After calculating the fracture strength and

toughness, the surfaces were analyzed and characterized using fracture surface analysis

and slit-island analysis. These analyses provided the size of the fracture initiating defect,

the geometry of the surrounding topography including the location of the regions of crack

branching and the fractal dimensions of selected areas on the fracture surface.

The { 100} plane is found experimentally to be the one with the highest toughness

of the three studied planes. The fracture plane on the {100} and {110} fracture plane

both initiate on the original plane and have the tendency to deviate to the { 111 } plane.










The fracture surfaces of Si have been analyzed and are found to be fractal both at

the atomic scale using the scanning tunneling microscope in a previous study and at the

micrometer scale using the optical microscope in this study. The irregularity of the

fracture surface is too complicated for Euclidean geometry to describe easily. Self-

similarity and scale invariance is suggested by the fracture surface appearance. The

fractal dimension is higher for the fracture plane with the higher toughness value. It is in

agreement with other studies which found higher toughness values associated with higher

fractal dimensions.

We also demonstrate the formation of fractal surfaces during fracture using a

molecular dynamics (MD) approach in a single crystal silicon structure and compare the

simulated results with experimental work. MD simulations using Stillinger-Weber

potential and Coulombic (Modified Born-Mayer) potential are performed in different

orientations to investigate the difference in toughness and fractal dimension with respect

to orientations. The close agreement between simulated and measured fractal structures

of the fracture surface suggests that this is a promising method for investigating atomic-

level processes during fracture.














CHAPTER 1
INTRODUCTION


The strength of brittle materials is determined by the stress required to

spontaneously propagate a crack. The strength is dependent on the local stress

distribution around a crack tip which is often treated using a stress-intensity factor

approach.1 The fracture toughness, which is a measure of the resistance to crack

growth, defines the work or mechanical energy expended in propagating the crack.

The fracture toughness can be represented by the critical stress intensity factor, the

fracture energy or the critical strain energy release rate. Techniques to determine the

fracture toughness of ceramic materials include cleavage,2 double cantilever beam

(DCB),3 crack indentation,4 strength-indentation,5 and fracture surface analysis (FSA)6

techniques. The mechanism of fracture in brittle materials is strongly influenced by

processes at the atomic level. The subject of fracture in brittle materials has long been

of interest; however, the underlying processes have remained difficult to investigate

directly. Some factors such as phonon-phonon interactions7 have been analyzed

recently, but the dynamics of bond breaking in brittle materials is only beginning to be

studied.8-11 Although it is intuitively obvious that there must be a relationship between

the bond breaking process and the macroscopic measures of fracture such as fracture

toughness and fracture topography, there has been little quantitative evidence of the

connection. Fractal geometry, which is a non-Euclidean geometry exhibiting self-

similarity and scale invariance, is a new tool that can be used to relate atomic scale

processes to macroscopic processes and features. The fracture surfaces of Si have been









analyzed and are found to be fractal both at the atomic scale12 using the scanning

tunneling microscope and at the micrometer scale13 using the optical microscope.

Fracture markings on glasses and polycrystalline brittle materials, known as

mirror, mist, and hackle, are precursors to crack branching14 and can be used to

describe the stress state6 and characteristics of crack propagation.15 These marking

have been observed for more than 50 years and were related quantitatively to the stress

condition in the 1950's.16 More recently, the repetition of these features was observed

and quantitatively related to stress intensity. Ravi-Chandar and Knauss14 also noted

that mist and hackle are self-similar; i.e., they appear to be physically similar and

produced in the same fashion. Their description, however, did not emphasize the self-

similar nature of the features.17 The distance to branching is directly related to the size

of the fracture initiating crack. Thus, one can imagine that the fracture surface features

can be related to the bond breaking processes at fractures.

Mandelbrot18 synthesized a branch of mathematics that provides a tool with

which to analyze self-similar and self-affmine processes. A self-similar process is one in

which a feature at one magnification is related to another at another magnification by a

scalar quantity. A self-affine process is one in which this magnification factor is a

vector quantity. Mandelbrot called this branch of mathematics "fractal geometry."

Mecholsky et al. 17 summarized that fracture in brittle materials can be described

as a self-similar process and thus, can be mathematically described by fractal geometry.

They derived an expression which relates the fracture of bonds at the atomic level (y) to

a characterization of the resulting morphology on the fracture surface of polycrystalline

materials, D*, which is the fractional part of the fractal dimension. They also derived

a relationship between the surface energy and the fracture energy of single crystals,

polycrystalline ceramics, and glass-ceramics. Their results suggest that modeling

brittle fracture as a fractal process may be useful in distinguishing toughening

mechanisms and in relating atomic bonding and fracture-surface morphology. The










present research was undertaken to study the relationship between fracture surface

morphology of single crystals and their resistance to fracture.

Unlike mathematically generated fractals, the self-similarity of fracture surface

geometry is bounded by measurement limitations. These limitations will be discussed

later. However, the self-similar nature of fractal surfaces offers a means of scaling

macro-scopically analyzable structures to microscopic processes on the atomic scale.

In addition, the fractal dimension has been shown to relate directly to fracture

toughness, thus forming a link between average macroscopic behavior and possible

atomic processes. 13'17

Single crystal silicon was selected as a model material in which to study the

correlation of the fracture surface features as characterized by their fractal dimension

for different orientations of fracture with the fracture toughness of the material as

measured using the strength-indentation, and fracture surface analysis techniques.

Single crystal Si was selected for several reasons: Si is a brittle, monoatomic material

that will obviate the complication of microstructure, i.e. grains, pores, etc. And Si has

been well studied so that many of its properties are well characterized. Flexure bars

were indented with a Vickers indent at various loads and fractured. After calculating

the fracture strength and toughness, the surfaces were analyzed and characterized using

fracture surface analysis and slit-island analysis. These analyses provided the size of

the fracture initiating defect, the geometry of the surrounding topography including the

location of the regions of crack branching and the fractal dimensions of selected areas

on the fracture surface.

In this study, we demonstrate the formation of fractal surfaces during fracture

using a molecular dynamics (MD) approach in a single crystal silicon structure and try

to compare the simulated results with the experimental works. The close agreement

between simulated and measured fractal structures of the fracture surface suggests that

this is a promising method for investigating atomic-level processes during fracture.










The use of molecular dynamics for studying the failure of materials has recently

become of great interest to investigators. Works by Paskin et al.,19 Dienes and

Paskin,20 and other collaborative works involving Paskin, have performed simulations

of crack growth in two-dimensional lattices. These works demonstrated the

applicability of the Griffith criteria in simple 2-D systems modeled by molecular

dynamics, and studied mechanisms of crack growth, including the formation of

dislocations and speed of crack propagation, as functions of applied stress, crack

length, and potential parameters. These works provide an atomistic basis to fracture

mechanics; however, they do not provide details on the fracture of particular materials.
271
Soules and Busby21 used molecular dynamics to study the rheological properties

and fracture behavior of a sodium silicate glass. Among the experiments in this work,

a uniaxial tension was applied to a system and the resulting behavior was studied. The

periodic boundary conditions used, i.e., on only two sides, made this experiment

equivalent to the fracture of a 50A diameter fiber, so bulk properties were not studied,

nor even approximated. This work was followed by works of Soules,8 Ochoa and

Simmons, and Ochoa et al.,10 which demonstrated the importance of dynamic effects

in the fracture of solids. Works by Ochoa, et al., 10 made use of full periodic boundary

conditions, i.e. on six sides, thus approximating fracture within a bulk material. These

simulations collectively studied the overall response of a system to an applied strain,

but did not study the atom-by-atom process in detail. A work by Simmons et al.11

examined the mechanisms involved in an individual bond fracture to describe a process

by which an entire fracture surface may be stabilized. Thus, these studies give possible

details on the fracture process for silicate glasses, and infer details of fracture for other

amorphous solids.

The difference in fracture between amorphous silica and single crystal silicon is

that silica has no preferred cleavage plane while single crystal silicon prefers fracturing

on the {111} and {110} plane.22 Thus, different loading orientations under different










loading histories in single crystal silicon would result in different fracture

topographies13'22 (although the fracture processes may be the same for different

orientations). The effect of elastic anisotropy on fracture provided the inspiration to

use single crystal silicon for the molecular dynamics simulation. From the simulated

loading history, the comparison of fracture toughness in different orientations can be

made. From the simulated fracture topography, the irregularity of the fracture surface

in different orientations can be analyzed using fractal analysis and the fractal dimension

can be compared with the toughness. Kieffer and Angell23 used the MD simulation

method to generate stable silica aggregates at various low densities similar to those of

experimental aerogels. They found that fractal dimensions and range of self-similarity

can be extracted from the radial distribution functions in those structure obtained from

MD simulation. They believed that a non-integer dimension is a characteristic feature

of the aerogel structure used in their study. However, because the way which they

used the change of the slope of the radial distribution functions with respect to the

density to obtain fractal dimension in their study was not reasonable, thus they got a

weird result which shows that a denser aerogel has a lower fractal dimension. It is

contradict to the fact that in the nature a full occupied volume has a dimension 3 while

a volume with porosity has dimension less than 3. Thus we must use a different

approach from theirs to obtained the fractal dimension in this study. Here the fractal

dimension on the generated fracture surfaces at the atomic scale from MD simulation

will be analyzed using slit-island technique and Richardson-plot.

This investigation will address several topics. First, there will be an

experimental determination of the fracture toughness in single crystal silicon as a

function of orientation of the crystal plane. Second, the fractal dimension of the

fracture surfaces of the single crystal silicon for the different orientations will be

determined experimentally. The fractal dimensions will be determined using

Richardson plots and slit-island analyses of contours of (horizontal) sections through







6


the fracture surfaces. Third, MD simulations using Stillinger-Weber24 and Coulombic

potentials will be performed in different orientations to investigate the difference in

toughness and fractal dimension with respect to orientations. Finally, there will be

discussions about the relationship between the fracture toughness and the fractal

dimensions of single crystal silicon for the different orientations and also comparisons

between experimental work and simulation results.













CHAPTER 2
FUNDAMENTALS


Structure of Single Crystal Silicon


Silicon is bonded by four covalent bonds which produce a tetrahedron. Each

silicon atom is bonded with four other silicon atoms due to the nature of the covalent

bonding. The covalent bond formed by two Si atoms sharing electrons is very

localized and directional. The space lattice of the diamond structure is face-centered

cubic with two atoms per lattice site, one at 0,0,0 and the other at 1/4, 1/4, 1/4 (Fig.

2.1). Each atom is tetrahedrally bonded to four nearest-neighbors due to the nature of

covalent bonding. As these tetrahedral groups are combined, a large cube is

constructed, which is called a diamond cubic (DC) unit cell. The large cube contains

eight smaller cubes that are the size of the tetrahedral cube but only four of the cubes

contain tetrahedrons. The lattice is a special face-centered cubic (FCC) structure,

which is shown in Fig. 2.1. The atoms on the corners of the tetrahedral cubes provide

atoms at each of the regular FCC lattice points. Four additional atoms are present

within the DC unit cell from the atoms in the center of the tetrahedral cubes.

Therefore, there are eight atoms per unit cell. The lattice parameter length is 5.43 A

and the unit cell diagonal length is 9.41 A as shown in Fig. 2.1. The Si bond distance

is 2.34 A.

Single crystal Si is generally classified as a brittle material whose atoms are

strongly covalently bonded. The primary cleavage plane is the {111} plane. The

principal factor for crack initiation is bond rupture.


















































(c)


Fig. 2.1 (a) Tetrahedron and (b) the diamond cubic (DC) unit cell. (c) The length
of AB is 5.43 A, AC is 9.41 A, and AD is 7.68 A.


J- I -- I --










Fracture of single crystal Si on the {100}, {110} and {111} plane has been

observed.13'22 Not only the toughness, but also the fracture surface topography were

found to be different on the different planes. These observations mean that elastic

anisotropy plays an important role in fracture; the bond breaking process is considered

the same since only one kind of Si-Si covalent bond exists.


Failure Analysis


The characterization of flaws and their relationship to strength plays an

important roll in the analysis of brittle materials. By comparing fracture mechanics

relationships with observed fracture surface markings, important information about the

fracture toughness can be determined. This section will present the basic principles of

fracture mechanics and fracture surface analysis of brittle materials.


Fracture Mechanics


One of the first analysis of fracture behavior of components that contain sharp

discontinuities was developed by Griffith.25'26 This analysis was based on the

assumption that incipient fracture in ideally brittle materials occurs when the magnitude

of the elastic energy supplied at the crack tip during an incremental increase in crack

length is equal to or greater than the magnitude of the elastic energy at the crack tip

during an incremental increase in crack length. By using a stress analysis developed by

Inglis, Griffith related fracture stress to the flaw size by means of an energy balance.

The relationship can be written as:

S=(2Ey1/2, (2.1)
whr -istC )

where arf is the failure stress,









E is the modulus of elasticity,

y is specific surface energy, and

c is the critical flaw size.

Although Griffith was the first to analyze the relationship between strength and

flaw size, it was Irwin who developed fracture mechanics into the present day form.

Irwin27-29 followed the work done by Griffith, Orowan,30 and Inglis31 to develop what

is known as linear elastic fracture mechanics. Irwin first analyzed the fracture of

flawed components using a stress analysis based on the Westergaard32 solution of an

elliptical crack in an infinite plate. For a surface cracked specimen under mode I

(tensile) loading, Irwin derived the following:

5f =, B (2.2)


where af is the failure stress,

Kic is the critical stress intensity factor for mode-I loading,

B is a geometrical factor which accounts for flaw shape, location,

and loading geometry, and

c is the critical flaw size.

Irwin also analyzed fracture in terms of the strain energy release rate, G. G is

defined as the elastic energy per unit crack length, U/c, and can be related to the

failure stress:27
= (Ecr' 1/2
CTf= -Ec (2.3)

where Gc is the elastic energy release rate at fracture.

The strain energy release rate can be related to surface energy, y, and it is as

follows for plane-stress condition:

G = 2y. (2.4)










In the above equation, only surface energy, y, is referred. Indeed during fracture

process, the released strain energy not only produce two free surface, but also produce

different terms like sound, light, electron emission. Thus, the y used here indeed is not

just as simple as surface energy and some other energy is also included.

The stress intensity factor can be related to the strain energy release rate and the

fracture surface energy. By combining those previous equations, it can be shown that
Kic= = EG= -2yE. (2.5)


Fracture Surface Analysis


Useful information on the mechanical and fracture behavior of a failed ceramic

can be gained from microscopic examination of the fracture surface.33-35 The fracture

surface of brittle materials exhibits distinct fracture markings surrounding the critical

flaw. These features shown in Figure 2.2 can be related by known fracture mechanics

relationships to provide additional information on the fracture process and can also be

used to locate the origin of failure.35

The fracture surface markings shown in Figure 2.2 can be used to describe the

stress state of a brittle material.14'33'35'36 Four different regions referred to as the

fracture mirror, mist, hackle and crack branching can be seen in Figure 2.2. These

regions are associated with particular stress intensity levels and crack velocities which

are responsible for each distinct region.34 As a crack propagates from the critical flaw,

a smooth region which is basically perpendicular to the tensile axis is formed. This

smooth region is known as the fracture mirror. The fracture mirror is typically

bounded by a region of small radial ridges known as mist and the mist is bounded by

an even rougher radial ridge region called hackle. Finally, the propagating crack

reaches a characteristic energy level and crack branching occurs. Crack branching is a

region where two or more cracks form from the primary crack front.



















































Fig. 2.2 Schematic of features found on the fracture surface of a brittle material
subjected to a constant load. Solid semi-elliptical line at center represents the initial
flaw size depth ai and width 2bi. Dashed line represents the outline of critical flaw
depth acr and width 2bcr. Mirror/mist, rl, mist/hackle, r2, and crack branching, r3,
radii are shown along the tensile axis.


r,2
--'cb7------ *










Once the failure origin is located after fracture, its size can be measured and

used to determined the toughness of the material. Figure 2.3 shows a semi-elliptical

crack in a plate. It can be shown through a Griffith/Irwin approach that the stress

intensity on a semi-elliptical surface crack of depth, a, and half-width, b, can be given
as:35,37

K, = 1-.--. _27 Y-aJa (2.6)

where a is the semi-minor axis for an elliptical crack,
ca is the applied stress, and

D is an elliptical integral of the second kind. D varies between 0 = 1

for a slit crack (a/b =0) to 0D = 1.57 for a semi-circular crack (a/b= 1).

The criterion for failure in a brittle material is that Ki _> Kic at which point the

initial flaw will propagate spontaneously, where KI, the stress intensity, is a measure of

the magnification of the external loading at the crack tip, and KIc is the critical stress

intensity factor or the fracture toughness of the material. The strength or failure stress,

cf, of a brittle material can be related to the flaw size:

a DK c (2.7)


For a semi-circular surface crack, which is stress free and which is small relative to the

thickness of the cross-section, ) = 1.57 and Kjc can be found as:

Kic=1.24 cyfVc, (2.8)

where c is the square root of a-b and is the size of a semi-elliptical crack which has the

size of an equivalent semi-circular crack.30

Thus, the fracture toughness of a material can be calculated if the failure stress

is determined and the crack dimensions a and b are measured.







14














a

CY














2b













Fig. 2.3 An semi-infinite plate under uniform tension and containing a semi-
elliptical crack.











Indentation Fracture Mechanics


Indentation fracture mechanics has been well established as an important method

for the study of the mechanical behavior of brittle materials.38-43 Indentation provides

a means of introducing flaws into a material with a controlled size, shape, and location

on the sample. This is in contrast to materials with naturally occurring flaws where the

flaws vary in size, shape, location and concentration. Indentation provides a means of

characterizing the fracture process by introducing a controlled flaw into a brittle

material. The following discussion will concentrate on fracture surface analysis and

strength indentation analysis.


Fractography-Indentation Analysis


Figure 2.4(a) shows a schematic of the indentation deformation/fracture pattern

for the Vickers diamond geometry: P is the peak load and a and b are characteristic

dimensions of the inelastic impression of the radial/median crack, respectively. Figure

2.4(b) shows the schematic of a fully developed Vickers indent. The flaw is taken to

possess a penny-like geometry. The following discusses the cases of zero and non zero

residual stress terms separately:40

(1) Zero residual contact stresses: If the indentation flaw were to be free of

residual stresses, the stress intensity factor for uniform tensile loading would have the

standard form

Ka = YfC12, (2.9)

where Ka is the applied stress intensity factor without residual stress,

Y is a unitless geometrical factor. For a semi-circular crack, Y = 1.24,











C
J


-1


I-


Fig. 2.4 Configuration of median/radial cracks for Vickers indentation showing:
(a normal indent load, P, generating median opening forces Pe (elastic component) and
Pr (residual component), (b) load removal eliminates the elastic component, and (c)
fully developed radial crack pattern.









cf is the failure strength, and
c is the critical flaw size.
(2) Nonzero residual contact stresses: With residual contact stresses present at
the flaw origin it becomes necessary to include an additional tensile term in the stress
intensity factor:44
KI = Ka + Kr = Ycac2 +XrP / c32, (2.10)

where Kr is the stress intensity factor with residual stress,
P is the peak load, and
Xr is a unitless geometry constant.
For large c values, the applied-stress term controls the fracture, as before, for

small c values, it is the residual-stress term that dominates. "Under equilibrium
fracture conditions the flaw will accordingly undergo a precursor stage of stable growth
as the tensile loading is applied; failure then occurs when the crack reaches a critical
size, at which point the applied stress is intense enough to cause spontaneous
propagation."40 This critical configuration is obtained by inserting K, = Kc (Kc is
used here instead of Kic because, strictly speaking, the toughness determined using
indentation is different than Kic; however, in practical terms, they are equivalent) into
the above equation and evaluating the instability condition dca/dc = 0:

= 4Xrp )2/3 =(3Kr_ )2 (.1
c Kc 4Ycrf(2.11)

from which Kc can be found as:45,46

4 1/2
Kc Yyfc2. (2.12)
3
For a semi-circular flaw, which is small relative to the thickness of the material into
which it is placed, Y = 1.24, and the above equation can be rewritten as:
Kcl = 1.65acfc12. (2.13)









Thus, the resistance to fracture can be determined from a measurement of the surface
crack size and the strength of bars that have been indented.


Strength-Indentation Technique5


Consider the Vickers diamond-induced radial cracks, which are shown in Fig
2.4. If the applied loading is uniaxial, the indentation is aligned with one set of
pyramidal edges parallel to the tensile axis.
It has been shown that the toughness may be derived using the following
expression:
-1/8 "/

V H)
K =nv T1{J (cafP1/'3) (2.14)

f')76 y23R1/4
where 11R = 256Y2/3 is a geometry constant,

E is the Young's modulus, and
H is the hardness.
Replacing rv by an average quantity, 0.59, would add no more than 10% to the error
in the Kc evaluation for a material whose elastic/plastic parameters are totally
unknown. So the above equation can be rewritten as
(gE1/8 )3/4
Kc = 0.59 E ((fP1/3) (2.15)

Thus, the resistance to fracture can also be measured from a flexural test in which the
specimens have been indented at different load levels without measuring the crack size.











Fractal Geometry


Fractal geometry is a non-Euclidean geometry that was rediscovered,

popularized and applied by B. B. Mandelbrot.18 The word fractal is derived from the

Latin "fractus" which means fragmented or broken. Nature exhibits diverse structures

having inherent irregularities. Some of these structures can be described using

Euclidean geometry but other structures are better described by using fractal geometry.

In his book, The Fractal Geometry of Nature,18 Mandelbrot describes the extensive

applications of fractal geometry. The concepts have been used to described the

geometry of clouds, soot aggregates, dielectric breakdown, coastlines, and other

natural phenomena, including fracture.

Fractal geometry exhibits self-similarity or self-affinity, shows scale invariance,

and is characterized by the fractal dimension. Self-similarity means that features in

different regions appear to be similar to one another. They can be related by a scalar

multiple. Features in a self-affine object are related by a vector quantity. Scale

invariance occurs if a feature on different scales appears to be the same. These

properties indicate that a fractal can be created by using a shape to be repeated and

reduced in size following a prescribed sequence.


Fractal Dimension


In Euclidean geometry, objects occupy integer dimensions, i.e., 1, 2, 3, and so

on. Fractal geometry admits to the description of objects that occupy fractional

dimensions. Fractal geometry is characterized by its fractal dimension which is related

to measurement theory.










Consider how we measure the length of a line. The measurement of a line

requires a measuring tool or ruler. A line is given in Fig. 2.5(a). The length of this

line is measured by covering it completely with as few of a given radius, R, discs as

possible, as in Fig. 2.5(b). The length is given by

length = (number of covering discs) x (2R). (2.16)

If we have a new supply of discs with a different radius, say R/2, then the line length is

determined in the same manner, and a new line length is computed, as shown in Fig.

2.5(c). Note the difference in the measured line length in Fig. 2.5(c). Although the

measuring disc was decreased to half of the original measure, the number required to

cover the line more than doubled. The "tortuous" nature of this curve gives a length

that is scale dependent. As the measuring scale decreases, the measured length

increases.17

The measured length of a line can be described by the following equation,

called Richardson's equation:

L = kSID, (2.17)

where L is the measured length,

S is the measured scale,

D is the dimension of the curve,

k is a proportionality constant.

If D is equal to one, the line is Euclidean and its measured length is not a

function of scale. If 1 < D < 2, the curve is said to be fractal with a dimension given by

D. Notice that for a given line segment, the length is not constant, but the dimension

is. Thus, the fractal dimension is the dimension in which a measure can be made, not

the measure itself.

The fractal dimension of a line can be computed from the above equation. The

line's length is computed over a range of scales. A log-log plot of length vs. scale

gives a straight line with slope equal to 1-D, as shown in Fig. 2.6.


















(a)






O diameter =2R



Fig. 2.5 Measurement of a line segment ( or signal). (a) A schematic for a line
which has texture. Measuring the length of a line requires a ruler. The (circle)
covering has a diameter, 2R. (b) The line is completely covered to determine its
length. (c) Changing the size of the measuring disc may produce a different measure of
the length.



































Continued.


Q diameter = 2R
length = # of disc x diameter 2R


O diameter = 2r
length -= # of disc x diameter 2r


Fig. 2.5
























Slope = 1-D


log L






LogE

Fig. 2.6 A log-log plot of length vs. scale gives a straight line with slope equal to
1-D.










Mathematical fractals will obey the above equation for all scales. However,

physical fractals may have a finite cutoff at both large and small scales. In the physical

world, the fluctuation of a line will find its limit in the smallest measurable feature.

The ruler length determines the smallest feature we can measure. If variations in

structure are below the sensitivity of the measurement scale, the line will appear to be

Euclidean. In that case, the longest line we could measure would depend on the finest

possible scale of measurement. We could not conclude that the line has no

fluctuations, only that we have no ability to measure the fluctuations.

On the contrary, a line may be composed of indivisible components. After our

measuring stick has become smaller than the smallest of these components, the line

would cease to exhibit scale-dependent length. This length, then, would represent a

true upper bound. Such a line could be called "fractal" over the range, it exhibited

scale-dependent length. It is not, however, a fractal in the strict mathematical sense.

The combination of the two limits would result in the Richardson plot as shown

in Fig. 2.7. The object would be described as a fractal over a bounded range of scales.

The cut-off scales are of special significance to the particular geometry of a specific

physical phenomenon.

A box is shown in Fig. 2.8(a)18'47 whose sides are of unit length. Next to the

box is a shape called a "generator," composed of line segments scaled to 1/4 of the

length of a side of the box. Each side of the square is replaced with this "scaled"

generator. The result is shown in Fig. 2.8(b). Again, the generator is scaled down to

the length of each straight line segment of this new object. The straight line segments

of Fig. 2.8(b) are replaced with the scaled generator. A portion of the resulting shape

is Fig 2.8(c). This process is continued, generating an object which is said to be scale

invariant and self-similar. A scaling fractal looks geometrically the same everywhere

and on all scales.























Fractal on bounded
range







log L







log E


Physical fractals have scale dependence over a bounded range of scales.


Fig. 2.7


















































The stepwise construction of a scaling fractal.


each side of the square
is replaced by the scaled
generator.


again, the generator is
scaled to replace each
of the segments above.


h-j


Fig. 2.8










The fractal dimension of this object, however, can be computed in another,

quite different fashion. Scaling fractals obey the following:

NrD = 1, (2.18)

where N is the number of elements in the generator,

r is the scale factor of an element,

D is the similarity or fractal dimension.

For the plot of Fig. 2.8 called the quadratic Koch curve, N=8, r= 1/4, Thus,

the fractal dimension D = log8/log4 = 3/2.


Fracture Surface Analysis by Fractal Geometry


Mecholsky et al.17'48 experimentally determined a relationship between fracture

toughness (Kc) and fractal dimension (D). They found that

Kc =A (D -1)1/2 =A D*1/2, (2.19)

where D* is the fractional part of the fractal dimension,

Kc = Kic Ko, where Ko is the toughness of the material for a smooth

(Euclidean) fracture surface, and

A is a constant.

Thus, as fractal dimension increases, fracture toughness increases. In some

papers (D-l) is referred to as D*. Thus the above equation can be rewritten as

Kc = AD*1/2, (2.20)

where A is a family parameter. The family parameter, A, identifies a line in the Kc

vs. D* plane. Within a family, an increase in fractal dimension corresponds to an

increase in fracture toughness.

Since D* is dimensionless, a dimensional analysis of this equation requires that

A have the dimensions of toughness. Mecholsky et. al.17 proposed that A is a product










of a characteristic length and Young's modulus, so that A = E(ao)11/2. Then the above

equation can be written as:

Kc = E ( ao D* )1/2, (2.21)

where Kc is the fracture toughness,

E is the Young's modulus,

D* is the fractal dimension, and

ao is the characteristic length.

Recalling that Kc = EG = -2yE for a plane stress condition, ao can be

obtained by the following equation,17

ao =- 2y (2.22)
ED*
In order to prevent getting an infinite values of ao calculated from the above

equation if D* is approaching zero. A modified equation from a different approach to

find ao is expressed as:

Y = o70 + aoEhkD*/2, (2.23)
where y is the fracture energy,

7o is the surface energy for an Euclidean fracture surface (D* = 0),
ao is a characteristic (atomic) length, and

Ehk is the Young's Modulus of the hkl fracture plane.

Using experimentally obtained values of y, E and D*, yo and ao can be

calculated. The exact interpretation is not clear at this time; however, it is reasonable

to assume that it is related to the characteristic length of the generator. In terms of

fracture, this would correspond to a characteristic bond length which is involved with

initial fracture at the crack tip.

Mecholsky et al.6 reported that the outer mirror constant, i.e., oa(r2)112 where r2

is as noted in Fig. 2.2, is related to Kc and independently to E. Kirchner49 shows a

similar relationship with the crack-branching constant and used this information to










support his idea of a constant strain intensity at branching. Tsai and Mecholsky50 used

a fracture energy concept to prove that the mirror-hackle boundary forms at a constant

energy value. Mecholsky51 points out that the rj/c = constant. Freiman et al.52

showed that the relationship could be represented as:
Kbj= (bj)1/2E, (2.24)

where bj is a characteristic dimension on the atomic scale, and

j = 1, 2, 3 is related to mirror-mist, mist-hackle and crack branching

boundary.

They further showed that if b1 = ao, where b1 corresponds to the mirror-mist

boundary, then the flaw-to-mirror-size ratio is equal to the fractal dimensional

increment:

c/rI = D*. (2.25)

This equation can provide a link between fractography and fractal geometry.

They found equation (2.25) in good agreement with experimental results. Thus, it

appears that the relationship between the crack size and the formation of the branching

boundaries is related to the dimensionality of the structure. Presumably, the

dimensionality of the structure is related to the bonding, i.e., the strength and length of

bonds on the atomic scale.


Molecular Dynamics Simulation


Molecular dynamics simulation is a computer technique to model material

structure at the atomic level in order to model or study the material properties with an

atomic view point.53'54 During the molecular dynamics simulation, a small box with a

limited number of atoms is generated. The small box is considered a simplified system

which is ideally a realistic subset of the real system. This MD technique uses

Newton's equations of motion at constant acceleration over very short time intervals.










Interaction forces generated from the potential energy generated between the atoms are

assumed. The acceleration of particles (or atoms) results from the interaction of those

interatomic forces. The interaction forces produce the fundamental momentum for the

particles inside the small and idealized system.

Molecular dynamics simulations can be applied to study the properties of

materials. During MD simulations, the position and velocity of each atom can be

calculated to show the dynamic properties at that instant of time. The power of the

MD simulation is that the rule of interaction can be changed at will to model any

complex sets of particles, and environments can be simulated at conditions which are

not accessible experimentally. The difficulty with the MD simulation is that the rule of

interaction, i.e., the selected potential, must be carefully chosen in order to be realistic

or meaningful.

Three major uses of MD simulation are the development of theoretical models

for materials, the testing of theoretical models, and the predictions of material behavior

under experimentally inaccessible or costly conditions.

Material models can be developed from molecular dynamic simulations to

observe what occurs during the simulation and be able to set the rules of a simulation.

If we want to model the fracture of silicon, we do this by observing individual atomic

motions as samples are fractured using identical interaction potentials. We can also

fracture at several strain rates to investigate the effects of allowing the material to relax

by thermal vibrations as compared to not allowing the material to relax. After

performing such simulations, we can develop a general idea of how single crystal

silicon fractures and those ideas will be valuable to those studying the strength of Si.

Thus, molecular dynamics simulations can be used to devise models.

Molecular dynamics can be also used to test theoretical models using the ability

to observe all modeled atomic motions and the ability to simulate a wide range of










environments. If the system behaves as predicted by a model under a range of

conditions, then the model can get greater credibility.

Extreme conditions also can be simulated using molecular dynamics. As one

might simulate any environment, one needs only to have a realistic simulation to

predict materials behavior or study possible modes of failure in extreme environments.

Thus, as molecular dynamics becomes better at simulating reality, one may expect this

application of molecular dynamics to find regular use.

