Quasiparticles and Fracture #2 (1)

The Fracture-Quasiparticle Connection Looking at Fracture in a New Light

Dann Passoja

New York, New York

2 [THE FRACTURE-QUASIPARTICLE CONNECTION LOOKING AT FRACTURE IN A NEW LIGHT]

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Table of Contents Introduction .................................................................................... Error! Bookmark not defined. Prime Objective of this Work ......................................................... Error! Bookmark not defined. Geometry in Fracture ........................................................................................................................ 5 Fractals ............................................................................................................................................. 5 Engineering Mechanics: Consideration of Linear Elastic Fracture Mechanics on the Macroscopic Scale ................................................................................................................................................. 9

The Atomic Structure and Force ................................................................................................ 13 From the Atomic Coordinates to the Laboratory Coordinates .................................................. 15 The Plastic Zone ......................................................................................................................... 22

The Physics of Fracture from the Quantum Scale to the Macro Scale ................................................................................................................. 26

A Historical Note ........................................................................................................................ 27

Insulators and Conductors ..................................................................... 27 The Structure of a Generic Crack ................................................................................................... 27

The Griffiths Equation - the “Gold Standard” ................................ 29 The Quantum/ Classical Boundary-Phonons ............................................................................. 30 Cracks are Fractal ..................................................................................................................... 31

Griffiths Equation with Phonons and Entropy ............................................................................... 34

Quantum Mechanics and the Schroedinger Equation ................ 38 Something Other than G Emerges from this Derivation ................................................................ 41

Surface Plasmons ....................................................................................................................... 44 The Crack as a Source of Electrons, Phonons and Surface Charges ........................................ 45 A Crack- A Source of Phonons and Electrons in Insulators ...................................................... 46

The Microstructure of a Crack ....................................................................................................... 48 The “Particle in The Tight Crack Tip” ...................................................................................... 50

Statistical Mechanics ................................................................................ 52 The Distribution of Broken Bonds at a Crack Front ................................................................. 54 Statistical Mechanics ................................................................................................................. 55 Another Example of Size Distributions ...................................................................................... 60

Insulators and Polarization .................................................................... 64 The Dipole Moment .................................................................................................................... 64

Fracture in Insulators ...................................................................................................................... 66 Statistical Physics of Bond Failure in Polar Insulators ............................................................. 70

Berry’s Phase .................................................................................................................................. 75 Applied Fields in Simple Geometries ......................................................................................... 77

Internal Fields ................................................................................................................................. 79

G,Γ and γ -Theory and Experiment .................................................... 84 Theoretical and Experimental Values of G .................................................................................... 85 Alkali Halides and Insulators ......................................................................................................... 85

On the many meanings of G,Γ and γ that will be encountered in this work .............................. 85 The Alkali Halide Data Base .......................................................................................................... 87

General Organization of the Analytical Work on the Alkali Halides ........................................ 99

The Fracture-Quasiparticle Connection Looking at Fracture in a New Light September 2015

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Bulk Modulus vs Polarizability Base-Line Relationship ......................................................... 101

Bulk Modulus vs 1ao3 Base-Line Relationship .......................................................................... 102

Band Gap Energy vs Ionization Energy Base-Line Relationship ............................................ 104 Band Gap Energy vs Polarizability Base-Line Relationship ................................................... 105 Band Gap Energy vs Electron Affinity Base-Line Relatioship ................................................. 106 Bulk Modulus ao vs Eionoz Base-Line Relationship ................................................................... 109 Γ vs Δ atom radius Theoretical Relationship ........................................................................... 112 Γ2 vs Eioniz Theoretical Relationship ......................................................................................... 114 Γ vs [ Ecohesion - (Eanion+Ecation)] Theoretical Relationship ........................................................ 117

Prime Objective of this Work1 The objectives of this work are to study how the solid state physics of fracture in insulators an delineate some of the fundamental influences that solid state phenomena have on a material’s fracture behavior. The fundamental concept that I hope to elucidate is to establish relationships existing between the various structural scales of matter and fracture

Fracture is complicated, wide ranging, technically challenging and ubiquitous. It is an extremely complicated physical process that takes place far from equilibrium. The physics is complicated because a crack concentrates force at quite small distances and couples to the atomic structure. Once the crack goes critical it grows quickly, while still being coupled to the atomic structure. The crack has surfaces that increase in time. In insulators the surfaces are active externally to the environment and internally to the band structure and to the solid’s polarization. The microstructure and its relationship to quantum behavior has been difficult to rationalize for various reasons. One in particular is that fractures are fractal and this, therefore, presents a substantial problem in dealing with the relationship between classical and quantum behavior Where does classical behavior end and quantum behavior begin?. Indeed a crack’s a macroscopic structure, an observable on the macro scale, is best described statistically by expected values, for example, <x> and similarly, expected values are at the heart of quantum mechanics <x>.

Another schism that invades this space is one of dealing with irreversible physical processes that are finite in contrast to ones that are defined on a continuum. Limits to integrals are not well defined, and when taking them, they

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can become discontinuous at certain scales requiring that a new set of boundary conditions be considered.

Some Preliminary Comments After preparing this work and then reading it several times I realized that I had presented two different physical viewpoints: classical and quantum. I had done this purposely , not to confuse the reader but to be honest and to include viewpoints of both the scientists and engineers. The viewpoints seemed to fall into- engineer macroscopic-classical, scientist- atomic quantum. I shouldn’t dwell too much on these classifications because they aren’t rigid. They do, however, reveal something that’s important about fracture: scaling phenomena play an important role in it. No longer will it be possible speak of the macro scale as though it is totally disconnected from the atomic scale.

Fracture’s Role in Our Lives Someone might think that something like fracture is quite far removed from

their lives. Nevertheless, perhaps they’ve seen signs of them in various places mostly when things get old and start falling apart. Maybe a sign of neglect and old age that we’d all like to ignore. I’d have to agree with them to a point, for the most part fractures aren’t too important in people’s lives, and they’re passive . However, sometimes they’re far from passive, and then they do play a role in some people’s lives, threaten them, in fact, they start earth quakes and Tsunamis. People notice them when they become threatened by them for example: when an aircraft crashes for mysterious reasons simply because a crack opened up in the fuselage, or when a ship had to be abandoned because a crack opened up in one of its bulkheads, or when one the cracked continental plates began sliding and earth quakes began to occur.

On larger scales, the earth’s surface and the seafloor remain extensively fractured but most of them are geologically quiet (the cracks are often many miles long) and planets’ surfaces and their moons are also extensively cracked showing signs of their history.

So, yes, we probably do notice fractures indirectly when we are threatened by their causative influences on our lives. By themselves, fractures usually aren’t a problem they just act as middlemen.

If someone takes the time to look carefully they’ll find evidence of fractures all around them in: street surfaces, sidewalks, buildings, dishware, windows there are many of them and most of them are associated with the degradation



of our environment. Cracking seems to be a sign of one of Nature’s aging mechanisms.

Geometry in Fracture

Fractals Fractures of all types have been found to be fractal i.e. they can be

described by fractal geometry. Fractals can be measured or created in various ways. A deterministic fractal is shown below:



A famous deterministic fractal known as the Koch “Snow Flake” is shown

above. It is constructed from triangles that have a certain size ratio relative to each other. By placing them at prescribed positions at each step of the construction a well define fractal construction can be constructed.. Scaling the sizes of the triangles is determined be the “self similarity ratio”.

A more loosely defined representation of a fractal starts by considering a space that collapses onto itself. It’s possible to consider it in this manner or that it’s an operation that an observer with a microscope makes as he/she increases the magnification as he/she makes his/her observations. In order for these observations to verify the existence of a fractal they must have the following mathematical properties:

where n= the number of objects observed at each magnification

m= the magnification Df= the fractal dimension that is related

to the self similarity ratio Additionally, the objects must be self similar (or self affine) in a manner that

can be directly observed or determined to be so mathematically. Statistical fractals are common in nature but it’s not possible to observe their dimensions directly. Such statistical fractals are frequently associated with fractures that show a wide range of scaling behavior. Fractures are simultaneously connected to: the atomic scale, the nano scale, the micro scale and the macro scale. It is for this reason that they obey fractal geometry.

nm−Df = 1



A Preliminary Peek at Fractals, Fourier Space and Wavenumber Space

Fourier space and the Fourier transform form the position<-> momentum gateway in quantum mechanics. It is what’s behind the scenes, a supporter of almost everything that quantum mechanics expresses. It’s possible (and helpful) to visualize how a fractal, (here the Serpinski carpet) might appear in wavenumber ( reciprocal, or “k” space)

Constructions such as those based on inverse length or “k” space are representations of physical operators, namely momentum, that are conjugate to length. Quite often fractal constructions seem to have a similarity to them. Shown above is a fractal known as the “Serpinski Carpet” that has eightfold symmetry that is downsized and repeated to infinity (that is, in physical terms “dilatational invariance”). The other construction is an adaption of the carpet- a Seripinski Carpet in k space. The construction is based on sixfold symmetry in which every point has a reduced copy of double diffraction ( commonly



observed in electron microscopy, where using the (000) beam for observation a a diffracted beam (say a (111 ) )is excited becoming a secondary source of diffraction spots having sixfold symmetry. To be a fractal, however, the secondary diffraction spacing mast be larger than the primary spacing.

Fractures too have been analyzed and described in k space. I’ve found that both metals and ceramics have scaling laws that form the relationship between the crack height amplitudes and wave vector, k that are all of the form:

YYo

⎛⎝⎜

⎞⎠⎟

2

= kok

⎛⎝⎜

⎞⎠⎟6−2Df

The fractal dimension determines the observed scaling behavior. Having the fracture information in the form of a spectrum has been

indispensible when it is used in terms of physics. Later in this paper it will be used to calculate the fracture entropy of a solid. Using spectra for describing macro, nano-scale and atomic phenomena facilitates extremely useful insights to be had when it comes to fracture because there’s a continuity of language.



Engineering Mechanics: Consideration of Linear Elastic Fracture Mechanics on the Macroscopic Scale

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[THE FRACTURE-QUASIPARTICLE CONNECTION LOOKING AT FRACTURE IN A NEW LIGHT]


To support the table above in order to maintain clarity:

σ = KIC

cφ

where

σ = the yield stress KIc= the fracture toughness c= the crack length



where G= the work of fracture ε= the strain E= the elastic modulus

Materials exhibit various stress-strain behavior upon loading, this is under the purview of many disciplines and has been for years. Mechanical Engineers, Materials Scientists, and others as well, study this (and have studied) material behavior because of its importance to us all.

When a solid is loaded by applying a force the bonds are stretched and a strain results. There’s reversible elastic deformation and permanent irreversible plastic deformation. It can be shown that the solid’s atomic bonds are responsible for what is observed. The energy that’s responsible for what’s observed is the strain energy density:

It is these measures of the mechanical quantities that can be related to fracture by

G=K2

E1−ν( )

K =σ c

Gc= Eε 2

The determination of these mechanical quantities takes place with equipment that makes measurements on an extremely large scale relative to the atomic scale nevertheless, the experimental outcomes are influenced by what transpires on the atomic scale.

G is also a materials constant for it is a measure of toughness. Toughness is determined by how much energy is absorbed by a material if it should foster a crack that’s under load. A material having a high toughness and a high yield strength is highly desirable. Tough materials like metals typically have a G~1000-2000Joules/meter2 with yield strengths ~ 300 GPa; G for insulators (“tough ones”) are a maximum of 50 Joules/meter^2. Metals plastically deform, stretch and absorb energy, but insulators don’t.

Loading and fracture takes place under non-equilibrium conditions non adibatically meaning that fracture takes place exchanging heat irreversibly with

ΕSED = Eε 2 = σ 2

2E= Gc= energyvolume

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its surroundings. Determining all of the energy lost during fracture can be a complicated affair. The input is the strain energy density introduced by an applied external load to a notched sample of material. The outputs are heat, sound, light, vibrations, fracture surfaces, disruption of the atomic and meta-atomic microstructures. Energy is lost in the outputs.

Everything begins with an applied load, a material, support and a notch either prepared or not. An example of a typical crack (shown below) is embedded in the microstructure and is acted upon by the applied load.

