Localization of Plastic Shear Events in Glassy

Materials

Qing Peng, Ph.D.

University of Connecticut, 2005

An algorithm is introduced for the molecular simulation of constant-pressure plas-

tic deformation in glassy materials at zero temperature. This allows for the direct

study of volume changes associated with plastic deformation (dilatancy) in glassy

materials. In particular, the dilatancy of polymer glasses is an important aspect

of their mechanical behavior. The new method is closely related to Berendsen’s

barostat, which is widely used for molecular dynamics simulations at constant

pressure. The new algorithm is applied to plane strain compression of a binary

Lennard-Jones glass. Conditions of constant volume lead to buildup of system

pressure with strain, and to a concommitant increase in shear stress. At con-

stant (zero) pressure, by contrast, the shear stress remains constant up to the

largest strains investigated (ε = 1), while the system density decreases linearly

with strain. The linearity of this decrease suggests that each elementary shear

relaxation event brings about an increase in volume which is proportional to the

amount of shear. In contrast to the stress-strain behavior, the strain-induced

structural relaxation, as measured by the self-part of the intermediate structure

factor, was found to be the same in both cases. This suggests that in order to

overcome the energy barriers, nucleation must continually grow in the case of con-

stant volume deformation, but remain the same if the deformation is carried out

at constant pressure.

The length scale of the elementary processes of plastic relaxation of amorphous

polymers is still an open question. The computer simulation of plastic deforma-

tion gives the details of the plastic relaxation events. To study the localization

of these events, a novel approach of the correlation of relative atomic strain is

invented, in which Delaunay tesselation and Fast Fourier Transforms techniques

are applied. Using this novel approach we have studied the localization of atomic

strain in discrete relaxation events during plastic deformation of glassy materials.

The strain in such relaxation events is highly localized in regions of atomic di-

mensions. The implications of the novel approach and our simulation results for

a universal theory of plasticity of amorphous polymers will be discussed.

Localization of Plastic Shear Events in Glassy

Materials

Qing Peng

B.S., Beijing University, Beijing, China, 1998

M.S., State University of New York at Binghamton, 2000

A Dissertation

Submitted in Partial Fulfillment of the

Requirements for the Degree of

Doctor of Philosophy

at the

University of Connecticut

2005

Copyright by

Qing Peng

2005

APPROVAL PAGE

Doctor of Philosophy Dissertation

Localization of Plastic Shear Events in Glassy

Materials

Presented by

Qing Peng,

Major Advisor

Marcel Utz

Associate Advisor

Steven Boggs

Associate Advisor

Douglas Hamilton

University of Connecticut

2005

ii

To my wife Zengyun Liu.

iii

ACKNOWLEDGEMENTS

I am grateful for the opportunity to participate in an exciting and interesting

research project in Dr. Marcel Utz’s group. I wish to sincerely thank all the

people who assisted me during my work. My deepest gratitude goes to my ad-

visor, Marcel Utz. His guidance, support and encouragement have made all of

this possible. As an advisor, he was always willing to discuss current successes

and difficulties and offer advice for future progress. In my research, he left me

complete freedom and provided enough computing power for my experiments and

calculations. His patience and guidance allowed me to surpass various struggles

and to maintain a proper focus. I also thank my associate advisor, Steven Boggs

and Douglas Hamilton for taking time to read and comment on this dissertation

and I am greatful for their assistance. Thanks to my group mates Michael Roz-

man, Magesh Nandagopal, Priyanga Bandara, Katsiaryna Prudnikova,

for the discussions of the material presented in this thesis. I especially appreciate

the opportunity to be an assistant of Dr. Michael Rozman in computer lab of

physics Department. I learned a lot of computer skills. I would like to thank

the Prof. Cynthia W. Peterson. It was my honor to be a teaching assistant

for her in PHYS 155 (Introduction to Astronomy). Thanks to Prof. Chandra

Roychoudhuri, for his great help and support. I am indebted to Prof. Edward

E. Eyler, Phillip Gould, William C. Stwalley for their help and support.

Thanks to Prof Philip D. Mannheim, M. Munir Islam, Gerald V. Dunne,

for their help in my study. I offer gratitude toward all of my teachers past and

present. You have helped shape and define my understanding of some of the most

iv

beautiful facets of the cosmos, a most precious gift. I owe enormous gratitude

to the University of Connecticut Physics Department for, without their support

in the form of assistantships, I simply would not have been able to pursue and

complete this dissertation. I especially want to thank my wife. She has patiently

waited for the completion of my research and has supported me continuously with

love and understanding.

v

TABLE OF CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Glassy Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2. Theory and Background . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Classify Glassy Materials . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Modeling the Physical System . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 Global Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.2 Infinitesimal Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.3 Atomic Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.4 Relative Atomic Strain . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 Computer Simulation at the Atomic Level . . . . . . . . . . . . . . . 33

2.4.1 Molecular Dynamics Method (MD) . . . . . . . . . . . . . . . . . . 35

2.4.2 Monte Carlo Method (MC) . . . . . . . . . . . . . . . . . . . . . . 37

2.4.3 Periodic Continuation Conditions . . . . . . . . . . . . . . . . . . . 40

2.4.4 Initial Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4.5 Energy Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.6 Constant Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.5 Characterization of Amorphous Structures and Dynamics . . . . . . . 43

2.5.1 Tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.5.2 Torsion Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

vi

2.6 Plastic Shear Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.7 Simulation of Plastic Deformation of Glassy Materials . . . . . . . . . 52

3. Athermal Simulation of Plastic Deformation at Constant Pres-

sure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3.1 Pressure Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3.2 Stress-Strain Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3.3 Strain-Induced Structural Relaxation . . . . . . . . . . . . . . . . . 70

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4. Quantification of Strain Localization Using the Relative Atomic

Strain Correlation Function . . . . . . . . . . . . . . . . . . . . . . 75

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2 Model and Simulation Method . . . . . . . . . . . . . . . . . . . . . . 79

4.2.1 Delaunay Tessellation . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2.2 Location Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.3 Working Materials and Units . . . . . . . . . . . . . . . . . . . . . 86

4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3.1 Plastic Deformation of Molecular Glasses at Constant Pressure . . 89

4.3.2 Dilatancy of Molecular Glasses . . . . . . . . . . . . . . . . . . . . 96

4.3.3 Stress-Strain Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.3.4 Pair Correlation Function and Structure Factor . . . . . . . . . . . 106

4.3.5 Strain-Induced Structural Relaxation . . . . . . . . . . . . . . . . . 108

vii

4.3.6 Localization of Plastic Shear Events . . . . . . . . . . . . . . . . . 114

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Bibliography 134

viii

LIST OF FIGURES

1.1 Four length scales in computer simulations: electronic structure, atomic,

Microstructural or Mesoscale and Continuum. Different scales have

different lengths and methods. . . . . . . . . . . . . . . . . . . . 11

2.1 A typical plot of the Lennard-Jones potential V (r) in equation (2.24) 21

2.2 Torsion Angle Definition . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1 System pressure P (in units of εAA/σ3AA) as a function of deformation

for ǫP = ∞ (dashed line) and ǫP = 0.0033 (solid line). The data

shown represents the average of 20 independent simulation runs. 64

3.2 Number density of particles ρ (in units of 1/σ2AA) as a function of de-

formation for ǫP =∞ (dashed line) and ǫP = 0.05 (solid line). The

data shown represents the average of 20 independent simulation

runs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3 Effect of different choices of the relaxation strain ǫP on the evolution

of pressure (top row) and density (bottom row) over the course of

a plane-strain deformation simulation from εzz = 0 to εzz = 1. For

each value of ǫP , results from a single simulation run are shown. . 66

3.4 Von Mises equivalent shear stress τ as a function of strain εzz for

relaxation strain ǫP =∞ (dashed line) and ǫP = 0.0033 (solid line).