When creating a molecular dynamics system, one would try to model the laws

of nature as close to reality as possible in order that the processes observed in the MD

simulation can be as realistic as possible, and that the resulting character of the

simulated system corresponds to the character of real materials. Because the laws of an

MD system are executed by high speed computer of limited computational power, there

must be limitations of the resulting MD system. These limitations are seen as

limitations in the spatial and temporal resolution in nature, as well as limitations on the

number of interactions possible within such a system.

The spatial dimensions of a molecular dynamics cell must be within a certain

order of magnitude of the computer precision of positions obtainable.53 The spatial

resolution of the computer calculation is generally 1 in 107 or 1 in 1016, depending on

whether single or double precision is used in the computer calculations. Over long

calculation periods, the use of single precision has resulted in a small error in the

system, while yielding a lot of time-saving is obtained. Thus, here the choice of spatial

resolution is not a problem in performing the MD simulations.

The length of time represented by the simulation must be within a certain order

of magnitude of the time step, the basic unit of time in a simulation. Generally the

duration of an experiment is between 10,000 to 100,000 time steps depending on

computational resources and the purpose of the experiment. The temporal resolution of

an MD simulation may be a serious limitation. The reason for this is that processes










relevant to the structural evolution of materials have characteristic times spanning about

19 orders of magnitude, far more than the four to five orders of magnitude spanned by

MD simulations. Generally a simulation which lasts for hundreds or thousands of

atomic thermal vibrations is needed in order to understand the properties of the system.

The mass of the electron is about one ten thousandth of those of atoms. The

motion of electrons cannot be modeled with the motions of atoms at the same time

since the characteristic time is so small compared with that of atoms. Thus electronic

motions cannot be used to provide true interatomic potentials of materials in such a

simulation and the motions studied here are limited to those of atoms only. The chosen

potential is necessary to accurately represent the thermal vibration motions of atoms in

order to model the properties of the simulated system. Thus the smallest time step used

must be a small fraction of the vibration period. A typical time step chosen in this

study is 0.5 x 10-15 sec. One generally can only run simulations for about 105 time

steps which results in a limitation in the duration of modeled experiments to hundreds

or thousands of atomic vibrations.

Another limitation is that the number of interactions between atoms cannot be

too big. The more atoms put in the system, the more computer CPU time will be

consumed. For example, if a system with n particles is created and only one-to-one

interaction is considered, then the total interactions between those atoms is n2. The

time to calculate a system with 10On atoms is 102 longer than that needed to calculate

one with n atoms. Thus the simulated system must be kept small and the chosen

interacted potential must be simple and easily applied.
Thus the differences between an MD system and the real universe are that the

length of studied time and the size of the studied system. The MD system has to be

small because of the limits on the spatial resolution of time and number of atoms used

in the system.














CHAPTER 3
MOLECULAR DYNAMICS TECHNIQUES


Overview


A technique for studying materials whose popularity and usefulness has arisen

within the past few decades is molecular dynamics (MD) computer simulation. This

technique can solve equations of motion for a system of particles which may be ions

(ionic bonding) or atoms covalentt bonding or metallic bonding). Each particle moves

according to the forces on it caused by all of the other particles in the system. The

inter-particle forces are derived from an assumed potential function which describes the

interactions between the atoms. Several types of systems have been simulated by MD

with various interatomic potentials. Those systems include hard sphere systems, soft

sphere systems, ionic systems and Lennard-Jones systems. These systems are

generally simulated using simple pair interaction potentials to obtain a close

approximation.

The covalent bond formed by two Si atoms sharing electrons is very localized

and directional. In ionic systems, Coulomb potentials which include short range

repulsive forces as well as attractive forces, are required. In covalent systems, bond-

directionality should be included. Two body interatomic potentials are not enough to

produce the diamond cubic structure. Since Si exhibits that structure as a solid, three

body potentials must be introduced in order to conduct a reasonable simulation. The

Stillinger-Weber potential24 has been used for Si solid simulations using MD and the

simulated pair correlation function agrees very well with the experimental data. It was










also generally found able to accurately simulate the elastic and thermodynamic

properties of Si.55'56 This potential has been adopted here.


Potential Determination


Single crystal silicon consists of atoms held in place (a diamond structure which

is relatively anisotropic from the atomic viewpoint) by strong and directional bonds. It

seems reasonable at first sight that the corresponding potential (D could be

approximated by a combination of pair and triplet potentials, (2 and cD3. And these

two potentials can be expressed as a function of energy, F, and length units, 5. Thus,

02(rij) = s f2(rij/5), (3.1)
and

't3(ri, rj, rk) = s f3(ri/5, rj/5, rk/6). (3.2)

We use a reduced pair potential for Si selected from the following five-

parameter family, a simplified Lennard-Jones potential, from the work of Stillinger and
Weber:24

f2(r) = A (B r-P r-q)exp[(r a)-], if r < a; (3.3)
= 0, if r>a.

Thus,

(D2(rij) = e f2(rij/5) (3.4)

= A (B ()-P (r)-q) exp[( r )-I], ifr 6 6 6 6
= 0, ifr>a.

This form automatically cuts off at r = a without discontinuities in any r derivatives,

which is a distinct advantage in any MD simulation application. Stillinger and Weber

use the same cutoff technique for the three-body interactions, f3(ri,rj,rk) = hjik + hijk
+ hkij. The function h is given :











h(rij, rik, 0jik) = ) exp[ri /(rij a) + ril /(rik a)] x (cosOjik + 1/3)2, if r < a;
(3.5)

= 0, ifr>a.

where 0jik is the angle between rj and rk. The function h exists when both rij and rik

are less than the previously introduced cutoff a.

The parameters which best satisfy the radial distribution function were

determined by Stillinger and Weber to be: A = 7.049556277, B = 0.6022245584, p =

4, q = 0, a = 1.8, 1 = 21.0, h = 1.20, d = 0.20951 nm, and s = 50 kcal/mole =

3.4723x10-12 erg/atom pair.

Although the Stillinger-Weber potential has been successfully used to model the

structure of silicon, it does not provide the expected repulsive forces following covalent

bond breakage which result from ion formation during the fracture process. The Si-Si

covalent bond is formed by an attractive force even if the distance between the two

atoms is larger than the equilibrium position as long as it is within the cutoff distance.

However, during fracture this condition is not realistic and makes the Si-Si bond

difficult to break. An unrealistically long fiber-like structure is observed to result if the

unmodified Stillinger-Weber potential is used. An improvement developed here

assumes that following bond breakage each Si atom will gain one positive charge for

each lost neighbor. Si will have two positive charges if it loses two of its four

neighbors and four positive charges if all four neighbors are displaced beyond the

cutoff distance.

A modified Born-Mayer potential1'57 is used for modeling the resulting ionic

forces. Consequently this approach assumes ionic repulsive forces when two Si atoms

both with positive charges fall inside the Born-Mayer cutoff distance. This potential

has the following form:









-r qiqj r
rij = Aij.exp() + ( ). erfc(-), (3.6)
p r

where erfc() is the complement error function,
E is a constant,

L is a linear dimension of the molecular dynamics cell,

q is the charge,

r is the distance between the i and j atoms, and

p is a hardness parameter.

After combining the Stillinger-Weber potential and the modified Born-Mayer

potential, the fiber-like structure disappears and potentially more realistic fracture

surfaces can be obtained.

Fundamentals About the MD Simulation



How the MD simulations are performed will be explained now. First, we

discuss how the system was set up, which includes the initial particle positions,

velocities and the potential described in the previous section. Then we discuss how the

system is operated, which includes the application of periodic boundary conditions, the

force summation on the particles, control of the velocities of particles, and control of

thermodynamic values. Finally the discussion of how the data from the simulation

system is collected will be given. A flow chart in Fig. 3.1 is given to illustrate how

the MD simulation is operated.


Initial Conditions


Initial particle positions are located at specific positions since a single crystal

structure is desired.



















Initial Velocity:


temperature


kinetic
k energy
Ek


Velocity (V)
random
distribution
function


Interactions between atoms:


apply strain


+(V+ AV) x At

t


new position
of atoms


force
between
atoms


I
acceleration D- AV
ax At


A flow chart for the MD simulation.


position
of atoms


Fig. 3.1







38

Initial velocities are given using the Poisson velocity distribution. The velocity
distribution is generally selected to correspond to a temperature at which the crystal is
stable. Here it corresponds to room temperature or to the temperature of the
simulation.
Random velocities are assigned to particles using the Poisson velocity
distribution. The formula giving the Maxwellian velocity distribution in three
dimensions are58
Vi 2kBT
Vi = 2kBT x ln(Randl) x cos(27' x Rand3) (3.7)
mi
I- M\i I
Vi= 2kBT x ln(Randl) x sin(27Tc x Rand3) (3.8)
Smi
V I- 2kBT

Vi= B x ln(Rand2) x sin(27 x Rand4) (3.9)
mi

where Vxi, Vyi, and Vzj are the components of the velocities of particle i,
Randl, Rand2, Rand3, and Rand4 are random numbers evenly
distributed between 0 and 1,
mi is the mass of particles i,
T is the temperature, and
kg is the Boltzmann's constant.
Particle velocities are applied by putting the velocities obtained above into
positions before and after the current time step, which is the basic unit of time during a
simulation. Thus the random velocities are effectively converted into average velocities
during the previous time step, and particles positions and velocities are given by the
difference of two sets of positions for each particles. The reason for this will become
clear in the discussion of Verlet's algorithm.










Interactions Between Atoms


Interaction forces are tabulated for all types of interactions possible over a range
of interparticle spacing. The interatomic potential assumed in this study is the
Stillinger-Weber potential and the modified Born-Mayer potential as illustrated above.
The derivative of these potentials is the force as a function of r. The two-body
potential is
2(rij) = 6 f2(rij/8)

=EA(B (6)-p ()q)exp[( 6 )-], ifr
= 0, ifr>a.
Thus, the two-body force is

f2(rij) = 2 (3.10)
ar

exp( )-1Y
8 r-a


(Bp (r)-p-1- (r)-q-1)+B(r)-P_( y 82)2
5 8 8 8 o (r -a)2J

,ifr = 0, ifr>a.
It is computer-time-effective to tabulate values of this function as a function of
r2 and divide by r so that one may use the square of the interaction distance in finding
interparticle forces in the table. One may find the Cartesian components of the force
between two particles by simply multiplying the appropriate tabulated value by the
difference in Cartesian coordinates of the two particles.







40

The energy of potential energies for interactions between particles are also

tabulated in terms of r2. This is used for the calculation of potential energies for the
system.
The three-body force is considered in the same way.

D3(ri, rj, rk) = s f3(ri/5 rj/5 rk/5).

f3(ri,rj,rk) = hjik + hijk + hkij.

The function h is given :

hjik = h(rij, rik, 0jik) (3.11)
= X exp[h /(rij a) + Tj /(rik a)] x (cosOjik + 1/3)2, if r = 0, ifr>a.

The following derivation related to the three-body potential will only use hjik for

convenience. Thus the three body force related to hjik is as follows:

a -, 1 a i
VO = gradD = er + 0 -- (3.12)
aT r aO

F=-V =-(e +e1 (3.13)

(A) At first, we differentiate r1 only.
F = Fr + F0 =-V(D (3.14)

At dr direction,

Fr- = --)2 exp[7y(rl -a)- +y7(r2 a)-l](coso+ )2 (3.15)
ar (r, -a) 3

At O direction,

F_=- 2X exp[7 (r1 a)- + y (r2-a)-1]sinO(cosO+-) (3.16)
r1 ao r1 3

Now transform the polarized coordinates into Cartesian coordinates.
Fr +Fe=Fx +Fy +Fz (3.17)







41

Consider Fr first. At first we have to find the transformation of u1 into Cartesian
coordinate.
U = UIx + uyj + ulzk (3.18)

FrUil = Fr(uixiJ+uiyj+uizk) (3.19)

Now consider F0. According to Fig. 3-1,
60 = fui x i3 = i1 x(i1 x i2) (3.20)
and because A x ( B x C) = (A.C)xB-(A.B)xC, thus
e0 =i x(il xu 2) (3.21)
=(i1.U2)x ii1 -(1il-i1)U2
=cosOUi1 iU2
Fo0e=Fo(cosOi1 -Ui2) (3.22)
= F(cos0uix U2x)i + FO(cos0uiy U2y)j + FO(cos0ulz -U2z)

Thus,
Fr +F0=Fx +Fy +Fz (3.23)
[FrUix + F0 (cosOulx- U2x)]1

+[FrUiy + F0(cos0Uly U2y)]j

+[Fruiz + F0 (cos Ouiz- U2z)]k

(B) We can consider the r2 term in the same way as (A).


MD Simulation Procedures


The explanation of how the simulations are performed will be given here.53
The concept of time step, the interaction between particles, the method used to update
particle positions, the application of periodical boundary conditions, and the control of
system temperature will be discussed as follows.











The Concept of Time Step


The time step is the basic unit of time during a molecular dynamics simulation.

As there are many atoms interacting in the simulated system, the motions of particles

cannot be solved analytically. Thus one can calculate the motions of atoms using

Newton's laws of motion where a time step dt is used in which the acceleration is kept

constant. A careful choice of time step is very important. If the time step is chosen to

be too small, a waste of computer running time results. If the time step is chosen to be

to big, nonlinear effects have to be considered and the atoms will move too far away to

reach unrealistic positions before the next time step is calculated. Following the work

of Soules, Ochoa, and Swiler, the time step used in our simulation is 0.5 x 10-15

second, and is about 1/40 of an atomic vibration period. A test to decide if a time step

is of sufficient length is to perform a simulation over about a thousand steps and check

to see if the system energy remains fairly constant. If not, the time step is too long. A

time step of 0.5 x 10-15 seconds was found to be of acceptable length and was applied

here.


The Method to Update Particle Positions


It is very important to handle particle interactions in a time-saving way.

Particle interactions are allowed only when particles are within a certain distance for

others. Beyond the distance, the effect of interaction is either none or negligible.

Cutoff distance is the distance in which interactions are allowed. Thus a neighbors-list

for all particles which lists all the neighbors which may be considered for the next 5 or

10 time steps will be made and updated at every 5 or 10 time steps depending on the

pulling rate. The fixed distance for the neighbors-list is chosen to be the sum of the

cutoff distance and the distance a particle may move in the next 5 or 10 time steps.










Only the neighbors listed on the neighbors-list will be considered when calculating

interactions.

The squares of the distances between particles are calculated using the

Pythagorean theorem for time-saving reasons. Square roots of these values are not

taken when calculating interaction forces because square root operations are time-

consuming and the force table is tabulated in terms of distance squared. Square roots

of the distances squared are only required when the neighbors-list are updated and

when pair correlation functions and bond angle distribution functions are accumulated.


Periodic Boundary Condition


Periodic boundary conditions are applied when finding inter-particle distances.

If periodic boundary conditions are applied, particles which leave the primary cell after

a time step will be translated back into it by subtracting the appropriate cell dimension.

Thus particles which leave through one side of the system cell will appear through the

opposite side of the cell.

Thus if the difference in the coordinate is greater than one-half the

corresponding dimension of the system cell, the corresponding image of the particle

located in the adjacent cell will be closer to the central particle than the particle in the

primary cell. The difference in the coordinate is then adjusted to reflect this by either

adding or subtracting the corresponding dimension of the cell to minimize the absolute

values of the difference. It is the resulting difference which is used in the Pythagorean

relation to find the square of distance between the particles. The algorithm by which

this is performed is as follows:
dijk = Xij Xik (3.24)


dijk = dijk si trunc(2 ) (3.25)









r2 ,2 d'2 +,2
2 =dljk + djk (3.26)
rjk Ij 2jk + 3jk

where i = 1, 2, 3 denote the Cartesian dimension of space,
Xij is the i Cartesian coordinate of atom j,
Xk is the i Cartesian coordinate of atom k,

dijk is the i coordinate difference between atoms j and k,

dijk is the i coordinate difference between atoms j and k after correction

for periodic boundary conditions,

si is the i dimension of the system cell,
trunc() is a function which returns the integer closest to the argument
between the argument and zero, and

rij is the distance between atoms i and j.


Interactions Due to Two-Body Potential


Two-body interactions between particles are handled in a four step process.53
The square of the interparticle distance is found for particles located in the neighbors

lists. The interparticle force is then found by looking up the appropriate value in the
force table. The components of the force are found by multiplying the value tabulated
in the force table by the difference in the coordinates for each dimension (dijk). The

components of force are then summed for each particle for all interactions.


Interactions Due to Three-Body Potential


Three-body interactions between particles are handled in a four step process.

The square of the interparticle distance is found for particles located in the neighbors
lists. Two atoms, j and k, within the cutoff potential range are found from the
neighbor list for the center atom, i. The angle 0jik between the center atom i and the







45

other two atoms j and k were calculated. The interparticle force is then found by the
derivative of the three-body potential. The radial force and the tangential force are
calculated separately. The components of the force in each dimension i, j, and k are
found by multiplying the previous calculated tangential and radial forces with respect to
the related vectors in the coordinates. The components of force are then summed for
each particle for all interactions.

Updating of the Atom Positions

After the forces on each atom are accumulated from either the two-body or the
three-body potential, the change of positions of particles are updated using Verlet's or
Gear's algorithm at each time step. The details of the Verlet's algorithm59 and Gear's
algorithm60 will be illustrated as follows.

Verlet's algorithm

The advantages of Verlet's algorithm59 are speed and small error. The
derivation for this algorithm is as follows: According to Taylor series expansion, the
position at time (t + At) calculated from time t is

Xi(t+ At)= X(t)+ (dXi(t) )At+ 1 dX(t)) At2 + 1 d3Xi(tAt3 + O(At4);
t 2/ dt2 ) 3! dt3

(3.27)
while the position at (t At) calculated from time t is

Xi(t-At)=Xi(t) d(d1 (t) A) -3At- 1 d3Xi(t)A3+
-' 2 t 3 t3 -+O(At4)"


(3.28)









Adding these two expansions, we can get the new position:

Xi(t +At) =2Xi(t)-Xi(t-At)+( d2x-(t)At2 + O(At4), (3.29)

Where i = 1, 2, 3 denote the Cartesian dimensions of space,
t is a particular time,
Xi(t) is the i Cartesian coordinate at time t,
Xi (t At) is the i Cartesian coordinate at time t At;
At is the length of a time step, and
O(At4) is the remainder terms.


Gear's algorithm


Gear's algorithm60 is another choice to calculate the change of positions of
particles due to two-body interaction. The advantage of Gear's algorithm is that it
gives less energy fluctuation; while the big disadvantage is that it is very time-
consuming. Gear's algorithm predicts molecular positions Xi at time (t + At) using a
fifth-order Taylor series based on positions and their derivatives at time t. Thus, the
derivatives X(i), X(, Xi, Xi(iv), Xi(v) are needed at each step; these are also

predicted at time (t + At) by applying Taylor expansions at time At.

X (t + At) = Xi (t) + X(i) (t)At + Xi(i) (t) (At)2 + Xi() (t) (At)3
2! 3!

+xi(iv) (t) (4t + Xi( (t)(t)5 (3.30)
4! 5!

Xi(i) (t + At) = Xi0) (t) + Xi(ii) (t)At + Xi(iii) (t) (At)2 + x(iv) (t) (At)3
2! 3!

+Xi(v) (t) (At)4 (3.31)
4 !








2 3^
1 tAt+X~iv)() (At X~()(t)(At) (3.2
X(ii) (t + At) = Xi(ii) (t) + Xi(iii) (t)At + Xi(iv) (t) (At + X(v) (t) (3.32)
2! 3

Xi(i") (t + At) = Xi(^) (t) + Xi(iv) (t)At + Xi(v) (t) 2-- (3.33)

Xi(iv) (t + At) = Xi(iv) (t) + Xi(v) (t)At (3.34)
Xi(v) (t + At) Xi(v) (t) (3.35)
In order to correct the predicted positions and their derivatives using the
discrepancy between the predicted acceleration and that given by the evaluated force.
The force at (t + At) obtained from Newton's second law can be used to determine the
acceleration X(ii)(t + At). The difference between the predicted accelerations and
evaluated accelerations is then formed,
AXi(ii) =Xi(ii) (t + At) Xi(ii)P (t + At). (3.36)
In gear's algorithm for second-order differential equations, this difference term
is used to correct all predicted positions and their derivatives. Thus,
Xi = X +ao0AR2 (3.37)
X(i)At = xi(i)PAt + alAR2 (3.38)

X X ") (A t)2
x(ii) (At)2! -x P (At + .X2AR2 (3.39)
2! 2!

xi(iii) (At)3 xi(iii)P (At)3
3!(ii 3!~' + c3AR2 (3.40)
3! 13!
(AtV'(_t_4
xi(iv) (A = X (V)P (At) + a4AR2 (3.41)
4! 4!

xi(V) (At)5 = Xi(V)P (At) + 5AR2 (3.42)
5! 5!cAR
l. A AY()(At)2
where AR2 = AXi' (At) 2
2!
The parameter aci promotes numerical stability of the algorithm. The cXi
depends on the order of the differential equations to be solved and on the order of the








48

Taylor series predictor. Gear determined their values by applying each algorithm to

linear differential equations and analyzed the resulting stability matrices. For a q order

predictor, the values of the ci were chosen to make the local truncation error of
O(Atq+1). Values of the aci for fifth-order predictors are UQ = 3/16, C1 = 251/360,

aC2 = 1, a3 = 11/18, ca4 = 1/6, oa5 = 1/60.


Temperature Calculation


The temperature of the system is controlled by assuming that the system

behaves as a classical statistical system, with average particle kinetic energies given by

the equipartition theorem. The temperature of the system is then calculated by solving

the expression Ek = 3kT/2 for T. The temperature is then adjusted by scaling the

velocities of all particles to give the proper average kinetic energy.


M.D. Video Presentation


Information is collected from the simulation. Any aspect of the simulation can

be recorded, but there are a few which are commonly recorded. These are periodic

snapshots of the structures, instantaneous or average thermodynamic properties, pair

correlation functions, and bond angle distribution functions.

Snapshots of the structure of the system are made by periodically saving all

positions of particles of the system in a data file. These snapshots of the structures

may then later be used to perform a free volume sphere analysis on the structure to

make graphic representations of the structure.

Thermodynamic properties, such as system temperature, energy, and pressure

are calculated. Average of these values may also be kept in a record file. Thus either

instantaneous thermodynamic properties or average properties may be made.











Potential Energy of the System


The potential energy for a system is calculated in a way similar to the

calculation of forces on individual particles. When the neighbors-list is updated,

interaction potentials are found using the potential energy table. The potential energies

for all interactions are then summed to find the potential energy for the system. The

system potential energy is added to the system kinetic energy as previously obtained to

find the system internal energy.


System Pressure Calculation


The system pressure or stress tensor is calculated using the Virial of Clausius,

which has the form


P=p.k.T t-ikT>(j VD(r)) (3.43)
3J J>i

When converted for use in simulations, it takes the form


P=-3 miv +ZiYj Fij, (3.44)


where P is the system pressure in Kbar,
V is the volume of the cell in A3,

v is the velocity of the particle in A/O'10-14s,

F is the interatomic force in 10-12 erg/A,

r is the interatomic separation in A, and

mi is the mass of the particle in 10-24 g.











Pair Correlation Function


Pair correlation function distributions are accumulated when updating neighbors

lists. These are done by incrementing arrays existing for each type of interaction in

which the elements of the arrays correspond to a segment of distance every time a pair

with some separation is found. The contents of the arrays are then converted into

distributions by normalizing using the function

2-nij(r).V
G(r) =- (3.45)
N ij 4rtr2 'Ar

where G(r) is the pair correlation function,
r is the separation distance,

Ar is the separation distance,

V is the volume of the cell,

Nij is the number of pairs found between the distances r and

Ar from the central atom,

Nij m. ni-. nj, (3.46)

where m is the number of accumulations,

ni is the number of atoms of type i,

nj is the number of atoms of type j, and

47rtr-Ar2 is recognized as the volume of the shell in which the non-

central atom may be located about the central atom.

This value is output to a normalized distribution to a data file.











Bond Angle Distribution


Bond angle distributions are found by saving the atom identification numbers of

the coordinating atoms about central atoms during updates of the neighbors lists, then

applying the definition of a dot product to find bond angles. The formula for doing

this is


0 = Cos-1 xxijxik + YijYik + ziJzik (3.47)
9=o--1 k Jl-- y- (3.47)
rij rik

where 0 is the bond angle,

i denotes the apex atom,

j, k denotes the end atoms,

xij, yij, zij are the vector components between i and j,

xik, yik, zik are the vector components between i and k,

rij is the distance between atoms i and j, and

rik is the distance between atoms i and k.

Periodic boundary conditions were used. An energy distribution function is

used to assign the energy (velocity) of each atom at the beginning. The temperature of

the sample can be varied by scaling the velocity of each particle with an appropriate

factor greater than one and letting the total kinetic energy of the particles be equal to

the enthalpy of the desired temperature. The temperature of the sample was kept

constant by scaling the velocity every time step to avoid heating (or cooling) effects.











Introduction of Strain


The determination of the stress-strain behavior will provide an insight into

material behavior for different loading conditions. Uniaxial strains will be applied by

scaling the x-component of all particle positions using the relation
xi=xoi(1 +Li/Sx), (3.48)

where xi and x0i are the scaled and initial x-components of the positions of
particle i, respectively;

Sx is the x-dimension of the cell at the time of scaling, and

Li is the desired incremental expansion factor of the cell in the x-

direction.

Various strain rates can be applied by varying both the value of Li and the

frequency with which the positions are scaled.10 In the calculation, the y and z

directions will be kept constant, the change in volume is directly proportional to the

change in the x direction, Sx.10













CHAPTER 4
EXPERIMENTAL AND SIMULATION PROCEDURES


Experimental Procedure


In this section, the experimental procedure is presented. Sample preparation,

fracture surface analysis, toughness measurements, and fractal dimension determination

will be given in detail.


Sample Preparation

Single crystal silicon was provided by AT&T and IBM. The Laue back

scattering method was used to determine the orientation of the crystal. Several low

index planes were chosen to be the desired fracture planes. They are the {100}, {110}

and {111} planes. After the orientation was determined, single crystal silicon was cut

into flexural bars with the desired orientations. The surface of the specimen was

polished to a l1pm finish.
A controlled flaw was introduced in the desired crystalline plane using a

Vickers diamond pyramid indentation at the center of the tensile surface on flexure

bars. Thus, the flaw was oriented perpendicular to the longitudinal axis of the

specimen. For the {110} orientation, 0.7, 0.9, 1.3, 1.5, 3, 4, and 5 Kgw indentation

loads were applied to produce a controlled flaw in each bar. For the {100} and {111}

orientations, 1 and 2 Kgw indentation loads were applied. Figure 4.1 shows a

schematic arrangement to demonstrate the location of the indentation on the tensile

surface. Flexural testing was performed on an Instron testing machine in air at room
temperature. Loads to failure were recorded. Fracture surfaces were examined to













(a) {1 00} fracture plane


100}


(b) { 11 O} fracture plane


{110}


{11}


{111} fracture plane


{ 10}(
{I12} '{111}

S {112}
110o} {111}


Figure 4.1 Three orientations are chosen for comparison.










insure the samples had failed at the indent site. The stress to failure was calculated

from the dimension of each sample:

a 3PL (4.1)


where Ga is the fracture strength,
P is the load to failure,

L is the load span,

b is the width of the bar, and

h is the height (thickness) of the bar.


Fracture Surface Analysis

The procedures for fracture surface analysis have been well described in the

literature.6 The size of the critical flaw was determined by modeling the flaw as an

idealized elliptical crack of depth a and half width b. The equivalent semi-circular

crack radius, c, is determined by c = ab and is shown in Fig. 4.2.30

The fracture surfaces were observed using a light microscope to locate the

flaws. Failure origins were located by observing particular fracture markings on the

fracture surfaces.61 Crack branching and mirror features were observed and were

used to locate the flaw.


Toughness Measurements

From the measurement of the critical flaw size, c, and fracture strength, aa, the

critical stress intensity factor, K., for those with residual stress caused from indentation

can be determined using

Kc = 1.65ac12. (4.2)































(b)


/a/
2b









Fig. 4.2 Micro-indentation cracked bar under bending. (a) A schematic diagram
of Vickers indentation on tensile surface. (b) An elliptical flaw on the fracture cross
section.










Furthermore, we can compare the obtained results from fracture surface analysis with

those from the strength-indentation technique by using:

Kc = 0.59(E / H)8(CaP13)34. (4.3)

Here, the fracture toughness is given in terms of the critical intensity factor,

Kc, instead of Kic. Kic is the critical stress intensity factor for mode I. Kic is usually

determined from a prescribed fracture mechanics test with no residual stress associated

with the crack. Kc is the resistance to crack propagation in the presence of a local

residual stress. Kic and Kc are expected to be close in value. During the test, the

mode one condition is not guaranteed because the fracture paths tend to go to the

easiest cleavage plane. Also, the stress condition on the fracture surface is not in a full

tension mode. Indeed, sometimes it is a combination of mode I and mode II. Thus, Kc

is an approximation of the mode I fracture toughness of the material.


Fractal Dimension Determination

Single crystal silicon fracture samples were carefully cleaned and coated with

nickel. The samples were then potted in epoxy. In general, there are two ways to get

contour lines: one is to polish the sample approximately parallel to the fracture surface

and the other is to polish the sample approximately vertical to the fracture surface.

Thus, the so-called horizontal contour (fractal) dimension or vertical profile dimension

can be obtained. The samples in this study were polished approximately parallel to the

fracture surface, as shown in Fig. 4.3. As the fracture surface is first encountered, a

section of the fracture surface appears in the polishing plane. These sections appear as

islands in the polishing plane, and are called slit-islands. The nickel coating performs

two functions: It provides good contrast during polishing and it helps to hold the

fracture surface together during polishing. As polishing proceeds, these islands begin













































potted in epoxy and

polished parallel to surface


top view


Fig. 4.3 Fractured samples are encapsulated in epoxy and polished approximately
parallel to the fracture plane. (Top View) Islands emerge in the polished plane.


fractured bar









to grow. The perimeter of the islands presents a line or section of a line that can be

measured according to Richardson's equation:

L(S) = k S1-D, (4.4)

where L(S) is the length of a section of a line along the perimeter of the slit

island and its value depends on the measuring scale, S;

S is the measuring scale used to measure the perimeter and its value

ranged from 5 to 100 utm;

D is the horizontal contour (fractal) dimension; and

k is a constant.
Richardson plots detail the change in measured length of a line as a function of

scale. Construction of a Richardson plot for a fracture surface requires a line that is

representative of the fracture surface and a range of scales for measuring that line.

At the first emergence of an island, polishing must be carefully performed. The

surface was polished to a 1 gm finish. Since the epoxy is transparent, the exact

location on the fracture surface can be observed, so that measurements can be taken for

selected segments of the perimeter, e.g., in the crack branching region. Polaroid

photographs were taken at a magnification of 400x and combined in a montage. This

montage was then measured with dividers set to various openings, as depicted in Fig.

4.4. The length of the section of perimeter was measured for each divider setting. In

this way, the line or perimeter length was computed as a function of scales. A log-log

plot of length vs. scale results in a line with slope equal to 1-D.

The scale is a measure of discernibility. As the scale becomes finer and finer,

we observe greater and greater detail. If the scale is larger than the largest features,

the observation is insensitive to those features. A curve will begin to look fractal only

after the scale becomes smaller than such features. The scale lengths used in this study

ranged from 1 unit length to 16 unit lengths (1 unit length is about 5 pin).






































A montage of the perimeter is measured with dividers.


Fig. 4.4











M.D. Simulation Procedure


Before performing the MD simulation, parameters24 for the potential and initial

conditions should be given beforehand. Two data files which contain these data are

needed in order to perform the MD simulation. One is called the input file, which

includes the initial conditions and the parameters of the potentials used. The other one

is the atom-position file which contains the initial position of each Si atom.

Several tasks were set to be accomplished using MD simulation. One was to

compare the toughness of Si for different orientations. The difference of stress-strain

curves for different strain rates could be obtained to compare the toughness as a

function of strain rate. Another was to obtain the fractal dimensions for different

fracture planes in order to compare them with the experimental results. Another goal

was to obtain the fracture strengths for each simulation. The comparison of fracture

strength due to different strain rates has been performed. The comparison of fracture

strength due to different crack sizes has also been performed.