This was an example of a brittle inclusion particles, MnSiO4 , in a welded

ductile steel matrix. This is an excellent example of ductile/brittle behavior that is easy to remember.

The crack initiated and failed below the yield stress of the matrix. The matrix around it held it in place until it too failed at a higher stress starting with the particle as a nucleus. The Griffith equation (derived later) can be used to obtain the yield stress. The stress on the particle at fracture was:

σ *inclusion =

Eγπc

γ = G = K

2

E1−ν( ) ! 10

5( )2 0.72x105

< c >! 0.5 µm ! 1.27x10−4inches σ * = 9,400 psi = 135.8 GPa



This was the stress in the ductile matrix that was related to the brittle fracture of the particle. The strain in the matrix exceeded the fracture strain in the brittle inclusion.

The Atomic Structure and Force Since “atomism” is behind all natural phenomena it seems pertinent to

begin by considering a force acting on the atomic scale. Energy vs distance relationships can be used to determine the cohesive energy of a great many of atomic structures such as ionic solids i.e. alkali halides . This equation is valuable since the force displacement relationship for alkali halides can be determined by differentiating the energy curve.

F(r) = ∂U∂r

= N 2αr

− Ze− rρ

ρ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Shown below is a theoretical force distance curve computed for NaCl an alkali halide. This was a simple well known structure that will be a member of the set of materials that will be used in this study.

Figure 1 Identity Spacing/Size Angstroms Comments Bohr radius 0.529177 A fundamental cut-off

spacing

U r( ) = N zλe− rρ − αe2

r⎛

⎝⎜⎞

⎠⎟

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Max in Force Curve 0.4987 The spacing at the force maximum

Atom Diameter 0.795 The atom diameter taken from ad Na /2+ Cl/2

Nearest Neighbor Spacing

2.820 The NN distance in the NaCl lattice

Lattice Spacing 5.426 The spacing of the NaCl lattice

Atom Diameter Na 1.9 The (uncoordinated) diameter of the Na atom

Atom Diameter Cl 0.79 The (uncoordinated) diameter of the Cl atom

Evidently the force, as calculated, acts on NaCl but it appears that there

needs to be a statement regarding how it does so because it’s not possible to identify a unique loading point with any confidence.

There are also some other considerations that need to be taken into account if a more realistic picture of load transfer between the macro and atomic scale is to be made. The arguments, as presented, are appealing and, for the most part, are correct but they too are incomplete. Since:

1. The presentation is in concepts and not in terms of materials-for example: if that is a force distance curve actually applies to atoms how did they prepare and hold the specimen?

2. Is the model used for E(x) the only one possible? Why use one over the other. I chose the one that I did because I intend to study the alkali halides in this work and I thought that it was correct and relevant.

However, there are others • The Morse Potential • The 6 12 Potential • The Rittner Potential • The V-S-4 Potential • The Modified Rittner Potential • The Gohel-Trivedi Potential • The V-S-3 Potential



And they all have their advantages/disadvantages. Some stem from educated guesses followed by empirical successes with scientific justifications for their success. Some are just accurate, tested, have been in use for many years and are easy to use. However these potential energy curves can be useful and, even though it might be difficult to get a reasonable answer without a struggle, there should always be a struggle when it comes to doing good science.

From the Atomic Coordinates to the Laboratory Coordinates

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This is a load displacement curve (in blue) that is observed

Q The atomic force-distance relationship doesn’t look anything like the load

displacement curve measured in the laboratory as shown above - how are they related to one another?

Part of the problem is associated with the fact that the force distance is for a single atom whereas in the laboratory the load is applied over the entire cross section. You can make an estimate how many atoms this is ~ for a 1 cm2 . Each atom has it’s own mechanics to follow. The atoms get “out of step “ (lose phase coherence) and so the lab test measures something of an ensemble average of things.

Rationalization of Classical Coordinates In attempt to make things more understandable and somewhat relevant I’ll

consider the atomic forces acting along a line in the plane of the test. I consider the coordinates along the line as being fixed and move the force curve along the line and in so doing form a convolution integral i.e.

F = kx = mω 2x

F x − xo( )Δ x( )xo=0

xm

∫ dx = mω 2x2 = E



Setting things up this way makes the right hand side of the equation equal to the energy (a constant) which makes everything tidy since energy is path independent. I’ d like to know what Δ is all about because that is what scales us from the atomic to the laboratory coordinates. By making the integration over large enough distances I’m doing just that.

I’d like everything to be neat and tidy but it’s not. I wouldn’t have expected it to be simple, nevertheless, although rudimentary, the mathematical approach is limited and probably is too much of an oversimplification. Shown below is the function Δ for ω and x.. It appears to be a spectrum (of sorts) along the ω =0 axis. As yet it is only an illustration of an interesting idea. The idea being that the force- distance curve is an incomplete description when working in terms of the laboratory coordinates.

It appears that the application of the load isn’t very well defined and it needs further definition. However, it’s not possible to specify the application of a load at a point on the atomic scale. Not being able to determine exactly where the load is ap

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plied is typical of the shortcomings of classical mechanics. Enter quantum mechanics that says ...

• “it’s just not possible to determine the load point on the atomic scale. Even if you reduce the scale somewhat you can only obtain statistical measures such as the expected value of force <F> and its standard deviation <σforce>.

• Where you intend to apply the load will depend on the Uncertainty Principle. Which says, in so many words, no one can be certain of where the load point is

So the force-distance curve gave us a feeling of confidence and a sense of

reality but it lacks credibility, it cannot be verified because it’s only a theoretical construct.

Mechanical and Electromagnetic Forces The force- distance curve doesn’t provide a number of things which need to

be seriously considered if we are to speak in macroscopic terms . I’m sure that as we work to encompass an even wider range of phenomena in mechanics it still won’t be enough, because it never is.

Here are some more realizations of reality:

1. If a load is applied to a material (ductile) sooner or later it will either bend permanently or break. Strain is evident in the form of plastic deformation and accompanies fracture

2. The force-distance curve shown for NaCl in Figure 1 doesn’t portray a realistic load displacement curve because the curve usually does some irreversible and unpredictable things that are non-linear.

3. Permanent strain and plastic deformation occurs beyond the elastic limit

4. An electron diffraction pattern of a metal under load (see #3 below ) shows that the crystal structure is intact with some additional streaking coming from another phase that’s forming and deformation that’s occurring along preferential planes.

5. In order to handle a number of small strains the approach is to use terms that are linear in strain and adopt the edict



“add small strains” ε i i

N

∑ but this can only be applied to

“small strains”. 6. Something that is important to remember: strain

determines what is happening to the microstructural coordinates of a material.

7. The load displacement behavior is not reversible meaning

that if you take a specimen and load it to particular point of load and strain and stop, record everything, then apply a new load moving to a new strain point, and record everything, then, it’s not possible to return to where you started. A path in load- strain space isn’t reversible. Basically, you did some work and couldn’t recover it.

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This is an interpretation of the above electron diffraction pattern.

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The Plastic Zone



Visualization of the macro constituents acting during the formation of a crack. This should put into perspective in relation to the ductile behavior of metals and to the brittle behavior of ceramics.

Two equations have been found, one applies to steels that exhibit ductile fracture by means of inclusions that are ~0,1 µ in diameter and the other applies to brittle fracture of insulators (ceramics).

Some of the relevant experimental work that has been done to date is summarized in the following table. Both equations are similar and show that the work of fracture G is related to some distance that dictates the fracture behavior in steels or ceramics. Ceramics G values are ~ 1000-5000 ergs/cm^2 and steels are about 20 times higher than that. The “characteristic length” in ceramics is in the 5-50 A range and the inclusion spacing on the fracture surface of steels is~1-10µ . So it would appear that by means of empirical observations that a (generic, non-descript) spacing of some kind is responsible for the observed G. These distances emerge due to statistical sampling at the crack tip and therefore they are best determined to be for steels- an average value:

x = Nsx2e−Nsx

2

∫ dx

and their statistical properties can be measured on a fracture surface.

Nevertheless, these experimental observations do indeed work in certain applications but they are quite a way from being incorporated into or establishing some new physical laws because they have not shown to be associated with the canonical variables of physics. In that respect the canonical coordinates are momentum and position.

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Material Conductor/Non-Conductor

Fracture Equation Structural Terms

Energy Terms

Steels

Ductile Fracture (Conductor)

Passoja-Hill

EDuctile = kεvλ

εV ! 5 ⋅10

13 ergscm3

The energy term εV could be determined from the fracture surface measurements

λ = directly measured mean inclusion spacing on a fracture surface

Mechanical measurements-

The area under the stress strain curve up to fracture was measured and it compared favorably with the values as determined from the fracture measurements

εV ! 1− 5 ⋅10

13 ergscm3

Ceramics

Brittle Fracture (non-Conductor)

Mackin-Mecholsky-Passoja

G = 12ED*ao

D*= the fractal dimension increment

E= the elastic modulus

a is a “characteristic length”

5A ≤ ao ≤ 50A

ao = not directly measured but inferred from experimental data

Equivalent energy density would be related to the strain energy density at fracture but it wasn’t measured



Specimen Dynamics of Fracture in a Nanoshell

1. The impulse response of the specimen-detector assembly before fracture

2. Representation of the collapse of a transverse wave due to fracture

3. Emission of longitudinal compression waves due to fracture

4. Reflection of emitted waves off the ends of the specimen and their interference with other reflected waves.

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The Physics of Fracture from the Quantum Scale to the Macro Scale

There’s an enormous amount of published information regarding the fracture of solids. Fracture phenomena displays wonderfully complicated behavior and to engage an analysis that tries to cover so many things is probably not a good idea. A limited view might help this problem but probably the best approach is a flexible one- one that uses what knowledge is available and realizes this as a limitation.

I think that it’s safe to say that even today, after all the work that’s been done on fracture the elusive relationship between a solid’s structure and the fracture behavior is only partly known. For example, at present, starting from first principles, it’s not possible to construct the fracture surface of 6061T6 aluminum. This is a practical problem of interest because this alloy is used in military aircraft. Given that task it would also be useful to predict the fracture toughness, and the behavior of this alloy in a fatigue test. To put it another way:

There’s an intelligent (hidden) beneficent being in all of us who might say:

• If you plan to fly in that aircraft will the wings fail when you’re aloft? I mean, honestly, the very wings on the aircraft that you’re planning to fly in. I’m not interested in any statistical data (quite obviously) for an explanation because you should know what’s going to happen for sure.

• Find someone could monitor the most important things, the ones that would most likely cause the plane to crash when you’re up!

Even if the wings were taken off the plane and scrutinized it would not be

possible to determine what would happen ...and if we were to use a microscope that allowed us to see things at the atomic level we’d know even less about everything (Heisenberg’s Uncertainty Principle) and would have to spend more and more time searching for some meaningful observations which may or may not be meaningful.

So the answers to problems such as the “Fracture Problem” certainly can’t be determined by a limited number of observations made at high



magnifications. Furthermore, observational “noise” enters analyses of all physical problems at many different scales. The noise usually is intrinsic and can’t be removed in a simple manner.

At present, these things can’t be done, but scientists and engineers are trying to find these things out because they are important problems.

A Historical Note Was improved toughness ever an important technological development in

history? Yes, Damascus steel swords come to mind. The metallurgy started in India and migrated to the Middle East where the fabrication of swords having superior hardness and toughness were produced. They had a decisive effect in many battles with the European conquerors of the Crusades.

Insulators and Conductors This work will be concentrated on insulators and alkali halides (which also

are insulators but have some wonderful and interesting properties in their own right). I chose to study insulators because of their extremely low electrical conductivity. Metals fail and absorb a great deal of energy and ehhibit ductile fracture. Any bond failures in metals are healed by the free electrons. In comparison a metal’s strain to failure makes insulators look anemic in comparison. The other property that interested me was that insulators are brittle and their work of fracture are notoriously low.

As someone with a background in solid state physics it seemed to me that the differences in the solid state properties could help to explain the fracture properties of insulators.