Each data set shown is the average of 20 independent simulation

runs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

ix

3.5 Self part of the isotropically averaged intermediate structure factor

Φs(k, εzz) as a function of strain εzz for constant volume (ǫP =∞,

dashed line), and constant pressure (ǫP = 0.0033, solid line). The

dashed curve has been displaced vertically by 0.1, otherwise the

two curves would coincide to within the line width. k = 7.251 in

both cases; corresponding to the location of the maximum in the

static structure factor. . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 Pair correlation functions gBB(r) for different degrees of deformation

(as indicated in the figure). Solid lines: constant pressure (ǫP =

0.0033); Dashed line: constant volume (ǫP =∞). The curves have

been vertically displaced in order to avoid coincidence. . . . . . . 70

4.1 Commonly observed stress-strain response in computer simulation of

glassy materials under plastic deformation (Ref [16]). . . . . . . . 76

4.2 Von Mises equivalent atomic strain observed in a computer model of

polypropylene undergoing a plastic relaxation event. Scale is indi-

cated by the size of circles around atoms (Ref [19]). . . . . . . . 77

4.3 The process of the method to study the localization of plastic relaxation

events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 A system of Ni80P20 containing 4000 atoms with box size 14.76 σNi−Ni

was tessellated by Delaunay Tessellation. . . . . . . . . . . . . . . 83

4.5 System pressure P of different materials as a function of deformation

for ǫP = 0.0706. The lines from bottom to top stand for C5, C10,

C20, C50, C100, C200 and C500 respectively. The curves have been

vertically displaced in order to avoid coincidence. . . . . . . . . . 90

x

4.6 The standard deviation σP of the system pressure of the seven glassy

materials, with the target value Pt = 0 with respect to their system

size, the total number of united atoms N . . . . . . . . . . . . . . 92

4.7 System pressure P of Pentane at different system size as a function of

deformation. The curves have been vertically displaced in order to

avoid coincidence. . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.8 The standard deviation σP of the system pressure of Pentane with

respect to their system size, the total number of united atoms N . 94

4.9 dP of the system pressure of Pentane with respect to their system size,

the total number of united atoms N . . . . . . . . . . . . . . . . . 95

4.10 Normalized number density ρ of particles of different glassy materials

as a function of deformation at constant pressure. . . . . . . . . 97

4.11 Normalized number density ρ of particles of pentane with different

system size as a function of deformation at constant pressure. The

curves have been vertically displaced in order to avoid coincidence. 98

4.12 The slopes k of system normalized number density linearly decreasing

with the strain are plotted for chain length l (solid line) with its

fitting (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.13 The normalized energy of the system of different glassy materials as a

function of deformation for values of εP = 0.0706. . . . . . . . . . 101

4.14 The von Mises equivalent stress τ of different materials as a function of

deformation at constant pressure. The curves have been vertically

displaced in order to avoid coincidence. . . . . . . . . . . . . . . 102

4.15 The number of drops of von Mises equivalent stress τ of different ma-

terials as a function of strain increment dεzz. . . . . . . . . . . . 104

xi

4.16 Von Mises equivalent stress of Pentane at different system size as a

function of deformation. The curves have been vertically displaced

in order to avoid coincidence. . . . . . . . . . . . . . . . . . . . . 105

4.17 The number of drops of von Mises equivalent stress τ of pentane with

different system size as a function of strain increment dεzz. . . . 106

4.18 Pair correlation function for pentane in un-deformed system. The

short-range order is expressed by two distinct peaks and a broad

peak in the short distances, following by a quite flat tail. The first

peak is centered near the bond length lbond = 1.53 Aand the second

peak is centered near 2.57 A, resulting from the bond angle of 114.

The third peak is very broad, about 0.5 Awide, centered near 4.12

A. The flat tail has the value of 1. This is very typical g(r) plot for

C − C chain structure. . . . . . . . . . . . . . . . . . . . . . . . . 107

4.19 The structure factor S(q) vs the scattering vector for pentane in un-

deformed system. The first peak is artificial for the limited size

of the simulation box. The second peak, which is centered at q =

1.6197 1/A, corresponds to the position of the maximum of the

static structure factor. This value of q = 1.6197 1/A is quite

typical for C − C chain structure. . . . . . . . . . . . . . . . . . . 109

4.20 Self part of the isotropically averaged intermediate structure factor

Φs(k, εzz) , for C5, C10, C20, C50, C100, C200 and C500, respectively,

as a function of strain εzz undergoing athermal plastic deformation

at constant pressure. k = 1.6197 1/A is corresponding to the

location of the maximum in the static structure factor. . . . . . . 110

xii

4.21 The stretching exponent β, isotropically and in x, y, z directions, ob-

tained from Φs for seven molecular glasses: C5, C10, C20, C50, C100,

C200 and C500, respectively, as a function of molecular chain length

L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.22 The correlation strain ε0, in x, y, z directions and isotropic, obtained

from Phis for seven molecular glasses: C5, C10, C20, C50, C100, C200

and C500, respectively, as a function of molecular chain length L. . 114

4.23 Shear relaxation events in pentane (C5) under althermal plastic defor-

mation at constant pressure, a part from −εzz = 0.15 to −εzz = 0.2. 116

4.24 Localization of the plastic shear events. (a) shows 4000 atoms in a

simulation box of Ni80P20. (b) shows the Delaunay tessellation

of the system of the deformation stage of elastic increment shown

in (e-a). (c) shows the Delaunay tessellation of the system of the

deformation stage of plastic shear events shown in (e-b). (d) shows

the DT with high εεεreq, 30 times greater than the average, of the

system (e-b). (e) shows the location of system a elastic increment

and b plastic shear event in the stress strain curve. . . . . . . . . . 119

4.25 (A)The simulation box with 50, 000 united atoms of glassy material

C500. (B) The elastic increment state and plastic shear event state

is illustrated in the von Mises equivalent shear stress-strain curve. 120

xiii

4.26 The distribution of distributions of Delaunay Tetrahedra of elastic in-

crement state at −εzz = 0.0703 of the glass material of C500. (A)

shows the the surface of the 3d distribution of distributions of De-

launay Tetrahedra with respect to the von Mises equivalent relative

atomic strain εreq and their volume. (B) shows the contour of the

distribution. (C) shows the DT distribution respect to the volume

only. (D) shows the DT distribution respect to εreq. . . . . . . . . 121

4.27 distribution of distributions of Delaunay Tetrahedra of elastic incre-

ment state at −εzz = 0.0704 of the glass material of C500. (A)

shows the the surface of the 3d distribution of distributions of De-

launay Tetrahedra with respect to the von Mises equivalent relative

atomic strain εreq and their volume. (B) shows the contour of the

distribution. (C) shows the DT distribution respect to the volume

only. (D) shows the DT distribution respect to εreq. . . . . . . . . 122

4.28 Autocorrelation function of the von Mises equivalent relative atomic

strain εreq for glassy materials. . . . . . . . . . . . . . . . . . . . 124

4.29 Covariance function of the von Mises equivalent relative atomic strain

εreq. for glassy materials . . . . . . . . . . . . . . . . . . . . . . . 126

4.30 System size effect on autocorrelation function (top) and covariance

function (bottom) of the von Mises equivalent relative atomic strain

εreq in plastic shear events of pentane. . . . . . . . . . . . . . . . . 128

4.31 cov(0) varies with respect to system size N of pentane (dotted line)

and cov(0) of the glassy materials plotted with respect to N (line). 129

xiv

Chapter 1

Introduction

1.1 Glassy Materials

Glassy materials are materials solidified from the molten state without crystal-

lization. Glassy materials are usually produced when a suitably viscous molten

material cools very rapidly, thereby not giving enough time for a regular crys-

tal lattice to form. Glassy materials have highly variable mechanical and optical

properties. Typical glassy materials are made by silicates fusing with boric oxide,

aluminum oxide, or phosphorus pentoxide [1], like window glass. Glassy materials

can also be made by any liquid subjected to a rapid cooling rate [2].

To understand glass, we should compare it with solids and liquids. Solids

have two major classes: crystalline, and amorphous [3]. Some solids, like ice

and salts, have a crystalline structure on microscopic scales. These materials are

called crystalline solids. Crystalline solids are characterized as rigid bodies, and

the molecules in these materials are arranged in a regular lattice. As the crystalline

solid is heated, the molecules vibrate about their position in the lattice until, at the

melting temperature Tm, the crystal breaks down, and the molecules start to flow.

This is the point at which the material changes phase from solid to liquid. For

such materials, there is a sharp distinction between the solid and the liquid state,

which is separated by a first order phase transition, i.e., a discontinuous change

1

2

in the properties of the material such as density. [4,5] Some solids, like window

glass, have non-ordered structure on the microscopic scale. These materials are

called amorphous solids, and have no long-range order, i.e., no regularity in the

arrangement of molecular constituents on a scale larger than a few times the size

of these groups [6].

If a liquid can be easily cooled below its melting temperature Tm without

crystallization, it is a good glass former. The viscosity of a liquid is a measure of

its resistance to flow. As a liquid is cooled, its viscosity normally increases, but

viscosity also has a tendency to prevent crystallization. Usually when a liquid is

cooled to below its freezing point, crystals form and it solidifies. But sometimes a

liquid can become supercooled and remain liquid below its freezing temperature,

from a lack of nucleation sites to initiate the crystallization. If the viscosity rises

enough as it is cooled further, it may never crystallize. The relaxation time of the

static and dynamical properties of such systems increase very quickly [7]. At a

certain temperature, the relaxation time exceeds the time scale of the experiment,

and therefore the system will fall out of equilibrium. This falling out equilibrium is

referred to as the glass transition. At temperatures well below the glass transition

temperature Tg, no relaxation appears to take place, and a material in this state

is called a glass.