Determination of Input Data


Before performing the MD simulation, several conditions should be decided

first. All those conditions should be illustrated in the input file for the MD simulation.

Those conditions are the desired orientation, temperature, strain rate, length of each

time step, length of the simulation, and the adiabatic or isothermal state.

At first, after choosing the desired orientation, a data file which contains the

number and the positions of those atoms should be constructed. The data file will

determine the initial position of each atom for the MD simulation. In this study, the

atom position for the ideal crystal structure will be given at first and the crystal will be









allowed to thermally vibrate for one unit time, 1 pica-second, before performing strain

pulling.

The temperature to perform the test should also be given. The chosen

temperature is important because it determines the energy state of the system. Here the

temperature or initial temperature is chosen to be 300 K.

The strain-rate should be given, also. Different strain-rates will give different

results. Basically, two categories will be given. One is higher than the speed of sound

while the other is slower than the speed of sound. Generally, results from different

strain-rate experiments will be compared for the same orientation. While for different

orientations, a strain rate of 0.2 was usually chosen. The reason for selecting 0.2 will

be explained in the results and discussion section.

A careful choice of the length of each time step is very important. It plays a

very important role not only on the stability of the system but also on the length of the

simulation time. If the length of the time step is chosen to be too long, atoms will

move too far in each time step. Thus the simulation will either be less realistic or

become unstable. If the time step is too small, atoms just move a very small distance at

each time step and waste a lot of computation time. For silicon, 0.005 pica seconds is

a good choice for the Stillinger-Weber and (ionic) Coulombic potentials because the

thermal vibration period is found to be 0.0766 ps for Si.24 The length of the

simulation is determined by the fracture of the specimen. For a lower strain-rate test, a

longer time should be given.

The choice of an adiabatic or isothermal condition is very important to do MD

simulations. If the adiabatic condition is used, the temperature will increase several

thousands Kelvin degrees and the system becomes unstable after fracture. If only the

Stillinger-Weber potential is used, the temperature after fracture is about 7,000 K. If

the ionic Coulombic potential is added in, the temperature after fracture will be about

50,000K. The reason is that the ionic Coulombic potential is always repulsive. Thus,









the energy state of the system will increase as more positive charged Si atoms are

produced during the fracture process. On the other hand, the isothermal condition will

result in a more stable thermal state for this study since the temperature remains

constant. Thus, the isothermal condition was used in this study. The temperature vs.

strain curves for both isothermal and adiabatic condition are shown in Fig. 4.5 and 4.6.

The combinations of the previous conditions generally should satisfy different

needs. After these conditions have been decided, one can begin to perform the MD

simulation.


Simulation Procedure

At first, toughnesses for different orientations were compared. Three different

orientations were chosen in this study. The fracture planes were chosen to be { 100},

{110} and {111} as shown in Fig. 4.7. A Periodic boundary condition was applied

during the simulation. A constant expansion rate in the x-direction was applied for

each simulation.

The stress-strain curves for different strain rates were compared in order to find

the dependence of strength on strain rate. The strain rates were chosen to be 0.1, 0.2,

0.5, 1, 2, 5 x-length/ps.

In the Griffith criterion the strength of brittle materials depends on the crack

size at which fracture begins. The introduction of a crack in the ideal crystal will cause

a decrease in the fracture strength. Larger cracks are expected to result in lower

strengths. A crack can be introduced in two ways. One type of crack simulates a void

by removing a cluster of atoms. The other type simulates a planar sharp crack by

removing a layer of atoms.






















315

.. 310

305

S300

g 295

290 I
0 0.2 0.4 0.6 0.8 1
strain

Fig. 4.5 The temperature vs. strain curve for isothermal condition. The
temperature remains at about 300K before the moment when fracture takes place at
strain 0.45. After then the temperature goes back to about 300 K.
























10000
9000
.' 8000
^ 7000 -
S6000 / ^
S5000
S4000
S3000
S2000
1000
0
0 -- ---- )--------
0 0.2 0.4 0.6 0.8 1
Strain

Fig. 4.6 The temperature vs. strain curve for adiabatic condition. The
temperature remains at about 300K before the moment when fracture takes place at
strain ; 0.45. After fracture, the temperature of the system rise very fast up to 7000
K, then slows down.




























fil










Fig. 4.7 The 11001, 11O0, and {11} planes are those fracture planes chosen in
the MD simulation.
/ :/ /no //{i
/{^\ /_^- /J -- J







Fig. 4.7 The {100}, {110}, and {111} planes are those fracture planes chosen in
the MD simulation.











Fractal Analysis Using Simulation Results


After the fracture has occurred in the simulation as shown in Fig. 4.8, an

important objective is to obtain the generated fracture surface from the MD

simulation.Each atom is assumed to be a sphere with an electron cloud of a prescribed

radius. The radius of each atom is assumed to be the radius of the potential field.

Thus the fracture surface appears as a surface with intersecting spheres, as shown in

Fig. 4.9. After obtaining the surface, a contour plane is used to intersect the spheres to

obtain a "slit-island." The obtained slit-island appears as a plane with lots of circular

disks on it as shown in Fig. 4.10. The slit-island technique as described in the fractal

dimension determination section in the experimental procedure was applied here to

obtain the fractal dimension. The perimeter of the selected island was measured using

different scales as described before and the fractal dimension can be obtained from the

Richardson plot.
















































The stereo-pair picture shows a frozen frame during fracture.


Fig. 4.8















































Fig. 4.9 If each atom is assumed to be as a potential field with sphere shape. The
fracture surface will look like a surface with intersecting spheres.
































.. . ..






~ __ ........ .,

I4e







Fig. 4.10 The slit-island obtained from the simulated fracture surface looks like a
plane with lots of circular disks on it.














CHAPTER 5
RESULTS AND DISCUSSION


Fracture toughnesses and fractal dimensions were obtained in two ways, one

was from experimental measurement and another was from molecular dynamics (MD)

simulation. Fracture toughnesses and fractal dimensions were investigated for three

different fracture planes. This chapter discusses the toughness results, the fractal

dimension results, and how the fractal dimensions for each orientation are related to the

toughness results. This chapter is divided into three sections: experimental results,

simulation results and comparison between measured and simulation results.

Relationships between variables will be discussed in each section.


Experimental Results


In this section, results obtained from the experimental measurements will be
given. These include fracture surface analysis, fracture toughness measurements,

fractal dimension measurements, and the relationship between fractal dimension and

fracture toughness.


Fracture Surface Analysis


After the specimens were broken, the fracture surfaces were examined. The
critical flaw size was measured to calculate the critical stress intensity factor using Eq.

(4.2). As in Fig. 4.1, three fracture planes, i.e., {100}, {110} and {111}, were chosen
and two different tensile surfaces in the {110} and {111} fracture planes were tested.









Thus, there were a total of 5 different groups of data. The fracture surfaces are shown
in Figs. 5.1, 5.2, 5.3. As shown in Fig. 5.1, the fracture surface of the {100} fracture

plane appears to be the most tortuous one. It has the smallest mirror region when
compared with the other two orientations. The fracture surface of the {110} fracture
plane with a {100} tensile surface has a "Batman"-shaped mirror region as shown in
Fig. 5.2(a). The fracture surface of the {110} fracture plane with a {110} tensile
surface has an inverted volcano-shaped mirror region as shown in Fig. 5.2(b), while
the fracture surface of the {111} fracture plane as shown in Fig. 5.3 is relatively
smooth compared with the other orientations. "River-marks", i.e., twist hackle or
cleavage marks, appear on most of the fracture surfaces of this fracture plane. "River-
marks" are twist hackle fracture features which appear to spread apart as the crack path
moves away from the origin of fracture, thus "pointing" back to the origin. The {111}
fracture plane is the easiest cleavage plane, as reported before.22 Shown in Fig. 5.3(a)
is the fracture surface of the {111} fracture plane with a {110} tensile surface while
that in Fig. 5.3(b) is the fracture surface of the same fracture plane but with a {112}
tensile surface. More data are listed in Tables 5.1, 5.2, and 5.3.

For fracture on the same plane, there should be no structural difference during
the fracture process. However, the loading, i.e., strain would be expected to be
different in different locations due to elastic anisotropy and a changing strain field,
which in turn is due to the type of strain applied here, i.e., bending. The fracture
surface can appear different due to a change in the tensile surface in the bending test.
Elastic anisotropy is the reason the fracture surfaces look different in the same fracture
plane without a structural change. Thus the fracture surfaces of the {110} fracture
planes as shown in Figs. 5.2(a) and (b) look different because the tensile surfaces are
different. An interesting result occurs if we rotate the fracture surface by 90 degrees
with a {110} tensile surface, as shown in Fig. 5.4; a half mirror boundary of this
fracture surface is similar to a half mirror boundary of the fracture surface on the













































A typical fracture surface on the {100} fracture plane.


Fig. 5.1



















{100} }













(b)


MOW,
_- d.. JB &. .


{110}
^--{


Fig. 5.2 (a) A typical fracture surface of the {100} tensile surface on the {110}
fracture plane. A "Batman"-like mirror is obvious on the fracture surface. And a
typical flaw is easily seen on the fracture surface. (b) A typical fracture surface of the
{110} tensile surface on the {110} fracture plane. An inverse volcano-like mirror is
obvious on the fracture surface, and a typical flaw is easily seen on the fracture
surface.


W








(a)





{11} }


(b)




F{11 2}


Fig. 5.3 (a) A typical fracture surface of the {110} tensile surface on the {111}
fracture plane. (b) A typical fracture surface of the {112} tensile surface on the {111}
fracture plane. It is found that both fracture surfaces look similar even though the
tensile surfaces are different.














Table 5.1 Data for Si samples fractured in the {100} plane


{100} fracture plane ,<110> tensile surface _______________
indent flaw load KIc* KIc**
S load widththickness b a size P strength 0.59(E/ 1.65*
No. Kg mm mm mm mm mm lb MPa MPam^l/2 MPam^l/2
al 0.5 5.70 5.35________
a2 0.5 6.75 5.00 0.068 0.068 0.068 96 102.5 1.31 1.40
a3 0.5 5.85 4.95 0.080 0.080 0.080 65 81.7 1.10 1.21
a4 0.5 6.70 5.40 0.056 0.052 0.054 118 108.8 1.37 1.23
_____average 1.26 1.28
--___ -- dev 0.11 0.08
bl 1 6.25 5.05 0.127 0.094 0.110 63 71.2 1.18 1.23
b2 1 5.90 5.25 0.080 0.080 0.080 78 86.4 1.37 1.28
b3 1 5.80 5.05 0.124 0.124 0.124 54 65.8 1.11 1.21
b4 1 5.90 5.00 0.114 0.072 0.091 63 77.0 1.25 1.21
b5 1 5.80 5.10 0.127 0.099 0.112 60 71.7 1.19 1.25
_____average 1.23 1.24
_____ dev 0.09 0.03
cl 2 .6.25 5.20 0.201 0.132 0.163 55 58.6 1.22 1.23
c2 2 5.90 5.15 0.159 0.131 0.144 57 65.6 1.32 1.30
____. average 1.27 1.27
___________dev 0.05 0.03

t____otal average 1.24 1.26
____________total dev 0.09 0.06

__three-point bending span=27mm_________________
KIc is calculated from Eq. (4.3) **KIIc is calculated from Eq. (4.2)










Table 5.2 Data for Si samples fractured in the {110} plane


{100} tensile surface ___ _____________________
indent flaw load ____ Kc* K 1c**
No. ToadT width thickness b a size P strength 0.59(E/ 1.65*
SKg mm mm um um urn N MPa MPam^l/2 MPamAl/2
1 0.9 7.44 7.03 66.2 84.8 74.0 322.6 103.1 1.46 1.46
2 0.9 7.31 7.02 70.2 66.3 68.0 0.0. 107.4 1.51T 1.46
3 0.9 7.31 7.00 U 66.2 79.5 72.0 249.2 81.8 1.23 1.14
4 0.9 73T 6.99 79.5 92.8 86.0 262.5 86.7 1.29 1.32
-53 0.9 7.53 7.15 -95T 98.0 96.5 266.9T 81.5 1.23 1.31
6 0.9 6.87 7.16 92.8 90.0 91.5 238.0 79.4 1.21 1.25
7 0.9 6.58 7.03 94.0 94.0 195.8' 70.8 1.10 1.13
8 0.9 6.89 7.01 97.0 97.0 242.6 84.2 1.26 1.35
9 0.9 7.07 7.07 102.0 f102.0 202.4 67.3 1.06 1.12
10 0.9 6.81 7.39 953. 89.8 89.8 233.6 73.9 1.14 1.15
11 0.9 75735 6.95 89.8 92.4 198.0 63.9 1.02 1.01
12 1.4 7.11 7.37 130.0 143.0 136.3 164.6 50.1 0.95 0.95
13 1.4 6.88 7.11 136.0 80.87 __
14 1.4 7.84 7.16 32.5 124.0 127.9 193.5 56.6 1.05 1.05
5 1.4 T 7.07 7.44 119.0 106.0 112.1 229.1 68.8 1.21 1.23
16 1.8 6.83 6.92 126.0 172.0 147.0 198.0 71.1 1.32 1.41
17 1.8 7.43 02242.5 77.8 1.41 _____
18 1.8 7.82 6.94 220.0 ___ 220.2 68.7 1.29 _____
19 1.8 6.9 7.35 185.0 185.0 200.2 63.2 1.21 1.41
20 0.53 6.87 7.28 63.4 52.8 57.8 269.2 87.0 1.12 1.09
21 0.5 7.36 6.60 352T 52.8 52.8 262-.5 96.3 1.20 1.15
22 2 7.57 6.90 132.0 142.6 137.2 204.3 66.8 1.29 1.29
23 2 7.70 7.32 190.0 158.4 173.5 191.3 543.5 1.11 1.18
24 3 7.41 7.24 T195.4 142.6 166.9 209.1 63.3 1.37 1.35
25 3 6.87 6.95 169.0 184.8 176.7 186.9 66.2 1.42 1.45
26 5 6.29 7.15 322.1 294.6 307.6 133.5 48.8 1.28 1.41
27 5 7.45 7.37 316.8 253.4 283.3 169.1 49.1 1.29 1.36
________ avg. 1.19 1.23~
________________dev. 0.08 0.08
{110} tensile surface _____ _____ ______
indent flaw load Kc1* K~ Klc**
No. load width thickness b a size P strength 0.59(E/ 1.65
- Kg mm mm umr um um N MPa MPam^ 1/2 MPamA1/2
41 0.9 7.35 6.97 T1584 105.6 129.3 160.2 52.76 0.89 0.98
42 0.9 7.55 7.39 95.0 95.0 95.0 278.1 79.30 1.20 1.26
43 0.9 7.10 7.56 89.8 100.3 94.9 233.6 67.71 1.07 1.07
44 0.9 6.93 7.84 1056 105.6 105.6 229.1 63.24 1.01 1.06
45 0.9 6.76 7.03 84.5 79.2 81.8 204.7 72.06 1.12 1.08
46 0.9 7.44 7.26 843. 95.0 89.6 215.8 64.73 1.04 1.01
47 5.0 7.08' 7.21 380.1 337.9 '3584 100.1 31.99 0.93 0.99
48 5.0TT739 7.26 269.3 316.8 292.1 142.4 43.00 1.17 1.21
______avg. 1.05 1.07
____ ____dev. 0.10 0.09
Klc is calculated from the Eq. (4.3) ** Klc is calculated from the Eq.(4.2)










Table 5.3 Data for Si samples fractured in the {111 } plane


110 ,tensile surface__________________
indent _____flaw load ____KIc* KIc**
load width thickness b a size P strength 0.59(E/ 1.65*
No. Kg mm mm nmm mm mm lb MPa MPamA^l/2MPam^l/2
cl 1 6.15 4.00 0.082 0.092 0.087 33 76.1 1.20 1.17
c2 1 5.80 4.95 0.093 0.102 0.098 46 73.4 1.17 1.20
c3 1 6.30 5.05 0.091 0.103 0.097 40 56.5 0.96 0.92
A1 1 6.15 5.60 0.110 0.112 0.111 82 70.9 1.14 1.23
A2 1 6.20 5.20 0.103 0.124 0.113 78 77.6 1.22 1.36
_____avg 1.14 1.18
________ dev 0.09 0.15
dl 0.7 6.15 4.65 0.078 0.074 0.076 44 75.1 1.09 1.08
d2 0.7 6.20 5.15 0.058 0.063 0.061 72 99.4 1.34 1.28
d3 0.7 6.10 4.90 0.080 0.086 0.083 48 74.4 1.08 1.12
d4 0.7 6.25 5.05 0.065 0.085 0.075 56 79.7 1.14 1.14
avg 1.16 1:15
_______________dev 0.11 0.07
el 2 6.30 4.80 0.171 0.200 0.185 34 53.2 1.09 1.19
e2 2 6.25 4.75 0.149 0.155 0.152 40 64.4 1.26 1.31
e3 2 5.70 4.70 0.172 0.173 0.172 29 52.3 1.08 1.13
e4 2 5.95 4.95 0.160 0.118 0.137 38 59.1 1.18 1.14
_________avg 1.15 1.19
_________dev 0.07 0.07

{112} tensile surface___________________
fl 2 6.15 6.25 0.108 0.115 0.111 68 64.2 1.26 1.12
f2 2 5.35 6.20 0.125 0.144 0.135 60 66.2 1.29 1.27
13 2 5.25 6.15 0.165 0.125 0.143 50 57.1 1.15 1.13
f4 2 5.80 6.10 0.165 0.183 0.174 48 50.5 1.05 1.10
avg 1.19 1.15
dev 0.09 0.07
gl 1 6.40 5.80 0.079 0.094 0.086 67 70.6 1.14 1.08
g2 1 5.50 6.20 0.105 0.105 0.105 70 75.1 1.19 1.27
g3 1 5.85 5.65 0.086 0.099 0.092 52 63.2 1.05 1.00
B1 1 6.30 6.05 0.109 0.098 0.103 101 73.1 1.17 1.23
B2 1 5.80 5.70 0.086 0.071 0.078 93 82.3 1.28 1.20
avg 1.16 1.16
____dev 0.07 0.10
KIc is calculated from Eq. (4.3) **KIc is calculated from Eq. (4.2)
A,B span = 25.5 mm J c,d,e,f, and span =34
three-point bending I Jf I | n









(a)


















(b)


4;


{100} 1 }
/{m /{


-i


K
-~ -~ -


Fig. 5.4 An interesting result is that if we rotate the fracture surface of the one
with a { 110} tensile surface 90 degrees, as shown in (a), then a half mirror boundary of
this fracture surface is similar to half a mirror boundary of the fracture surface on the
{110} fracture plane with a {100} tensile surface without rotation, as shown in (b).
This implies that a fracture mirror boundary for a specimen fractured in tension from
an internal defect would mimic the trace of the elastic constant values in that plane.









{110} fracture plane with a {100} tensile surface without rotation.50 This implies that
a fracture mirror boundary for a specimen fractured in tension from an internal defect

would mimic the trace of the elastic constant values in that plane.52

Freiman et al.52 found that the fracture mirror shapes in sapphire for pseudo-

cleavage type fracture have much more complex geometry, as shown Fig. 5.5, than the

similar fractures in glasses which yield more-or-less circular fracture mirrors. They

relate the mirror boundaries to the critical stress intensity factor. The complex mirror

shapes investigated in sapphire implies that the critical stress intensity factor along the

different crystallographic directions of the mirror boundary is not constant. Young's

modulus also plays an important role for the complex shape of mirror boundary. The

Young's modulus normal to the crack plane is a constant. However, the Young's

moduli either parallel or perpendicular to the mirror boundary in the crack plane can

vary as one traverses around the boundary.62 As seen in Fig. 5.5, The Young's

moduli for directions parallel and perpendicular to the crack front/mirror boundary

plotted as a function of angle of rotation about the tensile axis possesses maxima-

minima patterns which are similar to the fracture mirror geometry. A similar result for

single crystal silicon is expected and a schematic of the expected mirror boundary is

plotted in Fig. 5.5.

In contrast to the fracture surface of the {110} fracture plane, the fracture

surfaces of the {111} fracture plane look the same even when the chosen tensile

surfaces are different as shown in Fig. 5.3. Most probably, the fracture surface of the

easiest cleavage plane will not be affected by the chosen tensile surface because the

crack growth will remain on the easiest cleavage fracture plane as the lowest energy

path. Also, the size of the specimen is not large enough to show crack branching at

these tested fracture stress levels. Thus, the fractography would appear the same no

matter what the change in the tensile surface at these test specimen sizes. The available
fracture paths for the specific fracture planes are another important factor to determine


























<1014>
/I -- .


Fig. 5.5 (a) Anisotropy mirror shapes on fracture surfaces in single crystal
alumina; the corresponding relative magnitudes of Young's modulus for directions (in
the plane) parallel and perpendicular to the local mirror boundary are given. Larger
values of E are reflected as greater distances from the origin. (b) A schematic of the
expected mirror boundary for the {110} fracture plane is plotted.









the topography of the fracture surface. A more detailed discussion of macroscopic
crack branching follows.
The crack branching plane is an interesting topic in this study. From the
observation of the obtained fracture surfaces, the fracture surface of the { 111 } fracture
plane is relatively smooth such that no crack branching is observed as shown in Fig.
5.6. For fracture in the {110} plane, the crack either doesn't branch or tends to branch
to the available {111} planes as shown in Figs. 5.7 and 5.8. In Fig. 5.7, the crack
growth (shown by shaded regions) occurs on the {110} plane first and deviates to
{111} plane as shown in (c) and (d) of Fig. 5.7. In Fig. 5.8, the crack growth (shown
by shaded regions) occurs on the {110} plane first and deviates to {111} planes as
shown in (c) and (d) of Fig. 5.8.
Crack branching for the {100} fracture plane is complex. As seen in Figs. 5.9
and 5.10, the system of crack branching angles varies for one fracture plane as well as
for the loading direction changes, e.g. from the {100} tensile surface to the {110}
tensile surface.
For those fracture surfaces of the {100} fracture plane with a {100} tensile
surface, the crack easily branches to the {111} plane as shown in Fig. 5.9. If the
tensile surface is chosen to be {110}, the crack will remain on the {100} plane for
about 2 to 3 mm before branching to the {111 } plane if the indentation load is larger
than 2 Kgw, as shown in Fig. 5.10(b). If the indentation load is smaller than 2 Kgw,
the crack will branch in a distance smaller than 2mm as shown in Fig. 5.10(c).
It is also found that the fracture mirror region is not really very smooth.12 The
reason that we say it is "mirror-like" is because the tortuosity of the region is smaller
than the wave-length of visible light as shown in Fig. 5.11. Indeed, if a scanning
tunneling microscope (STM) is used to observe the mirror region, then the surface is
not smooth at the scale at which the STM is used. 12 As in Fig. 5.12, the surface on
the mirror region is quite rough at that magnification. Thus, the geometry in the
























,, {110}
{io
,{111}'{
, ;11 } L


i{\ {11:
{111} {111}
I :i!l 0!


Crack path for the { 111 } fracture plane.


Fig. 5.6






















{low,
1110}


fioo


Fig. 5.7 The crack path for the {110} fracture plane with a {100} tensile surface.
The crack growth (shown by shaded regions) occurred on the {110} plane first and
deviated to the {111} plane as shown in (c) and (d). (a) shows a schematic of the
fracture bar. (b), (c), and (d) show the different modes observed.



























{111}


{11


! 110


{100} .
I, I10:


1
f110}


*10I 10
*lO01. \ 'r ;
_ [110o


Fig. 5.8 The crack path for the {110} fracture plane with a {110} tensile surface.
The crack growth (shown by shaded regions) occurred on the {110} plane first and
deviated to the {111} planes as shown in (c) and (d). (a) shows a schematic of the
fracture bar. (b), (c), and (d) show the different modes observed.


SI 10k


:110:


>* ,



















7 L {100}
10
{l00}


Fig. 5.9 The crack paths (shown by shaded regions) for the {100} fracture plane
with a {100} tensile surface. (a) shows a schematic of fracture bar. (b) shows
different modes observed.










S {11O}
101
S{110}


Fig. 5.10 The crack paths (shown by shaded regions) for the {100} fracture plane
with a {110} tensile surface. (a) shows a schematic of fracture bar. If the tensile
surface is chosen to be {110}, the crack will remain on the {100} plane for about 2 to 3
mm before branching to the { 111} plane if the indentation load is larger than 2 Kgw as
shown in (b). If the indentation load is smaller than 2 Kgw, the crack will branch in a
distance smaller than 2 mm generally as shown in (c).























mirror 4-
region


mist
region


-M _- hackle
region


wavelength
of light


_4


Fig. 5.11 Schematic of vertical profile of the mirror, mist and hackle regions of a
fracture surface. The reason why the mirror region is mirror-like is because the tortuosity
of the region is smaller than the wave-length of visible light as shown above.














































Fig. 5.12 The fracture surface obtained from scanning tunneling microscopy.
(Courtesy of Mitchell and Bonnell12)









mirror region appears to be smooth to the unaided eye, but appears to be tortuous if the

STM is used as the tool of observation.

The reason that the fractographic features described above are important is that

the distances to crack branching have been shown to be related to different energy

levels at branching and the fracture toughness of the material.50 In addition, fracture

surface features have also been shown related to the fractal dimension of the surface.63
Thus, the fracture surface features are intimately connected to the fracture process as

described by the fracture toughness, branching constants and the fractal dimensions.

These inter-relationships will be discussed later in this chapter.


Fracture Toughness Measurement


Single crystal Si is anisotropic and the fracture toughness varies with the

fracture plane (and the tensile surface if a bending technique is used to fracture the Si).

Chen and Leipold22 determined the toughness for single crystal silicon tested in

different orientations. They found that the {111} fracture plane is the easiest cleavage

plane and the {100} fracture plane is the toughest one. They used an indentation

technique to control the crack sizes on the tested specimens before fracture. However,

in their work, they did not consider the residual stress caused by the indentation. The

equation which they used to calculate the critical stress intensity doesn't consider

residual stress:

Kc = YMB(na/Q)l/2, (5.1)

where Kc is the critical stress intensity factor,

cy is the maximum tensile stress,

a is the flaw size,

MB is the elastic stress intensity magnification factor, and









Q is the flaw shape parameter calculated from the elliptical integral of
the second kind.
From the given flaw size data in their paper,22 the MB value was calculated to
be 1.03 and Q was approximately 1.47. After substituting these data into equation
(5.1), the equation can be approximately rewritten as
Kc = 1.24oc/. (5.2)
If we modify Eq. (5.2) to include the effect of residual stress (due to the indentation
1/2404
process), the new modified equation becomes Kc = 1.65a fc i.e., Eq. (4.2).4044
Their data using Eq. (4.2) fall in the same range as those measured in this study. The
measured data and modified data from Chen and Leipold are presented in Table 5.4.


Table 5.4 Comparison of toughness values using different techniques


S.I. F.S.A. literature
fracture plane tensile surface toughness* toughness** toughness#
______ __________(MPaim) (MParim) (MPaJfm)
{100} {110} 1.24 0.09 1.26 0.06 1.26 0.07

{110} {100} 1.19 0.08 1.23 0.08 1.19 0.13
{110} 1.05 0.10 1.07 0.09
{111} {110} 1.19 0.10 1.17 0.09 1.09 0.09
{112} 1.21 0.09 1.16 0.08
*Calculated from the strength indentation (S.I.) technique,

KC =0.59E 1/8(afpl1/3 )3/4
1/82
Kc= 0.59 (fP1/3 .

**Calculated from fracture surface analysis (F.S.A.), Kc = 1.65afc12.
#By Chen and Leipold.22 Originally these values were calculated without considering
residual stress and were found to be 0.950.05 ({100} plane), 0.900.11 ({110}
plane), and 0.82+0.07 ({111} plane) MPaVm.









Gilman2, Jaccodine,64 Myers & Hillsberry,65 St. John,57 and Kalwani66 also
investigated the fracture toughness in single crystal silicon. Those data and applied

techniques are listed in Appendix A.
Single crystal Si is a diamond cubic structure which is considered to be one type
of face centered cubic structure. A simple approach67 which has been accepted to
estimate the surface energy of different planes in a FCC structure is to calculate the

number of dangling bonds on the specific plane for the FCC materials. The surface

energy can be obtained using the equation:

y = N "Nhkl} n{hkl}, (5.3)
2

where e is the energy needed to fracture the Si-Si bond,

is used because two surfaces are generated,
2
N{hkl} is the number of atoms per area on the {hkl} plane, and
n{hkl} is the number of dangling bonds per atom on the {hkl} plane.
By using the given equation, we can find the surface energy for the FCC
structure on the {100}, {110}, and {111} planes separately as shown in Fig. 5.13.
Since the number of atoms on the {100} plane is 1 + 1/4 4 as shown in Plot (b),
which is 2, and the area A where those atoms lie is a a where a is the length of the
side of a FCC unit cell, N{100} is found to be 2/a2. Since the number of dangling
bonds per atom on the {100} plane is 4 as shown in Plot (c), Y{100oo} is found to be

1 2 1 2
2 -E .- 4. In the same way, Y{1o10} is found to be 2- E 2 5, and Y{11 is found to
2 a-;2 x/2a-

1 2
be .6. .-3. Thus y{100} : Y{ll0} : Y{lll} = 8 : 7.07 : 6.92 = 1.16 : 1.02 :1.
2 "r3a 2/2

This ratio means that the {111} plane for the FCC structure has the lowest surface
energy and the {100} plane has the highest surface energy.


















(100) = a


A(1O) 2a2


~A(=32
A(llIl)=Vf3a /2


#. of atoms
= 4x1/4+1


#. of atoms
=4x1/4+2x1/2


#. of atoms
= 3x1/6+3x1/2


Fig. 5.13 The surface energy for FCC structures on the (100), (110), and (111)
planes can be found using a dangling bond calculation.


n(loo)= 4


00


00


n(110)= 5


n(11)= 3










However, single crystal Si has a diamond cubic (DC) structure which, in fact, is

different from FCC because it is not closed packed. We can find the surface energy for

the DC structure on the {100}, {110}, and {111} planes separately as shown in Fig.

5.14. However, the DC structure is not a close-packed structure, and the dangling

bonds will not be linked to all neighbors. Only some specific neighbors in the first

shell will share the dangling bond. The previous method cannot be completely

followed and some modifications must be made. As shown in the figure, the dangling

bond is expressed as a line. The surface energy can be obtained using the equation:

1 1 (5.3)
Y=-2"{hkl} "A' A5'3)
2A
where 6 is the energy needed to fracture the Si-Si bond,
1.
is used because two surfaces are generated,
2
n{hkl} is the number of dangling bonds on the {hkl} plane, and

A is the area where those atoms lie on.