The Structure of a Generic Crack A crack can be described as geometric object that is a surface that forms in a

physical continuum and reacts with the continuum in a number of ways. The crack owes its existence to the mechanical forces that created it. However, with a first glimpse a crack isn’t really isn’t a “thing”, it’s not a physical object because it has no mass.

The mechanics that created the crack operates on scales that can start with the atomic and can range upward to kilometers.

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That’s really not much of a surprise once you stop and think how fracture happens. The entire process can be pictured like this: on loading the mechanical state of an object (yet to obtain a propagating crack) is changed so that the stored elastic strain energy (from bending, shock loading etc.) reaches a critical value at some crack somewhere. This begins a cascade of events first damaging things on a local atomic scale and, as the crack grows, it soon envelops larger and larger scales. If the crack size is sub critical it may advance but not freely propagate, but should it ultimately reach criticality, failure and the formation of the fracture surface will occur. The fractal relationship is present since the crack is connected on the micro and the atomic scale by means of the crack front.

The surfaces that form must ultimately separate on the atomic scale after deformation with distributed bond breaking in the vicinity if the crack tip damage zone. Indeed the nano scale ↔ atomic scale interaction is an essential part of the fracture process.



An evaluation of the energies, momenta and displacements is in order but with a realistic set of eigenfunctions including eigenstates that includes everything involved with a fracture is daunting . One of the major problems that occur at the onset of fracture is that only a small number of atoms (compared to the entire volume containing the crack) are responsible for the crack’s behavior, in other words, a single bond, or a representative of a single bond, does not explain things but neither does a fairly large number of them. Mathematically there’s a problem of convergence. The analytical side of things is more like working with a DFT rather than a Fourier series.

The crack samples ~ 1016 atoms at the crack tip for every atomic extension it makes. Nevertheless, with some critical thought and by approaching fracture as being composed of a statistical ensemble of atoms followed by quantum mechanics and statistical mechanics it’s possible to make some realistic theoretical statements. Nevertheless a crack is a statistical representation of a geometric object embedded in a solid. It’s a finite thing that lends itself to counting what’s there in order to gain its secrets.

In developing theories of fracture that encompass a broad range of materials and phenomena fracto-emission, light emission, acoustic emission etc by making the direct observation of the fracture(ed) object and recording its mechanical properties often ends up short when it is attempted to include all the contributions of energy. That’s because the list of energy contributions is usually incomplete. Furthermore, evaluations are usually done after the fact by studying the fracture surfaces, the “burned out skeletons” of fracture. For example, in insulators a crack and its surfaces generate fields and dipoles that perturb the solid in a fundamental way. The fields disappear once the fracture is completed. Conductors and insulators fracture quite differently and they must be handled in a different manner. This type of behavior in either case isn’t included in the classic Griffiths Equation. But the Griffiths Equation is the “Gold Standard” and it’s the best place to start.

The Griffiths Equation - the “Gold Standard” Historically, the Griffiths equation forms the “Gold Standard” in fracture

mechanics It has withstood the test of time and even with the fruits coming from LEFM and the measurements of the toughness, KIc it still creates a valid scientific framework for understanding fracture.

Historically, fracture has relied on the Griffiths equation for its description. It has done well for the fracture of homogeneous materials. Fracture toughness and linear elastic fracture mechanics has superseded and improved on Griffiths

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considerably but it’s still possible to show that the fundamentals of the Griffiths are still part of LEFM.

I’m sure that someone has incorporated entropy into fracture equations before but the theory doesn’t seem to be taught this way. So, by using the Gibbs free energy in the standard manner, stating that there’s a balance between the surface created (a positive term) and the stored strain energy density lost (a negative term) that is associated with fracture it’s possible to write

(1 Under the assumption (here) that temperature is constant-(which it’s not!). The entropy is included, otherwise fracture is presented as a reversible process which isn’t realistic. The structural terms are included

The Quantum/ Classical Boundary-Phonons A crack is initiated, it propagates and then it stops usually when the part that

contains it comes apart. Consider an interval when the crack is moving at steady state velocity (some fraction of the speed of sound). In our case It’s an insulator so most of it’s behavior is elastic with a logic of the type “it’s either fractured or it’s not”, The incremental steps leading to fracture are in the Angstrom range so there’s nothing like plastic flow that would change the fracture’s logic. To be sure, quantized elastic waves of the lattice, phonons, are part of this process. As the crack generates them it does so incoherently but it can also do things coherently if the parasitic resonances of the fixture and other outside noise sources are suppressed. The crack acts as a phonon source that isn’t always keeping things in phase, instead the phonons wave shapes are shifted in time and interfere destructively with each other. Sometimes, that is. When there’s coherency at the source, there’s a potential for the existence of a phonon having a quantum number n. That’s related to the energy of the system:

Where υ is ≅ the Debye frequency ≅ 1013 Hz.

Like any other wave a phonon has a frequency and amplitude but it’s just quantized in terms of n. The running crack emits elastic waves of the form and it displaces a volume element a distance u of the material. There is ½ kinetic energy and ½ potential energy that is in the volume element. The kinetic energy density is

ΔEf = −ΔEv + ΔEa − TΔS( )

E = hω n + 12

⎛⎝⎜

⎞⎠⎟

u = uo cos Kx( )cos ωt( )



when it’s time averaged this becomes:

Which somewhat justifies the fact that a classical elastic waves are related to phonons that are quantized.

where the x terms are the amplitudes squared. This represents a boundary between classical and quantum mechanical behavior.

Cracks are Fractal The evidence has been accumulating cracks can be described by fractal

geometry. Whether the size of mountains and coastlines or as small as what cannot be seen by the unaided eye, “cracks are fractals” as the scientists would say.

There are many ways to describe the geometry of cracks and it certainly isn’t simple, so to have arrived at a time when we can have a precise and useful description of a crack’s geometry is certainly an enormous improvement of what we had before .

A crack spectrum on the macroscopic scale has a characteristic shape- it has a high amplitude low frequency with a high frequency low amplitude shape meaning that it resembles an echo, but one that, as time goes on, you would hear the whispers of something like Alvin the chipmunk.

KEρ =12ρ ∂u

∂t⎛⎝⎜

⎞⎠⎟2

KEρ = !∫

12ρ ∂u

∂t⎛⎝⎜

⎞⎠⎟2

dV = 14ρVω 2uo

2 sin2 ωt( )

18ρVω 2uo

2 = 12n + 1

2⎛⎝⎜

⎞⎠⎟ !ω

x2 = hmω

n + 12

⎛⎝⎜

⎞⎠⎟ = xq

2n + 1

2⎛⎝⎜

⎞⎠⎟

x2

x 2 = n + 12

⎛⎝⎜

⎞⎠⎟

32



This figure has some 1dimensional traces of potential crack paths.

The next figure reveals a slightly more sophisticated representation of “crack path amplitudes”

The lower figure shows a crack and some of the potential paths it could have

taken instead of the one it took. There’s a distribution function to the right of the crack showing a continuous distribution of crack paths that the crack could have taken.

This is a familiar problem in statistics and in Statistical Mechanics . The crack has the potential of making choices of where it ends up. There are N virtual cracks and k places to put them. The probability distribution is related to the entropy the probability distribution is shown above.



There are N virtual cracks

that fill a proportion of boxes in phase space. The trajectories of the boxes in phase space reveal the dynamics of the crack’s evolution (and virtual cracks) mapped on a set of canonical coordinates x and p. The virtual crack paths were drawn as random sets of straight lines. By inspection it’s possible to see the relationship between the crack’s tortuosity and entropy.

It is entropy that will allow us to scale between the atomic and the macroscopic scales. The number of paths that someone might take when traveling between points A and B is huge if someone cared to scrutinize every possibility by using an ultra high resolution microscope. At lower magnification some of the paths disappear (not really!) so choices become more limited, however, “if your stride widens” so that you can miss the gaps ( or use a ruler without a fine spacing on it so that you can shorten the path) . There’s entropy that goes along with this, so our walking along the paths needs to be accounted for in a different manner than one that’s dependent upon magnification.

There are a few fundamentally important ideas that emerge from arguments such as these:

1. Entropy is of fundamental importance in understanding physical behavior in general. Bridging the gap between the atomic and the macro scales would require that special statistical considerations be used to determine which features are the important ones in the set of the countably infinite lineaments that exist on the atomic scale.

2. Quantum mechanics doesn’t require us to see things like electrons in order to know that they are everywhere. In fact it negates the idea of calling a particle an “it”.

3. It is of fundamental importance that magnification be taken from it’s place of disregard and be established as an essential idea, hiding, unrecognized, behind all of our concepts of large and small. It is magnification that puts us in the driver’s seat as observers. I think that it’s about time

S = pi ln pii=1

k

∑

WN p1,p2 ...pk( ) = N!p1N( )! p2N( )!... pkN( )!

pk

34



that we realize that magnification might be as important as entropy is.

Griffiths Equation with Phonons and Entropy By using some of the ideas above it’s possible to make (superficial perhaps,

but it provides a valuable learning experience). Everything’s the same as it has been with the exception of entropy.

ΔG = − 4π3r3 σ 2

2E⎛⎝⎜

⎞⎠⎟+ 4πr2γ −TΔS

This leads to the standard equation for the critical crack size.

The out-of plane amplitudes that a crack has-i.e. its “roughness” is related to work of fracture, G for a host of materials-including ceramics, steels aluminum etc. This “roughness” can be determined analytically and is known as the fractal dimension.

The spectral method of measurement is able to characterize the out of plane fluctuations of the cracks. The fluctuations are related to the entropy since a crack having high toughness would have a high entropy and a crack that was flat would have low toughness and would have low entropy. These attributes of the fracture geometry can be used to evaluate the entropy in the following manner: as derived previously, in the quantum realm, the relationship between the y amplitudes and the quanta, n is

yoy

⎛⎝⎜

⎞⎠⎟

2

= n + 12

⎛⎝⎜

⎞⎠⎟

K, the wavevector is related to 1/x an inverse distance- it is an analytical measurement and not related to the momentum so that

koxkx

⎛⎝⎜

⎞⎠⎟

6−2Df

=

1xo1x

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

6−2Df

= xxo

⎛⎝⎜

⎞⎠⎟

6−2Df



The entropy is S = P[n]log[P[n]]

n∑

and using the above equations it is

S = n + 12

⎛⎝⎜

⎞⎠⎟

xxo

⎛⎝⎜

⎞⎠⎟

6−2Df⎛

⎝⎜

⎞

⎠⎟ log n + 1

2⎛⎝⎜

⎞⎠⎟ +

xxo

⎛⎝⎜

⎞⎠⎟

6−2Df⎛

⎝⎜

⎞

⎠⎟

n∑

Since this is an equation that is based on a quantized harmonic oscillator, the terms are summed over n instead of integrated.

Shown below is a 3D plot of the fractal Dimension, the number of quanta and the energy (in ergs on the y avis)

36



A High Value of the Work of Fracture is Dependent on the Presence of High Frequency Terms in the Fracture’s Geometry

Electrical Engineers know that there can be a huge amount of power in a seemingly insignificant electrical signal- the power is in much of low amplitude high frequency components. So too it might be that high toughness resides in the low amplitude high frequency spectral components.

A Surface and its Conjugate A glimpse of a fracture might give the impression that the very things that

created it were in a state of chaos: brittle fracture doesn’t look chaotic to the eye but it does at high magnification. That a fracture in a “crystalline” material shouldn’t look chaotic is just a matter of observation. It’s come to our knowledge that the very things that make fracture appear to be chaotic to the eye are also operable on the grand scale like the cracks that form and look chaotic in a glacier.

Unifying the observations such as those above is what fractographers do. They work with both quantitative and qualitative information hoping to find some commonality among their observations. As with any other science classification of structural details is where everyone begins.

Here are a few items involving fracture that are concerned with the nature of the structure of fracture surfaces.

Q: How well do the matching surfaces fit together after they’ve been fractured?