Crystalline materials are quite well understood [8] from X-ray diffraction

along with other diffraction experiments [9,3]. Most text books on solid state

physics emphasize the study of crystalline materials such as Introduction to Solid

State Physics, by Charles Kittel [3]. Only one out of eighteen chapters deal with

non-crystalline solids, probably because glassy materials are more complex than

crystals. The disordered molecules (or atoms) in glassy materials make it difficult

3

to investigate structures using the same methods (such as diffraction) used to

investigate crystalline solids. Diffraction techniques rely on the presence of long-

range order. Only scattering methods and several spectroscopic techniques are

available to investigate the structure of glassy materials [10]. As a result, much

less is known about the structure-property relationships of glasses than those of

crystalline solids.

Glassy materials keep their cohesive structure by van der Waals forces

or/and chemical bonds. Glassy materials can be divided into two classes by their

components: atomic glasses and molecular glasses. Atomic glasses are composed

of individual atoms or ions with no specific chemical bonds between the atoms,

while molecular glasses are composed of molecules which have chemical bonds

between atoms. If the molecules are large, we called them macromolecules or

polymers. Polymer glasses are of practical importance in our everyday life.

Polymers are giant molecules, composed of many atoms, and often in a

chain-like structure [11]. The chains are long, entangled and composed of monomer

units. A monomer unit is a small group consisting of a few atoms connected to

other monomers by covalent bonds. The manner in which atoms are connected to

form a polymer is called its primary or covalent structure. The covalent structure

of polymers used in experiments or computer simulations is always known in

advance [12]. However, the crucial aspect of polymers in simulations is their

three-dimensional shape, called conformation or tertiary structure [13]. Although

the atoms in the molecule are joined together by strong covalent bonds, their

position in space with respect to each other need not be fixed. The majority of

common synthetic polymers, as well as all protein molecules, have single C − C

chemical bonds along their backbone. Such molecules are flexible because parts of

4

a molecule can rotate around the single bonds. For example, a polyethylene chain

contains N monomers, and each monomer has two C − C bonds, which leads to

a total of 2N (freely) rotatable torsion angles per chain. This large number of

degrees of freedom makes the chain look like a random walk in vacuum or diluted

in a solvent [11].

Polymer substances are diverse, and include plastics, rubber, fibers, polymer

films and all the various polymers found in nature [14]. This variety arises be-

cause polymer substances are composed of very long, strongly entangled molecular

chains and because of the diversity in their chemical structure, ranging from sim-

ple linear homopolymers to branched polymers, hyper-branched polymers, stars,

H-shaped polymers and copolymers [11]. Depending on the kind and strength of

interactions between the monomers, polymers can exist in four different states:

melt state, elastic state, semi-crystalline state and glass state [12]. In all the states,

the material consists of densely packed, multiple entangled polymer chains. We

will focus mainly on polymer melts and glasses.

Since the glassy materials, including amorphous polymers, serve primar-

ily mechanical purposes such as cases for electronic and other devices, as beams,

gears, struts, supports and packaging material, the mechanical behavior, like plas-

tic deformation, of glassy material is very important and widely studied.

A force applied to a static structural material will cause the material to

change shape. This change in shape is called deformation. A temporary shape

change that is self-reversing after the force is removed is called elastic deformation.

Permanent deformation of the material is called plastic deformation. Plasticity

is the ability of a solid body to undergo plastic deformation in response to an

applied force, and it is a property of solids. Plastic deformation is of technological

5

importance in shaping and construction technologies.

Plastic deformation of crystalline solids is quite well understood from X-ray

diffraction, neutron diffraction and other experiments, such as transmission elec-

tron microscopy (TEM), which expose the elementary processes of plastic defor-

mation [15]. Crystal plasticity depends on nucleation, mobility, and interaction

of dislocations. Since dislocations are linear lattice defects with a long-ranged

strain field, they can be observed directly in transmission electron microscopy.

As a consequence, the relationship between plasticity and molecular structure of

metallic materials is well understood. This led to the development of many novel

alloys which have very high strength, such as stainless steel.

The plastic deformation of glassy materials is far less understood. Since the

molecular structure lacks of long range order, it can only be characterized in terms

of static distribution functions.

The research methods for plastic deformation are divided into three classes.

One class is phenomenological study, such as creep experiments, strain-rate or

calorimetric experiments. The second is computer simulations [16–23]. The third

is based on spectroscopic experiments, such as solid state nuclear magnetic res-

onance (NMR) spectroscopy [24–27]. Knowledge of plastic deformation is based

on the combination of all three methods.

The mechanism of plastic deformation is very important in practice and

research. It was found that the mechanism of plastic deformation is totally dif-

ferent in crystalline and glassy materials. In crystalline materials, the plasticity

is the result of mobility of lattice defects, such as dislocations and voids [28,29].

The mechanism of plasticity for glassy solids, on the other hand, is repeated and

spontaneous nucleation of stress-relaxation events [19], which lead to regions with

6

irreversible cooperative rearrangement of molecular segments [30]. These irre-

versible permanently deformed regions, called plastic shear zones (PSZ) [16], are

a crucial feature of plastic deformation in glassy materials. These regions also in-

dicate the stress-relaxation events are localized [19,31]. The localization of plastic

relaxation events will be discussed in greater detail in chapter 4.

Plastic Deformation controls many related properties, such as fracture and

impact behavior. Hence, its fundamental understanding could promote the de-

velopment of novel advanced materials and facilitate the optimization of non-

mechanical properties without compromising mechanical performance.

1.2 Objectives

The understanding of structure-property relationships is very important in the ap-

plications of materials and developing novel materials. Since crystalline materials

are well understood, many relationships between the structure and properties of

crystalline materials are known. These properties have been exploited technologi-

cally. But the glassy materials lack long range order in their molecular structure.

Thus characterizing their molecular structure is difficult, and it is hard to find

structure-property relationships. This is the challenge to study the molecular

glassy materials, especially polymer glasses, whose structures are complex due to

long chains. The properties of polymer glasses are highly variable. The problem

which we address in this thesis is how polymer glass properties are affected by the

chain length.

Amorphous structures can be investigated experimentally by spectroscopic

methods, X-Ray diffraction, neutron diffraction, NMR etc. Some models that de-

scribe phenomenologically the behavior of plastic deformation of molecular glasses

7

[16,32–40]. But the response of glassy polymers to large strain deformation below

their glass transition temperature (plastic deformation), has not been explained

convincingly on a molecular length scale.

An important question regarding the mechanism of plastic deformation is

the length scale of plastic relaxation. The length scale of the elementary processes

of plastic relaxation in glassy materials is the size of the region (localization)where

individual plastic relaxation events occurs, which is a key to understanding the

mechanism of plastic deformation of glassy materials. In polymer glass, there

are some length scales like bond length l, resistant length (correlation length

of bond vectors) [41], end-to-end length [42] etc. The resistant length is about

10 l. The end-to-end length is about 100 l. How much is the length scale of

elementary processes of plasticity in polymer glassy is still an open question. This

thesis try to quantify this length scale by correlation function of relative atomic

strain. Relative atomic strain is a strain in atomic lever, defined by the strain of

an atomic tetrahedron (tetrahedron formed by atoms) relative to the strain of the

system.

1.3 Methods

The work described in this thesis involves the study of localization of plastic

relaxation events in glassy materials by computer simulation. Computer simu-

lations become more and more important in physics. One reason follows from

the nonlinear nature of phenomena, which arises from the heterogeneous or non-

equilibrium within materials and implies that solutions are not possible from an-

alytical methods. The computer is very useful for exploring nonlinear, hetero-

geneous, non-equilibrated phenomena. Another reason is the growing interest in

8

complex systems that have many variables or many degrees of freedom. Also, dif-

ficult experimental conditions such as zero temperature, high temperature, high

pressure are trivial to model on a computer.

The glassy material is a nonlinear, non-equilibrated, sometimes heteroge-

neous system. Computer simulations provide a very good way to study glassy

materials since such simulations allow investigation of macroscopic properties of

experimental interest (the equation of state, transport coefficients, structural or-

der parameters, etc) starting from the microscopic details of a system (the masses

of the atoms, the interactions between them, molecular geometry etc). When the

results are compared with those of real experiments, computer simulation pro-

vides a test of theories or underlying models. Computer simulations can help in

interpretation of experimental data, test theoretical concepts, and provide insight

into behavior not easily accessible through experiment or theory.

The development of powerful computers made it possible to simulate and

analyze mathematical models. The first simulation of a liquid was carried out in

1953 by Metropolis, Rosenbluth, Teller and Teller at the Los Alamos National

Laboratories in the United States [43]. A survey of the history of polymer simu-

lation and today’s methods is given in reference [44].