Since the number of dangling bonds on the {100} plane is 4 as shown in Plot

(c), and the area A where those atoms lie is a a and a is the length of the side of a

silicon DC unit cell. y{100oo} is found to be I .E If s = 3.5-10-12 erg/pair-atom22
2 a2

and a = 5.43 A is substituted into the equation, 7{100oo} is found to be 2.3 J/m2. (The

Si-Si bond energy is found to be varied from 3.2 to 3.6 erg/pair-atom.24'64'68'69) In

1 4 _
the same way, y{110o} is found to be -*1 = 1.67 J/m2, and y{11} is found to be
2 72:a2

1 2
-.e. 1.36 J/m2. These values are close to the experimental value, 2.1
2 73a2/2

J/m2, which is obtained using double cantilever beam tests.70'71 Thus Y{100} : Y{110o}

Y{111} = 2.3 J/m2 : 1.67 J/m2 : 1.36 J/m2 = 1.69 : 1.23 : 1. So, from dangling bond




Full Text
35
h(ry, rik, Ojik) = X exp[ri /(ry a) + r| /(rik a)] x (cos0jik + 1/3)2, if r (3.5)
= 0, if r>a.
where 0jik is the angle between rj and rk. The function h exists when both ry and rik
are less than the previously introduced cutoff a.
The parameters which best satisfy the radial distribution function were
determined by Stillinger and Weber to be: A = 7.049556277, B = 0.6022245584, p =
4, q = 0, a = 1.8, 1 = 21.0, h = 1.20, d = 0.20951 nm, and s = 50 kcal/mole =
3.4723xl0~12 erg/atom pair.
Although the Stillinger-Weber potential has been successfully used to model the
structure of silicon, it does not provide the expected repulsive forces following covalent
bond breakage which result from ion formation during the fracture process. The Si-Si
covalent bond is formed by an attractive force even if the distance between the two
atoms is larger than the equilibrium position as long as it is within the cutoff distance.
However, during fracture this condition is not realistic and makes the Si-Si bond
difficult to break. An unrealistically long fiber-like structure is observed to result if the
unmodified Stillinger-Weber potential is used. An improvement developed here
assumes that following bond breakage each Si atom will gain one positive charge for
each lost neighbor. Si will have two positive charges if it loses two of its four
neighbors and four positive charges if all four neighbors are displaced beyond the
cutoff distance.
A modified Born-Mayer potential10,57 is used for modeling the resulting ionic
forces. Consequently this approach assumes ionic repulsive forces when two Si atoms
both with positive charges fall inside the Born-Mayer cutoff distance. This potential
has the following form:


EXPERIMENTAL AND MOLECULAR DYNAMICS DETERMINATION OF
FRACTAL FRACTURE IN SINGLE CRYSTAL SILICON
BY
YUEH-LONG TSAI
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


128
Toughness in Different Orientations
Stress-strain curves as shown in Fig. 5.31 have been obtained in three different
orientations, i.e., {100}, {110} and {111}. From the slope of the stress-strain curve,
the modulus is found to be 92 GPa for {100}, 130 GPa for {110} and 138 GPa for
{111}. For single crystal Si, s^, Sj2> and S44 are measured to be 0.768, -0.214, and
1.256 x 10'11 Pa1.75 Thus, the experimental E^i values are calculated to be 130 GPa
for {100}, 169 GPa for {110} and 188 GPa for {111}, respectively (Appendix C). The
simulated values are about three quarters of the measured values. Although the
simulated values are not the same as the actual values, the ratio of the values are about
the same.
The fracture strength from MD simulation is 38 GPa for {100}, 36 GPa for
{110} and 30 GPa for {111} if a fixed strain rate, 0.2 1/ps, is applied for each
orientation. Generally, the strength for ideal materials is about E/2ti. The simulated
value is approximately between E/4 to E/6 if the actual modulus is used or is about
between E/3 to E/4.5 if the simulated modulus is used. The reason for such high
fracture strengths in the simulation is that the Stillinger-Weber potential has a 3.76
cutoff distance. Since the cutoff distance is so large, the strain at the time of Si
fracture appears to be unreasonably high for brittle materials such as Si. However,
local stresses are expected to be near the theoretical values, which are high. The
simulation is considered to model local stresses.
The toughness can be estimated from the stress-strain curve, if we assume that
the area under the stress-strain curve at the point where it reaches the maximum stress
is the energy consumed to fracture the samples. Thus, the equation to calculate the
toughness is as follows:
toughness = jQmaxcr ds.
(5.10)


101
Fig. 5.16 Self-similar features shown on the fracture surface of the {110} fracture
plane with the {100} tensile surface. If a small area in the 100X picture is magnified to
1,000X, the same features are seen. When a small region in the 1,000X is magnified to
10,000X, once again the same features appear. Scale invariance is displayed, (a) A 100X
picture, (b) A 1,000X picture, (c) A 10,000X picture.


Relative Frequency
124
Fig. 5.27 The Si-Si-Si bond angle distribution during pulling. Initially the
distribution has a peak at about 109. After uniformly pulling, the distribution
undergoes a change which depends on the applied strain.


126
Fig. 5.29 At 47% extension, fracture occurs. The evidence for fracture is the
presence of a free volume sphere in the center region.


3
present research was undertaken to study the relationship between fracture surface
morphology of single crystals and their resistance to fracture.
Unlike mathematically generated fractals, the self-similarity of fracture surface
geometry is bounded by measurement limitations. These limitations will be discussed
later. However, the self-similar nature of fractal surfaces offers a means of scaling
macro-scopically analyzable structures to microscopic processes on the atomic scale.
In addition, the fractal dimension has been shown to relate directly to fracture
toughness, thus forming a link between average macroscopic behavior and possible
atomic processes.13,17
Single crystal silicon was selected as a model material in which to study the
correlation of the fracture surface features as characterized by their fractal dimension
for different orientations of fracture with the fracture toughness of the material as
measured using the strength-indentation, and fracture surface analysis techniques.
Single crystal Si was selected for several reasons: Si is a brittle, monoatomic material
that will obviate the complication of microstructure, i.e. grains, pores, etc. And Si has
been well studied so that many of its properties are well characterized. Flexure bars
were indented with a Vickers indent at various loads and fractured. After calculating
the fracture strength and toughness, the surfaces were analyzed and characterized using
fracture surface analysis and slit-island analysis. These analyses provided the size of
the fracture initiating defect, the geometry of the surrounding topography including the
location of the regions of crack branching and the fractal dimensions of selected areas
on the fracture surface.
In this study, we demonstrate the formation of fractal surfaces during fracture
using a molecular dynamics (MD) approach in a single crystal silicon structure and try
to compare the simulated results with the experimental works. The close agreement
between simulated and measured fractal structures of the fracture surface suggests that
this is a promising method for investigating atomic-level processes during fracture.


75
Fig. 5.3 (a) A typical fracture surface of the {110} tensile surface on the {111}
fracture plane, (b) A typical fracture surface of the {112} tensile surface on the {111}
fracture plane. It is found that both fracture surfaces look similar even though the
tensile surfaces are different.


109
a0 is a characteristic (atomic) length, and
Ejdd is the Young's modulus of the hkl fracture plane.
The plot of y vs. ED*/2 for different orientations of single crystal silicon is
shown in Fig. 5.18. The slope with a constant value suggests that a0 is a constant,
which means that fracture surface energy is the sum of the energy necessary to break
bonds and the energy which makes the flat surface into a tortuous surface. The latter
energy is consumed in the fracture process and contributes to the formation of a
fracture surface geometry. This meaning agrees very well with the fact that the
fracture surface is never ideally smooth even in the easiest cleavage plane. By using
Eq. (5.10), the a0 value is found to be 4.1 , which is interpreted to be the stretched
length of the Si-Si bond just before fracture.
As seen in Fig. 5.18, the value of y0 in Eq. (5.10) is the intersection of the line
with the y coordinate at ED* = 0 and is found to be 1.7 J/m2, which is very close to
the value of surface energy calculated from bond breaking as discussed in the fracture
toughness measurement section. Those calculated values are found to be 2.3 J/m2 for
the {100} plane, 1.67 J/m2 for the {110} plane, and 1.36 J/m2 for the {111} plane.
The assumption used to calculate the surface energy in the fracture toughness section is
that the bonds all break in one plane with no crack branching and no non-coplanar
crack growth. Thus, the calculated value should be lower than the measured value
from experimental work and should be close to the energy, y0, needed to produce a
fracture surface with Euclidean geometry.


115
the SW and the Coulombic modified Born-Mayer potential) as applied to single crystal
silicon in this study will be checked later with the pair correlation functions and system
pressure at p = 2.33 gm/cm3. Generally, before the fracture process takes place, only
the SW potential interacts between atoms because each Si atom has four neighbors.
Thus in doing evaluations of the system pressure and pair correlation functions, the SW
potential is the main factor at low temperature and before fracture.
Since the judgment of appropriate potentials rests on the fit with experiment, we
compared the pair correlation function obtained using the SW and modified Born-
Mayer potentials with measured pair correlation functions. Fig. 5.21 shows the pair
correlation function of single crystal Si at room temperature from MD simulation. Fig.
5.22 shows the same (simulated) function at a temperature a little lower than the
melting point. Fig. 5.23 shows the same function at a temperature a little over the
melting point. In Fig. 5.22, the curve from the simulation fits quite well with the
measured one at the first and second peaks.24 After the second peak, some differences
show up because only two-body and three-body potentials were considered in the SW
potential and thus, it is not really a long-range potential. So the agreement at the first
and second peaks is sufficient for local structure calculations.
Fig. 5.24 shows the pressure of the system at room temperature if the actual
density of silicon is selected in the input-data file. The system remains balanced slightly
above 0 GPa with some fluctuations caused by thermal vibrations. Cohesive energy,
crystal to liquid phase transformation, and structure factors obtained from MD
simulations have been studied to compare with the experimental values by Stillinger
and Weber24 and are found to fall in a reasonable range.


68
Fig. 4.8
The stereo-pair picture shows a frozen frame during fracture


157
Fractal Dimension
Fig. B.l The sample distribution for the fractal dimensions of the {100} fracture
plane with a {110} tensile surface, D = 1.16 + 0.04, and the {100} fracture plane
with a {110} tensile surface, D = 1.10 + 0.04.


107
equal to P x (4/3) and A x (1 + 1/3) separately after the first generation. The
r\
perimeter and area of the new object would be equal to P x (4/3) and A x {1 + 1/3 +
(1/3)2}, separately after the second generation. After the nth generation, the new
perimeter would become
Pnth = P x (4/3)n-l (5.8)
- infinity, if n is a very large number;
while the area of the generated object would become
Anth = A x ( 1 + 1/3 + (1/3)2 + (1/3)3 + ... + (i/3)n-l ) (5.9)
w A x ( 1.5 ), if n is a very large number.
Thus, the more tortuous the object, the more surface area generated. Using this
construction, an object with finite area but infinite perimeter could be generated and the
fractal dimension of this object is 1.26 ( = log 4 / log 3 as described in the fractal
dimension section in Chapter 2).
Table 5.7 Modulus, toughness, surface energy, and fractal dimension in
different orientations for single crystal silicon.
fracture
plane (hkl)
tensile
surface
Ehkl
(GPa)
K
(MPa m1/2)
Y
(J/m2)
D*
(average)
{110}
{110}
169
1.07
3.39
0.04
{111}
{112}
188
1.16
3.58
0.05
{111}
{110}
188
1.17
3.64
0.06
{110}
{100}
169
1.23
4.48
0.10
{100}
{110}
130
1.26
6.11
0.16
If this construction of an object between one and two dimensions is extended to
two and three dimensions, a solid body with infinite surface could be constructed.


63
the energy state of the system will increase as more positive charged Si atoms are
produced during the fracture process. On the other hand, the isothermal condition will
result in a more stable thermal state for this study since the temperature remains
constant. Thus, the isothermal condition was used in this study. The temperature vs.
strain curves for both isothermal and adiabatic condition are shown in Fig. 4.5 and 4.6.
The combinations of the previous conditions generally should satisfy different
needs. After these conditions have been decided, one can begin to perform the MD
simulation.
Simulation Procedure
At first, toughnesses for different orientations were compared. Three different
orientations were chosen in this study. The fracture planes were chosen to be {100},
{110} and {111} as shown in Fig. 4.7. A Periodic boundary condition was applied
during the simulation. A constant expansion rate in the x-direction was applied for
each simulation.
The stress-strain curves for different strain rates were compared in order to find
the dependence of strength on strain rate. The strain rates were chosen to be 0.1, 0.2,
0.5, 1, 2, 5 x-length/ps.
In the Griffith criterion the strength of brittle materials depends on the crack
size at which fracture begins. The introduction of a crack in the ideal crystal will cause
a decrease in the fracture strength. Larger cracks are expected to result in lower
strengths. A crack can be introduced in two ways. One type of crack simulates a void
by removing a cluster of atoms. The other type simulates a planar sharp crack by
removing a layer of atoms.


27
The fractal dimension of this object, however, can be computed in another,
quite different fashion. Scaling fractals obey the following:
Nr = 1, (2.18)
where N is the number of elements in the generator,
r is the scale factor of an element,
D is the similarity or fractal dimension.
For the plot of Fig. 2.8 called the quadratic Koch curve, N = 8, r=l/4, Thus,
the fractal dimension D = log8/log4 = 3/2.
Fracture Surface Analysis by Fractal Geometry
Mecholsky et al.17,48 experimentally determined a relationship between fracture
toughness (Kc) and fractal dimension (D). They found that
Kc = A ( D 1 )1/2 = A D*1/2, (2.19)
where D* is the fractional part of the fractal dimension,
Kc = KIc K0, where K0 is the toughness of the material for a smooth
(Euclidean) fracture surface, and
A is a constant.
Thus, as fractal dimension increases, fracture toughness increases. In some
papers (D-l) is referred to as D*. Thus the above equation can be rewritten as
Kc A D*1/2, (2.20)
where A is a family parameter. The family parameter, A, identifies a line in the Kc
vs. D* plane. Within a family, an increase in fractal dimension corresponds to an
increase in fracture toughness.
Since D* is dimensionless, a dimensional analysis of this equation requires that
A have the dimensions of toughness. Mecholsky et. al.17 proposed that A is a product


8
Fig. 2.1 (a) Tetrahedron and (b) the diamond cubic (DC) unit cell, (c) The length
of AB is 5.43 , AC is 9.41 , and AD is 7.68 .


121
Strain
!
2
The system pressure (stress) during 0.2 ps
-1
strain rate pulling along
Fig. 5.25
the x-axis.


123
Fig. 5.26 Pair correlation function during extension. The first peak in the pair
correlation functions all shift to greater distances during expansion of the sample.


18
Thus, the resistance to fracture can be determined from a measurement of the surface
crack size and the strength of bars that have been indented.
Strength-Indentation Technique5
Consider the Vickers diamond-induced radial cracks, which are shown in Fig
2.4. If the applied loading is uniaxial, the indentation is aligned with one set of
pyramidal edges parallel to the tensile axis.
It has been shown that the toughness may be derived using the following
expression:
(2.14)
where
E is the Young's modulus, and
H is the hardness.
Replacing by an average quantity, 0.59, would add no more than 10% to the error
in the Kc evaluation for a material whose elastic/plastic parameters are totally
unknown. So the above equation can be rewritten as
(2.15)
Thus, the resistance to fracture can also be measured from a flexural test in which the
specimens have been indented at different load levels without measuring the crack size.


52
Introduction of Strain
The determination of the stress-strain behavior will provide an insight into
material behavior for different loading conditions. Uniaxial strains will be applied by
scaling the x-component of all particle positions using the relation
xj = xoi( 1 + Lj / Sx), (3.48)
where Xj and x0 are the scaled and initial x-components of the positions of
particle i, respectively;
Sx is the x-dimension of the cell at the time of scaling, and
Lj is the desired incremental expansion factor of the cell in the x~
direction.
Various strain rates can be applied by varying both the value of Lj and the
frequency with which the positions are scaled.10 In the calculation, the y and z
directions will be kept constant, the change in volume is directly proportional to the
change in the x direction, Sx.10


50
Pair Correlation Function
Pair correlation function distributions are accumulated when updating neighbors
lists. These are done by incrementing arrays existing for each type of interaction in
which the elements of the arrays correspond to a segment of distance every time a pair
with some separation is found. The contents of the arrays are then converted into
distributions by normalizing using the function
G(r) =
2'njj(r)-V
Nij-47xr2-Ar
(3.45)
where G(r) is the pair correlation function,
r is the separation distance,
Ar is the separation distance,
V is the volume of the cell,
Ny is the number of pairs found between the distances r and
Ar from the central atom,
Ny = m nt nj, (3.46)
where m is the number of accumulations,
n¡ is the number of atoms of type i,
nj is the number of atoms of type j, and
4yir-Ar is recognized as the volume of the shell in which the non
central atom may be located about the central atom.
This value is output to a normalized distribution to a data file.


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
EXPERIMENTAL AND MOLECULAR DYNAMICS DETERMINATION OF
FRACTAL FRACTURE IN SINGLE CRYSTAL SILICON
By
Yueh-Long Tsai
December 1993
Chairman: Dr. J. J. Mecholsky
Major Department: Materials Science and Engineering
Single crystal silicon was selected as a model material in which to study the
correlation of the fracture surface features as characterized by their fractal dimension for
different orientations of fracture with the fracture toughness of the material as measured
using the strength-indentation, and fracture surface analysis techniques. Single crystal Si
was selected for several reasons: Si is a brittle, monoatomic material that will obviate the
complication of microstructure, i.e., grains, pores, etc. And Si has been well studied so
that many of its properties are well characterized. Flexure bars were indented with a
Vickers indent at various loads and fractured. After calculating the fracture strength and
toughness, the surfaces were analyzed and characterized using fracture surface analysis
and slit-island analysis. These analyses provided the size of the fracture initiating defect,
the geometry of the surrounding topography including the location of the regions of crack
branching and the fractal dimensions of selected areas on the fracture surface.
The {100} plane is found experimentally to be the one with the highest toughness
of the three studied planes. The fracture plane on the {100} and {110} fracture plane
both initiate on the original plane and have the tendency to deviate to the {111} plane.
v


85
(a)
(b)
(c)
(d)
Fig. 5.8 The crack path for the {110} fracture plane with a {110} tensile surface.
The crack growth (shown by shaded regions) occurred on the {110} plane first and
deviated to the {111} planes as shown in (c) and (d). (a) shows a schematic of the
fracture bar. (b), (c), and (d) show the different modes observed.


150
Table 5.12 Fractal dimension measured from experimental and simulation work
fracture
plane
tensile
surface
toughness
from
measurement
(MpaVm)
fractal
dimension
from
experiment
fractal
dimension
from
experiment
- the higher
fractal
dimension
from
simulation
{100}
{110}
1.26 0.06
1.16 0.04
1.16
1.16 0.06
{110}
{100}
{110}
1.23 0.08
1.07 0.09
1.10 0.04
1.04 0.03
1.10
1.11 0.05
{111}
{112}
{110}
1.16 0.08
1.17 0.08
1.05 0.00
1.06 0.02
1.06
1.09 0.04


Potential Energy of the System 49
System Pressure Calculation 49
Pair Correlation Function 50
Bond Angle Distribution 51
Introduction of Strain 52
4. EXPERIMENTAL AND SIMULATION PROCEDURE 53
Experimental Procedure 53
Sample Preparation 53
Fracture Surface Analysis 55
Toughness Measurement 55
Fractal Dimension Determination 57
M.D. Simulation Procedure 61
Determination of Input Data 61
Simulation Procedure 63
Fractal Analysis Using Simulation Results 67
5. RESULTS AND DISCUSSION 71
Experimental Results 71
Fracture Surface Analysis 71
Fracture Toughness Measurement 90
Fractal Analysis 100
Relationship Between Toughness and Fractal Dimension 104
Results From Molecular Dynamics Simulation Ill
Validity of the Applied Potential Ill
Fracture Using MD Simulation 120
Toughness in Different Orientations 128
Strength Dependence on the Strain Rate 130
Strength Dependence on the Crack Size 134
Fractal Analysis Using MD Results 142
Comparison Between Measured And Simulated Results 146
Comparison of Toughness 148
Comparison Between Experimental and Simulation Results for
Fractal Dimension 148
6. CONCLUSIONS 151
APPENDIX A. SUMMARY OF DATA FROM DIFFERENT INVESTIGATORS 154
APPENDIX B. STATISTICAL ANALYSIS 155
APPENDIX C. YOUNG'S MODULUS CALCULATION 158
REFERENCES 160
BIOGRAPF1ICAL SKETCH 164
iv


95
(a)
n(ioo) = 2 + 1/4 x 8
= 4
(i)
n(ni)= 3 1/2 + 3 1/6
=2
Fig. 5.14 The surface energy for DC structures on the (100), (110), and (111)
planes can be found using a dangling bond calculation.


39
Interactions Between Atoms
Interaction forces are tabulated for all types of interactions possible over a range
of interparticle spacing. The interatomic potential assumed in this study is the
Stillinger-Weber potential and the modified Born-Mayer potential as illustrated above.
The derivative of these potentials is the force as a function of r. The two-body
potential is
02(rij) = e f2(rij/5)
= sA (B (§)-P (bq)exp[( V], if r o o o o
= 0, if r>a.
Thus, the two-body force is
r-a
(3.10)
(BP dr11
f
M
-p
52
B

V
UJ
UJ J
(r-a)2
, if r = 0, if r>a.
It is computer-time-effective to tabulate values of this function as a function of
r2 and divide by r so that one may use the square of the interaction distance in finding
interparticle forces in the table. One may find the Cartesian components of the force
between two particles by simply multiplying the appropriate tabulated value by the
difference in Cartesian coordinates of the two particles.


17
Gf is the failure strength, and
c is the critical flaw size.
(2) Nonzero residual contact stresses: With residual contact stresses present at
the flaw origin it becomes necessary to include an additional tensile term in the stress
intensity factor:44
K¡ = Ka + Kr = Yaac1/2 + XrP / c3/2, (2.10)
where Kr is the stress intensity factor with residual stress,
P is the peak load, and
Xr is a unitless geometry constant.
For large c values, the applied-stress term controls the fracture, as before, for
small c values, it is the residual-stress term that dominates. Under equilibrium
fracture conditions the flaw will accordingly undergo a precursor stage of stable growth
as the tensile loading is applied; failure then occurs when the crack reaches a critical
size, at which point the applied stress is intense enough to cause spontaneous
propagation.40 This critical configuration is obtained by inserting Kj = Kc (Kc is
used here instead of KIc because, strictly speaking, the toughness determined using
indentation is different than KIc; however, in practical terms, they are equivalent) into
the above equation and evaluating the instability condition daa/dc = 0:
c =
4XrP
v Kc y
\2/3
3K
4Yg
(2.11)
fj
from which Kr can be found as:4546
Kc'lYcfC1'2.
(2.12)
For a semi-circular flaw, which is small relative to the thickness of the material into
which it is placed, Y = 1.24, and the above equation can be rewritten as:
Kc = 1.65afC
1/2
(2.13)


57
Furthermore, we can compare the obtained results from fracture surface analysis with
those from the strength-indentation technique by using:
Kc 0.59(E / H)1/8(aaP1/3)3/4. (4.3)
Here, the fracture toughness is given in terms of the critical intensity factor,
Kc, instead of K¡c. KIc is the critical stress intensity factor for mode I. Kic is usually
determined from a prescribed fracture mechanics test with no residual stress associated
with the crack. Kc is the resistance to crack propagation in the presence of a local
residual stress. KIc and Kc are expected to be close in value. During the test, the
mode one condition is not guaranteed because the fracture paths tend to go to the
easiest cleavage plane. Also, the stress condition on the fracture surface is not in a full
tension mode. Indeed, sometimes it is a combination of mode I and mode II. Thus, Kc
is an approximation of the mode I fracture toughness of the material.
Fractal Dimension Determination
Single crystal silicon fracture samples were carefully cleaned and coated with
nickel. The samples were then potted in epoxy. In general, there are two ways to get
contour lines: one is to polish the sample approximately parallel to the fracture surface
and the other is to polish the sample approximately vertical to the fracture surface.
Thus, the so-called horizontal contour (fractal) dimension or vertical profile dimension
can be obtained. The samples in this study were polished approximately parallel to the
fracture surface, as shown in Fig. 4.3. As the fracture surface is first encountered, a
section of the fracture surface appears in the polishing plane. These sections appear as
islands in the polishing plane, and are called slit-islands. The nickel coating performs
two functions: It provides good contrast during polishing and it helps to hold the
fracture surface together during polishing. As polishing proceeds, these islands begin


92
Gilman2, Jaccodine,64 Myers & Hillsberry,65 St. John,57 and Kalwani66 also
investigated the fracture toughness in single crystal silicon. Those data and applied
techniques are listed in Appendix A.
Single crystal Si is a diamond cubic structure which is considered to be one type
of face centered cubic structure. A simple approach67 which has been accepted to
estimate the surface energy of different planes in a FCC structure is to calculate the
number of dangling bonds on the specific plane for the FCC materials. The surface
energy can be obtained using the equation:
1 XT
y =-e.N{hkl} 'n{hkl};
(5.3)
where s is the energy needed to fracture the Si-Si bond,
is used because two surfaces are generated,
Nfhkl} is the number of atoms per area on the {hkl} plane, and
n{hki} is the number of dangling bonds per atom on the {hkl} plane.
By using the given equation, we can find the surface energy for the FCC
structure on the {100}, {110}, and {111} planes separately as shown in Fig. 5.13.
Since the number of atoms on the {100} plane is 1 + 1/4 4 as shown in Plot (b),
which is 2, and the area A where those atoms lie is a a where a is the length of the
side of a FCC unit cell, N|10o} is found to be 2/a^. Since the number of dangling
bonds per atom on the {100} plane is 4 as shown in Plot (c), Y{ioo} is found to be
g-y'4. In the same way, Y{no} is found to be -s
2 a. 2
and Y{i 11} is found to
b£ 1 S V/2 ~3' ThUS y{10} 1 y{11* : T{111* = 8 : 77 1 692 = 1 16 : 1-02 :L
This ratio means that the {111} plane for the FCC structure has the lowest surface
energy and the {100} plane has the highest surface energy.


74
(a)
/
Fig. 5.2 (a) A typical fracture surface of the {100} tensile surface on the {110}
fracture plane. A Batman-like mirror is obvious on the fracture surface. And a
typical flaw is easily seen on the fracture surface, (b) A typical fracture surface of the
{110} tensile surface on the {110} fracture plane. An inverse volcano-like mirror is
obvious on the fracture surface, and a typical flaw is easily seen on the fracture
surface.


77
Table 5.2 Data for Si samples fractured in the {110} plane
{100} tensile surface
indent
flaw
load
Klc*
Klc**
No.
load
width
thickness
b
a
size
P
strength
0.59(E/
1.65*
Kg
mm
mm
um
um
um
N
MPa
MPamAl/2
MPamAl/2
1
0.9
7.44
7.03
66.2
84.8
74.0
322.6
103.1
1.46
1.46
2
0.9
7.31
7.02
70.2
66.3
68.0
0.0
107.4
1.51
1.46
3
0.9
7.31
7.00
66.2
79.5
72.0
249.2
81.8
1.23
1.14
4
0.9
7.31
6.99
79.5
92.8
86.0
262.5
86.7
1.29
1.32
5
0.9
7.53
7.15
95.4
98.0
96.5
266.9
81.5
1.23
1.31
6
0.9
6.87
7.16
92.8
90.0
91.5
238.0
79.4
1.21
1.25
7
0.9
"638"
7.03
94.0
94.0
195.8
70.8
1.10
1.13
8
0.9
6.89
7.01
97.0
97.0
242.6
84.2
1.26
1.35
9
0.9
7.07
7.07
102.0
102.0
202.4
67.3
1.06
1.12
10
0.9
6.81
7.39
95.0
89.8
89.8
233.6
73.9
1.14
1.15
11
0.9
7.55
6.95
89.8
92.4
198.0
63.9
1.02
1.01
12
1.4
7.11
7.37
130.0
143.0
136.3
164.6
50.1
0.95
0.95
13
1.4
6.88
7.11
136.0
0.87
14
1.4
7.84
7.16
132.5
124.0
127.9
193.5
56.6
L05
1.05
l5
1.4
7.07
7.44
119.0
106.0
112.1
229.1
68.8
1.21
1.23
16
1.8
6.83
6.92
126.0
172.0
147.0
198.0
71.1
1.32
1.41
17
1.8
7.43
7.02
242.5
77.8
1.41
18
1.8
7.82
6.94
220.0
220.2
68.7
1.29
19
1.8
6.90
7.35
185.0
185.0
200.2
63.2
1.21
1.41
20
0.5
6.87
7.28
63.4
52.8
57.8
269.2
87.0
1.12
1.09
21
0.5
7.36
6.60
52.8
52.8
~5T8~
262.5
96.3
1.20
1.15
22
2
7.57
6.90
132.0
142.6
137.2
204.3
66.8
1.29
1.29
23
2
7.70
7.32
190.0
158.4
173.5
191.3
54.5
1.11
1.18
24
3
7.41
7.24
195.4
142.6
166.9
209.1
63.3
1.37
1.35
25
3
6.87
6.95
169.0
184.8
176.7
186.9
66.2
1.42
1.45
26
5
6.29
7.15
322.1
294.6
307.6
133.5
48.8
1.28
1.41
27
5
7.45
7.37
316.8
253.4
283.3
169.1
49.1
1.29
1.36
avg.
1.19
1.23
dev.
0.08
0.08
{110} tensi
e surface
indent
flaw
load
Klc*
Klc**
No.
load
width
thickness
b
a
size
P
strength
0.59(E/
1.65*
Kg
mm
mm
um
um
um
N
MPa
MPamAl/2
MPamAl/2
41
0.9
7.35
6.97
158.4
105.6
129.3
160.2
52.76
0.89
0.98
42
0.9
7.55
7.39
95.0
95.0
95.0
278.1
79.30
1.20
1.26
43
0.9
7.10
7.56
89.8
100.3
94.9
233.6
67.71
1.07
1.07
44
0.9
6.93
7.84
105.6
105.6
105.6
229.1
63.24
1.01
1.06
45
0.9
6.76
7.03
84.5
79.2
81.8
204.7
72.06
1.12
1.08
46
0.9
7.44
7.26
84.5
95.0
89.6
215.8
64.73
1.04
1.01
47
5.0
7.08
7.21
380.1
337.9
358.4
100.1
31.99
0.93
0.99
48
5.0
7.39
7.26
269.3
316.8
292.1
142.4
43.00
1.17
1.21
avg.
1.05
1.07
dev.
0.10
0.09
* Klc is calculated from the Eq. (A
k3)
** Klc is calculated from the Eq.(4.2)


34
also generally found able to accurately simulate the elastic and thermodynamic
properties of Si.55,56 This potential has been adopted here.
Potential Determination
Single crystal silicon consists of atoms held in place (a diamond structure which
is relatively anisotropic from the atomic viewpoint) by strong and directional bonds. It
seems reasonable at first sight that the corresponding potential O could be
approximated by a combination of pair and triplet potentials, d>2 and 3. And these
two potentials can be expressed as a function of energy, e, and length units, 5. Thus,
020-ij) = £ f2(rij/5), (3.1)
and
03(ri, rj, rk) = s f^q/S, rj/8, rk/5). (3.2)
We use a reduced pair potential for Si selected from the following five-
parameter family, a simplified Lennard-Jones potential, from the work of Stillinger and
Weber:24
f2(r) = A (B r'p r'q)exp[(r a)'1], if r = 0, if r>a.
Thus,
1'2'i'ij) = e f2(rij/S) (3-4)
= eA (B (§)- (fr^expK | V], if r o o o o
= 0, ifr>a.
This form automatically cuts off at r = a without discontinuities in any r derivatives,
which is a distinct advantage in any MD simulation application. Stillinger and Weber
use the same cutoff technique for the three-body interactions, f3(r¡,rj,rk) = hjik + h¡jk
+ hky. The function h is given :


25
Fractal on bounded
range
log L
Fig. 2.7
Physical fractals have scale dependence over a bounded range of scales.


16
Fig. 2.4 Configuration of median/radial cracks for Vickers indentation showing:
(a normal indent load, P, generating median opening forces Pe (elastic component) and
Pr (residual component), (b) load removal eliminates the elastic component, and (c)
fully developed radial crack pattern.