A: It’s been found that they don’t. It’s not a good match at all Q: How does this happen? A: Fracture takes place on different scales. But structural details might

not match on a large scale just by adding up many small things. Matching might occur coincidentally on a large scale; with many infinitesimal displacements coming together and making up the difference in a large scale displacement. The problem with such an assessment is associated with phase matching. There doesn’t appear to be coherent reinforcement of the small scale deformation and displacement on the atomic scale that can be readily associated with the macroscopic scale. Fractals has presented the first evidence of scaling of fracture that has the

possibility of bringing the different scales together. The autocorrelation



function of the profile shows nothing of merit. The cross spectrum and the cross correlation function show spectral lines and some periodic structure. This would indicate that both fractures contain information of interest. Once they’re separated it’s not possible to find the right conditions (including he right energy too) to put them back together with perfect matching.

The figures below show some aspects of this problem in an extremely simple way:

38



This is where we all begin; the fracture occurs and two pieces emerge

(normally) and additional surfaces- the fracture surfaces. The newly fractured surfaces are somehow related to each other. Attempting to fit them back together isn’t possible. It’s possible to add energy and rejoin them-weld them- but that would not satisfy our conditions.

There are several ways through the confusion but most of them don’t include fractography as their basis. This is not because fractography isn’t useful it’s only part of the answer. The frontier lies at the atomic level.

Quantum Mechanics and the Schroedinger Equation

Rather than present a lengthy exposition on quantum mechanics I’ll begin with an abbreviated one that I believe is relevant to fracture. I discovered this (path) relationship after many tries, some successful, some not. I have the wave function etc and can compute various quantities etc but I’d rather present something that’s useful and relevant to this work on fracture. Incidentally, some of this might appear to be obvious but I didn’t find it to be so. For example, why not use an altered version of the harmonic oscillator? I tried many of those.



Some worked, some didn’t but some of the successful ones presented unrealistic results. Some of my ruminations...

How about another totally different type of potential energy?

Which one applies to fracture? Fracture’s not a harmonic oscillator having “constants of the motion” so maybe it’s not the SWE at all. Fracture has a finite lifetime and is irreversible because you can’t fit the pieces back together after it’s happened. It’s an irreversible work phenomena. I finally found something, made a choice, and it seemed to be helpful. I’m not suggesting that it’s the only choice nor am I suggesting that this is the only correct choice because fracture’s too complex and others might work just as well.2 Is there any Connection to be Found Between Quantum Mechanics and the Griffiths Equation?

Yes, but more work is needed. Here is what I’ve found so far.

The Shroedinger Equation The Harmonic Oscillator has to be reconsidered in terms of G The term for the compliance in the harmonic oscillator should be changed in the following way:

The usual familiar terms involving k, the compliance are:

F = kx

ω = km

k = mω 2 (11

2 Γ is a provisional term for G that will be used throughout this work. The work will attempt to establish a scientific basis for Γ and, at some juncture prove that it is equal to G.

40



dividing the compliance k the numerator by area and multiplying by length squared leaves everything where it stands except

mω 2 αx2 mω 2 α

x2 x2 = Γx2

yields the term, Γ which can be used in the following manner.

Constructing a Hamiltonian that consists of a Coulomb term and a harmonic oscillator term establishes a useful basis and allows Γ to be found.1

h2

2m∂2ψ∂x2

− e2

xψ + Γx2ψ +mω 2x2 = Eψ

Γ = e2

4x3+mω 2

Γ* = 27e2

32aBohr3 +mω 2

and this results in an equation for Γ .

I’ll be using the length form of these equations in this work ie.

Γ x( ) = e2

Ω+mω 2



The Energy

The energy curve has a shallow minimum (for arbitrary constants) that appears

Figure 3 The plot of energy vs distance for the Hamiltonian that was used.

An Equation for G Emerges from this Derivation but there’s Something Else, Something Unexpected

I have found that the equation

mω 2 = e2

r 3

is associated with quasiparticles. Everything lines up, the units, the order of magnitudes but there’s nothing that is associated with surfaces when it comes to quasiparticles and vice versa. Without too much concern I’ll have to hypothesize that quasiparticles are indeed part of fracture but they just don’t stay around very long, they escape leaving the fracture behind. The work of fracture what this term is all about. Not being able to observe the fundamental particles of Nature in a scattering experiment is common in Physics. The best example of this is the electron; after all of these years we still can’t observe an electron. To put everything simply: the quasiparticles (are at least part) of the fracture process but they have a short lifetime and they scatter off of various

42



things in their environment phonons etc. They lose coherence with their source quickly and just cease to exist.

After going over some of the numbers in the equations above I’d have to say that they’re what the fracture story is all about. But rather than have you take this on faith (which I never do when it comes to Science). I’ll continue on in this work and reveal some supporting evidence.

iQuasiparticles

Plasmons-are based on the collective oscillations of electrons

Plasmons are oscillations of the free electron gas in conductors but are also present in semi conductors and insulators. The ion cores do not participate in this process due to the fact that they are more massive having more inertia and therefore they aren’t able to follow the electrons’ movements. The lattice ions are screened by the electrons to maintain stability but the ion masses are quite large compared to the electrons. They are free to move about but over very small distances. This is a classical calculation about plasmons using the ion cores for the masses.



This is based on the oscillations of the ion cores and it’s too low.

To use this for plasmons and electrons in an alkali halide NaCl:

o set the terms equal to each other o use the electron mass o calculate volume/electron in a unit cell o Find the frequency of an equivalent plasmon o Divide by the volume of the unit cell

F = ma

m d 2xdt 2 = −eEe− iωt

mawω2 = np/ve

2

ω =np/ve

2

maw

Using NaCl as an examplemaw = 4.683⋅10−23

np/v =4

5.62 ⋅10−8( )3 = 2.53⋅1022

ω = 2.53⋅1022 2.307 ⋅10−19

4.683⋅10−23

ω = 1.054 ⋅1013

Γ = e2

ao3 +mωD

2 vs Difference in first Ionization Energy

E = hω = 0.044eV

meω2 = e

2

Ω

ω = e2nvme

= 1.182 ⋅1015Hz

E = hω = 4.888eV

ΓNaCl = 2444ergscm2

44



There’s an additional, extremely interesting aspect to these expressions. These expressions are identical to the ones used to describe quasiparticles. The equation for :

(18 which describes the classical behavior of quasi particles eg. plasmons. The following figure expresses some general aspects of fracture and the possibility of its particle like behavior.

Surface Plasmons During fracturing, the fracture surface atoms lose their restoring Coulomb

field out of the plane. They are driven to oscillate by the fracture’s elastic waves, its phonons. The surface atoms no longer have a 3 dimensional support around them and, lacking that, their dipole fields become expanded out of the fracture plane. In this manner,the dipole ocsillate and generate a surface plasmon.

Γ

Γ = meω2 + e2nv



Fracture, Quasi Particle Behavior or both? Bringing quasiparticles into fracture, needs justification and some

experimental proof. I maintain that our familiarity with fracture is only a temporary and partial representation of what fracture is all about. A crack’s behavior in real time created the fracture surface but it’s probably only part of the story.

Quantum particles and their properties is the purview of quantum mechanics. Certainly, quantum mechanics has been very successful in a great number of scientific endeavors. As I suggested at the beginning of this work it’s sometimes not very easy to transcribe classical mechanics and into the “Quantum Language” . In particular, phenomena of higher dimensions are difficult to handle by means of the Shroedinger Equation. We appreciate that the fracture puzzle has several pieces that must be assembled with care starting with the most important ones, energy and geometry.

The Crack as a Source of Electrons, Phonons and Surface Charges

A propagating crack in insulators generates acoustic waves, stress waves, photons, fractured atomic bonds which cause electrons to enter the conduction band and holes to appear in the valence band and, perhaps, quasiparticles. There will be electrons having undefined states ( formed from fractured states)

46



within the band itself. The states in the band pose a problem because their numbers and type are not well defined.

As it propagates, the crack develops an effective mass. This is easy to understand in the case of insulators. Being unstable, the crack breaks bonds. Since this is an insulator any disrupted charges will show up on the crack’s surfaces . These charges interact with other lattice charges by means of a Coulomb interaction. Elastic loading displaces the lattice atoms from their equilibrium sites thus generating internal fields. This will cause an internal charge flow (if possible) so as to eliminate the field. The crack will now interact with its environment and become more massive.

This is a phonon that has just been initiated by the crack it is portrayed in a different color in order to distinguish it as being different. It’s easy to see that it has a wavelike character due to its amplitude but it also has a different solid state structure too.

A Crack- A Source of Phonons and Electrons in Insulators

Crack Direction

Phonon Direction



The involvement of a crack occurs by interactions of electrons with the uncompensated broken bonds on the crack’s surfaces. Then there are interactions with the polarization induced charged field from the uncompensated internal fields of the host material. But all of this stems from the

E r( ) = EC + EPE + EPC + EPc + E

C−a*c + E

C−c*c

48



fracture’s influence on everything and it’s not just the particles alone that behave in this manner-it’s the particles and, addition, the crack that’s responsible for this behavior. The crack is therefore a participant. It is an intermediary for several energy transfers.

The Microstructure of a Crack Crack’s aren’t “things” but they do develop an effective mass.This is

something that they acquire when they begin to propagate and become influenced by the complex microstructure. They have trouble growing easily and have a tendency to oscillate and to leave rough surfaces behind them.

There are scales where a crack’s onset of roughness falls into patterns. Such patterns can depend on a material, the sample geometry and loading geometry. These are regions where a crack becomes unstable relative to a plane. The ratio of the plastic zone size relative to the microstructural unit size can be related to these transitions.

Another destabilizing influence on the crack is related to the fact that the fracture ends up with two surfaces-(generally) at every point of its propagation that is, the crack leaves two surfaces behind (it’s actually just one surface until separation occurs) that influence its movements.

This is best described by the “Mexican Sombrero” example shown below. Consider an atom vibrating as a harmonic oscillator: sooner or later it has to

be in one or the other potential wells on either side of the fracture path when the two surfaces separate. It’s really not just one atom but all the atoms will undergo similar movements under the influence of the crack’s strain energy density field. There soon will be two different fields acting separately. In this case there would be two harmonic oscillator potentials that could be identified with the crack but joined together as the crack is propagating.



These potential energy curves are to be thought of as looking directly at the crack tip. The dancing ball is to be thought of as an atom that has to make a decision about which one of the harmonic oscillator PE curves it will be in once the crack has passed. This is part of a calculation known as the “anharmonic oscillator”.

50



The “Particle in The Tight Crack Tip” An extremely narrow space, such as the one that exists at the tip of a crack at

the onset of propagation, should give rise to quantum resonances. It is known that confinement of energy on the atomic scale gives rise to such resonances. The simplest one to envision is the particle in a box. Shown below is a depiction of this idea with the energy levels shifted in order to account for the crack’s being wedge shaped.

The resonance is a source of an electric field. Since this is an insulator, the crack sets up a field that ranges outward into the solid and polarizes the atoms throughout it. The solid attempts to eliminate the field ...I’ll leave this where it is because there’s more on this later.



Phonon Scattering Off of the Crack Tip The crack generates phonons as it moves and interacts with its environment.

The emitted phonons can react both globally and locally in the following way 1. Because the object that contains them is finite and their

lifetime is finite they make a limited number of traversals within the specimen before they are dissipated

2. During their lifetime they interact with electrons and holes that have resulted from fracture and reside, somewhat immobile, on the surfaces of the crack. Any field that the fracture creates in the volume of the solid emerges on the solid’s surfaces.

3. Complex electron->electric field->phonon->strain occur during fracture causing the crack tip to undergo some significant changes. Charge oscillations and strain oscillations similar to Friedel oscillations form and alter the fundamental structure of the crack tip.

52



Statistical Mechanics The use of statistical mechanics to determine the fate of bonds in the vicinity

of the crack tip is based on three assumptions: • that the number of bonds that fail at any one instant are

small, • that their lifetime out of equilibrium is short • that they lose their identity in the ensemble after they’ve

failed.

A crack’s influence takes place a small distance ahead of it. There bonds are broken or, alternatively, the material is overloaded below the applied stress due to the crack tip singularity. A unit fracture process that is large relative to the atomic spacing would be dependent on the averaging of statistical fluctuations on the atomic scale.