The four main length scales in computer simulations are electronic structure

scale, atomic scale, Microstructural or Mesoscale and Continuum. Different scales

involve different methods. For the electronic structure scale, the length is about

0.1 nanometer. In such a length scale, quantum mechanics is needed to describe

the motion of the particles [45]. By solving Schrodinger equation, it is possible

to determine the motion of the particles and the structure of the molecules, then

ultimately, the properties of the system. This method is referred to as the ab

9

initio method [46,47] and is widely employed. The term ab initio means from first

principles. It does not mean that we are solving the Schrodinger equation exactly.

It implies that we are selecting a method, that in principle, can lead to a reasonable

approximation to the solution of the Schrodinger equation. There is a sub-class

of the methods called the molecular orbital method (generally referred to as the

Hartree-Fock method) [47]. The essential idea of the Hartree-Fock or molecular

orbital method is that, for a closed shell system, the electrons are assigned two

at a time to a set of molecular orbitals. But the Hartree-Fock method is not

capable of giving the correct solution to the Schrodinger equation if a very large

and flexible basis set is selected, since the electrons are not paired in the way

that the Hartree-Fock method presupposes. This, in turn, suggests that the two

electrons have the same probability of being in the same region of space as being

in separate symmetry equivalent regions of space. The Hartree-Fock method only

evaluates the repulsion energy as an average over the molecular orbital. A large

number of methods has been used to improve the Hartree-Fock method. One of

them is density functional theory (DFT) [48] which has been around for many

years but only recently (1990) applied to chemical problems. In the first stage of

DFT, the energy is expressed as a functional of the density of a uniform electron

gas (E(ρ)). The function is then modified to express the electron density around

molecules.

The atomic scale is about 10 nano-meters (10−8 meters). In this length

scale, classical mechanics is sufficient for describing the motions of the heavy

atoms. When is classical mechanics a reasonable approximation? In Newtonian

physics, any particle may possess any one of a continuum of energy values. In

quantum physics, the energy is quantized, not continuous. That is, the system can

10

accommodate only certain discrete levels of energy, separated by gaps. At very

low temperatures these gaps are much larger than thermal energy, and the system

is confined to one or just a few of the low-energy states. Here, we expect the ‘dis-

creteness’ of the quantum energy landscape to be evident in the system’s behavior.

As the temperature is increased, more and more states become thermally accessi-

ble, the ‘discreteness’ becomes less and less important, and the system approaches

classical behavior. For a harmonic oscillator, the quantized energies are separated

by ∆E = hf , where h is Planck’s constant and f is the frequency of harmonic

vibration. Classical behavior is approached at temperatures for which kBT ≫ hf ,

where kB is the Boltzmann constant and kBT = 2.49 kJ/mol at 300 K. Setting

hf = 2.49 kcal/mol yields f = 6.25ps−1, or 209cm−1. So a classical treatment

will suffice for motions with characteristic times of a ps or longer at room tem-

perature. The classical Molecular Dynamics (MD) method [49] and Monte Carlo

(MC) method [43] are used in atomic scale for thermal average quantities, which

quantum mechanics does not usually yield. These two methods will be discussed

more in Chapter 2.

The mesoscale refers to a length of micrometers. The Finite Element Method

(FEM) [50] is best suited for this scale. The continuum is of the order of centime-

ters. The simulation in this scale has methods as Finite Element Analysis (FEA)

[51]and Computer Aided Design (CAD) [52]. The scale of the simulation can be

showed as in figure Fig. 1.1: In this research, the MD and MC method will be

used since the atomic scale is of interest.

The goal of this research is to understand plasticity in glassy materials at

the molecular level by studying the effect of increasing chain length and molecu-

lar structure on the localization of stress-relaxation events in glassy solids under

11

Electron

Structure

(ab initio)

Atomistic

(MD/MC)

Microstructure

(FEM)

Continuum

(FEA/CAD)

1 104

102

107

Fig. 1.1: Four length scales in computer simulations: electronic structure,atomic, Microstructural or Mesoscale and Continuum. Different scaleshave different lengths and methods.

plastic deformation. We use a novel approach to study the localization of relative

atomic strain in discrete relaxation events during plastic deformation of glassy

materials as a function of the chain length of linear molecules. The novel ap-

proach is based on the correlation of relative atomic strain is invented, in which

Delaunay tesselation and Fast Fourier Transforms techniques are applied. Our

research provides a framework for understanding recent experimental results on

the dependence of the shear activation volume on the entanglement density of

glassy polymers [53] and to understand the mechanism of plastic deformation of

polymer glasses.

1.4 Layout

The thesis is arranged as follows. In chapter 2 we provide the necessary back-

ground on glassy materials, polymers and computer simulations of glassy mate-

rials. We introduce two simulation methods, the Molecular Dynamics method

(MD) and the Monte Carlo Method (MC). Chapter 3 is devoted to the athermal

12

simulation of plastic deformation at constant pressure. A new approach to study

localization of plastic relaxation events and the relationships between mechanical

properties and chain length will discussed in chapter 4. Finally, chapter 5 includes

the results on the length scale of plastic shear events.

Chapter 2

Theory and Background

In this chapter, we introduce modeling of the glassy solids, general computer

simulation, amorphous structures, and shear transition zone theory.

2.1 Classify Glassy Materials

We need to classify glassy materials by chemical bond when we model and simulate

them. The reason is that the potential energy is the most important thing we need

to consider [44] and the potential energy between atoms connected by chemical

bond is much different from those without chemical bonds. Formation of chemical

bonds, especially between elements with a large difference in electronegativity, can

cause large changes in energy which often overwhelm all other effects [54]. For

example, the heat of formation of CuO is -1.63 eV [55]. By comparison, chemical

effects in metallic alloying are much smaller. For example, the heat of mixing

of equiatomic Cu-Ag is 20 meV [56]. Based on the nature of chemical bonds,

glassy materials are divided into two classes, atomic glasses and molecular glasses.

Atomic glasses lack chemical bonds between atoms. The atoms in molecular

glasses are connected to other atoms with chemical bonds to form molecules. If

the molecules are huge, we called them macromolecules or polymers. Polymers

are giant molecules, composed with many atoms, often formed into a chain-like

13

14

structure and play an important role in the biolog and technology [14].

2.2 Modeling the Physical System

When we simulate a material system, the first step is to specify the model. Ma-

terials are composed of atoms, and atoms are composed of electrons and nuclei

[57]. A system of simulation for a certain material composed of electrons and

nuclei can be described by a total wave function, whose form is determined by its

Hamiltonian with the Schrodinger equation [58]:

ıh∂

∂tφ(r, t) = Hφ(r, t) (2.1)

where t is the time, r is the coordinates of the particles, and the operator H is the

Hamiltonian operator. The true Hamiltonian must deal with electrons and nuclei

including their various Columbic and magnetic interactions. In principle, one

should solve the Schrodinger equation for the total wave function of the system,

and then everything about the system can be known [46,47]. Of course this

is impossible to carry out in practice, and approximation schemes have to be

employed.

Born-Oppenheimer Approximation

Born and Oppenheimer noted in 1923 [59] that nuclei are much heavier than

electrons and move on a time scale which is about two orders of magnitude longer

than that of the electrons:

ωele

ωnuc∼

√

Mnuc

mele∼ 100 (2.2)

15

The nuclei can therefore be regarded as fixed as far as the electronic part of the

problem is concerned. Born-Oppenheimer approximation separates the electronic

motion and the nuclear motion in molecules. That is, a molecular wave function

in terms of electron coordinates r and nuclear coordinates R can be written as

φmolecule(r,R) = φelectrons(r,R)φnuclei(R). (2.3)

The full Hamiltonian for a molecular system is

Hmolecular(r,R) = Kelectrons(r) +Knuclei(R) + Velectrons(r) + Vnuclei(R)

+ Velectrons−nuclei(r,R),

(2.4)

where Kelectrons is the kinetic energy of electrons, Knuclei is the kinetic energy of

nuclei, Velectrons is the potential energy of electrons, Vnuclei is the potential energy

of nuclei, Velectrons−nuclei is the potential energy between electrons and nuclei.

Electrons react instantaneously to changes in the position of the nuclei since

it is much faster than that of nuclei (about 100 times). The ”electronic” Hamil-

tonian (one that neglects the kinetic energy term for the nuclei) is

Helectrons(r,R) = Kelectrons(r) + Velectrons(r) + Vnuclei(R)

+ Velectrons−nuclei(r,R).