94
However, single crystal Si has a diamond cubic (DC) structure which, in fact, is
different from FCC because it is not closed packed. We can find the surface energy for
the DC structure on the {100}, {110}, and {111} planes separately as shown in Fig.
5.14. However, the DC structure is not a close-packed structure, and the dangling
bonds will not be linked to all neighbors. Only some specific neighbors in the first
shell will share the dangling bond. The previous method cannot be completely
followed and some modifications must be made. As shown in the figure, the dangling
bond is expressed as a line. The surface energy can be obtained using the equation:
1 1
Y = -£'n{hkl}---
where s is the energy needed to fracture the Si-Si bond,
(5.3)
is used because two surfaces are generated,
n{hki} is the number of dangling bonds on the {hkl} plane, and
A is the area where those atoms lie on.
Since the number of dangling bonds on the {100} plane is 4 as shown in Plot
(c), and the area A where those atoms lie is a a and a is the length of the side of a
J 4
silicon DC unit cell. 7/ioo> is found to be £ If s = 3.510-12 erg/pair-atom22
2 a2
and a = 5.43 is substituted into the equation, 7{ioo} *s found to be 2.3 J/m2. (The
Si-Si bond energy is found to be varied from 3.2 to 3.6 erg/pair-atom.24646869) In
1 4
the same way, Y{no} is found to be e- = 1.67 J/m2, and Y{m} is
2 y 2a
found to be
1 2
£ _-- = 1.36 J/m2. These values are close to the experimental value, 2.1
2 V3a2/2
J/m2, which is obtained using double cantilever beam tests.70,71 Thus Y{ioo} : Y{110} :
Y{i11} = 2.3 J/m2 : 1.67 J/m2 : 1.36 J/m2 = 1.69 : 1.23 : 1. So, from dangling bond


13
Once the failure origin is located after fracture, its size can be measured and
used to determined the toughness of the material. Figure 2.3 shows a semi-elliptical
crack in a plate. It can be shown through a Griffith/Irwin approach that the stress
intensity on a semi-elliptical surface crack of depth, a, and half-width, b, can be given
35 37
as:
Vl.27TtfVa
K:= (2.6)
where a is the semi-minor axis for an elliptical crack,
o is the applied stress, and
is an elliptical integral of the second kind. varies between for a slit crack (a/b=0) to = 1.57 for a semi-circular crack (a/b= 1).
The criterion for failure in a brittle material is that K( > KIc at which point the
initial flaw will propagate spontaneously, where Kj, the stress intensity, is a measure of
the magnification of the external loading at the crack tip, and KIc is the critical stress
intensity factor or the fracture toughness of the material. The strength or failure stress,
Of, of a brittle material can be related to the flaw size:
Cf = 7ET (2'7)
For a semi-circular surface crack, which is stress free and which is small relative to the
thickness of the cross-section, = 1.57 and Kic can be found as:
KIC = 1.24cfVc, (2.8)
where c is the square root of a b and is the size of a semi-elliptical crack which has the
on
size of an equivalent semi-circular crack.
Thus, the fracture toughness of a material can be calculated if the failure stress
is determined and the crack dimensions a and b are measured.


54
(a) {100} fracture plane
(c) {HI} fracture plane
{110}
{112} [X
'{111}
^ {112}
{110}
{111}
Figure 4.1 Three orientations are chosen for comparison.


113

Fig. 5.19 Potential energy and force curves for the Stillinger-Weber potential.
From the curve, it is found that the equilibrium position for Si is about 0.34, and the
distance for cut-off potential is 3.78.


156
Here n=12, Thus
x2cjx = 1.16 2-2= 1.16 + 2-^= = 1.16 0.02
vl2 vl2
Thus, about 0.95 of all specimen should have values between 1.15 and 1.17.
For the {110} fracture plane with a {100} tensile surface,
X
the average fractal dimension, x = = 1.10, and
n
IX! fx X)
the standard deviation, s = J 0.04.
V n-1
To form the interval of 2 standard deviations around x, we calculate
x2ox =1.102-^=1.102-i=
Vl2 y/l2
Here n=12, Thus
x + 2(Jx = 1.10 2^=
1.10 2
y/\2
1.100.02
Thus, about 0.95 of all specimen should have values between 1.09 and 1.11.
The distribution plots for these two cases can be plotted together to see whether
these two sets of data are overlapping or separating as seen in Fig. B.l.


88
mirror
region
mist
region
-4 hackle
region
wavelength
of light
Fig. 5.11 Schematic of vertical profile of the mirror, mist and hackle regions of a
fracture surface. The reason why the mirror region is mirror-like is because the tortuosity
of the region is smaller than the wave-length of visible light as shown above.


134
rate, as shown in Fig. 5.34, the position of the first peak only changes very little but
the width of the first peak changes very much. In addition, the tip of the first peak is
very sharp, which means that the expansion rate is too fast for the system to smooth the
first peak. The shape and position of the second and third peaks change very much
also.
The Si-Si-Si bond angle distribution for the 0.2 ps'1 and the 2 ps'1 strain rates
can be obtained and are shown in Figs. 5.35 and 36. Initially the distribution has a
peak at about 109. After uniform expansion, the shape of the bond angle distribution
undergoes a change which depends on the applied strain. From Figs. 5.35 and 36, we
can see that the peak of the bond angle distribution is at 109 at first then it resolves
into two peaks at higher strain. While comparing the shape of the peak, those from the
0.2 ps'1 are much smoother than those from 2 ps'1. The system with the slower strain
rate can reach its equilibrium faster to produce a smoother curve than that with higher
strain rates. Since the system with the slow strain rate can reach its equilibrium in a
more realistic manner than the faster strain rates, the simulations used for determining
the strength dependence on the crack size, and toughness were all performed at 0.2 ps'1
strain rate.
Strength Dependence on the Crack Size
The Griffith criterion implies that if the material possesses larger cracks, it
would fracture at a lower stress level. Here a controlled crack will be introduced in the
specimen. Specimens with 1, 2, 4, 8, and 16 atoms removed, respectively, will be
added to the simulation criteria. Two types of cracks are introduced here, one is void
like, and another is plane-like.
The stress-strain curves for specimens with a void-like crack are shown in Fig.
5.37. It is seen that the fracture strength varies with the size of the cracks. The crack


5
loading histories in single crystal silicon would result in different fracture
topographies13,22 (although the fracture processes may be the same for different
orientations). The effect of elastic anisotropy on fracture provided the inspiration to
use single crystal silicon for the molecular dynamics simulation. From the simulated
loading history, the comparison of fracture toughness in different orientations can be
made. From the simulated fracture topography, the irregularity of the fracture surface
in different orientations can be analyzed using fractal analysis and the fractal dimension
can be compared with the toughness. Kieffer and Angel23 used the MD simulation
method to generate stable silica aggregates at various low densities similar to those of
experimental aerogels. They found that fractal dimensions and range of self-similarity
can be extracted from the radial distribution functions in those structure obtained from
MD simulation. They believed that a non-integer dimension is a characteristic feature
of the aerogel structure used in their study. However, because the way which they
used the change of the slope of the radial distribution functions with respect to the
density to obtain fractal dimension in their study was not reasonable, thus they got a
weird result which shows that a denser aerogel has a lower fractal dimension. It is
contradict to the fact that in the nature a full occupied volume has a dimension 3 while
a volume with porosity has dimension less than 3. Thus we must use a different
approach from theirs to obtained the fractal dimension in this study. Here the fractal
dimension on the generated fracture surfaces at the atomic scale from MD simulation
will be analyzed using slit-island technique and Richardson-plot.
This investigation will address several topics. First, there will be an
experimental determination of the fracture toughness in single crystal silicon as a
function of orientation of the crystal plane. Second, the fractal dimension of the
fracture surfaces of the single crystal silicon for the different orientations will be
determined experimentally. The fractal dimensions will be determined using
Richardson plots and slit-island analyses of contours of (horizontal) sections through


163
71. S. M. Wiederhom, in Fracture Mechanics of Ceramics, Vol. 2, edited by R. C.
Bradt, D. P. H. Hasselman, and F. F. Lange, Plenum, New York, 613-646 (1974).
72. C. Z. Wang, C. T. Chen, and K. M. Ho, Phys. Rev. Lett., 66[2] 189-92 (1991).
73. T. P. Swiler, Ph.D. dissertation, University of Florida, 1994.
74. L. A. K'Singam, J. T. Dickinson, and L. C. Jensen, J. Am. Ceram. Soc., 68[9] 510-14
(1985).
75. H. J. McSkimin and P. Andreatch, Jr., J. Appl. Phys., 35 3312 (1964).
76. F. Ebrahimi, Inter. J. Fracture, 56 61-73 (1992).
77. M. L. Williams, J. Appl. Mech., 24 109-14 (1957).


30
Interaction forces generated from the potential energy generated between the atoms are
assumed. The acceleration of particles (or atoms) results from the interaction of those
interatomic forces. The interaction forces produce the fundamental momentum for the
particles inside the small and idealized system.
Molecular dynamics simulations can be applied to study the properties of
materials. During MD simulations, the position and velocity of each atom can be
calculated to show the dynamic properties at that instant of time. The power of the
MD simulation is that the rule of interaction can be changed at will to model any
complex sets of particles, and environments can be simulated at conditions which are
not accessible experimentally. The difficulty with the MD simulation is that the rule of
interaction, i.e., the selected potential, must be carefully chosen in order to be realistic
or meaningful.
Three major uses of MD simulation are the development of theoretical models
for materials, the testing of theoretical models, and the predictions of material behavior
under experimentally inaccessible or costly conditions.
Material models can be developed from molecular dynamic simulations to
observe what occurs during the simulation and be able to set the rules of a simulation.
If we want to model the fracture of silicon, we do this by observing individual atomic
motions as samples are fractured using identical interaction potentials. We can also
fracture at several strain rates to investigate the effects of allowing the material to relax
by thermal vibrations as compared to not allowing the material to relax. After
performing such simulations, we can develop a general idea of how single crystal
silicon fractures and those ideas will be valuable to those studying the strength of Si.
Thus, molecular dynamics simulations can be used to devise models.
Molecular dynamics can be also used to test theoretical models using the ability
to observe all modeled atomic motions and the ability to simulate a wide range of


14
k
Fig. 2.3 An semi-infinite plate under uniform tension and containing a semi
elliptical crack.


70
Fig. 4.10 The slit-island obtained from the simulated fracture surface looks like a
plane with lots of circular disks on it.


41
Consider Fr first. At first we have to find the transformation of u¡ into Cartesian
coordinate.
U] =ulxi+ulyj+ulzk (3.18)
Fri =Fr(u]xi+uiyj + u,zk) (3.19)
Now consider Fq. According to Fig. 3-1,
e0 = ! x 3 = , x (j x 2) (3.20)
and because Ax(BxC) = (AC)xB-(AB)xC, thus
e0 = j x (] x 2) (3.21)
= (r2)x1-(r1)2
= COS0] -2
F0e0 = F0(cos91 -2) (3.22)
= F0(cos9ulx u2x)i + F0(cos9uly u2y) j + F0(cos9ulz u2z)k
Thus,
Fr+F0 = FX+Fy + Fz (3.23)
= [FrUix + Fe(cos0ulx-u2x)]I
+[Fruiy + F0(cos9uly u2y)] j
+[Frulz + Fe(cos0ulz u2z)]k
(B) We can consider the r2 term in the same way as (A).
MD Simulation Procedures
The explanation of how the simulations are performed will be given here.53
The concept of time step, the interaction between particles, the method used to update
particle positions, the application of periodical boundary conditions, and the control of
system temperature will be discussed as follows.


114
Force
Distance
Fig. 5.20 The magnitude of the Coulombic modified Born-Mayer potential is
related to the two-body part of the SW potential. It is found the magnitudes of the two
potentials used in this study are of the same order.


141
here if the simulated systems are considered a combination of a continuum solid away
from the crack region and individual atoms around the crack region. The absolute
value of the calculated stress intensity factor is not really important here. The
important observation here is that a constant value of critical stress intensity factor is
obtained using MD techniques for different crack sizes with all other parameters the
same.
Table 5.9 Crack size and fracture strength for planar cracks from MD
# of atoms removed
flaw size*
()
fracture stress
(GPa)
apparent stress
intensity factor**
(MPa m1/2)
0
3
38
0.7
1
6
29
0.8
2
7
26
0.8
4
8
23
0.7
8
11
20
0.7
12
13
18
0.7
18
14
16
0.7
*Rlaw size, 2c, is the diagonal length of the removed rectangular or square plane of
atoms.
**Stress intensity factor is calculated using Kc = ajnc = 1.13aVc ,77 which is used
n
for the embedded penny-shaped crack condition.


42
The Concept of Time Step
The time step is the basic unit of time during a molecular dynamics simulation.
As there are many atoms interacting in the simulated system, the motions of particles
cannot be solved analytically. Thus one can calculate the motions of atoms using
Newton's laws of motion where a time step dt is used in which the acceleration is kept
constant. A careful choice of time step is very important. If the time step is chosen to
be too small, a waste of computer running time results. If the time step is chosen to be
to big, nonlinear effects have to be considered and the atoms will move too far away to
reach unrealistic positions before the next time step is calculated. Following the work
of Soules, Ochoa, and Swiler, the time step used in our simulation is 0.5 x 10'15
second, and is about 1/40 of an atomic vibration period. A test to decide if a time step
is of sufficient length is to perform a simulation over about a thousand steps and check
to see if the system energy remains fairly constant. If not, the time step is too long. A
time step of 0.5 x 10"15 seconds was found to be of acceptable length and was applied
here.
The Method to Update Particle Positions
It is very important to handle particle interactions in a time-saving way.
Particle interactions are allowed only when particles are within a certain distance for
others. Beyond the distance, the effect of interaction is either none or negligible.
Cutoff distance is the distance in which interactions are allowed. Thus a neighbors-list
for all particles which lists all the neighbors which may be considered for the next 5 or
10 time steps will be made and updated at every 5 or 10 time steps depending on the
pulling rate. The fixed distance for the neighbors-list is chosen to be the sum of the
cutoff distance and the distance a particle may move in the next 5 or 10 time steps.


96
calculations for the DC structure, the {111} plane has the lowest surface energy and the
{100} plane has the highest surface energy, just as before.
A direct measurement of surface energy of single crystal silicon using a
cleavage technique was done by Jaccodine.64 The measured surface energy depends on
determining the force necessary to just move a crack that is already present in the
specimen under test. Determining this force and the lever arm to the crack tip, along
with other specimen geometry measurements, allows one to calculate the surface
energy. Jaccodine calculated the strength of the Si-Si bond to be 45.5 Kcal/mole,
which is equal to 3.2T0"12 erg/pair-atom. From this value and the bond density, the
surface energies for different planes were found to be 2.1 J/m2 for the {100} plane, 1.5
J/m2 for the {110} plane, and 1.2 J/m2 for the {111} plane.
Hesketh et al.69 also calculated the surface free energy for single crystal silicon
in different planes. Using the unit cell dimension of 5.432 and a Si-Si bond energy
of 2.9 1012 erg/pair-atom, the surface energies for different planes were found to be
2.0 J/m2 for the {100} plane, 1.4 J/m2 for the {110} plane, and 1.2 J/m2 for the {111}
plane.
A table of comparison for the surface energy obtained by the different authors
are listed in Table 5.5. Any difference in values is due to the variance of Si-Si bond
energy obtained from the different techniques. However, the values are very similar.
Thus, dangling bond calculations show that the {100} plane is the most difficult
cleavage plane while the {111} plane is the easiest cleavage plane. The trend for the
toughness from dangling bond calculations is the same as experimental measurements,
but the numerical ratio for different fracture planes from the dangling bond calculations
is not the same as the experimentally measured results.


BIOGRAPHICAL SKETCH
Yueh-Long Tsai finished his high school degree at Chia-Yi High School in 1982.
After that he entered National Taiwan University and graduated in 1986. He served as an
officer in the Taiwan Army for two years. He came to America in 1988 and was a
graduate student in the Department of Engineering Mechanics at Pennsylvania State
University (PSU). He achieved a 3.97 GPA at PSU and got his M.S. under the guidance
of Dr. J. J. Mecholsky in engineering mechanics in 1990.
He followed Dr. J. J. Mecholsky to join the Department of Materials Science and
Engineering at the University of Florida in 1990. Within four years there, he obtained an
M.S. degree and Ph.D. degree in materials science and engineering, achieved a 3.83 GPA
for graduate courses, published 7 papers and made 7 presentations.
164


28
i /o
of a characteristic length and Young's modulus, so that A = E(a0) Then the above
equation can be written as:
Kc = E ( a0 D* )1/2, (2.21)
where Kc is the fracture toughness,
E is the Young's modulus,
D* is the fractal dimension, and
a0 is the characteristic length.
Recalling that Kc = VEG = JlyE for a plane stress condition, a0 can be
17
obtained by the following equation,
2y
ED* '
(2.22)
In order to prevent getting an infinite values of a0 calculated from the above
equation if D* is approaching zero. A modified equation from a different approach to
find a0 is expressed as:
Y = Yo + aoEhklD*/2> (2.23)
where y is the fracture energy,
y0 is the surface energy for an Euclidean fracture surface (D* = 0),
a0 is a characteristic (atomic) length, and
Ehifi is the Young's Modulus of the hkl fracture plane.
Using experimentally obtained values of y, E and D*, y0 and a0 can be
calculated. The exact interpretation is not clear at this time; however, it is reasonable
to assume that it is related to the characteristic length of the generator. In terms of
fracture, this would correspond to a characteristic bond length which is involved with
initial fracture at the crack tip.
Mecholsky et al.6 reported that the outer mirror constant, i.e., cr(r2)1/2 where r2
is as noted in Fig. 2.2, is related to Kc and independently to E. Kirchner49 shows a
similar relationship with the crack-branching constant and used this information to


CHAPTER 4
EXPERIMENTAL AND SIMULATION PROCEDURES
Experimental Procedure
In this section, the experimental procedure is presented. Sample preparation,
fracture surface analysis, toughness measurements, and fractal dimension determination
will be given in detail.
Sample Preparation
Single crystal silicon was provided by AT&T and IBM. The Laue back
scattering method was used to determine the orientation of the crystal. Several low
index planes were chosen to be the desired fracture planes. They are the {100}, {110}
and {111} planes. After the orientation was determined, single crystal silicon was cut
into flexural bars with the desired orientations. The surface of the specimen was
polished to a 1pm finish.
A controlled flaw was introduced in the desired crystalline plane using a
Vickers diamond pyramid indentation at the center of the tensile surface on flexure
bars. Thus, the flaw was oriented perpendicular to the longitudinal axis of the
specimen. For the {110} orientation, 0.7, 0.9, 1.3, 1.5, 3, 4, and 5 Kgw indentation
loads were applied to produce a controlled flaw in each bar. For the {100} and {111}
orientations, 1 and 2 Kgw indentation loads were applied. Figure 4.1 shows a
schematic arrangement to demonstrate the location of the indentation on the tensile
surface. Flexural testing was performed on an Instron testing machine in air at room
temperature. Loads to failure were recorded. Fracture surfaces were examined to
53


CHAPTER 5
RESULTS AND DISCUSSION
Fracture toughnesses and fractal dimensions were obtained in two ways, one
was from experimental measurement and another was from molecular dynamics (MD)
simulation. Fracture toughnesses and fractal dimensions were investigated for three
different fracture planes. This chapter discusses the toughness results, the fractal
dimension results, and how the fractal dimensions for each orientation are related to the
toughness results. This chapter is divided into three sections: experimental results,
simulation results and comparison between measured and simulation results.
Relationships between variables will be discussed in each section.
Experimental Results
In this section, results obtained from the experimental measurements will be
given. These include fracture surface analysis, fracture toughness measurements,
fractal dimension measurements, and the relationship between fractal dimension and
fracture toughness.
Fracture Surface Analysis
After the specimens were broken, the fracture surfaces were examined. The
critical flaw size was measured to calculate the critical stress intensity factor using Eq.
(4.2). As in Fig. 4.1, three fracture planes, i.e., {100}, {110} and {111}, were chosen
and two different tensile surfaces in the {110} and {111} fracture planes were tested.
71


104
plane which consumes the lowest energy. Here, the easiest cleavage plane might not be
necessarily the {111} plane because the {111} plane is not always available during the
fracture process. The crack path may follow a plane which contains a resolved stress
high enough to fracture and at the same time consumes the least energy after the
external stress is applied. Thus, if the orientation changes, the path of fracture can
change. Since the fracture path or angle relative to the designed fracture plane changes,
the topography of the fracture surface will also change and the fractal dimension will be
different for those different orientations. Self-similarity is exhibited on the fracture
surface for the same fracture plane and loading directions, while self-affinity is exhibited
as the loading direction changes.
Relationship Between Toughness and Fractal Dimension
The relationship between the toughness and fractal dimension is one of the main
objectives in this study. From Table 5.7, we can see that the moduli of the fracture
planes do not relate to the toughnesses as the orientation changes. Generally, materials
with a higher modulus are stiffer and have higher toughness. But this is not the case
here for single crystal Si. Indeed, the easiest cleavage plane with the lowest toughness
which is the {111} plane possesses the highest modulus. The same condition occurs
with Ge. Thus, modulus is not the only determinant factor for the toughness in single
crystal silicon or anisotropic materials.
From Table 5.7, we can see that the fractal dimension increases as the critical
stress intensity factor increases, which means that a fracture surface possessing higher
tortuosity indicates that the fracture produces more surface during crack propagation.
Here, an example is given to illustrate this point of view. If there is a triangle
originally with area, A, and perimeter P and each side is replaced by one specific
generator as shown in Fig. 5.17, the perimeter and area of the new object would be


132
vibration period is found to be 0.0766 ps for Si. Thus a 0.2 ps"1 strain rate translates
to a 0.0152 expansion rate (unitless) in terms of the vibration periods, and a 2 ps'1
strain rate translates to a 0.152 expansion rate. Those numbers mean that for the
slower strain rate, the sample is elongated 1.52% per vibration period, while at the
higher strain rate, the sample is elongated 15.2% per vibration period. Although these
strain rates sound fast, relaxation by thermal vibrations are allowed at the lower strain
rates because the elongation per vibration period is relatively quite small, while for the
higher strain rate, the time for relaxation is not long enough. If each tensile
deformation is small enough for the system to relax before the next extension, the
system can reach its equilibrium condition during the period between two loadings.
Reaching the equilibrium condition means that the lowest energy state and, hence, the
stress state for the system at that moment is relieved. Thus, a system with a slower
strain rate will produce a lower fracture strength.
After reaching the maximum stress, the stress in the stress-strain curve drops
faster for the slow strain rate than the fast strain rate. The reason for this rapid drop is
that the distance increment at each time step for the slower strain rate is small enough
for the system to reach equilibrium at a shorter time. In the fast strain rate case, the
distance increment between two atoms at each extension increment is so large that the
system cannot respond fast enough. Thus, the stress drops faster if the strain rate is
slower.76
The structural changes during elongation and fracture can be studied for the
above two cases, i.e., for the 0.2 ps'1 and the 2 ps'1 strain rate. The first peaks for
10%, 20%, 30% and 40% strain in the pair correlation functions for the 0.2 ps"1 strain
rate all shift to greater distances and there is no big shape change during expansions of
the sample, as shown in Fig. 5.33. However, the shape and position of the second and
third peaks change greatly as compared to the first peak. This large change means that
the long range order has changed greatly. As for the condition with a 2 ps"1 strain


20
Consider how we measure the length of a line. The measurement of a line
requires a measuring tool or ruler. A line is given in Fig. 2.5(a). The length of this
line is measured by covering it completely with as few of a given radius, R, discs as
possible, as in Fig. 2.5(b). The length is given by
length = (number of covering discs) x ( 2R). (2.16)
If we have a new supply of discs with a different radius, say R/2, then the line length is
determined in the same manner, and a new line length is computed, as shown in Fig.
2.5(c). Note the difference in the measured line length in Fig. 2.5(c). Although the
measuring disc was decreased to half of the original measure, the number required to
cover the line more than doubled. The tortuous nature of this curve gives a length
that is scale dependent. As the measuring scale decreases, the measured length
increases.17
The measured length of a line can be described by the following equation,
called Richardson's equation:
L = kS1D, (2.17)
where L is the measured length,
S is the measured scale,
D is the dimension of the curve,
k is a proportionality constant.
If D is equal to one, the line is Euclidean and its measured length is not a
function of scale. If 1 < D < 2, the curve is said to be fractal with a dimension given by
D. Notice that for a given line segment, the length is not constant, but the dimension
is. Thus, the fractal dimension is the dimension in which a measure can be made, not
the measure itself.
The fractal dimension of a line can be computed from the above equation. The
line's length is computed over a range of scales. A log-log plot of length vs. scale
gives a straight line with slope equal to 1-D, as shown in Fig. 2.6.


29
support his idea of a constant strain intensity at branching. Tsai and Mecholsky50 used
a fracture energy concept to prove that the mirror-hackle boundary forms at a constant
energy value. Mecholsky51 points out that the rj/c = constant. Freiman et al.52
showed that the relationship could be represented as:
Kbj= (bj)1/2 E, (2.24)
where bj is a characteristic dimension on the atomic scale, and
j = 1, 2, 3 is related to mirror-mist, mist-hackle and crack branching
boundary.
They further showed that if bj = a0, where bj corresponds to the mirror-mist
boundary, then the flaw-to-mirror-size ratio is equal to the fractal dimensional
increment:
c/r2 = D*. (2.25)
This equation can provide a link between fractography and fractal geometry.
They found equation (2.25) in good agreement with experimental results. Thus, it
appears that the relationship between the crack size and the formation of the branching
boundaries is related to the dimensionality of the structure. Presumably, the
dimensionality of the structure is related to the bonding, i.e., the strength and length of
bonds on the atomic scale.
Molecular Dynamics Simulation
Molecular dynamics simulation is a computer technique to model material
structure at the atomic level in order to model or study the material properties with an
atomic view point.5354 During the molecular dynamics simulation, a small box with a
limited number of atoms is generated. The small box is considered a simplified system
which is ideally a realistic subset of the real system. This MD technique uses
Newton's equations of motion at constant acceleration over very short time intervals.


62
allowed to thermally vibrate for one unit time, 1 pica-second, before performing strain
pulling.
The temperature to perform the test should also be given. The chosen
temperature is important because it determines the energy state of the system. Here the
temperature or initial temperature is chosen to be 300 K.
The strain-rate should be given, also. Different strain-rates will give different
results. Basically, two categories will be given. One is higher than the speed of sound
while the other is slower than the speed of sound. Generally, results from different
strain-rate experiments will be compared for the same orientation. While for different
orientations, a strain rate of 0.2 was usually chosen. The reason for selecting 0.2 will
be explained in the results and discussion section.
A careful choice of the length of each time step is very important. It plays a
very important role not only on the stability of the system but also on the length of the
simulation time. If the length of the time step is chosen to be too long, atoms will
move too far in each time step. Thus the simulation will either be less realistic or
become unstable. If the time step is too small, atoms just move a very small distance at
each time step and waste a lot of computation time. For silicon, 0.005 pica seconds is
a good choice for the Stillinger-Weber and (ionic) Coulombic potentials because the
thermal vibration period is found to be 0.0766 ps for Si.24 The length of the
simulation is determined by the fracture of the specimen. For a lower strain-rate test, a
longer time should be given.
The choice of an adiabatic or isothermal condition is very important to do MD
simulations. If the adiabatic condition is used, the temperature will increase several
thousands Kelvin degrees and the system becomes unstable after fracture. If only the
Stillinger-Weber potential is used, the temperature after fracture is about 7,000 K. If
the ionic Coulombic potential is added in, the temperature after fracture will be about
50,000K. The reason is that the ionic Coulombic potential is always repulsive. Thus,


161
22. C. P. Chen and M. H. Leipold, J. Am. Ceram. Soc. 59[4] 469-72 (1980).
23. J. Kieffer and C. A. Angel, J. Non-Cryst. Solids., 106 336-42 (1988).
24. F. H. Stillinger and T. A. Weber, Phys. Rev. B 31 [8], 5262 (1985).
25. A. A. Griffith, Phil. Trans. Roy. Soc., A221 [4] 163-198 (1920).
26. A. A. Griffith, First Inter. Congr. Appl. Mech., Ed. by C. B. Bienzo and J. M.
Burger, Delft, 55-83 (1924).
27. G. R. Irwin, Fracturing of Metals, ASM, Cleveland, OH (1949).
28. G. R. Irwin, Ninth Inter. Congr. Appl. Mech., Brussels, (1957).
29. G. R. Irwin, J. Appl. Mech., 20 651-654 (1962).
30. E. Orowan, Weld, J. Res. Suppl., 20 1573-92 (1955).
31. C. E. Inglis, Trans. Int. Naval Archit., 55 219-41 (1913).
32. H. W. Westergaard, Trans., ASME, J. Appl. Mech., 61 [6] A49-53 (1939).
33. J. J. Mecholsky, Jr. and S. W. Freiman, in Fracture Mechanics Applied to Brittle
Materials, ASTM STP 678, Ed. by S. W. Freiman, 136-50, ASTM, Philadelphia,
PA (1979).
34. H. P. Kirchner and J. W. Kirchner, J. Am. Ceram. Soc., 62[3-4] (1979).
35. J. J. Mecholsky, Jr., in Nucleation and Crystallization in Glasses, Advances in
Ceramics, Vol. 4, 261-76, Am. Ceram. Soc., Columbus, OH (1982).
36. R. W. Rice, in Fractography of Ceramic and Metal Failures, ASTM STP 827, Ed.
by J. J. Mecholsky and S. R. Powell, 5-103, ASTM, Philadelphia, PA (1984).
37. P. N. Randall, in Plane Strain Crack Toughness Testing of High Strain Metallic
Materials, ASTM STP 410, Ed. By W. F. Brown and J. E. Srawely, 88-126, ASTM,
Philadelphia, PA (1967).
38. B. R. Lawn and M. V. Swain, J. Mater. Sci., 10[1] 113-22 (1975).
39. B. R. Lawn and A. G. Evans, J. Mater. Sci., 12[11] 2195-99 (1977).
40. D. B. Marshall and B. R. Lawn, J. Mater. Sci. 14[8] 2001-12 (1979).
41. J. T. Hagan, J. Mater. Sci., 14[12] 2975-80 (1979).
42. B. R. Lawn, A. G. Evans, and D. B. Marshall, J. Am. Ceram. Soc., 63[9-10] 574-81
(1980).
43. D. B. Marshall, J. Am. Ceram. Soc., 66[2] 127-31 (1983).
44. B. R. Lawn and D. B. Marshall, J. Am. Ceram. Soc., 62[l-2] 106-08 (1979).
45. R. F. Cook, and B. R. Lawn, J. Am. Ceram. Soc., 66[11] C200-201 (1983).
46. E. D. Kragness, Master's Thesis, Pennsylvania State University, 1988.