There must be a transition region where some of the bonds have started to fail but have not done so as yet. In this region it makes no sense to speak of a single bond that is responsible for the fracture “event” any more than it should be called an “event” because these things are under the purview of statistical mechanics and quantum mechanics where such phenomena are statistical in nature.



Analyzing a crack is usually done with a certain scale in mind (as shown above). Should it be deemed necessary the volume of analysis can be adjusted.

It’s very useful to establish the number of atoms that a straight crack touches in a bar of silicon that’s 10 centimeters wide with (as above) an analytical area 50 ao silicon: Item Number of atoms Plane of silicon 6.78x1017

Crack tip 7.13x1015

The mean atomic spacing at the crack front is ~ 1.184 A. This is presented as a reference.

Essential Aspects of Bond Failures

Lattice Construction Atoms , at first are isolated but experience interactions which change the

symmetries and energies of their wave functions. As more atoms become involved broadening and overlap of the wave functions occurs. Due to filling of the states the interatomic distances decrease. At a certain point it is energetically favorable to have the entire process emerge as a condensate- a periodic soild. This indicates a well defined lattice spacing and a build-up of the crystalline structure. With a further decrease in this distance, energy bands form. The presence of periodicity gives rise to energy gaps which interfere with electron propagation

Fracture a Deconstruction Deconstruction of a crystal is a far more complicated process because there

are far more choices that can be made by an atom in order to make the deconstruction- in other words the entropy is quite high in this process. The figure below summarizes the

54



The Distribution of Broken Bonds at a Crack Front The energy of bond failure will be discussed later in this work. Until then and particularly in this section, failed bonds are simply considered to be no bonds or bonds. A distribution (sequence) along a line taken perpendicular to the fracture plane and perpendicular to the crack front might appear as Broken Bonds............ Broken/Unbroken Bonds..........Unbroken Bonds 00000000000000101011011110111101110101011101111111111

In the following figure I’ve included a distribution of stretched bonds in order to appreciate the complexity of the statistics at the crack front.



The width of the transition region will be considered to be a result of a

balance between crack speed and the bond failure rate. This has never been determined so the width that I used was based on an assumption.

The bluish part of the figure below is where everything begins (no bonds broken) and the reddish area is where all the bonds are broken. For this reason it’s the cumulative probability distribution that describes the distribution of bond breaking.

There are two different ways to approach this problem. One is in the purview of statistical mechanics that takes a broad approach and scrutinizes everything as being part of a thermodynamic system. The other is just a statistical counting method that could be plugged into the statistical mechanics construct if was desired. I wanted to keep these two methods separate so that the statistics could be clearly understood.

Statistical Mechanics Following the figure below assume that there are two bond energies

broken, Ub and unbroken Uu. Let the energy of the system be U then U/ε atoms

56



are unbroken and (N-U/ε) are broken.The number of ways of choosing U/ε atoms from the total number N is

Ω = N!Uε

⎛⎝⎜

⎞⎠⎟ ! N −U

ε⎛⎝⎜

⎞⎠⎟ !

S = kB lnΩ = kB ln N!( )− ln Uε

⎛⎝⎜

⎞⎠⎟ !− ln N −U

ε⎛⎝⎜

⎞⎠⎟ !

⎡⎣⎢

⎤⎦⎥

⎛⎝⎜

⎞⎠⎟

S = Uε− N⎛

⎝⎜⎞⎠⎟ kB ln 1−

UNε

⎛⎝⎜

⎞⎠⎟ −

UεkB ln

UNε

⎛⎝⎜

⎞⎠⎟

F =U −TS

F =U −T [ kBUε− N⎛

⎝⎜⎞⎠⎟ ln 1−

UNε

⎛⎝⎜

⎞⎠⎟ − kB

Uεln U

Nε⎛⎝⎜

⎞⎠⎟ ]

The energy of the system is U = Nε

1+ eεkBT

This probability is the same as the Fermi Dirac probability distribution.

The Binomial Distribution The distribution function for the figure below is:

The coefficients are of the Binomial distribution function.

The statistics that I’ve introduced is that of the Binomial distribution. Its applicable to bond breaking in the zone around the crack front ~ 50 ao (ao is the lattice parameter) . It should apply to brittle ceramics.

In order to see what the binomial distribution offers this type of problem, an expansion to the third power of ↓ ↑ 0 in other words the downward arrow represents a failed bond, the upward arrow is one that has also failed and the zero is one that has survived. The following expression is an excellent way to visualize everything ( with z=0)

z+ ↑ + ↓( )3 =↑↑↑ +3↑↑ z + 3↓↑↑ +3↑ z2 + 6 ↑↓ z + 3↓↓↑ +z3 + 3↓ z2 + 3↓↓ z+ ↓↓↓

p=probability of successq=probability of failure

n!k! n − k( )! p

kqn−k



Where z=0 and the terms that survive are:

0+ ↑ + ↓( )3 =↑↑↑ +3↓↑↑ +3↓↓↑ + ↓↓↓ It’s easy to show that the coefficients are the binomial coefficients and can be found in the third row of Pascal’s triangle. It’s quite interesting that the arrows replace the term’s powers.

Pascal’s Triangle

There are several advantages of using Pascal’s triangle along with the

Binomial distribution. 1. Entropy and the binomial coefficients can all be found on Pascal’s

triangle since row

column⎛⎝⎜

⎞⎠⎟ =

r!c! r − c( )! . This is the way almost every

entropy problem is developed.

2. The value of the row sums on the triangle is 2n = 2 2m −1( )n=1

m

∑ for

any m, so this is a cumulative term. Choosing an m is like integrating to the limit, m.

58



1. The maximum value of the even numbered rows on the triangle is2n!( )2n!

this is a most probable value. Usually in statistical

mechanics problems it’s not possible to include all the states in an ensemble so what’s used is the most probable value of a distribution instead. With this maximum term on the Triangle the most probable value is ready made for you.

2. Another interesting about the triangle is that it is robust. It is capable of handling some amazing numbers. The rows of the triangle are all 2Row and if you’ve ever worked with the Triangle you seldom have cause to go much beyond 28 or so, simply for practical reasons. A power of 2 looks innocent nothing suprising 220 , 240 well now you’re getting into computer managed territory. How about 279? how big can that be? Looks innocent... It’s so close to Avagadro’s number that it’s just unbelievable 1% or so I think. 279 just doesn’t look that large!

This is a Binomial distribution for a sample size of 25 and for a lattice spacing of one Angstrom.



This is the cumulative distribution for the above probability distribution. It is a map of the broken bonds from the region in front of the crack tip. This is only a model calculation made to see how things work.

The mean value of a binomial distribution is <x>=np where n = the sample size and p = the probability of a success on a draw.

In brittle materials it has been found that the following equation applies:

γ = 12ED*ao

where:

γ= the work of fracture E= the elastic modulus

For a binomial distribution the average value of x is

<x>=np where n is the sample size and p is the probability of success. In terms of this model,

ao= the characteristic distance ao=n alattice n= the sample size

If the distribution is binomial then

<a>=nalattice ( n=probability of a broken bond) So it is n that is related to the observed value of <a> on a statistical basis.

60



Another Example of Size Distributions The above figure shows there can be three different microstructural fracture

unit distributions. In this case fracture occurs in the vicinity around the crack tip. I’ve included three different probability curves just to make the problem interesting.

There’s also another assumption that brings out some nice aspects of the problem: the cracks that compose the crack “atmosphere” depend upon the strain energy density. That sets a position on the probability curves which have to be integrated from d* upward in order to get the number of fracture units that will collaborate in the “main event”

Other Forms of Fracture Fracture is a very complex process and it takes on many forms. Indeed figures below represent uncommon and difficult fractures that are commonly seen in ceramics and high strength metal alloys. They can also be seen in fractures that take place in foams or porous metals or molecular sieves. All the failures have something in common -they failed by fracturing- but in the following example the fracture wa



s wonderfully complex:

62



The ancillary figures show how the crack profile was deconstructed in order perform an analysis of strain vs position. This is an unwrapping technique had to be used in order to flatten the edges. The next figure shows more unwrapping with the detachment of the microstructural units. There were still some atoms out of place ( in red) that had to be dealt with. Our goal is shown in the last figure. Once the structure is flattened and unwrapped it is embedded with circles which are the basis set. The difference between the circle and the microstructure is then digitized entered into a computer and with an FFT a spectrum is found. These techniques are often difficult and challenging because unlike crystallography, many of the analyses aren’t ordered.

64



Insulators and Polarization Fracture takes place in a lattice of atoms surrounded by

electrons some are free, others are not Increasing the strain energy density applied to an object will cause a crack

within it to approach criticality. The influential (positive) energy inputs to the crack comes in several forms: increased atomic vibrational frequencies and amplitudes, scattering of Fermi electrons off the periodic lattice, changes in the ground state energy due to elastic strain, scattering of phonons and electrons off of the crack tip ...

Decreasing this heightened strain energy state comes from crack advance: bond breaking and the formation of new surface (the change in the system’s energy from the surface is positive but it’s released by having the surface grow) , new atomic states on the surfaces, changes in the vibrational frequencies, scattering of the Fermi electrons off of the new surfaces, phonon emission and phonon-electron interaction (polarons), field disruption and dipole interactions, surface disruption from field changes on the crack’s surfaces. In light of the previous paragraph I’d say “you gain some energy and you lose some energy” there are a variety of different paths to be taken and some are more probable than others.

The Dipole Moment When an insulator, a polar material, is subject to an applied field changes

occur in its charge distribution. The modern theory of polarization considers such changes to occur from the adiabatic current flow coming from deformation or some other modification of the crystal. Fracture most likely can be part of this. Borrowing on some essentials of the “modern theory”, where possible, I’ll try to establish how fracture might fit into this picture.

I’ll state two of the most important problems for trying to fit the fracture of insulators into the picture

1. The modern theory requires that there be an adiabatic path in order to evaluate its most important term the Berry Phase, in other words, everything happens slowly

2. Everything happens under null field conditions



Even with these limitations the modern theory offers an excellent chance to fit fracture into the picture because it clarifies some of the physics involved.

Fracture separates the bulk of a crystal and introduces new surfaces. The polarization that ensues would be expected to be:

Pf =1Vf

1 dr rρ1f −1∫ r( ) + 1

Vf2 dr rρ2 r( )f −2∫

This is just a formal statement that the fracture, f , separates the crystal into two parts. Continuity conditions.

The fracture introduces two new surfaces with surface charge densities:

−Δσ on the left and Δσ on the right. There’s a charge build-up on the surfaces and that can be ascribed to the amplitudes of the wavefunctions with the current being ascribed to the phase of the wave functions This changes the polarization but it isn’t a change identified by Pf.. It can, in all probability, be identified with fracture.

In order to establish the role of the fracture surface let there be an integral over a volume element η on the surface

Pη =

1Vη

drη!∫ rρ r( )

This term is far from exact but is, most likely,

Pη These issues indicate that they fall outside the definition of a sample’s bulk

polarization. The way this problem was solved for a crack free sample was to consider the change in ΔPSample that occurs due to an outside influence. Apply a field and measure the difference in polarization after the field is switched off. It is the difference in the ground state wave functions that is behind this behavior. Evidently, fracture would cause changes ΔP1 and ΔP2 . The charge buildup that occurs on the surfaces due to fracture must now be considered as part of the change in polarization of the bulk electronic structure

It is the charge flow during fracture that is important and that is

ΔPf = dt 1Vη

∫ dr j r,t( )η∫

66



The instantaneous change in polarization is

dP t( )dt

= j t( )

ΔP = P t( )− P 0( ) = dt j t( )0

Δt

∫

At this point the new theory adopts an adiabatic limit letting j go to zero and Δt→∞ . Certainly the influence of fracture could indeed approach these limits. After fracturing, the pieces would be protected and allowed to remain at rest for a given period of time.

In the theory a dimensionless time is introduced for convience:

ΔP = dλ dPdλ0

1

∫

In order to maintain ΔP = P Δt( )− P 0( ) which, basically establishes the definition of polarization. It is time dependent as is fracture. In order to maintain stability there must be a flow of current through the sample and it is for this reason that polarization is considered to be a bulk phenomena.