(2.5)

The Schrodinger equation describing the motion of electrons in a field of fixed

nuclei is:

Helectrons(r,R)φelectrons(r,R) = Eeff(R)φelectrons(r,R), (2.6)

16

where Eeff is the effective nuclear potential function. Then equation (2.4) can be

rewritten as

Hmolecular(r,R) = Knuclei(R) + Eeff (R) (2.7)

= Hmolecular(R). (2.8)

Classical Mechanics Approximation

Consider a particle with mass m in an equilibrium system at temperature T ,

where it will have a Maxwellian velocity distribution (in each direction) with

< p2 >= 3mkBT . The particle’s width σx is the standard deviation of its wave

function distribution and has the relationship with the uncertainty in momentum

prescribed by Heisenberg’s uncertainty principle as

∆σx∆p ≥ h

2. (2.9)

∆σx ≥h

2√

3mkBT. (2.10)

A critical quantum width would be 0.1 A. For a carbon atom at room temperature,

T = 300 Kelvins, the quantum width is 0.058 A.

There will be quantum effects if the forces acting on the particle vary appre-

ciably over the width of the particle. The momentum change is the expectation

value of the force as

F =d < p >

dt(2.11)

= −〈dV

dx〉. (2.12)

17

Expanding the potential in a Taylor series, we see that the leading correction term

on the force is proportional to the second derivative of the force times the variance

of the wave packet:

〈dV

dx〉 = (

dV

dx)<x> +

1

2(d3V

dx3)<x> < (x2− < x >2)2 > + · · · . (2.13)

The motion is classical if the gradient of the force does not vary much over the

quantum width of the particle. For electrons near point charges the force varies

enormously over the quantum width and the classical approximation fails com-

pletely.

Classical Model

To study the plastic deformation of glassy materials, we need to assume that the

atomic motion can be described by classical mechanics. We also usually assume

that the atoms are spherical, chemically inert, and their internal structure can

be ignored. Suppose the system consists of N classical particles. The positions

and momenta of the constituent set of particles can be expressed as generalized

coordinates q and the generalized momentum p, where:

q = (q1,q2, . . . ,qN) (2.14a)

p = (p1,p2, . . . ,pN) (2.14b)

and qi is the generalized coordinate of particle i and pi is its generalized mo-

mentum. The microscopic state of the system can be specified in terms of the

positions and momentum. The Hamiltonian H of the system can be expressed as

18

(2.15):

H(q,p) = K(p) + V(q), (2.15)

where K is kinetic energy, which takes the form

K =∑

i

P2i

2mi

=N

∑

i

1

2mi|qi(t)|2;

(2.16)

V is the potential energy, and takes the form

V = V (q). (2.17)

Once V is known, we can find the forces on each particle:

F = − V (q1, . . . ,qN) (2.18)

From the Hamiltonian, H, an equation of motion which governs the entire

time-evolution of the system and all its mechanical properties can be constructed.

H also dictates the equilibrium distribution function for molecular positions and

momenta. The Hamiltonian is the basic input for a computer program. If the

system temperature T is known, we just need to provide V since

K =3

2NkBT, (2.19)

where, kB is the Boltzmann constant. For particle i, the momentum, Pi, takes

the form

Pi = mivi, (2.20)

19

where vi is the velocity of particle i, and its magnitude obeys the Boltzmann

distribution [57]. The initial speed of each particle can be assigned statistically.

The potential energy between atoms connected by chemical bond is much

different from that between free atoms. The atoms in the system could be bonded

and non-bonded. We can separate them as bonded potential energy U b and non-

bonded potential energy Unb. Then the total potential energy U is:

U = Unb + U b (2.21)

For non-bonded atoms, we break up the potential energy into terms which

depend on the coordinates of individual atoms, pairs, triplets, etc.:

V(r) =∑

i

v1(ri) +∑

i

∑

j>i

v2(ri, rj) +∑

i

∑

j>i

∑

k>j>i

v3(ri, rj, rk) + . . . (2.22)

Here v1(ri) is the individual potential, which represents the effect of an exter-

nal field on the system; v2(ri, rj) is the pair potential, most important part;

v3(ri, rj, rk) is the triplet potential, which has magnitudes up to 10 percent of

the v2. Four-body (and higher) terms in equation (2.22) are expected to be small

in comparison with v2 and v3. Three-body and higher terms are only rarely in-

cluded in computer simulations, because the contribution to the total energy is

small. In addition, the calculation is time consuming. For example, summation

over v3 is about N times longer time than v2. The average three body effects

can be partially included by defining an “effective” pair potential. We can then

rewrite the (2.22) as [44] :

V(r) =∑

i

v1(ri) +∑

i

∑

j>i

veff2 (rij), (2.23)

20

where rij = |ri − rj|.

In principle, the form of V for electrically neutral molecules can be con-

structed from first principles. But such a calculation is very complex and difficult,

which limits such an approach to small systems (about one thousand atoms only).

It usually is sufficient to choose a simple phenomenological form for V. The most

important features of V are a strong repulsion for small separation and a weak

attraction at large separation. In describing the interaction of two atoms, we dis-

tinguish between non-bonded (intermolecular) and bonded (intramolecular) atom

pairs. One of the most common phenomenological forms of pair potential of two

non-bonded atoms is the Lennard-Jones potential:

V LJ(r) = 4ǫ[(σ

r)12 − (

σ

r)6], (2.24)

where r is the distance between two atoms, and ǫ and σ represent the energy and

length parameters, respectively. The term∼ 1/r6 dominates at large distances and

is the attractive component, which results from van der Waals dispersion forces

(also called London Force), caused by dipole-dipole interactions from fluctuating

dipoles. The ∼ 1/r12 term is dominant at short distances and is the repulsive

component. Its physical origin is related to the Pauli principle (The electron

wave functions of two molecules must distort to avoid overlap, causing some of

the electrons to be in different quantum states. The net effect is an increase

in kinetic energy and an effective repulsive interaction between the electrons).

Its form should be exponential but it can be approximately fit with 1/r12. The

Lennard-Jones potential is the standard potential for all the investigations where

the focus is on fundamental issues rather than studying the properties of a specific

material. A typical plot of the Lennard-Jones potential is shown in Fig. 2.1.

21

ǫ

V

σ r

V LJ(r) = 4ǫ[(σr )

12 − (σr )6]

Fig. 2.1: A typical plot of the Lennard-Jones potential V (r) in equation (2.24)

The potential energy between atoms in the same chain but separated by

more than three bonds or in a different chain is considered as the non-bonded

potential energy Unb.

Unb = V LJ(r). (2.25)

The corresponding forces between a pair of non-bonded atoms in the system

are:

FLJ (r) = −∇V LJ(r)

= 24ǫr

r2[2(

σ

r)12 − (

σ

r)6],

(2.26)

where r = rj−ri is the position vector from position of atom i to position of atom

j.

22

Chemical bonds are, in principle, also inter-atomic potential energy terms.

The potential energy of bonded atoms U b depends on the geometry and types of

bonds and atoms. For bonded atoms in the polymer glasses, there are three kinds

of potentials that need to be considered: harmonic bond potential ubond for two

beads connected by a bond, bond-angle potential uang for adjacent two bonds,

and torsion angle potentials utors for three successive bonds. That is:

U b = ubond + uang + utors. (2.27)

The harmonic bond potential ubond for two atoms connected by a bond is:

ubond(l) = kl(l − l0)2 (2.28)

where kl is the spring constant, l and l0 are the distance between two consecutive

atoms and the equilibrium bond length, respectively.

The bond-angle potential uang is given by

uang(θ) = kθ(θ − θ0)2 (2.29)

where kθ is the bond-angle-potential coefficient, θ is the bond angle between ad-

jacent bonds, and θ0 is the equilibrium bond angle.

The torsion angle potential utors is defined by

utors(φ) =∑

ck cosk(φ) (2.30)

where φ represents the torsion angle for three successive bonds and ck are co-

efficients for the potential. Torsion motions involve energy changes comparable

with normal thermal energies. Moreover, torsion motions are the main degrees of

23

freedom allowing a single molecule to change its 3-dimensional shape.

2.3 Strain

Deformation is a change of shape or volume of a physical body. Strain is the

geometrical expression of deformation caused by the action of stress on a phys-

ical body [60]. If strain is equal over all parts of the body, it is referred to as

homogeneous strain; otherwise, it is inhomogeneous strain.

Four kinds of strain in a physical body are discussed in this thesis. They are

global strain, infinitesimal strain atomic strain and relative atomic strain. Global

strain and infinitesimal strain are concepts used in continuous matter, including

both solids and fluids (i.e., liquids and gases). Global strain is the strain of a

whole physical body. Infinitesimal strain is the strain of a infinitesimal part of the

physical body. Atomic strain is the strain of atomic tetrahedron, which formed

by 4 atoms at the vertices. Relative atomic strain is the atomic strain relative to

global strain.