55
insure the samples had failed at the indent site. The stress to failure was calculated
from the dimension of each sample:
3PL
aa 9 >
2bh2
where aa is the fracture strength,
P is the load to failure,
L is the load span,
b is the width of the bar, and
h is the height (thickness) of the bar.
(4.1)
Fracture Surface Analysis
The procedures for fracture surface analysis have been well described in the
literature.6 The size of the critical flaw was determined by modeling the flaw as an
idealized elliptical crack of depth a and half width b. The equivalent semi-circular
crack radius, c, is determined by c = Vab and is shown in Fig. 4.2.30
The fracture surfaces were observed using a light microscope to locate the
flaws. Failure origins were located by observing particular fracture markings on the
fracture surfaces.61 Crack branching and mirror features were observed and were
used to locate the flaw.
Toughness Measurements
From the measurement of the critical flaw size, c, and fracture strength, aa, the
critical stress intensity factor, Kc, for those with residual stress caused from indentation
can be determined using
Kc = 1.65aac1/2.
(4.2)


127
Fig. 5.30 At 50% elongation, fracture continues. The free volume spherical
region gets larger and bonds break.


TABLE OF CONTENTS
Pages
ACKNOWLEDGMENTS ii
ABSTRACT v
CHAPTER
1. INTRODUCTION 1
2. FUNDAMENTAL 7
Structure of Single Crystal Silicon 7
Failure Analysis 9
Fracture Mechanics 9
Fracture Surface Analysis 11
Indentation Fracture Mechanics 15
Fractography-Indentation Analysis 15
Strength-Indentation Technique 18
Fractal Geometry 19
Fractal Dimension 19
Fracture Surface Analysis by Fractal Geometry 27
Molecular Dynamics Simulation 29
3. MOLECULAR DYNAMICS TECHNIQUES 33
Overview 33
Potential Determination 34
Fundamentals About the MD Simulation 36
Initial Conditions 36
Interactions Between Atoms 39
MD Simulation Procedures 41
The Concept of Time Step 42
The Method to Update Particle Positions 42
Periodic Boundary Condition 43
Interactions Due to Two-Body Potential 44
Interactions Due to Three-Body Potential 44
Updating of the Atom Positions 45
Verlet's Algorithm 45
Gear's Algorithm 46
Temperature Calculation 48
M.D. Movie 48
iii


145
Fig. 5.41 The slit-island obtained from the simulated fracture on the {111} fracture
plane.


162
47. R. Koch, Dept, of Mathematics, Univ. of Oregon, Macintosh Public Domain
Software Package, Fractal.
48. J. J. Mecholsky, Jr., D. E. Passoja, and K. S. Feinberg-Ringel, J. Am. Ceram. Soc.,
72[1] 60-65 (1989).
49. H. P. Kirchner, Eng. Fract. Mech., 10 283-88 (1978).
50. Y. L. Tsai and J. J. Mecholsky, Jr., Inter. J. Fracture, 57 167-82 (1992).
51. J. J. Mecholsky, Jr., in Fractography of Glasses and Ceramics II, Ed. by Frechette
and Varner, Ceramic Transactions, 17, Am. Ceram. Soc., Columbus, OH (1991).
52. S. W. Freiman, J. J. Mecholsky, Jr. and P. F. Becher, in Fractography of Glasses and
Ceramics II, Ed. by Frechette and Varner, Ceramic Transactions, 17, Am. Ceram.
Soc., Columbus, OH (1991).
53. T. P. Swiler, Master's Thesis, University of Florida, 1988.
54. J. M. Haile, Molecular Dynamics Simulation Elementary Method, Wiley, New
York (1992).
55. S. R. Phillpot, J. F. Lutsko, D. Wolf, and S. Yip, Phys. Rev. B, 40[5], 2831 (1989).
56. J. Q. Broughton, and X. P. Li, Phy. Rev. B, 35[17], 9120-27 (1987).
57. C. St. John, Philos. Mag. 31, 1193-1212 (1975).
58. T. F. Soules, J. Non. Cryst. Sol., 49 29 (1982).
59. L. Verlet, Phys. Rev., 159 98 (1967).
60. C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations,
Prentice-Hall, Englewood Cliffs, NJ (1971).
61. J. J. Mecholsky, Jr., in Ceramics and Glasses, Chairman: J. J. Mecholsky, Jr.,
Engineered Materials Handbook, Vol. 4, ASTM, Philadelphia, PA (1991).
62. P. F. Becher, J. Am. Ceram. Soc., 59 59-61 (1976).
63. J. J. Mecholsky, Jr. and S. W. Freiman, J. Am. Ceram. Soc., 74[12] 3136-38 (1991).
64. R. J. Jaccodine, J. of Electrochem. Soc., 110[6] 524-27 (1963).
65. R. J. Myers and B. M. Hillsberry, Fracture 1977, Vol. 3, 19-24 (1977).
66. L. Kalwani, Master's Thesis, University of Florida, 1993.
67. F. Ebrahimi, private communication, The University of Florida, Dept, of Materials
Science and Engineering.
68. R. Honig, RCA Rev., 18[2] 203 (1957).
69. P. J. Hesketh, C. Ju, S. Gowda, E. Z., E. Zanoria, and S. Danyluk, J. Electrochem.
Soc., 140[4] 1080-85 (1993).
70. A. Kelly, Strong Solids, 2nd ed., Claredon, Oxford, England, 5-8 (1973).


CHAPTER 3
MOLECULAR DYNAMICS TECHNIQUES
Overview
A technique for studying materials whose popularity and usefulness has arisen
within the past few decades is molecular dynamics (MD) computer simulation. This
technique can solve equations of motion for a system of particles which may be ions
(ionic bonding) or atoms (covalent bonding or metallic bonding). Each particle moves
according to the forces on it caused by all of the other particles in the system. The
inter-particle forces are derived from an assumed potential function which describes the
interactions between the atoms. Several types of systems have been simulated by MD
with various interatomic potentials. Those systems include hard sphere systems, soft
sphere systems, ionic systems and Lennard-Jones systems. These systems are
generally simulated using simple pair interaction potentials to obtain a close
approximation.
The covalent bond formed by two Si atoms sharing electrons is very localized
and directional. In ionic systems, Coulomb potentials which include short range
repulsive forces as well as attractive forces, are required. In covalent systems, bond-
directionality should be included. Two body interatomic potentials are not enough to
produce the diamond cubic structure. Since Si exhibits that structure as a solid, three
body potentials must be introduced in order to conduct a reasonable simulation. The
Stillinger-Weber potential24 has been used for Si solid simulations using MD and the
simulated pair correlation function agrees very well with the experimental data. It was
33


69
Fig. 4.9 If each atom is assumed to be as a potential field with sphere shape,
fracture surface will look like a surface with intersecting spheres.
The


CHAPTER 6
CONCLUSIONS
This dissertation studied the fracture of anisotropic single crystal silicon using
both experimental and analytical techniques. I first will summarize the main
conclusions of the experimental results, then summarize the main conclusions of the
analytical results from molecular dynamics simulations. I will then summarize the
comparison of the experimental and analytical results.
Fractal fracture behavior was observed in single crystal silicon. Self-similarity
and scale-invariance is suggested by the fracture surface appearance as shown in Fig.
5.16. Fracture toughnesses and fractal dimensions can be related to the atomic scale
of silicon using fractal analysis. Using fractal geometric principles, the fractal
dimension on the fracture surface allows us to connect the macroscopic topography to
the atomic process from the calculation of the characteristic length, a0.
The fractal dimensional increment, D*, is higher for the fracture plane with the
higher toughness values in agreement with other studies which found higher toughness
values associated with higher fractal dimensions.13,17,48
The crack propagation path for cracks starting on the {100} and {110} fracture
planes both initiate on the original plane and have the tendency to deviate to the {111}
plane as seen in Figs. 5.7, 5.8, 5.9 and 5.10. This observation suggests that fracture
would be more likely to deviate or branch on to those fracture planes with lower
fracture energy during crack propagation.
In this dissertation, the fracture strength dependence on the crack size, the
fracture energy for different orientations, the fractal dimension obtained from the
simulated fracture surface, and the strength dependence on the strain rate have been
151


31
environments. If the system behaves as predicted by a model under a range of
conditions, then the model can get greater credibility.
Extreme conditions also can be simulated using molecular dynamics. As one
might simulate any environment, one needs only to have a realistic simulation to
predict materials behavior or study possible modes of failure in extreme environments.
Thus, as molecular dynamics becomes better at simulating reality, one may expect this
application of molecular dynamics to find regular use.
When creating a molecular dynamics system, one would try to model the laws
of nature as close to reality as possible in order that the processes observed in the MD
simulation can be as realistic as possible, and that the resulting character of the
simulated system corresponds to the character of real materials. Because the laws of an
MD system are executed by high speed computer of limited computational power, there
must be limitations of the resulting MD system. These limitations are seen as
limitations in the spatial and temporal resolution in nature, as well as limitations on the
number of interactions possible within such a system.
The spatial dimensions of a molecular dynamics cell must be within a certain
CO
order of magnitude of the computer precision of positions obtainable. The spatial
n 1
resolution of the computer calculation is generally 1 in 10 or 1 in 10 depending on
whether single or double precision is used in the computer calculations. Over long
calculation periods, the use of single precision has resulted in a small error in the
system, while yielding a lot of time-saving is obtained. Thus, here the choice of spatial
resolution is not a problem in performing the MD simulations.
The length of time represented by the simulation must be within a certain order
of magnitude of the time step, the basic unit of time in a simulation. Generally the
duration of an experiment is between 10,000 to 100,000 time steps depending on
computational resources and the purpose of the experiment. The temporal resolution of
an MD simulation may be a serious limitation. The reason for this is that processes


64
Fig. 4.5 The temperature vs. strain curve for isothermal condition. The
temperature remains at about 300K before the moment when fracture takes place at
strain 0.45. After then the temperature goes back to about 300 K.


CHAPTER 1
INTRODUCTION
The strength of brittle materials is determined by the stress required to
spontaneously propagate a crack. The strength is dependent on the local stress
distribution around a crack tip which is often treated using a stress-intensity factor
approach.1 The fracture toughness, which is a measure of the resistance to crack
growth, defines the work or mechanical energy expended in propagating the crack.
The fracture toughness can be represented by the critical stress intensity factor, the
fracture energy or the critical strain energy release rate. Techniques to determine the
fracture toughness of ceramic materials include cleavage,2 double cantilever beam
(DCB),3 crack indentation,4 strength-indentation,5 and fracture surface analysis (FSA)6
techniques. The mechanism of fracture in brittle materials is strongly influenced by
processes at the atomic level. The subject of fracture in brittle materials has long been
of interest; however, the underlying processes have remained difficult to investigate
directly. Some factors such as phonon-phonon interactions have been analyzed
recently, but the dynamics of bond breaking in brittle materials is only beginning to be
studied.8"11 Although it is intuitively obvious that there must be a relationship between
the bond breaking process and the macroscopic measures of fracture such as fracture
toughness and fracture topography, there has been little quantitative evidence of the
connection. Fractal geometry, which is a non-Euclidean geometry exhibiting self
similarity and scale invariance, is a new tool that can be used to relate atomic scale
processes to macroscopic processes and features. The fracture surfaces of Si have been
1


before pulling
after pulling
displacement between two atoms = dx
X 9
summed energy = E dx
(b)
before pulling
after pulling
displacement between two atoms = -J(x + dx)2 + y2 -yjx2 +y2 < dx
1 2
consumed energy < E dx
Fig. 5.15 The energy needed to elongate the Si-Si bond is dependent on the angle
between the dangling bond and the pulling direction.


10
E is the modulus of elasticity,
y is specific surface energy, and
c is the critical flaw size.
Although Griffith was the first to analyze the relationship between strength and
flaw size, it was Irwin who developed fracture mechanics into the present day form.
Irwin followed the work done by Griffith, Orowan, and Inglis to develop what
is known as linear elastic fracture mechanics. Irwin first analyzed the fracture of
flawed components using a stress analysis based on the Westergaard32 solution of an
elliptical crack in an infinite plate. For a surface cracked specimen under mode I
(tensile) loading, Irwin derived the following:
CJf =
K
Ic
(2.2)
Ba/ttc
where of is the failure stress,
KIc is the critical stress intensity factor for mode-I loading,
B is a geometrical factor which accounts for flaw shape, location,
and loading geometry, and
c is the critical flaw size.
Irwin also analyzed fracture in terms of the strain energy release rate, G. G is
defined as the elastic energy per unit crack length, U/c, and can be related to the
failure stress:27
( EGC ^
1/2
(2.3)
V tic ;
where Gc is the elastic energy release rate at fracture.
The strain energy release rate can be related to surface energy, y, and it is as
follows for plane-stress condition:
G = 2y.
(2.4)


120
Fracture Using MD Simulation
Analysis of fracture by MD is performed by recording system properties and
structural-related data during expansion. The primary interest in the system properties
is focused on the system pressure. The pressure data can be used to determine the
stress-strain curve. During simulations, the location of fracture varied for different
simulations even when the conditions, like strain rate, number of atoms, temperature,
thermal properties, and density, are set to be the same. However, the stress-strain
curve obtained is almost exactly the same for different simulations with the same
conditions. This means that although the locations for fracture may vary due to random
thermal vibrations, statistically the whole system's properties will be the same despite
the local differences.
In order to demonstrate the use of these tools in the investigation of fracture by
MD, an example of a system at 0.2 ps'1 strain rate along the x-axis was examined.
The system pressure during 0.2 ps'1 strain rate pulling along the x-axis is shown in
Fig. 5.25. The system pressure increases initially then decreases after reaching some
point. This indicates that the tensile stress is developed in the system as the system is
elongated, as would be expected. Note that the system pressure drops when the
elongation reaches 45 %. This indicates that the material has some kind of structural
transition. The maximum tensile pressure attained by the material is assumed to be the
strength of the material and at that moment the material has begun to fracture.
However, the material does not fracture in one instant initially, the bonds break
primarily individually and the pressure decreases as the number of broken bonds
increases. The next generation of broken bonds will occur near the already broken
bonds to form the crack surface.


37
Initial Velocity:
temperature kinetic velocity (V)
energy ,
E, random
k
distribution
function
Interactions between atoms:
apply strain
position
of atoms
force
between
atoms
+ (V + A V) x A t
- acceleration
new position
of atoms
AV
ax At
Fig. 3.1
A flow chart for the MD simulation.


100
area under the stress strain curve) for the {100} plane will be higher than those of the
other two planes based on surface energy calculation and bond angle.
Fractal Analysis
The fractal dimension for each orientation is measured using several different
slit-island lines. Generally, about 8 or more different sets of slit-islands for each
orientation are obtained. A section of the perimeters were photographed using an
optical microscope at 400X. Different scales are used to measured the length of a
portion of the perimeter of the obtained slit-island as shown in Fig. 2.5. For each slit-
island, two sets of data of length vs. scale are obtained. Each set can be used to find
one fractal dimension using Richardson's technique (Eq. 4.4). A log-log plot of length
vs. scale results in a line with a slope equal to 1-D as shown in Fig. 2.6. The
measured fractal dimension is determined valid if the correlation coefficient (r2) is
higher than 0.8. The fractal dimension for the chosen orientation is the average value
of those obtained fractal dimensions for different sets.
The average fractal dimension measured in different orientations13 is shown in
Table 5.6. The statistical analysis of the data will be explained in Appendix B. From
the data, it is seen that the fractal dimension varies not only with the fracture planes but
also with the tensile surfaces under flexure. Thus the change in orientations includes
both the choice of the fracture plane and that of the tensile surface. This implies that the
fractal process varies with the fracture plane and tensile surface plane, which further
implies that the fracture process in single crystal Si is self-affine. Features in a self-
affine object are related by a vector (scaling) quantity in different directions.
The {111} fracture plane is relatively smooth, while the {100} is tortuous. The
{110} is somewhere in between. Fig. 5.16 shows features throughout the {110} fracture
surface. If a small area in the 100X picture is magnified to 1,000X, similar features are


148
Comparison of Toughness
The data obtained from the experimental and simulation work is shown in Table
5.11.The fracture energy, y, calculated from the stress-strain curve (Fig. 5.31) from
the MD simulation depends on the dl unit length chosen. However, the unit length is
unknown here. If the unit length dl is chosen as the Si- Si bond length, 2.34 , the
calculated y is about half of the experimental value. The cutoff distance and the
characteristic length, a0, are candidates to be proposed here as the unit length dl in
order to calculate the y value. A comparison of the calculated values are listed in Table
5.11. From those data, those y value calculated from the characteristic length are
relatively close to the measured value. Although we cannot say the characteristic
length calculated from the fractal dimension is definitely the unit length during fracture,
the similarity of values in Table 5.11 probably means that the characteristic length is
related to the atomic process during fracture in some way.
Comparison Between Experimental and Simulation Results for Fractal Dimension
The data obtained from experimental and simulation results are shown in Table
5.12. The simulation results show that the {100} fracture plane has the highest fractal
dimension, while the {110} and the {111} fracture planes have lower values.
In the experimental work, the fractal dimensions have been measured for two
tensile surfaces in the {110} and {111} planes. This made comparison more difficult.
In the MD simulations, only tension tests are performed.


56
(a)

A
Fig. 4.2 Micro-indentation cracked bar under bending, (a) A schematic diagram
of Vickers indentation on tensile surface, (b) An elliptical flaw on the fracture cross
section.


89
Fig. 5.12 The fracture surface obtained from scanning tunneling microscopy.
(Courtesy of Mitchell and Bonnell12)


97
Table 5.5 Comparison of surface energy obtained from different authors.
fracture plane
Si-Si bond
{100}
{110}
{in}
(erg/pair-atom)
(J/m2)
(J/m2)
(J/m2)
this work
3.5 10-12
2.3
1.7
1.4
Jaccodine64
3.2 10-12
2.1
1.5
1.2
Hesketh et al.69
2.9 10-12
2.0
1.4
1.2
From dangling bond calculations, the fracture energy on the {100} plane is
almost twice as high as that on the {111} plane. The reason may be that the dangling
bond calculations do not consider the angle between the bonds and the specific fracture
plane. For example, the angle between the dangling bond and the {111} plane is 90
while the angle between the dangling bond and the {100} plane is about 40. If the
angle between the bond and fracture plane is 90, more energy is needed when two
atoms are pulled a distance dx away from each other as compared to the case for an
angle less than 90. An example is illustrated in Fig 5.15(a), in which two atoms are
pulled a distance dx in the {111} plane. The energy consumed for this case is 1/2-K-
dx since the angle between the force and displacement is 0, where K is the force
(spring) constant between the two atoms. But for the {100} plane, as shown in Fig.
5.15(b), the angle between the force and the displacement is not 0. Thus if two atoms
were pulled a distance dx away from each other, the actual movement (distance)
between the two atoms is ->/(x + dx)2 +y2 -yj\2 + y2 The quantitative relation
between dx and -J(x + dx)2 + y2 ^x2 + y2 can be determined by the following
procedure:


142
Fractal Analysis Using MD Results
The fractal dimension from the MD simulation can be obtained as shown in
Chap. 4. After the fracture has occurred in the simulation as shown in Fig. 4.8, we
would try to obtain the fracture surface from the simulation. Each atom is assumed to
be a sphere surrounded with an electron cloud of diameter 3.76 (the potential energy
cut-off distance). The radius of each atom is assumed to be the radius of the potential
field. Thus, the fracture surface looks like a surface with intersecting spheres, as
shown in Fig. 4.9. After obtaining the surface, the slit-island technique as described in
the experimental procedure section in Chap. 4 is applied here to obtain the fractal
dimension. In order to get a slit-island, a cut at constant height is applied at the
fracture surface. After the cut, the obtained slit-island looks like a plane with lots of
circular disks on it as shown in Fig. 4.10. The perimeter of the selected island is
measured using different scales as described before and the fractal dimension can be
obtained using a Richardson plot. The bubble-like fracture surface and the slit-islands
for each fracture plane are shown in Figs. 5.39, 40, and 41. It is easy to see that in
Fig. 5.39, which is taken from the {100} fracture plane, the slit-island looks much
more irregular than those from the {110} and {111} fracture planes as shown in Figs.
5.40 and 5.41.
The fractal dimension was measured with respect to different orientations, and
those data are shown in Table 5.10. From the data, it was found that the {100}
fracture plane has the highest fractal dimension, while the {111} fracture plane has the
lowest fractal dimension. If we compare the fractal dimensions with the toughness
obtained by calculating the area under the stress-strain curve at the point it reaches the
maximum stress level, i.e., the fracture strength, from MD simulation as in Table


79
Fig. 5.4 An interesting result is that if we rotate the fracture surface of the one
with a {110} tensile surface 90 degrees, as shown in (a), then a half mirror boundary of
this fracture surface is similar to half a mirror boundary of the fracture surface on the
{110} fracture plane with a {100} tensile surface without rotation, as shown in (b).
This implies that a fracture mirror boundary for a specimen fractured in tension from
an internal defect would mimic the trace of the elastic constant values in that plane.


138
Fig. 5.37 Strength dependence on the crack size. The shape of the crack is void
like.


22
(b)
(c)
Fig. 2.5
Continued.


59
to grow. The perimeter of the islands presents a line or section of a line that can be
measured according to Richardson's equation:
L(S) = k S1-0, (4.4)
where L(S) is the length of a section of a line along the perimeter of the slit
island and its value depends on the measuring scale, S;
S is the measuring scale used to measure the perimeter and its value
ranged from 5 to 100 pm;
D is the horizontal contour (fractal) dimension; and
k is a constant.
Richardson plots detail the change in measured length of a line as a function of
scale. Construction of a Richardson plot for a fracture surface requires a line that is
representative of the fracture surface and a range of scales for measuring that line.
At the first emergence of an island, polishing must be carefully performed. The
surface was polished to a 1 pm finish. Since the epoxy is transparent, the exact
location on the fracture surface can be observed, so that measurements can be taken for
selected segments of the perimeter, e.g., in the crack branching region. Polaroid
photographs were taken at a magnification of 400x and combined in a montage. This
montage was then measured with dividers set to various openings, as depicted in Fig.
4.4. The length of the section of perimeter was measured for each divider setting. In
this way, the line or perimeter length was computed as a function of scales. A log-log
plot of length vs. scale results in a line with slope equal to 1-D.
The scale is a measure of discernibility. As the scale becomes finer and finer,
we observe greater and greater detail. If the scale is larger than the largest features,
the observation is insensitive to those features. A curve will begin to look fractal only
after the scale becomes smaller than such features. The scale lengths used in this study
ranged from 1 unit length to 16 unit lengths ( 1 unit length is about 5 pm).


2
analyzed and are found to be fractal both at the atomic scale12 using the scanning
tunneling microscope and at the micrometer scale using the optical microscope.
Fracture markings on glasses and polycrystalline brittle materials, known as
mirror, mist, and hackle, are precursors to crack branching14 and can be used to
describe the stress state6 and characteristics of crack propagation.15 These marking
have been observed for more than 50 years and were related quantitatively to the stress
condition in the 1950's.16 More recently, the repetition of these features was observed
and quantitatively related to stress intensity. Ravi-Chandar and Knauss14 also noted
that mist and hackle are self-similar; i.e., they appear to be physically similar and
produced in the same fashion. Their description, however, did not emphasize the self-
similar nature of the features. The distance to branching is directly related to the size
of the fracture initiating crack. Thus, one can imagine that the fracture surface features
can be related to the bond breaking processes at fractures.
i o
Mandelbrot synthesized a branch of mathematics that provides a tool with
which to analyze self-similar and self-affine processes. A self-similar process is one in
which a feature at one magnification is related to another at another magnification by a
scalar quantity. A self-affine process is one in which this magnification factor is a
vector quantity. Mandelbrot called this branch of mathematics fractal geometry.
Mecholsky et al.17 summarized that fracture in brittle materials can be described
as a self-similar process and thus, can be mathematically described by fractal geometry.
They derived an expression which relates the fracture of bonds at the atomic level (y) to
a characterization of the resulting morphology on the fracture surface of polycrystalline
materials, D*, which is the fractional part of the fractal dimension. They also derived
a relationship between the surface energy and the fracture energy of single crystals,
polycrystalline ceramics, and glass-ceramics. Their results suggest that modeling
brittle fracture as a fractal process may be useful in distinguishing toughening
mechanisms and in relating atomic bonding and fracture-surface morphology. The


6
the fracture surfaces. Third, MD simulations using Stillinger-Weber24 and Coulombic
potentials will be performed in different orientations to investigate the difference in
toughness and fractal dimension with respect to orientations. Finally, there will be
discussions about the relationship between the fracture toughness and the fractal
dimensions of single crystal silicon for the different orientations and also comparisons
between experimental work and simulation results.


21
(a)
Fig. 2.5 Measurement of a line segment ( or signal), (a) A schematic for a line
which has texture. Measuring the length of a line requires a ruler. The (circle)
covering has a diameter, 2R. (b) The line is completely covered to determine its
length, (c) Changing the size of the measuring disc may produce a different measure of
the length.


146
5.10, then we see that the fracture plane with higher toughness has a higher fractal
dimension.
Table 5.10 Fractal Dimensions Determined from MD Simulations
Fracture
Plane
(MD)
Fractal Dimension
(unitless)
(MD)
Toughness^
{100}
1.16 0.06
0.90 dl
{110}
1.11 0.05
0.61 dZ
{111}
1.09 0.04
0.43 dZ
'The toughness is estimated from the stress-strain curves from MD simulation, if
toughness is assumed as the area under the stress-strain curve at the point where it
reaches the maximum stress, i.e., fracture strength. Toughness = j^max a ds.
Comparison Between Measured And Simulated Results
Before the results are compared, the difference in the experimental and
simulation techniques should be pointed out. In the experimental procedure, the
specimens were tested using three-point flexure. Thus, not only the fracture plane, but
also the tensile surface will have some effect on the measured toughness and fractal
dimension value. In the MD simulation, the specimens were tested in tension instead
of bending. Thus, the stress states were different between the experiment and
simulation. In the bending test, there will be a neutral plane on the specimen. Thus,
while part of the sample is under tensile stress, the rest is under compressive stress.
But in the tension test, only tensile stress exists on the cross-section. Since a periodic
boundary condition is applied in the MD simulation, the only difference in the chosen


135
r ()
Fig. 5.34 The pair-correlation function for a 2 ps'1 strain rate. The first peak in
the pair correlation functions all shift to greater distances during expansion of the
samples.



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I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
\\o%O
r-4
'Joseph lj
T Sin
imons
Professor of Materials Science
and Engineering
This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
December 1993
Winrd M. Phillips
Dean, College of Engineering
Karen A. Holbrook
Dean, Graduate School


137
Angle
Fig. 5.36 The Si-Si-Si bond angle distribution during pulling for the 2 ps"1 strain
rate. Initially the distribution has a peak at about 109. After uniformly pulling, the
distribution undergoes a change which depends on the applied strain.


87
(a)
Fig. 5.10 The crack paths (shown by shaded regions) for the {100} fracture plane
with a {110} tensile surface, (a) shows a schematic of fracture bar. If the tensile
surface is chosen to be {110}, the crack will remain on the {100} plane for about 2 to 3
mm before branching to the {111} plane if the indentation load is larger than 2 Kgw as
shown in (b). If the indentation load is smaller than 2 Kgw, the crack will branch in a
distance smaller than 2 mm generally as shown in (c).


60
Fig. 4.4
A montage of the perimeter is measured with dividers.


15
Indentation Fracture Mechanics
Indentation fracture mechanics has been well established as an important method
for the study of the mechanical behavior of brittle materials.38'43 Indentation provides
a means of introducing flaws into a material with a controlled size, shape, and location
on the sample. This is in contrast to materials with naturally occurring flaws where the
flaws vary in size, shape, location and concentration. Indentation provides a means of
characterizing the fracture process by introducing a controlled flaw into a brittle
material. The following discussion will concentrate on fracture surface analysis and
strength indentation analysis.
Fractography-Indentation Analysis
Figure 2.4(a) shows a schematic of the indentation deformation/fracture pattern
for the Vickers diamond geometry: P is the peak load and a and b are characteristic
dimensions of the inelastic impression of the radial/median crack, respectively. Figure
2.4(b) shows the schematic of a fully developed Vickers indent. The flaw is taken to
possess a penny-like geometry. The following discusses the cases of zero and non zero
residual stress terms separately:40
(1) Zero residual contact stresses: If the indentation flaw were to be free of
residual stresses, the stress intensity factor for uniform tensile loading would have the
standard form
Ka=Yafcl/2, (2.9)
where Ka is the applied stress intensity factor without residual stress,
Y is a unitless geometrical factor. For a semi-circular crack, Y = 1.24,


The fracture surfaces of Si have been analyzed and are found to be fractal both at
the atomic scale using the scanning tunneling microscope in a previous study and at the
micrometer scale using the optical microscope in this study. The irregularity of the
fracture surface is too complicated for Euclidean geometry to describe easily. Self-
similarity and scale invariance is suggested by the fracture surface appearance. The
fractal dimension is higher for the fracture plane with the higher toughness value. It is in
agreement with other studies which found higher toughness values associated with higher
fractal dimensions.
We also demonstrate the formation of fractal surfaces during fracture using a
molecular dynamics (MD) approach in a single crystal silicon structure and compare the
simulated results with experimental work. MD simulations using Stillinger-Weber
potential and Coulombic (Modified Born-Mayer) potential are performed in different
orientations to investigate the difference in toughness and fractal dimension with respect
to orientations. The close agreement between simulated and measured fractal structures
of the fracture surface suggests that this is a promising method for investigating atomic-
level processes during fracture.
vi


84
Fig. 5.7 The crack path for the {110} fracture plane with a {100} tensile surface.
The crack growth (shown by shaded regions) occurred on the {110} plane first and
deviated to the {111} plane as shown in (c) and (d). (a) shows a schematic of the
fracture bar. (b), (c), and (d) show the different modes observed.


144
F. 5.40
plane.
The slit-island obtained from the simulated fracture on the {110} fracture


23
log L
Fig. 2.6
1-D.
A log-log plot of length vs. scale gives a straight line with slope equal to


51
Bond Angle Distribution
Bond angle distributions are found by saving the atom identification numbers of
the coordinating atoms about central atoms during updates of the neighbors lists, then
applying the definition of a dot product to find bond angles. The formula for doing
this is
9 = cos
-l
X;;
ijxik + Y ijy ik + zijzik
rij -rik
(3-47)
where 0 is the bond angle,
i denotes the apex atom,
j, k denotes the end atoms,
xy, yjj, Zy are the vector components between i and j,
xik, y¡k, Zjk are the vector components between i and k,
r¡j is the distance between atoms i and j, and
rjk is the distance between atoms i and k.
Periodic boundary conditions were used. An energy distribution function is
used to assign the energy (velocity) of each atom at the beginning. The temperature of
the sample can be varied by scaling the velocity of each particle with an appropriate
factor greater than one and letting the total kinetic energy of the particles be equal to
the enthalpy of the desired temperature. The temperature of the sample was kept
constant by scaling the velocity every time step to avoid heating (or cooling) effects.


139
sizes, fracture strengths and calculated critical stress intensity factors are listed in Table
5.8.
Table 5.8 Void sizes and fracture strengths from MD
# of atoms
removed
flaw size*
0
(A)
fracture stress
(GPa)
apparent stress
intensity factor**
(MPa m1/2)
0
2
38
0.7
1
3
29
0.6
2
4
27
0.7
4
5
24
0.7
8
6
22
0.7
16
7
20
0.7
*Flaw size is considered as the diameter of the void flaw, 2c.
**Stress intensity factor is calculated using Kc = 1.24 af c1/2.
The stress-strain curves for the specimens with a planar sharp crack are shown
in Fig. 5.38. It is seen that the fracture strength varies with the size of the cracks.
The crack sizes, fracture strengths and calculated critical stress intensity factors are
listed in Table 5.9.
Although a periodic boundary condition is applied during the simulation, the
simulated results agree quite well with linear elastic fracture mechanics (similar to the
Griffith criterion) for void-like crack or plane-like crack (sharp crack). The strength of
the specimen decreases as the size of the crack increases and the calculated critical
stress intensity factors for both cases are approximately constant. Although the stress
intensity factor is a value to describe the toughness of continuum solids, it is applied


78
Table 5.3 Data for Si samples fractured in the {111} plane
{110} tensile sur
face
indent
flaw
load
KIc*
KIc**
load
width
thickness
b
a
size
P
strength
0.59(E/
1.65*
No.
mm
mm
mm
mm
mm
lb
MPa
MPamAl/2
MPamAl/2
cl
1
6.15
4.00
0.082
0.092
0.087
33
76.1
1.20
1.17
c2
1
5.80
4.95
0.093
0.102
0.098
46
73.4
1.17
1.20
c3
1
6.30
5.05
0.091
0.103
0.097
40
56.5
0.96
0.92
A1
1
6.15
5.60
0.110
0.112
0.111
82
70.9
1.14
1.23
A2
1
6.20
5.20
0.103
0.124
0.113
78
77.6
1.22
1.36
avg
1.14
1.18
dev
0.09
0.15
dl
0.7
6.15
4.65
0.078
0.074
0.076
44
75.1
1.09
1.08
d2
0.7
6.20
5.15
0.058
0.063
0.061
72
99.4
1.34
1.28
d3
0.7
6.10
4.90
0.080
0.086
0.083
48
74.4
1.08
1.12
d4
0.7
6.25
5.05
0.065
0.085
0.075
56
79.7
1.14
1.14
avg
1.16
1.15
dev
0.11
0.07
el
2
6.30
4.80
0.171
0.200
0.185
34
53.2
1.09
1.19
e2
2
6.25
4.75
0.149
0.155
0.152
40
64.4
1.26
1.31
e3
2
5.70
4.70
0.172
0.173
0.172
29
52.3
1.08
1.13
e4
2
5.95
4.95
0.160
0.118
0.137
38
59.1
1.18
1.14
avg
1.15
1.19
dev
0.07
0.07
{112} tensi
e surface
fl
2
6.15
6.25
0.108
0.115
0.111
68
64.2
1.26
1.12
f2
2
5.35
6.20
0.125
0.144
0.135
60
66.2
1.29
1.27
3
2
5.25
6.15
0.165
0.125
0.143
50
57.1
1.15
1.13
f4
2
5.80
6.10
0.165
0.183
0.174
48
50.5
1.05
1.10
avg
1.19
1.15
dev
0.09
0.07
S1
1
6.40
5.80
0.079
0.094
0.086
67
70.6
1.14
1.08
§2
1
5.50
6.20
0.105
0.105
0.105
70
75.1
1.19
1.27
g3
1
5.85
5.65
0.086
0.099
0.092
52
63.2
1.05
1.00
B1
1
6.30
6.05
0.109
0.098
0.103
101
73.1
1.17
1.23
B2
1
5.80
5.70
0.086
0.071
0.078
93
82.3
1.28
1.20
avg
1.16
1.16
dev
0.07
0.10
* KIc is calculated from Eq.
.3)
**KIc is calculated from Eq. (4.2)
A,B span =
25.5 mm
c,d,e,f, and g span =
= 34
three-point bending


110
Fig. 5.18 The plot of y vs. ED*/2. The slope of the fitted line is found to be 4.1.
The line intersects with the y coordinate at y0, 1.70 J/m .