Fracture in Insulators The following comments on polarization and how it relates to fracture will

help to follow the next few sections of this paper: 1. Charge screening of fracture at initiation. during propagation and

in its final state are important in insulators. 2. The charge build up on the fracture surfaces of insulators makes

them environmentally sensitive 3. The high band gaps found in insulators makes intrinsic electron

compensation limited. 4. The surface and bulk charges are intertwined by time dependence

- the surface state being influenced by charge i.e.the amplitude of the electron’s wave functions and the bulk being influenced by electron current, the phase of the electron’s wavefunctions.



5. Berry’s phase plays an important role in the fundamental understanding of polarization of insulators, however, it’s not clear that it will be helpful in fracture

6. The changes in the charge distribution brought about by fracture undergoes changes followed by relaxation into a static polarization configuration owing to the reorganization of the ground state wavefunctions. This is a fundamental process that applies to both insulators and conductors. The polarization difference is related to this.

Dipole Moments and Strain A polarizable insulator will remain in an equilibrium state until it is exposed

to an electrical field or to a mechanical load. Being polarizable means that a solid will respond to an electrical field by having its core charges shifted. The protons shift a little and the electrons shift more giving rise to a charge dipole that aligns itself with the electrical field. The fundamental Laws of Electromagnetism are in play here since any internal fields introduced by the electrical field will be eliminated by carrier redistribution i.e. electrons will act to remove any internal fields. Now polarizable insulators can have band gaps ~10-15 eV meaning that they can’t have very much internal carrier flow at room temperature. But researchers inject carriers in order to get the properties that they want.

This discussion will be concerned with intrinsic carrier compensation at room temperature, mechanical loading and fracture.

The reason that there has to be elimination of any internal fields is that there can’t be any internal unsupported carrier current, there will be current flowing to eliminate the fields.

An external applied field aligns the dipoles. With its removal the structure relaxes back into an equilibrium state. The expected values of the dipoles’ properties reach zero. They can be changed again but there’s no way to determine a starting point only an end point. to a strain the spacing between its atoms change. Everything appears as elastic strain until fracture occurs. When the spacing between the bonds change the equilibrium dipole distribution acts in response to this changing environment . They will do so as long as there is current flow within the structure, otherwise, their response is limited and they will act to oppose the applied stress. The <E> of an insulator in equilibrium is zero otherwise there would be current flow.

68



Removing the material from this capacitor will leave charged surfaces. This is the nature of a crack in insulators.

The crack introduces internal fields that need to be eliminated on grounds that are based on the Laws of Electromagnetism. The internal field can be eliminated by internal current flow however, an excess of charge carriers are not to be expected in an insulator that might have ~ 10 eV band gap or so.. The presence of internal fields will cause a small internal current to flow but probably not enough to eliminate the fields. The uncompensated fields will respond to the applied load as reactive forces acting against it.



Should the developing reactive forces increase to extremely high levels and the internal fields remain uncompensated they can be eliminated by propagating a crack . This would allow the dipole distribution to return to equilibrium and to eliminate the internal fields.

Should the developing reactive forces increase to extremely high levels and the internal fields remain uncompensated they can be eliminated by propagating a crack . This would allow the dipole distribution to return to equilibrium and to eliminate the internal fields.

Before the load was applied the nuclei were relaxed throughout the entire solid, relaxed in some collective equilibrium state and the internal field was equal to 0. Upon loading the stress altered the atomic structure which caused internal changes in the field. Some charge migration occurred but due to the limited carrier concentration in insulators only so much elimination of the field could occur. Once the internal current flow stops but the applied load continues to rise, the internal field will become reactive and resist the applied load.

70



Statistical Physics of Bond Failure in Polar Insulators In polar insulators the dipole distribution without any outside extrinsic

influence is in some equilibrium state. Upon loading, the physical structure of a material is strained and the material undergoes changes that disrupt the internal field. Fundamentally, this will soon be mitigated by internal current flow since internal fields can’t be maintained in a polar material.

As was mentioned before, the addition of an external field will cause a change in the atomic charge distribution, basically, small changes in the atoms’ electron and nuclear charge distributions will occur thus creating a dipole. The other way to create dipoles is to distort and to break bonds as what happens at a fracture surface. The figure below is an example of what is to be expected in the vicinity of a crack face. The electric fields are shown in red lines. It’s possible to find the average spacing for the array of atoms. However, wherever there’s a spacing that’s deviated from the average such as those shown in yellow, a dipole has developed, in fact whenever an atomic displacement occurs like those in the vicinity of a crack an extraordinary dipole is formed . This doesn’t require piezoelectric properties to be present for their existence because the bonds that have been affected have been destabilized and are far from equilibrium. In other words, the bonds that are strained an extraordinary amount have become dipoles and will interact with their environment differently than they did before. The yellow “atoms” in the figure are to be associated with the anions and the white “atoms” are to be associated with the cations. It’s not expected for them to interact in the same way due to size differences and electronegativity differences .



That bonds become stretched and broken in the vicinity of a crack is not surprising. Nevertheless it’s useful to explore the occurrence of bond breaking and field alterations in order to understand some of the the crack’s properties. It’s important to realize that the fracture itself becomes an active partner influencing the fundamental electronic structure of the material . Both of these depictions show different things.

72



1. The first one is an idealialized crack tip for which strains can be calculated and ( most likely a periodic Bloch wavefunction could be merged with the crack tip singularity in k space.

2. The second one is a distorted structure, derived from, perhaps, 6 well -defined units which don’t have any repetitious behavior i.e.brittle fracture of insulators exhibit very few high frequency structures having long range correlations.

3. It’s possible to first understand things on a more fundamental scale, that is, quantum mechanically but chemically too. What I mean is that ( which is not to say that chemists don’t use quantum mechanics!!) it would be nice to have some practical rules that just made sense when it came to fracture. Indeed we have some but I’d like to have more! Tune in for the next section of this paper!!

A more de-magnified image of the crack, idealized of course, showing field lines (in red) where bonds have broken and dipoles have formed irreversibly and arrows in green showing atomic shifts, dipoles once again, but with a different origin than the ones that originated on the broken bonds.



Mathematics Concerning the Crack The equations that describe the stress field around the crack have a

singularity at a crack length =0. Essentially the equations of interest are:

Gc= Eε 2

This equation is in terms of

Gc= Eε 2 Energy

Volume

This singularity can be evaluated in the Complex − Discreteplane in the following manner:

1. First evaluate the region around the crack so that the crack is isolated this is important only insofar as it isolates the crack.

2. Next the Bloch periodic function u(x) describing the lattice potential must be considered. It is present as an ancillary function in the complex plane.

3. Integrating around the crack in the complex plane yields:

f z( )dz!∫ = 2πi0 = 0

4. Integrating within the crack so as to include the singularity yields =G2πi 5. The atoms within the crack have been removed so they can’t influence

their neighbors, however, their neighbors will have an influence on them. The Bloch function must be changed to reflect this. Therefore there are virtual atoms within the crack.

6. Integrating within in crack using virtual atomic fields yields ( there are n atoms in the crack (in the figure n=6 for example)

7. Since zz = x using real coordinates within the crack and letting the virtual atoms’ potentials be replaced by delta functions. Let g z( ) = δ (x)

Then gn z( )

n∑!∫ dz = δ x( )

n∑

δ n x( )n∑ =

n∑ e− ikx

0

∞

∫

U k( ) =n∑ ∫ un x( )e− ikxdx

The crack's opening is Δτ

74



making the exponential factor = Δτk x

u x( )∫ e− iΔτ kxdx =U k( ) This is a Fourier transform of u(x). Which is a one dimensional k space representation of the “Bloch function”. Its properties include broadening in the

x direction with decreasing crack length. The Gc

function maps as a flat straight

line in k space . It shouldn’t be a problem to bring Bloch’s function into the analysis in order to find out what influence lattice periodicity has on the stored energy per unit volume.

The upper figure is a representation of a periodic structure and the bottom

figure is a representation of a cracked plate in the complex plane. This development was needed because the analysis of Berry’s phase takes

place in k space and also due to the fact that polarization is quantized.



This is a K space representation of all the relevant physical dimensions of the fracture problem. The coordinates are scaled on a log basis so that all of the dimensions can be represented in a compact form. The circular figure on the right is an expanded view of things with 1/(crack size) and 1/(specimen size) shown in the middle. The ordered purple circles represent the 1/(lattice constant) of the solid being fractured.

Berry’s Phase Berry’s phase is a quantum mechanical analysis that takes place in k space

for determining the geometric influence on wavefunctions. The concept has awakened an enormous amount of interest and has found Michael Berry with a Nobel Prize

There was a great deal of confusion about the polarization of insulating solids. In the last 10 years things have changed for the better. The Claussius-Mossotti equation, while still being used, has taken a back seat to solid state physics and the definition of a quantum of polarization.

Starting with the Bloch form of eigenfunctions:

ψ nk = eikrunk r( )

Hk unk = Enk ψ nk getting down to the Shroedinger equation.

λ is a convenient parameter that takes the place of time. After much development the equation that emerges that is related to the polarization is:

76



dPdλ

= ie2π( )3

dk ∇kunk ∂λunk∫ +C

Peff = ΔPion + [Pel (1)− Pel (0)] This shows that the polarization has electron and ion core contributions.

Pel λ( ) = e2π( )3

Im ∫ dk unk ∇k unkn∑

this equation is at the heart of Berry’s phase . It is the central result of the modern theory of polarization. The integral takes place over a closed manifold-the Brillouin zone, in k space. The sum is done on a discrete mesh of k points that span the Brillouin zone. The operator ∇k is a derivative taken in k space.

Pn =

12π

eΩ

φn, jRjj∑

Where Rj and G j are the real space translation vector that corresponds to the reciprocal lattice vector G.

φn = Im ln un,k j un,k j+1j=0

M−1

∏

Since this is an imaginary term it refers to time also. The subscripts indicate that it’s discrete time and it’s only well defined as MOD 2π which leads to the conclusion that

Pn! = Pn +

eRΩ

since R is a lattice constant. The polarization as defined results in the definition of a polarization quantum. The quantization comes about due to the MOD term as it depends on the lattice constant R.

ΔP ! Pλ=1 − Pλ=0( ) MOD eR

Ω



Significance to Fracture It’s possible to say that Berry’s Phase has restrictions on it e.g. adiabatic

conditions, parallel transport, a well defined Brillouin zone etc which are to be considered before using it in a fracture application. The quantization of polarization is real and readily applicable to everyday problems. One has to consider the fact that a crack, in an insulator, is formed in a polar solid and it changes it. Without a doubt screening provides the mechanism for charge build-up on the newly fractured surfaces. The time dependent electron current that flows after the fracture is finished helps to neutralize the internal fields that the fracture created. It appears that Berry’s phase would applicable after fracture has occurred and adiabatic (?) conditions apply.

Applied Fields in Simple Geometries In the presence of an electric field, the induced dipole moment in an ionic

crystal may result from several mechanisms. Part of the induced moment is a result of distortions of the symmetric charge distributions relative to the nucleus of each ion; this induced moment defines the electronic polarizability. However part of the induced moment can come from relative displacement of the entire ion away from its normal lattice site; this is known as ionic polarizability. The ionic polarizability arises from the relative displacement of the various ions in the crystal which occurs because cations and anions move in opposite directions under the applied field. For this reason ionic polarizability depends on the nearest neighbor repulsive forces in a crystal. If a crystal has permanent dipoles they may reorient and contribute to the polarizability.

The units of polarizability are in terms of atomic volume and are typically in units of ~10--24 cm3. There are several types of induced polarizations that are known to occur in insulators. It’s not certain that precisely the same mechanisms would apply to fracture, however, it’s probably a good place to start looking. When a crack induces its physical characteristics into a solid insulator several things are known to happen including: bond failure, phonon generation, electron liberation, ionic displacements and the formation of a space charge that would come about from the disruption of the internal field.

Polarizability is of great interest because in insulators it is an indication of how the atomic environments respond to an applied field. It’s to be expected that some of the induced polarizability that is observed would just as well be induced by a crack..