2.3.1 Global Strain

Global strain is the strain of a whole physical body. Given that strain results

in the deformation of a body, it can be measured by calculating the change in

length of a line or by the change in angle between two lines (where these lines are

theoretical constructs within the deformed body). The change in length of a line

dl is termed the stretch. The strain may be given by

dε =dl

l0, (2.31)

24

where l is the length after deformation, l0 is the original undeformed length.

equation (2.31) can be rewritten as

ε = ln

(

l

l0

)

. (2.32)

The strain ε defined by equation (2.32) is referenced as true strain. If ε is

positive, the body has been lengthened; if it is negative, it has been compressed.

In structural engineering the strain is given as:

ε =∆l

l0(2.33)

where ∆l is the change in length. The strain defined by equation (2.33) is refer-

enced as engineering strain. This definition is only valid for small deformation. It

is an approximation of equation (2.32) as ∆l = l − l0 is small compare to l0.

When an object undergoing a 3-dimensional deformation, the global strain

is a tensor εεε, represented by a 3× 3 matrix. In this thesis, we use εεεg to distingue

it with other strains.

Strain Increment

In the plastic deformation simulations in this thesis, the large deformations are

simulated by many small deformations, quantified as the strain increment δεεε, [61].

A plane strain deformation mode was used with a strain increment

δεεε =

δεxx 0 0

0 0 0

0 0 δεzz

, (2.34)

25

and a constant deformation step δεzz. This corresponds to a plane strain com-

pression. Under conditions of conserved sample volume, or for an ideally incom-

pressible material, εxx = −εzz.

2.3.2 Infinitesimal Strain

Infinitesimal strain is the strain of a infinitesimal part of the physical body and

is used to describe the strain field. As a physical body is deformed globally, the

strain at each point may vary dramatically during the deformation. To track the

strain at each point, further refinement in the definition is needed.

When we discuss deformations, a very typical notation is to use X′ = X ′iei

to indicate the configuration (the set of spatial positions of the material points of

the object) of undeformed state, which sometimes called reference state, the state

at time t = zero, and use X = Xiei to indicate the configuration of deformed

state at time t. We assume that reference configuration is undeformed. In other

words, the strain of reference configuration is zero. In the absence of singularities

such as rupture and crack, X is continuously differentiable. And we can express

X as a differentiable function:

X = X(X′, t). (2.35)

Then the state of deformation can be characterized by the derivative of X with

respect to X′, which is called the deformation gradient tensor:

Fij =∂xi

∂x′j

(2.36)

If we have a small vector dx′ in the deformed body, then the corresponding vector

26

in the deformed body dx can by calculated as

dxi = Fijdx′j (2.37)

The change in volume dV during the deformation is given by the determinant of

F as

dV

V0

= detF, (2.38)

where V0 is the volume of reference state. The deformation gradient tensor pro-

vides information about both the true deformation of the body, and solid body

rotation. The deformation gradient tensor F can be expressed as a product of a

symmetric tensor V for true deformation and a unitary tensor R for rotation as

F = V·R, (2.39)

where V = VT. As superposition of rotation and the inverse rotation leads to

no change (R · RT = 1) we can exclude the rotation by multiplying F by its

transpose,

B = F · FT = V ·VT . (2.40)

This tensor is named the Finger tensor [62]. We can rewrite the Finger tensor in

Cartesian form as

Bij =∑

k=1..3

∂xi

∂x′k

∂xj

∂x′k

. (2.41)

Physically speaking, the Finger tensor gives us the local changes in area within a

sample:

µ2 = nT ·B · n, (2.42)

27

where µ is the ratio of un-deformed surface to the deformed surface and n is the

normal vector to the undeformed surface.

If we reverse the order of multiplication in the formula for the Finger tensor

we would get the Cauchy-Green tensor C as:

C = FT · F (2.43)

Physically, the Cauchy-Green tensor gives the local change in distances due

to deformation:

α2 = n′T ·C · n′ (2.44)

where α is the ratio of lengths of a vector in deformed and undeformed states and

n′ is the direction of the vector in undeformed state.

The rate of deformation is given by the gradient of the velocity field v = X

with respect to X as:

L =∂v

∂X= F · F−1. (2.45)

The rate of strain tensor D is given by the symmetric part of L as:

D =1

2(L + LT ). (2.46)

The strain tensor εεε can be given by the gradient tensor F as:

εεε =1

2ln(FT · F), (2.47)

where FT is the transpose F. If we define the relative displacement u as

u = X−X′, (2.48)

28

Then we have

X = X′ + u, (2.49)

we write Fij as

Fij = δij +∂ui

∂X ′j

. (2.50)

Then

(FT · F)ik = (δji +∂uj

∂X ′i

)(δjk +∂uj

∂X ′k

) (2.51)

= δik,j +∂uk

∂X ′i

+∂ui

∂X ′k

+∂uj

∂X ′i

∂uj

∂X ′k

. (2.52)

For small deformation,

∂uj

∂X ′i

∂uj

∂X ′k

≪ 1. (2.53)

We can have:

(FT · F)ik = δik,j +∂uk

∂X ′i

+∂ui

∂X ′k

(2.54)

= (FT + F)ik. (2.55)

Then we have εεε for small deformations as

limF→1

εεε =1

2(FT + F− 1). (2.56)

Considering a cube with an edge length a. It is a quasi-cube after the

deformation with the volume V :

V = a · (1 + ε11)× a · (1 + ε22)× a · (1 + ε33), (2.57)

29

and V0 = a3, thus the relative variation of the volume ∆V/V0 is:

∆V

V0=

(1 + ε11 + ε22 + ε33 + ε11 · ε22 + ε11 · ε33 + ε22 · ε33 + ε11 · ε22 · ε33) · a3 − a3

a3

(2.58)

If we consider small deformations, where,

1 >> |εii|, (2.59)

the relative variation of the volume ∆V/V0 becomes the trace of the tensor :

∆V

V0= ε11 + ε22 + ε33 (2.60)

In case of pure shear, that is

ε11 + ε22 + ε33 = 0, (2.61)

we can see that there is no change of the volume.

Tensor Invariants

The components of a tensor matrix, such as the infinitesimal strain tensor, change

with the choice of coordinate system. That is if Cartesian coordinates are used

for cylinder’s deformation, different tensor matrix components will result than if

cylindrical coordinates are used. Additionally, if the reference frame even using

Cartesian coordinates is rotated then the matrix will completely change. If the

tensor itself does not change then there are some features of the tensor that

can be calculated from any coordinate system that remain unchanged on change

of coordinate system. These features are called the tensor invariants. Tensor

30

invariants are scalars that are intimately related to the tensor. A 3 × 3 tensors

has 3 invariants, which can be defined in several ways. If M is a 3 × 3 tensor

matrix and Mij are the components. The three invariants can be defined as

IT =∑

i

Mii (2.62)

IIT =∑

i

∑

j

MijMji (2.63)

IIIT =∑

i

∑

j

∑

k

MijMjkMki (2.64)

IT is referred as first rank invariant, IIT is called second rank invariant, and IIIT

is called third rank invariant. The system pressure is a first rank invariant of the

stress tensor. Volume change is a first rank invariant of the strain tensor. The

von Mises equivalent stress is a second rank invariant of the stress and von Mises

equivalent strain is a second rank invariant of the strain. Their general definition

will be given in the following.

Von Mises Equivalent Tensor

Von Mises equivalent tensor Meq is a second rank invariant of the tensor M. Meq

is defined as

Meq =

√

3

2tr (Md ·Md), (2.65)

where Md is the deviatoric part of the tensor

Md = M + MI, (2.66)

31

with the identity tensor I and M is a first rank invariant, defined as

M = −1

3trM. (2.67)

The von Mises equivalent strain εeq, a second rank invariant of strain tensor

εεε, is of special interest and is defined as

εeq =

√

3

2Tr(εεε− εI)2, (2.68)

where I is the unity tensor and ε is a first rank invariant of strain tensor εεε, defined

as

ε =1

3Trεεε (2.69)

2.3.3 Atomic Strain

Strain is a continuum concept. continuum strain includes global strain and in-

finitesimal strain. There are many ways to generalize it to the atomic level as

atomic strain. Integration of atomic strain in a finite volume leads to continuum

strain. Mott et al. [63] define atomic strain as the strain of the voronoi polyhedron

around the atom.

In this study, we define the atomic strain as the strain of Delaunay tetrahe-

dron formed by 4 atoms. The atomic strain εεεa can be calculated by the equation

(2.47). The atomic strains are distributed homogenously inside the tetrahedron

and discontinuous in the boundary of the tetrahedron (vortices, edges, faces),

however, the infinitesimal strain is continuum everywhere in the system. Before

the calculation of atomic strain, we should tell the system apart into groups of

atoms, which form the tetrahedra to fill the whole simulation cell. This procedure

32

is tessellation [64], which will discuss more in the following section.