APPENDIX B
STATISTICAL ANALYSIS
The average fractal dimension measured for different orientations are shown in
Table. B. 1. Those data are found to be very close to each other.
Table B. 1 Lractal dimension for different orientations
fracture plane
tensile surface
fractal dimension
{100}
{110}
1.16 0.04
{110}
{100}
1.10 0.04
{110}
1.04 0.03
{111}
{110}
1.06 0.02
{112}
1.05 0.00
Lor the {100} fracture plane,
a
the average fractal dimension, x2a^ = 1.162 r=, and
Vn
y (x_x)2
the standard deviation, s =, = 0.04.
V n-1
To form the interval of 2 standard deviations around x, we calculate
x2dx = 1.162 ? 1.162p=
x Vl2 V2
155


APPENDIX C
YOUNG'S MODULUS CALCULATION
Since a single crystal material, in general, is anisotropic, Young's moduli are
different for different orientations. The young's modulus for different orientations was
calculated from the stiffness matrix measured by McSkimin and Andreatch.75 Single
crystal silicon has a diamond cubic structure. The stiffness matrix for a cubic system is
as follows:
S11
Sl2
s13
s14
s15
Sl6
S11
s12
s12
0
0
0
S21
s22
s23
s24
s25
s26
Sl2
S11
S12
0
0
0
S31
s32
S33
S34
s35
s36
s12
s12
S11
0
0
0
S41
s42
s43
S44
s45
s46
0
0
0
S44
0
0
S51
s52
s53
S54
s55
s56
0
0
0
0
s44
0
S61
s62
s63
s64
s65
s66
0
0
0
0
0
s44
0.768
-0.214
-0.214
0
0
0
-0.214
0.768
-0.214
0
0
0
-0.214
-0.214
0.768
0
0
0
x 10'11 Pa'1

0
0
0
1.256
0
0
0
0
0
0
1.256
0
0
0
0
0
0
1.256
The modulus of elasticity for a cubic system in any given < hkl > direction may
be calculated by the following equation in terms of three independent elastic
158


143
Fig. 5.39 The slit-island obtained from the simulated fracture on the {100} fracture
plane. The slit-island looks much more irregular than those from the {110} and {111}
fracture planes as shown in Figs. 5.40 and 5.41.


125
Fig. 5.28 Unstrained single crystal silicon which is designed to fracture in the
{100} plane.


48
Taylor series predictor. Gear determined their values by applying each algorithm to
linear differential equations and analyzed the resulting stability matrices. For a q order
predictor, the values of the oq were chosen to make the local truncation error of
0(Atq+1). Values of the a¡ for fifth-order predictors are ocq = 3/16, a1 = 251/360,
a2 = 1, a3 = 11/18, 0,4 = 1/6, 0C5 = 1/60.
Temperature Calculation
The temperature of the system is controlled by assuming that the system
behaves as a classical statistical system, with average particle kinetic energies given by
the equipartition theorem. The temperature of the system is then calculated by solving
the expression = 3kT/2 for T. The temperature is then adjusted by scaling the
velocities of all particles to give the proper average kinetic energy.
M.D. Video Presentation
Information is collected from the simulation. Any aspect of the simulation can
be recorded, but there are a few which are commonly recorded. These are periodic
snapshots of the structures, instantaneous or average thermodynamic properties, pair
correlation functions, and bond angle distribution functions.
Snapshots of the structure of the system are made by periodically saving all
positions of particles of the system in a data file. These snapshots of the structures
may then later be used to perform a free volume sphere analysis on the structure to
make graphic representations of the structure.
Thermodynamic properties, such as system temperature, energy, and pressure
are calculated. Average of these values may also be kept in a record file. Thus either
instantaneous thermodynamic properties or average properties may be made.


119
Pressure
(GPa)
Fig. 5.24
Pressure of the unstrained system at room temperature at p = 2.33.


4
The use of molecular dynamics for studying the failure of materials has recently
become of great interest to investigators. Works by Paskin et al.,19 Dienes and
Paskin,20 and other collaborative works involving Paskin, have performed simulations
of crack growth in two-dimensional lattices. These works demonstrated the
applicability of the Griffith criteria in simple 2-D systems modeled by molecular
dynamics, and studied mechanisms of crack growth, including the formation of
dislocations and speed of crack propagation, as functions of applied stress, crack
length, and potential parameters. These works provide an atomistic basis to fracture
mechanics; however, they do not provide details on the fracture of particular materials.
Soules and Busby21 used molecular dynamics to study the rheological properties
and fracture behavior of a sodium silicate glass. Among the experiments in this work,
a uniaxial tension was applied to a system and the resulting behavior was studied. The
periodic boundary conditions used, i.e., on only two sides, made this experiment
equivalent to the fracture of a 50 diameter fiber, so bulk properties were not studied,
nor even approximated. This work was followed by works of Soules,8 Ochoa and
Simmons,9 and Ochoa et al.,10 which demonstrated the importance of dynamic effects
in the fracture of solids. Works by Ochoa, et al.,10 made use of full periodic boundary
conditions, i.e. on six sides, thus approximating fracture within a bulk material. These
simulations collectively studied the overall response of a system to an applied strain,
but did not study the atom-by-atom process in detail. A work by Simmons et al.11
examined the mechanisms involved in an individual bond fracture to describe a process
by which an entire fracture surface may be stabilized. Thus, these studies give possible
details on the fracture process for silicate glasses, and infer details of fracture for other
amorphous solids.
The difference in fracture between amorphous silica and single crystal silicon is
that silica has no preferred cleavage plane while single crystal silicon prefers fracturing
on the {111} and {110} plane. Thus, different loading orientations under different


45
other two atoms j and k were calculated. The interparticle force is then found by the
derivative of the three-body potential. The radial force and the tangential force are
calculated separately. The components of the force in each dimension i, j, and k are
found by multiplying the previous calculated tangential and radial forces with respect to
the related vectors in the coordinates. The components of force are then summed for
each particle for all interactions.
Updating of the Atom Positions
After the forces on each atom are accumulated from either the two-body or the
three-body potential, the change of positions of particles are updated using Verlet's or
Gear's algorithm at each time step. The details of the Verlet's algorithm59 and Gear's
algorithm60 will be illustrated as follows.
Verlet's algorithm
The advantages of Verlet's algorithm59 are speed and small error. The
derivation for this algorithm is as follows: According to Taylor series expansion, the
position at time (t + At) calculated from time t is
Xj (t + At) X(t) +
(dX;(t)V 1
lV At +
2
dt
d2Xj(t)Y 2 1 d3X,(tp
/
V.
df
Ar+
3!
dr
At3 +0(At4);
(3.27)
while the position at (t At) calculated from time t is
Xj(t At) = Xj (t) -
rdx¡(t)-
U.A
fd2Xi(t)']
At2
d3Xj (t)
V dt
J 2
l dt2 J
3!
l dt3 J
At3 +0(At4).
(3.28)


118
2
1.8 -
1.6 -
1.4 ..
1.2
1
0.8
0.6 -
0.4 -
0.2
0
r()
Fig. 5.23 Pair correlation function for single crystal silicon at a temperature
slightly higher over the melting point.


11
In the above equation, only surface energy, y, is referred. Indeed during fracture
process, the released strain energy not only produce two free surface, but also produce
different terms like sound, light, electron emission. Thus, the y used here indeed is not
just as simple as surface energy and some other energy is also included.
The stress intensity factor can be related to the strain energy release rate and the
fracture surface energy. By combining those previous equations, it can be shown that
KIc = VEG=V2yE- (2.5)
Fracture Surface Analysis
Useful information on the mechanical and fracture behavior of a failed ceramic
OO O.C
can be gained from microscopic examination of the fracture surface. The fracture
surface of brittle materials exhibits distinct fracture markings surrounding the critical
flaw. These features shown in Figure 2.2 can be related by known fracture mechanics
relationships to provide additional information on the fracture process and can also be
used to locate the origin of failure.35
The fracture surface markings shown in Figure 2.2 can be used to describe the
stress state of a brittle material.14,33,35,36 Four different regions referred to as the
fracture mirror, mist, hackle and crack branching can be seen in Figure 2.2. These
regions are associated with particular stress intensity levels and crack velocities which
are responsible for each distinct region.34 As a crack propagates from the critical flaw,
a smooth region which is basically perpendicular to the tensile axis is formed. This
smooth region is known as the fracture mirror. The fracture mirror is typically
bounded by a region of small radial ridges known as mist and the mist is bounded by
an even rougher radial ridge region called hackle. Finally, the propagating crack
reaches a characteristic energy level and crack branching occurs. Crack branching is a
region where two or more cracks form from the primary crack front.


80
{110} fracture plane with a {100} tensile surface without rotation.50 This implies that
a fracture mirror boundary for a specimen fractured in tension from an internal defect
would mimic the trace of the elastic constant values in that plane.52
Freiman et al.52 found that the fracture mirror shapes in sapphire for pseudo
cleavage type fracture have much more complex geometry, as shown Fig. 5.5, than the
similar fractures in glasses which yield more-or-less circular fracture mirrors. They
relate the mirror boundaries to the critical stress intensity factor. The complex mirror
shapes investigated in sapphire implies that the critical stress intensity factor along the
different crystallographic directions of the mirror boundary is not constant. Young's
modulus also plays an important role for the complex shape of mirror boundary. The
Young's modulus normal to the crack plane is a constant. However, the Young's
moduli either parallel or perpendicular to the mirror boundary in the crack plane can
vary as one traverses around the boundary.62 As seen in Fig. 5.5, The Young's
moduli for directions parallel and perpendicular to the crack front/mirror boundary
plotted as a function of angle of rotation about the tensile axis possesses maxima-
minima patterns which are similar to the fracture mirror geometry. A similar result for
single crystal silicon is expected and a schematic of the expected mirror boundary is
plotted in Fig. 5.5.
In contrast to the fracture surface of the {110} fracture plane, the fracture
surfaces of the {111} fracture plane look the same even when the chosen tensile
surfaces are different as shown in Fig. 5.3. Most probably, the fracture surface of the
easiest cleavage plane will not be affected by the chosen tensile surface because the
crack growth will remain on the easiest cleavage fracture plane as the lowest energy
path. Also, the size of the specimen is not large enough to show crack branching at
these tested fracture stress levels. Thus, the fractography would appear the same no
matter what the change in the tensile surface at these test specimen sizes. The available
fracture paths for the specific fracture planes are another important factor to determine


19
Fractal Geometry
Fractal geometry is a non-Euclidean geometry that was rediscovered,
popularized and applied by B. B. Mandelbrot.18 The word fractal is derived from the
Latin fractus which means fragmented or broken. Nature exhibits diverse structures
having inherent irregularities. Some of these structures can be described using
Euclidean geometry but other structures are better described by using fractal geometry.
In his book. The Fractal Geometry of Nature.18 Mandelbrot describes the extensive
applications of fractal geometry. The concepts have been used to described the
geometry of clouds, soot aggregates, dielectric breakdown, coastlines, and other
natural phenomena, including fracture.
Fractal geometry exhibits self-similarity or self-affinity, shows scale invariance,
and is characterized by the fractal dimension. Self-similarity means that features in
different regions appear to be similar to one another. They can be related by a scalar
multiple. Features in a self-affine object are related by a vector quantity. Scale
invariance occurs if a feature on different scales appears to be the same. These
properties indicate that a fractal can be created by using a shape to be repeated and
reduced in size following a prescribed sequence.
Fractal Dimension
In Euclidean geometry, objects occupy integer dimensions, i.e., 1, 2, 3, and so
on. Fractal geometry admits to the description of objects that occupy fractional
dimensions. Fractal geometry is characterized by its fractal dimension which is related
to measurement theory.


CHAPTER 2
FUNDAMENTALS
Structure of Single Crystal Silicon
Silicon is bonded by four covalent bonds which produce a tetrahedron. Each
silicon atom is bonded with four other silicon atoms due to the nature of the covalent
bonding. The covalent bond formed by two Si atoms sharing electrons is very
localized and directional. The space lattice of the diamond structure is face-centered
cubic with two atoms per lattice site, one at 0,0,0 and the other at 1/4, 1/4, 1/4 (Fig.
2.1). Each atom is tetrahedrally bonded to four nearest-neighbors due to the nature of
covalent bonding. As these tetrahedral groups are combined, a large cube is
constructed, which is called a diamond cubic (DC) unit cell. The large cube contains
eight smaller cubes that are the size of the tetrahedral cube but only four of the cubes
contain tetrahedrons. The lattice is a special face-centered cubic (FCC) structure,
which is shown in Fig. 2.1. The atoms on the corners of the tetrahedral cubes provide
atoms at each of the regular FCC lattice points. Four additional atoms are present
within the DC unit cell from the atoms in the center of the tetrahedral cubes.
Therefore, there are eight atoms per unit cell. The lattice parameter length is 5.43
and the unit cell diagonal length is 9.41 as shown in Fig. 2.1. The Si bond distance
is 2.34 .
Single crystal Si is generally classified as a brittle material whose atoms are
strongly covalently bonded. The primary cleavage plane is the {111} plane. The
principal factor for crack initiation is bond rupture.
7


32
relevant to the structural evolution of materials have characteristic times spanning about
19 orders of magnitude, far more than the four to five orders of magnitude spanned by
MD simulations. Generally a simulation which lasts for hundreds or thousands of
atomic thermal vibrations is needed in order to understand the properties of the system.
The mass of the electron is about one ten thousandth of those of atoms. The
motion of electrons cannot be modeled with the motions of atoms at the same time
since the characteristic time is so small compared with that of atoms. Thus electronic
motions cannot be used to provide true interatomic potentials of materials in such a
simulation and the motions studied here are limited to those of atoms only. The chosen
potential is necessary to accurately represent the thermal vibration motions of atoms in
order to model the properties of the simulated system. Thus the smallest time step used
must be a small fraction of the vibration period. A typical time step chosen in this
study is 0.5 x 10'^ sec. One generally can only run simulations for about 105 time
steps which results in a limitation in the duration of modeled experiments to hundreds
or thousands of atomic vibrations.
Another limitation is that the number of interactions between atoms cannot be
too big. The more atoms put in the system, the more computer CPU time will be
consumed. For example, if a system with n particles is created and only one-to-one
interaction is considered, then the total interactions between those atoms is n The
time to calculate a system with lOn atoms is 102 longer than that needed to calculate
one with n atoms. Thus the simulated system must be kept small and the chosen
interacted potential must be simple and easily applied.
Thus the differences between an MD system and the real universe are that the
length of studied time and the size of the studied system. The MD system has to be
small because of the limits on the spatial resolution of time and number of atoms used
in the system.


105
(a)
Area = A
Perimeter = P
(b) After the first generation
Area = A ( 1 + 1/3)
Perimeter = P (4/3)
After the second generation
Area = A (1 + 1/3 + (l/3)2)
Perimeter = P (4/3)2
Fig. 5.17
Construction of a fractal object.


133
Fig. 5.33 The pair-correlation function for a 0.2 ps'1 strain rate. The first peak in
the pair correlation functions shift to greater distances during expansion of the samples.


43
Only the neighbors listed on the neighbors-list will be considered when calculating
interactions.
The squares of the distances between particles are calculated using the
Pythagorean theorem for time-saving reasons. Square roots of these values are not
taken when calculating interaction forces because square root operations are time-
consuming and the force table is tabulated in terms of distance squared. Square roots
of the distances squared are only required when the neighbors-list are updated and
when pair correlation functions and bond angle distribution functions are accumulated.
Periodic Boundary Condition
Periodic boundary conditions are applied when finding inter-particle distances.
If periodic boundary conditions are applied, particles which leave the primary cell after
a time step will be translated back into it by subtracting the appropriate cell dimension.
Thus particles which leave through one side of the system cell will appear through the
opposite side of the cell.
Thus if the difference in the coordinate is greater than one-half the
corresponding dimension of the system cell, the corresponding image of the particle
located in the adjacent cell will be closer to the central particle than the particle in the
primary cell. The difference in the coordinate is then adjusted to reflect this by either
adding or subtracting the corresponding dimension of the cell to minimize the absolute
values of the difference. It is the resulting difference which is used in the Pythagorean
relation to find the square of distance between the particles. The algorithm by which
this is performed is as follows:
dyk-Xjj-Xik (3.24)
f 2 d i 51^;
dijk=dijk-si'trunc(~)
(3.25)


122
The structural changes during elongation and fracture can also be studied from
the simulated data. The first peak in the pair correlation functions all shift to greater
distances during expansion of the sample, as shown in Fig. 5.26.
The Si-Si-Si bond angle distributions can be obtained from simulations and are
shown in Fig. 5.27. Initially the distribution has a peak at about 109 in agreement
with the ideal DC structure. After uniformly pulling, the distribution undergoes a
change which depends on the strain rate.
The structure-related data recorded during simulations are like frame-by-frame
pictures of the fracture process at the atomic level. The MD simulations used in this
work are based on a program written by Swiler.73 From these data, pair correlation
functions, bond strain distributions and a 3-D presentation of the structure can be
obtained. Stereo pairs of video frozen frames give a visual presentation of the fracture
process. The location of the atomic sites in an unstrained single crystal silicon in the
{100} fracture plane is shown in Fig. 5.28. At 45% strain, fracture is in progress as
evidenced by the presence of a free volume sphere in the center region as shown in
Fig. 5.29. At 50% elongation, the fractured region increases as evidenced by the
presence of the large free volume and several broken bonds as shown in Fig. 5.30.
Thus, the ability of the material to tolerate stress is not immediately lost when fracture
occurs in part of the specimen during the MD simulation. During the fracture event in
MD simulation, it has been observed that atom particles are emitted during fracture.
This phenomenon agrees with the experimental work done by K'Singam et al.74
I have just presented a brief discussion of MD analysis. I will next present the
values of toughness and fractal dimensions obtained from the simulations for different
orientations. The strength dependence on the strain rate, and the strength dependence
on the crack size will be discussed later.


129
Fig. 5.31
{llll-
Stress-stram curve for three different orientations, {100}, {110} and


90
mirror region appears to be smooth to the unaided eye, but appears to be tortuous if the
STM is used as the tool of observation.
The reason that the fractographic features described above are important is that
the distances to crack branching have been shown to be related to different energy
levels at branching and the fracture toughness of the material.50 In addition, fracture
surface features have also been shown related to the fractal dimension of the surface.63
Thus, the fracture surface features are intimately connected to the fracture process as
described by the fracture toughness, branching constants and the fractal dimensions.
These inter-relationships will be discussed later in this chapter.
Fracture Toughness Measurement
Single crystal Si is anisotropic and the fracture toughness varies with the
fracture plane (and the tensile surface if a bending technique is used to fracture the Si).
99
Chen and Leipold determined the toughness for single crystal silicon tested in
different orientations. They found that the {111} fracture plane is the easiest cleavage
plane and the {100} fracture plane is the toughest one. They used an indentation
technique to control the crack sizes on the tested specimens before fracture. However,
in their work, they did not consider the residual stress caused by the indentation. The
equation which they used to calculate the critical stress intensity doesn't consider
residual stress:
Kc = aMB(7ia/Q)l/2, (5.1)
where Kc is the critical stress intensity factor,
a is the maximum tensile stress,
a is the flaw size,
MB is the elastic stress intensity magnification factor, and


9
Fracture of single crystal Si on the {100}, {110} and {111} plane has been
observed.13,22 Not only the toughness, but also the fracture surface topography were
found to be different on the different planes. These observations mean that elastic
anisotropy plays an important role in fracture; the bond breaking process is considered
the same since only one kind of Si-Si covalent bond exists.
Failure Analysis
The characterization of flaws and their relationship to strength plays an
important roll in the analysis of brittle materials. By comparing fracture mechanics
relationships with observed fracture surface markings, important information about the
fracture toughness can be determined. This section will present the basic principles of
fracture mechanics and fracture surface analysis of brittle materials.
Fracture Mechanics
One of the first analysis of fracture behavior of components that contain sharp
discontinuities was developed by Griffith.25,26 This analysis was based on the
assumption that incipient fracture in ideally brittle materials occurs when the magnitude
of the elastic energy supplied at the crack tip during an incremental increase in crack
length is equal to or greater than the magnitude of the elastic energy at the crack tip
during an incremental increase in crack length. By using a stress analysis developed by
Inglis, Griffith related fracture stress to the flaw size by means of an energy balance.
The relationship can be written as:
2Ey
\l/2
K. nc J
(2.1)
where
erf is the failure stress,


131
Strain rate: 1/112 sec
Fig. 5.32 The strength dependence on the strain-rate. Strength increases as the
strain rate increases.


36
4 y = Ay exp() + (^-) erfc(-i-), (3.6)
J J p r sL
where erfc() is the complement error function,
s is a constant,
L is a linear dimension of the molecular dynamics cell,
q is the charge,
r is the distance between the i and j atoms, and
p is a hardness parameter.
After combining the Stillinger-Weber potential and the modified Born-Mayer
potential, the fiber-like structure disappears and potentially more realistic fracture
surfaces can be obtained.
Fundamentals About the MD Simulation
How the MD simulations are performed will be explained now. First, we
discuss how the system was set up, which includes the initial particle positions,
velocities and the potential described in the previous section. Then we discuss how the
system is operated, which includes the application of periodic boundary conditions, the
force summation on the particles, control of the velocities of particles, and control of
thermodynamic values. Finally the discussion of how the data from the simulation
system is collected will be given. A flow chart in Fig. 3.1 is given to illustrate how
the MD simulation is operated.
Initial Conditions
Initial particle positions are located at specific positions since a single crystal
structure is desired.


116
12 T
10
8 -
S 6-
M
4 -
2 -
0
0
Fig. 5.21
simulation.
Pair correlation function of silicon at room temperature from MD


93
(a)
(b)
Q
Q
O
O-
o
(c)
#. of atoms
= 4 x 1/4 + 1
n(i oo) 4
(d)
(e) (0
#. of atoms _
= 4xl/4 + 2xl/2 11(1 lor
\n])~j3a /2
(h) (i)
#. of atoms n(lu) ^
= 3 x 1/6 + 3 x 1/2
Fig. 5.13 The surface energy for FCC structures on the (100), (110), and (111)
planes can be found using a dangling bond calculation.


12
Fig. 2.2 Schematic of features found on the fracture surface of a brittle material
subjected to a constant load. Solid semi-elliptical line at center represents the initial
flaw size depth a¡ and width 2bi. Dashed line represents the outline of critical flaw
depth acr and width 2bcr. Mirror/mist, rl5 mist/hackle, r2, and crack branching, r3,
radii are shown along the tensile axis.


46
Adding these two expansions, we can get the new position:
V l /
Where i =1, 2, 3 denote the Cartesian dimensions of space,
(3.29)
t is a particular time,
Xj(t) is the i Cartesian coordinate at time t,
Xj (t At) is the i Cartesian coordinate at time t At;
At is the length of a time step, and
0(At4)is the remainder terms.
Gear's algorithm
Gear's algorithm60 is another choice to calculate the change of positions of
particles due to two-body interaction. The advantage of Gear's algorithm is that it
gives less energy fluctuation; while the big disadvantage is that it is very time-
consuming. Gear's algorithm predicts molecular positions Xj at time (t + At) using a
fifth-order Taylor series based on positions and their derivatives at time t. Thus, the
derivatives X¡, X, x/ul\ x/iv\ x/v^ are needed at each step; these are also
predicted at time (t + At) by applying Taylor expansions at time At.
2 3
Xt (t + At) = X¡ (t) + X¡ (t) At + X¡0¡) (t) + X¡(t)
+X¡(iv) (t) + X¡(v) (t)
4! 5!
(3.30)
2 3
Xj(i) (t + At) = X¡(i) (t) + Xj(ii) (t)At + Xi(iii) (t) + Xi(iv) (t)
(3.31)


91
Q is the flaw shape parameter calculated from the elliptical integral of
the second kind.
From the given flaw size data in their paper,22 the MB value was calculated to
be 1.03 and Q was approximately 1.47. After substituting these data into equation
(5.1), the equation can be approximately rewritten as
Kc=1.24gVc. (5.2)
If we modify Eq. (5.2) to include the effect of residual stress (due to the indentation
process), the new modified equation becomes Kc = 1.65afc1/2 i.e., Eq. (4.2).40,44
Their data using Eq. (4.2) fall in the same range as those measured in this study. The
measured data and modified data from Chen and Leipold are presented in Table 5.4.
Table 5.4 Comparison of toughness values using different techniques
fracture plane
tensile surface
S.I.
toughness*
( MPaVm)
F.S.A.
toughness**
( MPaVm)
literature
toughness^
( MPaVm)
{100}
{110}
1.24 0.09
1.26 0.06
1.26 0.07
{110}
{100}
1.19 0.08
1.23 0.08
1.19 0.13
{110}
1.05 0.10
1.07 0.09
{111}
{110}
1.19 0.10
1.17 0.09
1.09 0.09
{112}
1.21 0.09
1.16 0.08
Calculated from the strength indentation (S.I.) technique,
1 / 2
^Calculated from fracture surface analysis (F.S.A.), Kc = 1.65cjfC
it 99
"By Chen and Leipold. Originally these values were calculated without considering
residual stress and were found to be 0.950.05 ({100} plane), 0.90+0.11 ({110}
plane), and 0.820.07 ({111} plane) M?a^[m.


40
The energy of potential energies for interactions between particles are also
tabulated in terms of r2. This is used for the calculation of potential energies for the
system.
The three-body force is considered in the same way.
03(ri, r¡, rk) = s f3(rj/8 rj/8 rk/8).
^3(rirjrk) Fjik ^ijk ^kij-
The function h is given :
hjik h(ry, rjk, 0jjk) (3.11)
= X exp[p /(ry a) + r| /(% a)] x (cos0jik + 1/3)2, if r = 0, if r>a.
The following derivation related to the three-body potential will only use hjik for
convenience. Thus the three body force related to hjjj^ is as follows:
(3.12)
(3.13)
(A) At first, we differentiate r¡ only.
F = Fr + Fq = VO
At er direction,
(3.14)
(3.15)
At eg direction,
Fe=-
^ 1 60 2X
Fe =-vr =
rj 60 r¡
(3.16)
Now transform the polarized coordinates into Cartesian coordinates.
Fr + F0 Fx + Fy + Fz
(3.17)


81
(b)
Fig. 5.5 (a) Anisotropy mirror shapes on fracture surfaces in single crystal
alumina; the corresponding relative magnitudes of Young's modulus for directions (in
the plane) parallel and perpendicular to the local mirror boundary are given. Larger
values of E are reflected as greater distances from the origin, (b) A schematic of the
expected mirror boundary for the {110} fracture plane is plotted.


67
Fractal Analysis Using Simulation Results
After the fracture has occurred in the simulation as shown in Fig. 4.8, an
important objective is to obtain the generated fracture surface from the MD
simulation.Each atom is assumed to be a sphere with an electron cloud of a prescribed
radius. The radius of each atom is assumed to be the radius of the potential field.
Thus the fracture surface appears as a surface with intersecting spheres, as shown in
Fig. 4.9. After obtaining the surface, a contour plane is used to intersect the spheres to
obtain a slit-island. The obtained slit-island appears as a plane with lots of circular
disks on it as shown in Fig. 4.10. The slit-island technique as described in the fractal
dimension determination section in the experimental procedure was applied here to
obtain the fractal dimension. The perimeter of the selected island was measured using
different scales as described before and the fractal dimension can be obtained from the
Richardson plot.


153
is estimated from MD to be 3.8, the cutoff distance of the Stillinger-Weber potential.
From this work and previous work, we know that fracture energy is the sum of the
energy necessary to breaks bonds, plus the energy to generate light, heat and sound,
plus the energy to produce a tortuous surface. This latter energy is consumed in the
fracture process and contributes to the formation of a fracture surface which is not
Euclidean. The results shown here imply that the fracture surface is never ideally
smooth even in the easiest cleavage plane.
The experimental and simulation data are summarized in Table 6.1.
Table 6.1 Summarized results of this study
fracture
plane
tensile
surface
(Lit.)
E
(GPa)
(Exp.)
K
(MPa m1/2)
(Exp.)
Y
(J/m2)
(MD)
Y
(J/m2)
(Exp.)
D*
MD)
D*
{110}
{110}
169
1.07
3.39
t
0.04
t
{111}
{112}
188
1.16
3.58
2
0.05
0.09
{111}
{110}
188
1.17
3.64
f
0.06
t
{110}
{100}
169
1.23
4.48
3
0.10
0.11
{100}
{110}
130
1.26
6.11
4
0.16
0.16
t
Only the choice of the fracture plane is important for those data from the MD
simulation.


83
(a)
Fig. 5.6
Crack path for the {111} fracture plane.


130
It is easily seen from Fig. 5.31 that the {100} plane has the highest value, while
the {111} plane has the lowest value. The values of toughness are found to be 0.90d/,
0.61d/, and 0.43d/ for the {100}, {110} and {111} planes, respectively, where dl is the
unit length. The choice of the unit length determines the value of the actual toughness
and will be illustrated after the fractal analysis has been discussed.
Since the simulated moduli are lower than the actual values, it is reasonable to
assume that the fracture strength and toughness will be lower than the actual values
also. However, the simulated strength is a little higher than the actual value and the
toughness is about the same, if a reasonable unit length, dl, is chosen. Thus, the actual
values for the simulated moduli do not appear to be as important as other variables
such as the (SW) potential truncation and the two potential approach.
Strength Dependence on the Strain Rate
The dependence of the strength on the strain-rate was investigated for the {100}
fracture plane. The stress-strain curves for strain rates of 0.1, 0.2, 0.5, 1, 2, and 5 ps'
1 are shown in Fig. 5.32. As the strain rate increases, the strength increases in
agreement with the experimental work of Ebrahimi.76 There are two reasons for this.
When specimens are elongated at the higher strain rate, more cracks initiate and several
fracture planes are formed after final fracture. So, at higher strain rates, more energy
has to be consumed in order to generate more cracks and fracture surfaces. Another
reason is that structural relaxation is allowed to occur at lower strain rates, but not at
higher strain rates. This relaxation allows for the relief of stress. Thus, the selection
of strain rates relative to the time scale of the material is critical to the type of behavior
that is observed.
A useful unit of time at the atomic scale is the period of thermal vibration,
which is an atom's response time to a change in the environment. The thermal


47
2 3
Xi(ii) (t + At) = X{(ii) (t) + Xi(iii) (t) At + Xi(iv) (t) + X¡(v) (t) (3.32)
Xi(iii) (t + At) = Xi(iii) (t) + X¡(iv) (t) At + X¡(v) (t) (3.33)
Xi(iv) (t + At) Xj(iv) (t) + Xj(v) (t)At (3.34)
Xj(v)(t + At) = X¡(v)(t) (3.35)
In order to correct the predicted positions and their derivatives using the
discrepancy between the predicted acceleration and that given by the evaluated force.
The force at (t + At) obtained from Newton's second law can be used to determine the
acceleration x/n^(t + At). The difference between the predicted accelerations and
evaluated accelerations is then formed,
AXj(ii) = X¡(ii) (t + At) Xi(ii)P(t + At). (3.36)
In gear's algorithm for second-order differential equations, this difference term
is used to correct all predicted positions and their derivatives. Thus,
Xj=XiP+a0AR2 (3.37)
Xi(i)At = Xi(i)PAt + a1AR2 (3.38)
Xi(ii)^- = Xi(ii)P^- + a2AR2 (3.39)
Xi(iii)^y--Xi(iii)P^- + a3AR2 (3.40)
x (1V) (At/ = x (iv)P (At/ a AR (3.4!)
4! 4!
x.(v) (^L = X.(v)P ^L + (X5ar2 (3.42)
2
where AR2 = AXj*-- .
2!
The parameter cq promotes numerical stability of the algorithm. The cq
depends on the order of the differential equations to be solved and on the order of the


73
Fig. 5.1
A typical fracture surface on the {100} fracture plane.


136
Angle
Fig. 5.35 The Si-Si-Si bond angle distribution during pulling for the 0.2 ps'1 strain
rate. Initially the distribution has a peak at about 109. After uniformly pulling, the
distribution undergoes a change which depends on the applied strain.