Polarizability is an indication of how the bonds are able to respond to the field. It should be possible to find a relationship between some mechanical variables and polarizability since polarizability is also related to the ease of

78



stretching atomic bonds. An electrical force acts on the system and is related to the field F=eE that imposes a stress on the system as well as a strain on the atomic level.

The ability of bonds to align and stretch at the atomic level is dependent upon small movements of the electrons and the ion cores. Putting it another way it’s the system’s ability to adjust itself on a basic level, i.e. an adjustment that takes place just above the ground state but below the first ionization level. It’s a subtle adjustment but not much of one.

In order to proceed with a derivation for this let there be two distances δ and do that are shown below ( used in a strain)

|----|-----------| do d

Γ = e

2

α The units are correct and this rendition of Γ is a form of the equation:

Γ = e2

x3

but instead of the variable x it has a specific definition. This relationship will be of interest in the alkali halide section.

E⌢= the electric field

Eelastic = the elastic modulus

d = eE!

doEelastic

pe = de =

e2E!

doEelastic

pe = no po =

noe2E!

doEelastic

α = e2

Eelasticd0



Internal Fields

Fields within an insulator propagate from positive to negative surfaces. Internal surfaces, like fracture surfaces carry a charge. A previously charged surface interacts with the internal field by means of a Coulomb interaction.

This gives an indication of what the field around a short internal crack looks like.

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These are the fields around an unpolarized atomic site. Notice that the field

lines exit at the head on the major arrow and return at its back. Although not noted in the figure, the back of the arrow is negative.



In more complicated structures such as three dimensional crystal lattices it is very difficult to visualize the atomic bonds and the field lines together in computer models or in 3 Dimensional models.

There are some rules regarding the crystal lattice and the number of bonds per atom that help visualizing the crystallographic structure but for the electromagnetic field lines there are few rules. There’s no restriction on the number of field lines emenating from an atom. One rule is an imperative though; the field lines must be drawn from positive to negative sites.

The Null Particle I have introduced something new in these figures that helps keep track of

the field lines - a Null particle. The particle has a zero charge and no mass. This is a fictional particle that helps separate the + and - sides of the field lines which can be removed in order to count lines etc.The reason that this has importance is due to the Heisenberg Uncertainty Principle. The HUP makes it impossible to know exactly when a bond fails. The Null particle avoids such a conflict by setting things aside and recognising that a bond wil indeed fail due to many possible conditions - it’s only a matter of time. Accumulating a number of Null particles only means that a network is in jeapordy. There’s no problem mapping field lines onto it and from it because its potential is below positive but above negative.. They don’t interact with eath other only with field lines. It’ also possible to study bond failure which is extremely difficult to ascertain sometimes.. With the Null particle it’s possible to supress extension of the

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dipoles and separate bonds Their use is somewhat like playing a game of chess.

84



This is a visualization of an incipient crack (upper) where the bonds are

stretched and dipoles have formed. The lower figure is a situation where the bonds have broken and the fields loop back to the neatest neighbor sites because they must be terminated somewhere.

This type of evaluation becomes extremely complicated with only a few more atoms included to make things 3 Dimensional. It’s not clear that it’s fruitful to follow this any further. What’s called for is a probabilistic model accompanied by statistical mechanics.

G,Γ and γ -Theory and Experiment At the present time finding a general wave function for fracture that includes

most materials and using it in Schroedinger’s equation just isn’t possible. I’m sure that it’s a goal that many of the fracture community dreams about. It isn’t possible to do this at the present time principally due to the uncertainty about scales-is it atomic?-is it macroscopic? or is it both?

The other problem is that fracture occurs in a great many different materials and, usually, each class of fractures with their accompanying mechanical properties appear to be unconnected. In other words, unification has a long way to go when it comes to fracture.



Theoretical and Experimental Values of G

Alkali Halides and Insulators

Structure G Joules/m2 Theory G Joules/m2 Exptl NaCl 1.155 *** KCl 0.758 1.40 CsCl 0.676 *** CaF2 3.179 0.51-3.6 MgO 13.704 7.9-35 CaO 8.335 *** BaO 4.274 *** FeO 4.885 6.1-8.1 CoO 7.449 *** NiO 9.964 *** ZnO 5.599 *** TiO2 12.445 68 Al2O3 19.250 25-54

On the many meanings of G,Γ and γ that will be encountered in this work

The expressions for the tentative G, - Γ has been adopted in this work because Γ is considered to be a tentative theoretical expression. This work has revealed there might be several possible expressions for G; some will be useful others will not.

I found that as this work progressed their numbers proliferated. This was somewhat unexpected but not something that was uninvited. This is Science so the proliferation of expressions required some “sorting out”.

To avoid confusion, I’ve defined the problem carefully and outlined everything below.

Fracture and Surfaces Fracture is an irreversible process that includes a loss of energy and the

formation of fracture surfaces. The experimental measurement of this is G, the

86



energy lost per area fractured that is representative of the mechanical-material event

G The energy absorbed/area from fracture is related to G, the work of fracture or WOF, through K the fracture toughness. G is determined by means of a test that is performed under well defined conditions. The mechanical variables for the test are reproducible having the usual statistical variations.

γ The equilibrium surface free energy based upon thermodynamics. In this case it is truly the energy/area that is related to . Measurements of must be taken under near-equilibrium conditions. The surfaces that form are not due to fracture. They are often vicinal and stepped on the atomic level. The energies that are measured are usually 1-2 orders of magnitude below the values of G.

The Theory has No Surfaces The derivations that came earlier in this work didn’t have the presence of

physical surfaces in them. Instead, quasi particles apparently played a role which resulted in energies that were reasonable but there was no means to verify them.

Γ will be used for the analytical section of this paper. Γ represents a term that is a hypothetical symbol, Gamma one that requires development and verification before it can be compared to G.

The reasons for this approach are simple and straight forward: the relationship between Γ and other solid state variables will be determined establishing a base for Γ . After all, as it stands, Γ is a theoretical creature and nothing else. The relationships, if any, will indicate what the properties of Γ are perhaps, new and some undiscovered behavior can be found.

The fact that these expressions might express something different than fracture toughness involving surfaces is intriguing. It’s not surprising because the expression for Γ came from the derivation using the Shroedinger equation.:

Γ = e2

x3+mω 2

and although the units are right, there isn’t the possibility of having a physical surface emerge from it.

Q: In your analysis Shouldn’t there be accommodation for a surface? A: No Quantum mechanics deals with mechanics things such as the

harmonic oscillator and atomic structures which don’t emerge as independent physical things but they can be analyzed by means of quantum

γ γ



mechanics. Surfaces are no different that that. They must be brought in as independent things and embedded in 3 space.

Q So how did you end up with a Γ and what does it mean? A: I merely did the following transformation for the spring constant in the

harmonic oscillator

ω = km

mω 2 = k

mω 2x2

α= Energy

Area= Γ

Next, I used two terms for the potential energy, the harmonic oscillator

and the Coulomb energy. I envisioned a surface atom bound to an atom with an electrostatic Coulomb field and vibrating around some closed path. I didn’t have to do much with Shroedinger’s equation because I got what I was looking for.

Q So now what? A: Several things. In this work I’ve presented some very specific and

detailed information about real surfaces and perhaps now it’s clearer why I did it. These are observations done on real surfaces so their physics can be incorporated in a surface knowledge base. The geometry of a material surface needs to be approached mathematically so that a physical surface is clearly defined- in other words, we need a matter based surface metric.

Q Why correlate the properties of alkali halides with Γ? A: Simple To find out where Γ fails and where it’s successful. It’s really

related to the fact that there’s no surface in the physics so far. This will establish a reality for the theoretical construct.

The Alkali Halide Data Base THEORY AND EXPERIMENTAL VALUES OF KCL I computed the Γ terms for the alkali halides that I intended to use. The only experimental value that I had for comparison was for KCl KCl Experimental 1400 ergs/cm2 Theoretical #1 758 ergs/cm2 My Theoretical 995 ergs/cm2

It appeared that, although the amount of evidence was minimal, what I had was reasonable.

88



I did some further computations on Γ using the alkali halides that I would use in the study.

The Computed Theoretical values of the Alkali Halides Γ = e2

Ω+mω 2

Alkali Halide Computed ergs/cm2

LiF 3029 LiCl 1068 LiBr 1474 LiI 1078 NaF 2542 NaCl 1397 NaBr 1152 NaI 915 KF 1476 KCl 995 KBr 846 KI 697 RbF 1330 RbCl 863 RbBr 746 RbI 603

A More Inclusive Table of Alkali Halides for Comparative Purposes

Γ



Alkali Halides Computed

Gamma ergs/cm2 Energy- Gamma ao

2 eV Plasmon Energy eV

LiF 3029 3.676 7.106 LiCl 1068 1.950 5.23 LiBr 1474 2.76 5.13 LiI 1078 2.47 4.411 NaF 2542 1.47 6.62 NaCl 1397 3.38 4.94 NaBr 1152 2.53 4.54 NaI 915 2.38 4.008 KF 1476 2.60 5.36 KCl 995 2.449 4.18 KBr 846 2.286 3.899 KI 697 2.168 3.50 RbF 1330 2.64 4.91 RbCl 863 2.30 3.93 RbBr 746 2.178 3.679 RbI 603 2.01 3.323

90



Alkali Halide Band Gap

Energy eV LiF 12.5 LiCl 9.5 LiBr 8.0 LiI 6.2 NaF 11.0 NaCl 8.6 NaBr 7.7 NaI 5.9 KF 10.80 KCl 8.5 KBr 7.8 KI 6.5 RbF 10.0 RbCl 8.2 RbBr 7.5 RbI 6.5

The Alkali Halide Data The data that I used came from the following resources:

1. Introduction to Solid State Physics, Charles Kittel, John Wiley and Sons, 2005, pages 50,52,54,63,66 and 71

2. Salem,M.A., “Studies on the Opto-Electronic Constants of Alkali Halide Crystals”, Indian Journal of Pure and Applied Physics,Vol 41,May 2003,pp.384-388

3. Wacke,S., Gorecki,K , Ksaizek,K, “Relations Between the Cohesive Energy, Molar Volume, Bulk Modulus and Sound Velocity in Alkali Halides” VISNYK LVIV UNIV. Ser. Physics. 2009, Is. 43, P.87-92

4. Reddy,RR., Nazeer,A., Gopal K Ramma, P Abdul Azeem, “Opto-electronic properties of semiconductors and alkali halides” Vol.40, July, 2002,pp. 471-475

5. Reddy, R.R., Viswanath, R., Rao,T.V.R, “ Correlation Between Bond energies and Forbidden Gaps of Alkali



Semiconductors”, Indian Natural Scientific Academy, 56,A, No. 5, 1990, pp. 421-425

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Alkali Halide Chemistry

The literature provided information on the chemical properties of the alkali halides and indicated that their chemical properties were influenced by the electronegativity and sizes of both the cations and the anions. Furhermore, certain energies were of importance: the cohesive energy and the first ionization energy. These data are shown below:

• First Ionization energy • First Ionization Energy for Period 3 • Atomic radius • Calculated atomic radius • Electron affinity • Magnitude of electron affinity • Electronegativity • Pauling’s electronegativity

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Anion Atomic Radius A

Lithium 1.67 Sodium 1.90 Potassium 2.43 Rubidium 2.65

Cation Atomic Radius A

Fluorine 0.42 Chlorine 0.79 Bromine 0.94 Iodine 1.15

96



98



This is a commonly held construct that’s used to understand the operation of semi-conductors and insulators. These structures are physically stable and their properties are reversible, in other words everything here is based on the fact that the properties are state variables. Meaning that everything is reversible and the systems are able to return to where they started.

Fracture isn’t reversible isn’t associated with a state variable and is notoriously unstable. However some of these ideas are probably useful.

The depiction is color coded green for anions chlorine and blue for sodium. The hexagonal structures represent the p orbitals. The anion loses an electron to the cation in the conduction band leaving a hole in the valence band.



General Organization of the Analytical Work on the Alkali Halides

Next Page Please....