In the plastic deformation simulations in this thesis, the large deformations

of the system (global) εεεg are simulated by many strain increment δεεε, as

εεεg = n δεεε, (2.70)

where n is the number of the deformation steps and strain increment is defined in

equation (2.34). Average of the atomic strain in ith step of deformation weighted

by the tetrahedra volume will be equal to the strain increment of ith step, as

δεεε =

∑

vjεεεaj

∑

vj

, (2.71)

where vj is the volume of j tetrahedron and εεεaj is the atomic strain of j tetrahedron.

2.3.4 Relative Atomic Strain

The atomic strain εεεa depends on the strain increment δεεε with equation (2.71) and

the strain increment depends in the number of deformation steps n if the global

strain εεεg is fixed. As a consequence, atomic strains resulting from two successive

deformation steps may vary with the number of deformation steps n. To cancel

such artificial effect of n, we define a new variable, εεεr, as a derivative of atomic

strains εεεa in response to global strain εεεg as

εεεr =∆εεεa

∆εεεg(2.72)

=1

δεεε

∆εεεa

∆n(2.73)

=1

δεεε

εεεa − 0

1(2.74)

=εεεa

δεεε. (2.75)

33

We name this variable εεεr relative atomic strain.

Keep in mind that relative atomic strain εεεr is a forth rank tensor, which can

be expressed as :

εεεrnmkl =

εεεanm

δεεεkl(2.76)

In this study, the deformation step δεzz is a constant in strain increment

tensor δεεε in equation (2.34). From equation (??), only one component of strain

increment δεεε is independent. Then equation equation (2.76) can be simplified as

a second rank tensor:

εεεrnm =

εεεanm

δεzz

. (2.77)

The first rank invariant of εεεr is

εr =

1

3Trεεεr. (2.78)

And the Von Mises equivalent relative atomic strain εεεreq is defined as:

εεεreq =

√

3

2Tr(εεεr − εr

I)2, (2.79)

where I is the unitary tensor.

2.4 Computer Simulation at the Atomic Level

In this thesis, we simulate the plastic deformation at the atomic level with classical

mechanics. According to statistical mechanics, the probability that a given state

with energy E is occupied in equilibrium at constant particle number N , volume

V, and temperature T (constant NVT , the ‘canonical’ ensemble) is proportional

34

to e−E/kBT , the ‘Boltzmann factor’ as [65]

ρ ∝ e−E/kBT , (2.80)

wherec ρ is the probability, kB is Boltzmann constant. The equilibrium value of

any, observable O is therefore obtained by averaging over all states accessible to

the system, weighting each state by this factor.

Quantum mechanically, this averaging is performed simply by summing over

the discrete set of microstates as [66]

〈O〉 =

∑

n One−En/kBT

Z, (2.81)

where Z is the partition function:

Z =∑

n

e−En/kBT , (2.82)

and On is the expectation value of the quantity O in the nth energy eigenstate:

On =

∫

Ψ∗nOΨndr. (2.83)

Classically, a microstate is specified by the positions and velocities (mo-

menta) of all particles, each of which can take on any value. The averaging over

states in the classical limit is done by integrating over these continuous variables:

〈O〉 =∫

Oe−E/kBT dpdr∫

e−E/kBT dpdr, (2.84)

where the integrals are over all phase space (positions r and momenta p) for the

35

N particles in 3 dimensions.

When all forces (the potential energy V ) and the observable O are velocity-

independent, the momentum integrals can be factored and canceled:

〈O〉 =∫

e−K/kBT dp∫

Oe−V/kBT dr∫

e−K/kBT dp∫

e−V/kBT dr

=

∫

Oe−V/kBT dr∫

e−V/kBT dr,

(2.85)

where K =∑N

i=1 p2i /2mi is the total kinetic energy, and E = K + V . As a result,

Monte Carlo simulations compare V ’s, not E’s.

2.4.1 Molecular Dynamics Method (MD)

The most obvious way to simulate a many-particle system is to compute its real

world time evolution by solving the equations of motion (Newton’s equations)

derived from the potential energy function V (r) in phase space (positions ~r and

momenta ~p) for the N particles in 3 dimensions. In most cases, this cannot be

done analytically. To solve the equations of motion for a set of Lennard-Jones

particles (particles interact by LJ potential only), an approximate, step-by-step

procedure is needed, since the force changes continuously as the particles move.

Molecular dynamics (MD) is the technique used to compute the time evolution and

the dynamic properties of many-particle systems. Computer simulation generates

detailed information at the atomic level. Methods of statistical mechanics are

used to convert this information into macroscopic terms as pressure and internal

energy. Suppose that we can write the instantaneous value of some property F (r)

(for example the end-to-end distances of the chains) such as a function of the

actual conformation r. The system evolves in time, so that r, and hence F (r)

36

will change. The experimentally observable “macroscopic” property Fobs is really

the time average of F (r) taken over a long time interval tobs. Since the equations

of motion are solved on a discrete step-by-step basis, the time average may be

written in the form

Fobs ≈< F >time

=1

tobs

tobs∑

t=1

F (r(t)),(2.86)

where t is the discrete step-by-step time and tobs is the total observation time

(steps). A practical question regarding this method is whether or not a sufficient

region of the conformational space is explored by the system trajectory to yield

satisfactory time averages within a reasonable amount of computer time. Since

the system is not likely to cross high energy barriers, the Molecular Dynamics

method will only explore a rather small region of conformational space. In other

words, a system of entangled, densely packed polymer chains is not likely to change

its 3-dimensional shape significantly. Consequently < F >time will converge very

slowly towards the observable value Fobs.

The ‘Leap Frog’ algorithm is one method commonly used to numerically

integrate Newton’s second law. We obtain all atomic positions ri at all times

tn and all atomic velocities ~vi at intermediate times tn+1/2. This method gets

its name from the way in which positions and velocities are calculated in an

alternating sequence, ‘leaping’ past each other in time:

vi(tn+1/2) = vi(tn−1/2) +Fi(tn)

mi∆t, (2.87)

37

ri(tn+1) = ri(tn) + vi(tn+1/2)∆t. (2.88)

Initial velocities are assigned so as to reflect equilibrium at the desired tem-

perature T (a Maxwellian distribution), without introducing a net translation or

rotation of the system.

2.4.2 Monte Carlo Method (MC)

Suppose an irregularly shaped pond is in a field of known area A. The area of the

pond can be estimated by throwing stones so that they land at random within

the boundary of the field and counting the number of splashes that occur when

a stone lands in a pond. The area of the pond is approximately the area of the

field times the fraction of stones that make a splash. This simple procedure is an

example of a Monte Carlo method.

Instead of evaluating forces to determine incremental atomic motions, Monte

Carlo simulation simply imposes relatively large motions on the system and deter-

mines whether or not the altered structure is energetically feasible at the temper-

ature simulated. The system jumps abruptly from conformation to conformation,

rather than evolving smoothly through time. It can traverse barriers without

feeling them; all that matters is the relative energy of the conformations before

and after the jump. Because MC simulation samples conformation space without

a true ‘time’ variable or a realistic dynamics trajectory, it cannot provide time-

dependent quantities. However, it may be much better than MD in estimating

average thermodynamic properties for which the sampling of many system con-

figurations is important. When the potential energy V and observables to be

calculated from the simulation are velocity-independent (as is typical), an MC

38

simulation need only compare potential energies V , not total energies E.

Monte Carlo method is to replace the time average in equation (2.86) by the

ensemble average. Instead of following the path r(t) governed by the equations

of motion, we choose a sequence of randomly chosen conformations r1 . . . rN to

compute the observable Pobs. But now we have to be careful because the confor-

mations do not appear with equal probability along the time trajectory r(t). The

probability for a conformation r to appear at a temperature T is

ρ(r) ∼ exp−V (r)/kBT . (2.89)

Thus, Pobs must be computed as the expected value

Fobs ≈< F >ensemble

=

∑Nk=1 F (rk)ρ(rk)∑N

k=1 ρ(rk).

(2.90)

Unfortunately, a randomly chosen conformation is most likely to have a very high

energy and thus, very low probability, because there will be pairs of atoms that are

very close to each other, resulting in high Lennard-Jones energy terms. In equa-

tion (2.90) we sum properties of non-relevant conformations and < F >ensemble

converges very slowly towards Fobs. The technique that solves this problem is

called importance sampling. The sequence of conformations r1 . . . rN is chosen

randomly, but according to the probability distribution ρ. In this case we may

rewrite equation (2.90) as a simple average

Fobs =1

N

N∑

k=1

F (rk) (2.91)

39

The most common method to generate a sequence of conformations accord-

ing to a probability distribution ρ is the Metropolis acceptance-rejection algorithm

[43]:

procedure MC;

1: begin

2: Choose a starting conformation r;

3: Loop

4: r′ ←− locally changed r;

5: probability ←− min(1, ρ(r′)/ρ(r));

6: If random() < probability, then

7: r ←− r′;

8: end if;

9: end loop

10: end

where random() returns a random number between zero and one. The current

conformation r is changed locally by a move. Examples for Monte Carlo moves

are the displacement of a single atom or the modification of a torsion angle inside a

molecule. The new conformation r′ is accepted with probability min(1, ρ(r′)/ρ(r))

or rejected otherwise. A Monte Carlo move should not completely change the

current conformation because such a move would have a very low acceptance

probability. Monte Carlo moves therefore perform local changes keeping most of

the particles frozen. With this restriction, the problem of locality arises as it

did within the molecular dynamics method. The sequence of conformations is

not likely to cross high energy barriers and the starting conformation cannot be

changed completely.