61
M.D. Simulation Procedure
Before performing the MD simulation, parameters24 for the potential and initial
conditions should be given beforehand. Two data files which contain these data are
needed in order to perform the MD simulation. One is called the input file, which
includes the initial conditions and the parameters of the potentials used. The other one
is the atom-position file which contains the initial position of each Si atom.
Several tasks were set to be accomplished using MD simulation. One was to
compare the toughness of Si for different orientations. The difference of stress-strain
curves for different strain rates could be obtained to compare the toughness as a
function of strain rate. Another was to obtain the fractal dimensions for different
fracture planes in order to compare them with the experimental results. Another goal
was to obtain the fracture strengths for each simulation. The comparison of fracture
strength due to different strain rates has been performed. The comparison of fracture
strength due to different crack sizes has also been performed.
Determination of Input Data
Before performing the MD simulation, several conditions should be decided
first. All those conditions should be illustrated in the input file for the MD simulation.
Those conditions are the desired orientation, temperature, strain rate, length of each
time step, length of the simulation, and the adiabatic or isothermal state.
At first, after choosing the desired orientation, a data file which contains the
number and the positions of those atoms should be constructed. The data file will
determine the initial position of each atom for the MD simulation. In this study, the
atom position for the ideal crystal structure will be given at first and the crystal will be


26
(a)
generator
(b)
each side of the square
is replaced by the scaled
generator.
(c)
Fig. 2.8
The stepwise construction of a scaling fractal.


66
Fig. 4.7 The {100}, {110}, and {111} planes are those fracture planes chosen in
the MD simulation.


99
yj(x + dx)2 + y2 -y]x2 + y2 2 dx. (5.4)
If we adding -y/x2 + y2 at both sides, then the above equation changes to
yj(x + dx)2 +y2 2 yjx2 +y2 + dx. (5.5)
If we square both sides, the above equation changes to
x^ + dx^ + + 2xdx 2 x^ + + dx^ + 2-dx-^x2 +y2 (5.6)
If we eliminate x + dx + y on both sides, then the mark can be found to be
< , and the relation for the above equation is
yj(x + dx)2 + y2 -y]x2 + y2 < dx. (5.7)
Eq. (5.7) means that the energy needed to pull a section dx in the {100} plane is
smaller than 1/2-K dx as for the {111} plane. Thus, if only one bond is considered,
the energy consumed to pull one bond in the {100} plane is smaller than that for the
{111} plane for the same pulling distance. Although the ratio of toughness between
{100} and {111} is calculated to be 2.3/1.36, the actual value should be less than that.
In the same way, if dx is pulled in the x direction, the elongation along the
dangling bond is longer for atoms laid on the {111} plane than those laid in the {100}
plane since the {111} plane is perpendicular to the x direction. If a bond will break
after a specific elongation, then when one expands the single crystal Si perpendicular to
the {100} plane (or along the [100] direction), more deformation is needed to fracture
the specimen.
Thus, when we observe the stress-strain curve for the {111}, {HO}, {100}
orientation, we can find that the fracture strain for the {100} plane is the highest. (The
result, which will be discussed later, is the same as the stress-strain curve obtained
from the MD simulation, which shows the fracture strain for the {100} fracture plane is
larger than the other two orientations.) Although the modulus in the {100} plane is
lower than the other two planes, the toughness (here the toughness is defined as the


147
orientations are the fracture planes. The choice of the orientation of the tensile surface
does not affect the results of the simulations.
In this study, the important factor is the fracture process near the crack tip. In
the experimental work, the stress state on the bending specimen near the crack tip is
almost uniform tension. Since the region near the crack tip is nearly the same for the
two conditions of loading, it is reasonable to compare the data between the
experimental and simulation results even though the testing methods were different.
Another difference between the experimental and simulation procedure is the
size of the specimen. In the experimental work, the size of the specimen cross-section
varies between about 3mm 5mm to 6mm 6mm. The scale used to measure the
fractal dimension is in the order of pm's. But in the MD simulation, the size of the
specimen cross-section is about 20 30 to 50 50 , which is about
1/1,000,000 of the size used in the experimental specimen. The difference in sizes are
considerable. One may say that since a periodic boundary condition is applied in the
MD simulation, then the size of the simulated system is infinite. If this assumption is
correct then the difference in scale will not be that important. The scale used to
measured the fractal dimension is on the order of Angstroms, which is about 1/1,000 of
the scales used in the experimental work. Thus, if the fractal dimensions measured in
the experimental work and simulation agree with each other, then the fractal dimension
is the same at the pm level and at the level. The agreement between experiment and
simulation implies that the valid scale range to measure fractal dimension is at least
from the Angstroms level to the pm level. Thus, the fractal phenomenon in the
fracture process exists for a large scale range.


102
Fig. 5.16
Continued.


149
Table 5.11 The toughness value from experimental and simulation measurement
Fracture plane
y (experiment)
(J/m2)
y* (MD)
(J/m2)
y** (MD)
(J/m2)
{100}
6
4
4
{110}
4
3
2
{111}
3
2
2
* the unit length, c
/, is the average characteristic length a0
= 4.1 ,
calculated from the measured data.
**the unit length is the cutoff distance of the SW potential (3.8 ).
It is very clear that both the simulation results and the experimental results have
the same trend and fall in a reasonable range. The simulated and experimental values
for the fractal dimension are about the same even though the difference in measuring
scale is large between the experimental and simulation results. The MD technique
provides a means to estimate the fractal dimension. Here the highest dimension was
selected if two tensile surfaces were tested for one fracture plane. Notice that better
agreement is obtained if we compare the column with the higher fractal dimension and
the one obtained from MD simulation in the Table 5.12. The results suggest that the
highest fractal dimension for any given plane is the true dimension.


103
seen. When a small region in the 1,000X is magnified to 10,000X, once again similar
features appears. This is a suggestion of self-similarity and scale invariance. Self
similarity means that features in different regions appear to be similar to one another.
They can be related by a scalar multiple. Scale invariance occurs if a feature on
different scales appears to be the same. These properties indicate that a fractal can be
created by using a shape to be repeated and reduced in size followed by a prescribed
sequence. Thus, those photographs of the fracture surfaces of single crystal silicon
display fractal patterns.
Table 5.6 Fractal dimension for different orientation
fracture plane
tensile surface
fractal dimension
{100}
{110}
1.16 0.04
{110}
{100}
1.10 + 0.04
{110}
1.04 0.03
{111}
{110}
1.06 0.02
{112}
1.05 0.00
Fracture and fractal process varies with the fracture plane and tensile surface
plane. The fracture process in single crystal is self-affine when the fracture plane or the
tensile surface are different. Self-similarity and scale invariance just occurs in the same
fracture plane with the same tensile surface.
If the fracture surfaces for different orientations are carefully compared, the
topography looks different. It suggests that self-similar behavior just occurs in the same
orientation. It is relatively easy to explain this phenomenon from the easiest cleavage
plane point of view. The fracture path tends to follow the least energy path and thus the
crack will try to find the easiest cleavage plane at the time of propagation or choose a


106
(c) After nth construction
Pnth =Px (4/3)n
= infinity, if n is a very large number;
Anth = A x (1 + 1/3 + (1/3)2 + (1/3)3 + ... + (l/3)n)
= A x ( 1.5 ), if n is a very large number.
Fig. 5.17 Continued.


86
(a)
(b)
Fig. 5.9 The crack paths (shown by shaded regions) for the {100} fracture plane
with a {100} tensile surface, (a) shows a schematic of fracture bar. (b) shows
different modes observed.


72
Thus, there were a total of 5 different groups of data. The fracture surfaces are shown
in Figs. 5.1, 5.2, 5.3. As shown in Fig. 5.1, the fracture surface of the {100} fracture
plane appears to be the most tortuous one. It has the smallest mirror region when
compared with the other two orientations. The fracture surface of the {110} fracture
plane with a {100} tensile surface has a Batman-shaped mirror region as shown in
Fig. 5.2(a). The fracture surface of the {110} fracture plane with a {110} tensile
surface has an inverted volcano-shaped mirror region as shown in Fig. 5.2(b), while
the fracture surface of the {111} fracture plane as shown in Fig. 5.3 is relatively
smooth compared with the other orientations. River-marks, i.e., twist hackle or
cleavage marks, appear on most of the fracture surfaces of this fracture plane. River-
marks are twist hackle fracture features which appear to spread apart as the crack path
moves away from the origin of fracture, thus pointing back to the origin. The {111}
fracture plane is the easiest cleavage plane, as reported before.22 Shown in Fig. 5.3(a)
is the fracture surface of the {111} fracture plane with a {110} tensile surface while
that in Fig. 5.3(b) is the fracture surface of the same fracture plane but with a {112}
tensile surface. More data are listed in Tables 5.1, 5.2, and 5.3.
For fracture on the same plane, there should be no structural difference during
the fracture process. However, the loading, i.e., strain would be expected to be
different in different locations due to elastic anisotropy and a changing strain field,
which in turn is due to the type of strain applied here, i.e., bending. The fracture
surface can appear different due to a change in the tensile surface in the bending test.
Elastic anisotropy is the reason the fracture surfaces look different in the same fracture
plane without a structural change. Thus the fracture surfaces of the {110} fracture
planes as shown in Figs. 5.2(a) and (b) look different because the tensile surfaces are
different. An interesting result occurs if we rotate the fracture surface by 90 degrees
with a {110} tensile surface, as shown in Fig. 5.4; a half mirror boundary of this
fracture surface is similar to a half mirror boundary of the fracture surface on the


44
r^=di;i. + d;i. + d,
jk
(3.26)
1 jk 2 jk 3 jk
where i = 1, 2, 3 denote the Cartesian dimension of space,
Xjj is the i Cartesian coordinate of atom j,
Xjjj is the i Cartesian coordinate of atom k,
dyk is the i coordinate difference between atoms j and k,
djjk is the i coordinate difference between atoms j and k after correction
for periodic boundary conditions,
Sj is the i dimension of the system cell,
trunc() is a function which returns the integer closest to the argument
between the argument and zero, and
r¡j is the distance between atoms i and j.
Interactions Due to Two-Bodv Potential
Two-body interactions between particles are handled in a four step process.53
The square of the interparticle distance is found for particles located in the neighbors
lists. The interparticle force is then found by looking up the appropriate value in the
force table. The components of the force are found by multiplying the value tabulated
in the force table by the difference in the coordinates for each dimension (dyk). The
components of force are then summed for each particle for all interactions.
Interactions Due to Three-Bodv Potential
Three-body interactions between particles are handled in a four step process.
The square of the interparticle distance is found for particles located in the neighbors
lists. Two atoms, j and k, within the cutoff potential range are found from the
neighbor list for the center atom, i. The angle 0jik between the center atom i and the


140
Fig. 5.38 Strength dependence on the crack size. The shape of the crack is planar
and simulates a sharp crack.


38
Initial velocities are given using the Poisson velocity distribution. The velocity
distribution is generally selected to correspond to a temperature at which the crystal is
stable. Here it corresponds to room temperature or to the temperature of the
simulation.
Random velocities are assigned to particles using the Poisson velocity
distribution. The formula giving the Maxwellian velocity distribution in three
CO
dimensions are
Vxi
V,
yi
x in(Rand 1)
mi
2kRT
x ln(Randl)
mi
x cos(27T x Rand3)
x sin(27t x Rand3)
(3.7)
(3.8)
V.
Z1
X ln(Rand2)
x sin(27i x Rand4)
(3.9)
where Vxi, Vyi, and Vzi are the components of the velocities of particle i,
Randl, Rand2, Rand3, and Rand4 are random numbers evenly
distributed between 0 and 1,
mj is the mass of particles i,
T is the temperature, and
kg is the Boltzmann's constant.
Particle velocities are applied by putting the velocities obtained above into
positions before and after the current time step, which is the basic unit of time during a
simulation. Thus the random velocities are effectively converted into average velocities
during the previous time step, and particles positions and velocities are given by the
difference of two sets of positions for each particles. The reason for this will become
clear in the discussion of Verlet's algorithm.


49
Potential Energy of the System
The potential energy for a system is calculated in a way similar to the
calculation of forces on individual particles. When the neighbors-list is updated,
interaction potentials are found using the potential energy table. The potential energies
for all interactions are then summed to find the potential energy for the system. The
system potential energy is added to the system kinetic energy as previously obtained to
find the system internal energy.
System Pressure Calculation
The system pressure or stress tensor is calculated using the Yirial of Clausius,
which has the form
(3.43)
When converted for use in simulations, it takes the form
(3.44)
where P is the system pressure in Kbar,
0 O
V is the volume of the cell in A ,
0 14
v is the velocity of the particle in A/10 s,
F is the interatomic force in 10"12 erg/,
r is the interatomic separation in , and
mj is the mass of the particle in 1CT24 g.


76
Table 5.1 Data for Si samples fractured in the {100} plane
{100}
fracture plane ,<110> tensile surface
indent
flaw
load
KIc*
KIc**
load
width
thickness
b
a
size
P
strength
0.59(E/
1.65*
No.
Kg
mm
mm
mm
mm
mm
lb
MPa
MPamAl/2
MPamAl/2
al
0.5
5.70
5.35
a2
0.5
6.75
5.00
0.068
0.068
0.068
96
102.5
1.31
1.40
a3
0.5
5.85
4.95
0.080
0.080
0.080
65
81.7
1.10
1.21
a4
0.5
6.70
5.40
0.056
0.052
0.054
118
108.8
1.37
1.23
average
1.26
1.28
dev
0.11
0.08
bl
1
6.25
5.05
0.127
0.094
0.110
63
71.2
1.18
1.23
b2
1
5.90
5.25
0.080
0.080
0.080
78
86.4
1.37
1.28
b3
1
5.80
5.05
0.124
0.124
0.124
54
65.8
1.11
1.21
b4
1
5.90
5.00
0.114
0.072
0.091
63
77.0
1.25
1.21
b5
1
5.80
5.10
0.127
0.099
0.112
60
71.7
1.19
1.25
average
1.23
1.24
dev
0.09
0.03
cl
2
6.25
5.20
0.201
0.132
0.163
55
58.6
1.22
1.23
c2
2
5.90
5.15
0.159
0.131
0.144
57
65.6
1.32
1.30
average
1.27
1.27
dev
0.05
0.03
total
average
1.24
1.26
total
dev
0.09
0.06
three-point bending span=27mm
* KIc is calculated from Eq. (4.3)
**KIc is calculated from Eq. (4.2)


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
and Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is folly adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Robert T. DeHoff ¡/
Professor of Materials Science
and Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is folly adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
f
dLJL
eresteh Ebrahimi
Associate Professor of Materials
Science and Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is folly adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Peter A.
Assistant Pfef&sor of Aerospace
Engineering, Mechanics, and
Engineering Science


Ill
Results From Molecular Dynamics Simulation
Validity of the Applied Potential
In this study, before we assume that molecular dynamics (MD) is an applicable
method to study atomic fracture behavior of single crystal silicon, the accuracy and
appropriateness of the numerical procedure and the interpretation of the simulation
results should be discussed. Here we concentrate on the study of the fracture process
at the atomic level and the fractal analysis of the generated fracture surfaces. Earlier
works8"11,21 in MD simulation of fracture concentrated on silicate glasses. Soules and
Busby21 used molecular dynamics to study the rheological properties and fracture
behavior of a sodium silicate glass. The work was followed by works of Soules,8
Ochoa and Simmons,9 and Ochoa et al,10 which demonstrated the importance of
dynamic effects in the fracture of silicates. A work by Simmons et al.11 examined the
mechanisms involved in an individual bond fracture to describe a process by which an
entire fracture surface may be stabilized. The MD technique has been used to perform
the study of fracture behavior and should be useful for the study of silicon. However,
the previous studies only studied ionically bonded materials. This study includes
covalently bonded materials. Thus, a different potential had to be developed and
Stillinger and Weber24 suggested a potential to be used for silicon.
Kieffer and Angel23 proposed a way to relate the fractal dimension with the
density function of aerogel materiel using the MD technique. They claimed that a non
integer dimension is a characteristic feature of the aerogel structure used in their study.
The fractal dimension can be associated with dynamic properties in relation to many
kinetic processes and rate phenomena. However, because of the way in which they
obtained the fractal dimension by using the change of the slope in the radial distribution


152
modeled using molecular dynamics. For the first time a method to determine the
fractal dimension on the MD generated fracture surface was proposed here and found
to be applicable.
The size of the introduced crack is an important factor for the simulated
strength in MD simulation. If a crack is introduced in the simulation, the fracture
strength decreases as the flaw size increases. The simulated results agree quite well
with the Griffith criterion for those specimens with either void-like flaws or the plane
like flaws. The critical stress intensity factor obtained from simulations with a plane
like crack (sharp crack) agreed with the macroscopic experimental values.
The strength increases as the applied strain rate increases in MD simulation,
which agrees with the experimental work done by Ebrahimi. One probable reason
for this behavior is that more cracks initiate in the high strain rate case and these
initiations consume more energy. Another possible reason is that relaxation is not
allowed or equilibrium is not reached in the high strain rate case.
The fracture energy results from MD simulation agree quite well with those
from the experimental work. The relation of the simulated surface energy on different
planes is in the same range with that from experimental work. The present results
suggest that using MD simulation to model fracture behavior is applicable.
The fractal dimension measured from MD simulation is about the same with
that from experimental work. It suggests that using MD simulation to model fractal
analysis is acceptable. The MD results implied that the fractal dimension can be
measured using a scale from level in MD simulation to pm level on the fracture
surface.
From the plot of y vs. ED*/2 for different orientations in single crystal silicon,
a constant slope suggests that a0 is a constant for single crystal silicon. The
characteristic length, a0, was calculated from experimental data to be 4.1. This
value is very close to the stretched length of the Si-Si bond just before fracture which


58
fractured bar
potted in epoxy and
polished parallel to surface
top view
Fig. 4.3 Fractured samples are encapsulated in epoxy and polished approximately
parallel to the fracture plane. (Top View) Islands emerge in the polished plane.


65
10000
9000
£ 8000 ..
-3 7000
& 6000 -
| 5000 -
| 4000
ft 3000
I 2000
H 1000
0
0
0.2
Strain
Fig. 4.6 The temperature vs. strain curve for adiabatic condition. The
temperature remains at about 300K before the moment when fracture takes place at
strain 0.45. After fracture, the temperature of the system rise very fast up to 7000
K, then slows down.


24
Mathematical fractals will obey the above equation for all scales. However,
physical fractals may have a finite cutoff at both large and small scales. In the physical
world, the fluctuation of a line will find its limit in the smallest measurable feature.
The ruler length determines the smallest feature we can measure. If variations in
structure are below the sensitivity of the measurement scale, the line will appear to be
Euclidean. In that case, the longest line we could measure would depend on the finest
possible scale of measurement. We could not conclude that the line has no
fluctuations, only that we have no ability to measure the fluctuations.
On the contrary, a line may be composed of indivisible components. After our
measuring stick has become smaller than the smallest of these components, the line
would cease to exhibit scale-dependent length. This length, then, would represent a
true upper bound. Such a line could be called fractal over the range, it exhibited
scale-dependent length. It is not, however, a fractal in the strict mathematical sense.
The combination of the two limits would result in the Richardson plot as shown
in Fig. 2.7. The object would be described as a fractal over a bounded range of scales.
The cut-off scales are of special significance to the particular geometry of a specific
physical phenomenon.
A box is shown in Fig. 2.8(a)18,47 whose sides are of unit length. Next to the
box is a shape called a generator, composed of line segments scaled to 1/4 of the
length of a side of the box. Each side of the square is replaced with this scaled
generator. The result is shown in Fig. 2.8(b). Again, the generator is scaled down to
the length of each straight line segment of this new object. The straight line segments
of Fig. 2.8(b) are replaced with the scaled generator. A portion of the resulting shape
is Fig 2.8(c). This process is continued, generating an object which is said to be scale
invariant and self-similar. A scaling fractal looks geometrically the same everywhere
and on all scales.


112
functions with respect to the density, their results were not reasonable. Their results
show that a denser aerogel has a lower fractal dimension. This results contradicts the
fact in nature that a fully occupied volume has a dimension 3 while a volume with
porosity has a dimension less than 3. Since it was concluded that their approach was
incorrect, we decided to develop a different approach from theirs to obtained the fractal
dimension in this study.
As discussed in Chapter 3, a modification of the Stillinger-Weber (SW)
potential is needed to correctly predict fracture behavior in covalently bonded
materials. Before fracture, silicon possesses a DC structure which can be simulated
using the SW potential which performs very well in covalently bonded and tetrahedral
structures. However, after fracture, the local structure changes and those Si atoms
around the crack do not have 4 neighbors anymore and some electric charge must be
added to those atoms. A different potential must be introduced after fracture in order
to describe the local changes. A modified Born-Mayer potential works very well in
silica crystal and is used here. Thus, the SW potential and the modified Born-Mayer
potential were applied in the system for this study. In Chapter 3 the potential and the
detailed simulation procedures have been described. The two-body potential and force
curves for single crystal silicon are shown in Fig. 5.19. From the curves, it is found
that the equilibrium position for Si is about 2.34 , and the distance for the cut-off
potential is 3.78 . Fig. 5.20 shows the magnitude of the Coulombic modified Born-
Mayer potential related to the two-body part of the SW potential. Notice that the
magnitudes of the two potentials used in this study are of the same order.
The validity of the numerical procedure and the interpretation of the simulation
results can be checked by obtaining physical properties from the MD results and
comparing these with the experimental results. The SW potential has been accepted
widely in several applications such as thermodynamics properties,24-56 phase
diagrams,56 and defect study.72 The validity of using this potential (the combination of


ACKNOWLEDGMENTS
I would like to express my sincerest gratitude to Dr. J. J. Mecholsky, my
supervisory committee chairman, for his invaluable guidance, support and encouragement
throughout the research portion of this project, and for his numerous suggestions and
corrections during the preparation of this dissertation. I am also thankful to the rest of my
committee members, Dr. R. T. DeHoff, Dr. F. Ebrahimi, Dr. P. A Mataga, and Dr. J. H.
Simmons, for their encouragement and advice throughout the research of this project.
I would like to express my special thanks to T. P. Swiler, from whom I learned a
lot about the MD technique and UNIX computer operation system. Ele also allows me to
use his computer codes to operate the MD simulation and MD movies.
I would also like to extend my thanks to Dr. T. J. Mackin, Mr. Z. Chen, Mr. L.
Hehn, Mr. J. Niaouris and A. aman, with whom I had many precious discussions on
various aspects of this research. Special thanks must also be given to Ms. J. Y. Chan for
her help in obtaining some of the experimental data.
Finally, I am deeply indebted to my parents for their love and encouragement
throughout my entire life and to my wife, Yu, for her patience, understanding, and
support during the course of this work. I am especially grateful at this time because my
daughter, Gina, arrived in the world and she makes my school life more fruitful.
n


REFERENCES
1. T. L. Anderson, Fracture Mechanics, CRC Press, Inc., Boca Raton, FL (1991).
2. J. J. Gilman, J. Appl. Phys. 31 (12), 2208, (1960).
3. S. W. Freiman, R. Mulville, and P. W. Mast, J. Mater. Sci. 8 (11), 1527 (1973).
4. G. R. Anstis, P. Chantikul, B. R. Lawn, and D. B. Marshall, J. Am. Ceram. Soc. 64
(9), 533 (1981).
5. P. Chantikul, G. R. Anstis, B. R. Lawn, and D. B. Marshall, J. Am. Ceram. Soc. 64
(9), 539 (1981).
6. J. J. Mecholsky, Jr., S. W. Freiman, and R. W. Rice, J. Mater. Sci. 11, 1310-1319
(1979).
7. C. Kittel, Introduction to Solid State Physics, 5th ed., J. Wiley & Son, Inc., New
York (1976).
8. T. F. Soules, J. Non. Cryst. Solids 73, 315 (1985).
9. R. Ochoa, J. H. Simmons, J. Non-Cryst. Solids 75, 413 (1985).
10. R. Ochoa, T. P. Swiler and J. H. Simmons, J. Non. Crystal. Sol. 128, 57 (1991).
11. J. H. Simmons, T. P. Swiler, R. Ochoa, J.N.C.S. 134, 179-182 (1991).
12. M. W. Mitchell and D. A. Bonnell, J. Mater. Res. 10, 2244 (1990).
13. Y. L. Tsai and J. J. Mecholsky, Jr., J. Mater. Res. 6, 1248 (1991).
14. K. Ravi-Chandar and W. G. Knauss, Int. J. Fract., 26, 65-80 (1984).
15. J. J. Mecholsky, Jr., S. W. Freiman, and R. W. Rice, in Fractography in Failure
Analysis, ASTM STP 645, 363-79, ASTM, Philadelphia, PA (1978).
16. E. B. Shand, J. Am. Ceram. Soc., 42[10] 474-77 (1959).
17. J. J. Mecholsky, Jr., T. J. Mackin, and D. E. Passoja, in Fractography of Glasses and
Ceramics, Advances in Ceramics, Vol. 22, 127-34, Am. Ceram. Soc., Columbus,
OH (1988).
18. B. B. Mandelbrot; The Fractal Geometry of Nature, J. Wiley & Son, Inc., New York
(1976).
19. A. Paskin, A. Gohar, G. J. Dienes, Phys. Rev. Lett. 44, 940 (1980).
20. G. J. Dienes, and A. Paskin, J. Phys. Chem. Solids 48, 1015-133 (1987).
21. T. F. Soules, R. F. Busby, J. Chem. Phys. 78, 6307 (1983).
160


159
(compliance) constants, Sy, and the direction cosines of the crystallographic direction
under study:
(sll -S12)~T
s44
+ l22l23 +
(C.l)
where /j, l2, l3 = directions cosines. The elastic modulus is calculated to be 130 GPa
in the <100> direction, 169 GPa in the < 110 > direction, and 189 GPa in the
< 111 > direction.


APPENDIX A
SUMMARY OF DATA FROM DIFFERENT INVESTIGATORS
Those fracture toughnesses measured by Gilman2, Chen and Leipold,22 St.
John,57 Jaccodine,64 Myers & Hillsberry,65 and Kalwani66 are listed in Table A.l.
Table A.l Comparison of data obtained from different techniques and investigators.
reference
techniques
applied
{ioo¡
fracture plane
(MPa m1/2)
{110}
fracture plane
(MPa m1/2)
{in}
fracture plane
(MPa m1/2)
Jaccodine64
double cantilever
0.75
0.71
0.62
Chen &
Leipold22
indentation &
four point bend
1.26*
1.19*
1.09*
Kalwani66
indentation
{110}$ 1.25
{100}t 0.86
{110}$ 0.83
{100}$ 1.01
{111}$ 0.81
{110}$ 1.22
Gilman2
double cantilever
0.62
Myers &
Hillsberry65
notched four
point bend
1.34-2.85
St. John57
Double
cantilever
{110}$ 0.93
this work
strength
Indentation
{110}$ 1.24
{100}$ 1.19
{110}$ 1.05
{110}$ 1.19
{112}$ 1.21
this work
three point bend
{110}$ 1.26
{100}$ 1.23
{110}$ 1.07
{110}$ 1.17
{112}$ 1.16
*Modified data. (of. p. 90-91)
^Indentation plane.
$Tensile surface.
154


117
Fig. 5.22 Pair correlation function for single crystal silicon at
slightly lower than the melting point.
9 10
a temperature


108
However, if we limit the scale of generation of fractal objects to atomic size, i.e.,
about several Angstroms, the steps of the generation, n, would not be very large and
the fracture surface with higher fractal dimension would have higher surface area than
one with lower fractal dimension. Here, this idea is applied to the generation of
fracture surfaces. Some fracture surfaces are smoother than others, like the {111}
fracture plane is smoother (D = 1.05) than the {110} (D = 1.08, the higher of the two
chosen tensile surfaces) and {100} (D = 1.16, the higher of the two chosen tensile
surfaces) fracture plane in single crystal silicon. In single crystal Si, there is one kind
of bond and the energy of the covalent bond is the same for different orientations.
More surface area means more bonds are broken and more energy is consumed during
the fracture process. Thus, it is reasonable to say that the {111} plane is the fracture
plane with the least toughness since it has the smoothest fracture surface with fewest
bonds broken, while {110} and {100} planes are tougher since their fracture surfaces
show more irregularity and have more bonds broken.
Note that the area beyond crack branching should not be considered in the
fractal analysis at the scale for which we examine the surface. Thus, only the fracture
surfaces generated prior to crack branching were considered and compared with the
tortuosity or fractal dimensions. If we said that the fracture surfaces on the {100}
plane are more tortuous than those on the {111} plane, then the fracture surfaces to
which we refer are in the area before crack branching. The areas beyond crack
branching always look very smooth, because the mirror region repeats itself after
branching, because of the self-similar nature of the fracture process.
In order to prevent getting an infinite value of a0 calculated from Eq. (2.22) if
D* approaches zero, we modify Eq. 22:
y = Yo + a0EhkiD*/2, (5.10)
where y is the fracture energy,
y0 is the surface energy for an Euclidean fracture surface (D*=0),


82
the topography of the fracture surface. A more detailed discussion of macroscopic
crack branching follows.
The crack branching plane is an interesting topic in this study. From the
observation of the obtained fracture surfaces, the fracture surface of the {111} fracture
plane is relatively smooth such that no crack branching is observed as shown in Fig.
5.6. For fracture in the {110} plane, the crack either doesn't branch or tends to branch
to the available {111} planes as shown in Figs. 5.7 and 5.8. In Fig. 5.7, the crack
growth (shown by shaded regions) occurs on the {110} plane first and deviates to
{111} plane as shown in (c) and (d) of Fig. 5.7. In Fig. 5.8, the crack growth (shown
by shaded regions) occurs on the {110} plane first and deviates to {111} planes as
shown in (c) and (d) of Fig. 5.8.
Crack branching for the {100} fracture plane is complex. As seen in Figs. 5.9
and 5.10, the system of crack branching angles varies for one fracture plane as well as
for the loading direction changes, e.g. from the {100} tensile surface to the {110}
tensile surface.
For those fracture surfaces of the {100} fracture plane with a {100} tensile
surface, the crack easily branches to the {111} plane as shown in Fig. 5.9. If the
tensile surface is chosen to be {110}, the crack will remain on the {100} plane for
about 2 to 3 mm before branching to the {111} plane if the indentation load is larger
than 2 Kgw, as shown in Fig. 5.10(b). If the indentation load is smaller than 2 Kgw,
the crack will branch in a distance smaller than 2mm as shown in Fig. 5.10(c).
It is also found that the fracture mirror region is not really very smooth.12 The
reason that we say it is mirror-like is because the tortuosity of the region is smaller
than the wave-length of visible light as shown in Fig. 5.11. Indeed, if a scanning
tunneling microscope (STM) is used to observe the mirror region, then the surface is
not smooth at the scale at which the STM is used.12 As in Fig. 5.12, the surface on
the mirror region is quite rough at that magnification. Thus, the geometry in the