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Analysis of Propertiesof Solid Alkali Halides

Bulk Modulus vsPolarizability

Linear R =0.9377 Generic relationship

Bulk Modulusvs 1/ao^3

Linear R=0.9633 Generic Relationship

Band Gap vs ao LinearR=0.6991

Poor CorrelationGeneric Relationship

Unexpected Result for Alkali halidesPoor correlation R=0.69

The band gap in semiconductors varies with the lattice parameter

Band Gap vs FirstIonization Energy Linear R=0.9722 Generic Relationship

Band Gap vsPolarizability Linear R=0.6381 Generic Relationship

Band Gap vsElectron Affinity

Linear4 Separate lines

Discovered somethingnew in properties

Properties were: a seriesof straight lines

associated with anionshaving negative slopes

New unexpected result that wasto be observed frequently for both

anions and cations

Gamma vsDispersion

EnergyLinear

4 Separate linesCation Alignment

4 Linear Relationships Positive Slopes

Cohesive Energy vsDispersion Energy

Linear 4 separatelines

Cation alignment4 Linear Relationships Positive slopes

Gamma vsElectron Affinity

Linear 4 separatelines

Anion alignment4 linear relationships

Negative slopes

Bulk Modulus x ao vsIonization Energy

Linear 4 SeparateLines

Cation Alignment4 linear relationships Positive Slopes

Gamma #1 #2 vs(ao1-ao2)

Linear 4 separatelines Cation Alignment

4 linear relationships

Discoveredsomething new

Anions were aligned along cation lines

according to electronegativity

Positive slopes

Gamma#1 e^2/ao^3 vsfirst ionization energy

Ean +EcatLinear 4 Separate

linesCation Alignment

4 Linear relationships

Gamma #2 vs firstIonization energy Linear 4 separate lines

Cation alignment4 Linear relationships Positive slopes

Gamma vselectron Affinity Linear 4 separate lines Anion alignment

4 Linear relationships

Discovered something newcations were aligned according

to electronegativity alongcation lines

Negative slopes

Gamma vsCohesive Energy-(E(ion an)+E(ion cat))

Linear 4 Separate linesCation Alignment

4 Linear relationships Positive Slopes Calculated

slopes werefound to be 17 A and 25.3 A

This data was exceptionally

uniform with minimum scatter.Calculate

Properties fromD(Energy)

Calculate theslopes of 3of the lines

Cation Alignment

I found the (uncoordinated ) atomicradius of each cation and plotted

it vs the (1/slopes)^(1/2)

Linear plots of 3 lines(2 were parallel) Size plays an important

role in this relationship

Since this is anenergy relationship it standsto reason that the size term

would be involved through PV

Goal: To Establisha Fundamental Understanding of How

Ionic Solids (Alkali Halides) are Relatedto their Properties



Bulk Modulus vs Polarizability Base-Line Relationship This relationship showed that the bulk modulus had an inverse dependence

on the polarizability and was coupled with an anion dependence. The anion dependence manifested itself as clusters of points that were aligned in regions of the curves. This type of anion/cation influence was unexpected .

The most electro negative anion, fluorine, was at one end of the plotted data and at the other end was iodine, the least electronegative anion. Anion electronegativity created the ordering of the groups of data.

This is an example of a series of “dry runs” that needed to be done before the theoretical tests were attempted . Furthermore, it was important a) to know whether there were any artifacts that may accompany the data i.e

alkali halide “junk” b) find anything in other relationships that might help in the interpretations of

the theoretical work-in other words “borrowing from reality”

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Bulk Modulus vs 1ao3 Base-Line Relationship

This was another “dry run” the determination of the relationship between

the bulk modulus and 1ao3 . I had done similar analyses for ceramics and had

found a 1ao3 dependence but I didn’t know what to expect for alkali halides.

The relationship showed excellent linearity and was generic- that is it wasn’t sensitive to the kind of structure that was involved.. The R2 value was 0.963 meaning that it was a good fit and that the bulk modulus was dependent

on 1ao3



Band Gap Energy vs Lattice Parameter Base-Line Relationship This is an example of a poor correlation, one between the band gaps and

lattice parameters. This was unexpected since it is common knowledge that in silicon the bandgap is dependent on the lattice constant.

104



Band Gap Energy vs Ionization Energy Base-Line Relationship

This is an example of an excellent generic relationship existing between the bandgap energy and the first ionization energy



Band Gap Energy vs Polarizability Base-Line Relationship This was another experiment investigating the dependence of the bandgap

energy on the polarizability. The fit is good and is anion dependent. The line’s

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slope is negative. The organization of data is associated with the anions. The positions of the anion’s influence is dependent in the anion’s electronegativity.

Band Gap Energy vs Electron Affinity Base-Line Relatioship This plot again showed anion sensitivity but this time it was something

different. There were four sets of lines and each was associated with a different



anion. This was a case of “anion convergence”, in fact, there were three different ones with the forth one being a convergence of chlorine and flourine.

None of these lines had to be fitted with straight lines. In fact most of the data that I computed throughout this phase of the work needed no fitting whatsoever. They were straight enough so that I could use a line positioned by hand. This was quite surprising.

These slopes were all negative.

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Γ Theoretical vs Electron Affinity Theoretical Relationship The behavior for the theoretical Γ shows the same type of behavior that the

real data does. There is anion convergence once again, the slopes are negative and the positions of the alignment takes place with respect to the electron affinity of the anion.



Bulk Modulus ao vs Eionoz Base-Line Relationship This a relationship found for real data, between the bulk modulus and the

first ionization energy. There is cation covergence and along each line the data points are ordered from high electron affinity at the upper right to low electron affinity at the lower left. The slopes are all positive.

110



Γ vs Band Gap Theoretical Relationship This is the relationship between the theoretical Γ and the band gap. It’s a generic relationship having a good fit R2=0.9422



Γ vs Electronegativity Theoretical Relationship This some more theoretical Γdata plotted vs the electronegativity. There is

cation convergence once again and the slopes are all positive.

112



Γ vs Δ atom radius Theoretical Relationship This is a plot of the theoretical gamma once again vs the atom radius. This

clearly shows the four cation convergent lines having positive slopes. The anions are also convergent along perpendicular lines that can be connected by lines having negative slopes.. Both cations and anions are ordered according to their electronegativity.



Γ1 vs Eion Theoretical Relationship This is a cation convergent situation with positive slopes.Gamma 1 is the first

term in Gamma. I wanted to see if there was any difference between the two terms of the equation.

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Γ2 vs Eioniz Theoretical Relationship The second term of the equation is similar to the first: cation convergent with

a positive slope.



Γ vs Electron Affinity - Energy of Ionization Theoretical Relationship

This relationship was far more complicated than the others. This is how the gap varies within the alkali halide group. It shows that on one side the relationships are anion convergent and on the other side they are cation convergent. Furthermore, the anions have a positive slope on the left hand side and the cations have a negative slope on the right hand side. The cation wavefunctions make up the conduction band and the anions’ wave functions make up the valence band. The left hand side and the right hand side of this relationship must meet in the middle, at the Fermi energy.

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This is a figure showing the relevant energies for the above figure.



Γ vs [ Ecohesion - (Eanion+Ecation)] Theoretical Relationship

This was the relationship that had the best linear relationships having cation

convergence. The cohesive energy is the energy that is required to remove an atom from a solid and take it to an infinite distance away to an (external) energy free environment. ΔEsfc = nEcoh − EI+ + EI−( ) This is related to the x axis.

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The slopes are shown as lengths but I think that they should be changed to areas (see Appendix). The next graph is a strong argument for using something like scaled lengths but that needs to be considered carefully

α Lower = 6.4 ⋅10

−14 cm2

αUpper = 2.89 ⋅10−14 cm2

Comparisons between atom radii (published) and atom sizes taken from slopes of the experimental data

Cations



A Summarization of the Behavior of the Anions and Cations in this Study

This work shows that, without a doubt, there are consistent behaviors of the cations and the anions and it doesn’t matter whether the data are base line or theoretical Type Slope Convergence Cation Positive Yes usually 4 Anion Negative Yes usually 4

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This is an “Electronegativity Triangle” that organizes the observations made on the data. On the left hypotenuse is the electronegativity of the cations on the right hypotenuse is the electronegativity of the anions. The left and the right sides have to be adjusted to bring them into coincidence at two points. The Fermi energy = 0 lies in the middle. The scale for displaying the log of the ionization energy is shown at the bottom The energy along the x axis is the ionization energy

The slopes are a result of the band positions relative to the Fermi energy ie,

ΓA =EAVal − EFermi

α

Γc =EFermi − EC

Cond

α

Summary and conclusions A summary of this work won’t be forthcoming except to say that the relationships presented, although then may look disjointed are not, because prior to this work most of the fundamentals and applications of the fracture work was widely distributed and therein, narrowly focused. LEFM has helped to consolidate many of the ideas and standardize the tests making more unification possible. This work is not narrowly focused, but it is because it deals with the fracture of insulators-and that’s (kind of) narrow but then it addresses how the structural details of insulators might influence fracture. The appearance of quasiparticles was not injected intentionally for effect, but rather came about as I began to do some derivations with the Shroedinger equation. I have tried very many different approaches (hundreds!!) on the fracture problem with limited success. I had realized that mω2 had the proper units for G but in the case of the harmonic oscillator it wasn’t’ treated as a fracture toughness but rather as a compliance. I next found this same term to be used along another term that could be thought of as the Bohr energy per unit area

mω 2 = e2

ao3

I “played around” with these things ( as I tend to do) and using some reasonable constants I got some results that were interesting



Passoja’s “Fooling Around” Results Hydrogen Atom Item of Interest Result Bohr radius A 0.52917 A Frequency hZ 4.134x1016 hZ Hartree eV 27.211 eV mω2 quasiparticle

1556Jm2

e2

ao3quasiparticle 1556

Jm2

Although I wasn’t using a solid etc I was interested in these results

because they were in the “ballpark”. Also there wasn’t a surface in the picture so I was doubtful.

I then obtained some encouraging results from calculations done with the Shroedinger equation. Both terms were forthcoming so I was mildly interested. I worked out my analytical scheme using the alkali halide data and I got some positive results once again!

Then I asked myself:

• “How come no one has ever seen anything like this?” Answer: because no one has looked for it.

• “Where should they look?” Question before an answer was forthcoming: What are the energies of the fractures (not the surface fracture energy) ~2-10 eV

• “I said that I’d check things out” The uncertainty principle allows a check on a particle’s lifetime since E t~h and the lifetime came out to be ~1.6 10-16 seconds

• The answer was obvious- the particles would decohere quickly with respect to the source so that in order to see them there would have to be some special experiments to be done.

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• Yes indeed the quasi particles could be responsible, in part, for the observed fracture energy.

• The LEFM technology is too useful and ,I feel, technically correct to just ignore it so there’s some exciting new work to be done in rationalizing these results.

Appendix

A Search for the Missing Area The equations I’ve derived by means of quantum mechanics have the

correct units but are missing the description of physical surfaces that should be in association with them.



There were several different ways to proceed • The details of the fracture and the fact that the fracture was

a fractal • Euclidean Geometry with a complex area • Definition of an area metric and consideration of

embedding it in a lattice with a periodic potential

I chose the third way first because it offered the opportunity of developing the concept of an area and therefore I used Lagrange’s Identity in matrix form. I needed to determine the concept of a length existing between two areas ( I realized that this idea didn’t exist in any of our thinking but seemed quite relevant to the fracture problem). This isn’t a distance measured between two pre existing planes. The term on the right of this equation is what I’m looking for:

Using Lagrange’ s identity it can be easily seen that it is the outer product of vectors a and b lying two different planes.

ak2

k=1

n

∑⎛⎝⎜⎞⎠⎟

bkk=1

n

∑2⎛

⎝⎜⎞

⎠⎟− akbk

k=1

n

∑2

= aibj − ajbi2

j=n+1

n

∑i=1

n−1

∑

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i j kai aj akbi bj bk

2

Which can be shown to be the magnitude of 3 cross products;

By additional consideration a crack can be thought of as something that has a displacement (x-y)2

x + iy( ) x − iy( ) = x2 + y2 − i x, y[ ] where : x, y[ ] is the commutator: xy-yx xy = x ⋅ y + x ∧ y where xy is the geometric product

Quasiparticles and Fracture #2 (1)

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