40

2.4.3 Periodic Continuation Conditions

The goal of the computer simulation is to describe the behavior of macroscopic

systems that contain an order of 1023 particles. Obviously, this cannot be done

by the molecular dynamics technique with any currently envisaged computer.

The maximum number of atoms that participate in the present simulation is 105.

Consequently, placing the boundary atoms at some fixed sites will influence the

atoms in the bulk after a short time, giving rise to undesired results.

One way to overcome this problem is to use Periodic Continuation Condi-

tions (PCC), which sometimes referenced as periodic boundary conditions. When

PCC is applied, a particle that crosses a face of the simulation box, is reinserted

at the opposite face. The primary simulated box is then periodically replicated in

all directions to form a macroscopic sample. Thus, the neighbors that surround it

and the forces applied on it would be different to those in the case of fixed bound-

ary conditions. One of the main consequences of this kind of boundary conditions

is that it will give rise to energy reflections from the boundaries, which can be

helpful in the case of amorphous crystal simulations [67,68].

2.4.4 Initial Configuration

To start the molecular dynamics simulation, we assign initial positions and veloci-

ties to all atoms in the system. For polymer simulations, the appropriate choice of

initial conditions is very important, because the results of computer simulations

can be affected greatly by this choice. A efficient way to build initial configu-

rations is suggested by Matthias Muller [69] and will discussed more detailed in

chapter 4.

In simulations of atomic glasses, the choice of initial conditions does not

41

affect the structures. The initial configuration of the system is chosen to be a

perfect cube crystal, which is heated up to a high temperature, at which it melts.

The simulation of the melting process is carried out until the equilibrium state is

reached. So in the atomic glasses obtained by quenching the melt, the memory of

the initial state is completely lost.

The initial velocities can be chosen randomly with a Maxwell distribution

[65]:

f(v) = 4π(m

2πkBT)3/2v2e

− mv2

kBT , (2.92)

where f(v) is the probable number of molecules which have velocities from v to

v + dv. Then the can be rescaled so that they can be related to the ambient

temperature T . The following relation should hold:

3

2NkT =

∑

i

1

2mv2

i . (2.93)

2.4.5 Energy Minimization

The energy landscape of a glassy material possesses an enormous number of min-

ima, or conformational substates. Nonetheless, the goal of energy minimization is

simply to find the local energy minimum, i.e., the bottom of the energy well occu-

pied by the initial conformation. The energy at this local minimum may be much

higher than the energy of the global minimum. Physically, energy minimization

corresponds to an instantaneous freezing of the system; a static structure in which

no atom feels a net force corresponds to a temperature of 0 K.

One of the more intuitive methods of finding the minimum energy structure

is to gradually extract kinetic energy (thus eventually find a 0 K configuration) by

quenching the system, i.e, removing kinetic energy by stopping the atoms when

42

the force exerted on them is in the opposite direction to their velocities. This

method will continue until the forces are sufficiently small, where the ions are in a

local minimum of energy. In practice, this can be slow and therefore more efficient

methods are employed. There are more efficient methods such as steepest descents

or conjugate gradients [70] which is able to find local minima of functions (such

as the energy as a function of atomic coordinates) much faster than a molecular

dynamics technique.

2.4.6 Constant Temperature

An intuitive method for stabilizing the temperature of the system in computer

simulation is to scale the velocities of the particles at every time step so that the

total kinetic energy is conserved, and the temperature is constant. Although this

ensures that the temperature is constant at every time step by adding or removing

energy from the system, it is not physically correct. Temperature is a statistical

quantity, and therefore an average over many time steps is required to define

the temperature. A method which allows the instantaneous temperature, T , to

fluctuate but where < T > remains constant was suggested by Nose [71]. This

couples the temperature to an external heat reservoir, allowing the temperature

to become a dynamical variable of the system. This is analogous to attaching a

“spring” to the temperature so that T can fluctuate around a mean value. This

implies that once the system has reached thermal equilibrium, the total energy

remains constant although kinetic and potential energies vary.

To use the Nose thermostat, Newton’s equations of motion have to be inte-

grated at each time step, requiring knowledge of the instantaneous force on each

particle. Calculation of the forces by ab initio methods is extremely computation-

43

ally intensive, whereas the use of empirical potentials requires only the evaluation

of simple functions. Therefore ab initio molecular dynamics is very limited, so

that even for very small unit cells, it is only practical to simulate a real time of

1 − 10 ps. For this reason, empirical potentials are the most practical method

for simulating longer timescales. A comparable simulation using an empirical

potential can extend to several tens of nanoseconds.

A Nose thermostat is used in the finite temperature molecular dynamics

simulations and all zero temperature configurations have been found by using

conjugate gradients to move the atoms.

2.5 Characterization of Amorphous Structures and Dynamics

In order to describe an amorphous structure the following characteristics can

be used: a coordination number, a radial distribution function and an angular

distribution function.

The coordination number z is the number of nearest neighbor atoms. For

example, z is 4 for the diamond structure or 12 for the FCC structure. For perfect

lattices, the coordination number has no real significance, but for more complex

structures, like amorphous lattices, it plays a crucial role in the determination of

the amorphous structure type. It will be shown in the next section that z for the

amorphous material is very close to that of the corresponding crystal.

The radial distribution function g(r) is a generalization of the coordination

number. Instead of looking at the first nearest neighbors only, one counts the

number of atoms that lie within the distance r → r + dr from a specific atom,

averaging over all the atoms of the lattice. When normalized, g(r)dr is precisely

the probability of finding a neighboring atom at distance r. The coordination

44

number can be measured by determining the integrated area under the first peak.

It is clear that for a perfect lattice, g(r) will give delta functions at characteristic

distances of the lattice. The g(r) function, as a coordination number, can be

very useful to cahracterize more complicated structures. For example, short-range

order is expressed by one of two distinct and broad peaks in the shortest distances,

followed by a quite flat tail, which is characteristic to the g(r) of amorphous

structure. For the C − C structure in polyethylene, for instance, the first peak is

centered near the bond length (1.53 A), and the second peak is centered near the

second neighbor (2.57 A). The radial distribution function g(r) can be calculated

directly from a sample created by a computer simulation and compared with

the distribution function obtained from experiment. In the latter case, g(r) is

derivable, via Fourier transformation, from the results of diffraction experiments,

i.e. from the intensity distribution as a function of the wave number [3,72]. If

ρ(r) is the concentration of atoms at distance r from a reference atom, and ρ0 the

average concentration, then the radial distribution function is given by [?]:

g(r) = 1 + G(r), (2.94)

where G(r) is the pair distribution function. It is defined as

G(r) =1

2π2ρ 0r

∞∫

dq [S(q)− 1] q sin(qr), (2.95)

where S(q) is the measured structure factor and q is the scattering vector. q is

defined as

q = (4π/λ) sin φ, (2.96)

45

where λ is the wavelength of the probing radiation and 2φ the scattering angle.

The bond angle distribution function g(θ) is defined for angles between

nearest neighbors atoms. For a diamond crystal, g(θ) is a delta function centered

at θ = 109.47. For an amorphous crystal, g(θ) is centered at θ = 114 for the

C−C structure. Large angle distortions occur in these structures, as is indicated

by the significant width of the bond angle distribution. Experimentally, the bond

angle can be obtained from the ratio of the first and second neighbor distances

(r and r1 respectively) as in reference [73]:

θ = 2 arcsin(r1/2r).

2.5.1 Tessellation

In order to calculate the relative atomic strain (a strain in atomic level, more

detail in chapter 4), we need to divided the space containing the atoms into small

pieces. This procedure of partition space called tessellation. The basic object

having volume in space is tetrahedron. Any three dimensional object can be

divided into one or more tetrahedron. Delaunay Tessellation [64,74] forms a series

Delaunay tetrahedra tiling the space.

An objective definition of nearest neighbors in three-dimensional space can

be obtained by applying the methods of statistical geometry. The statistical ge-

ometry approach for studying structure of disordered systems was introduced by

Bernal [75]. He suggested characterization of structural disorder using statistical

analysis of irregular polyhedra obtained by a specific tessellation in the three-

dimensional space. The method, including the design and implementation of prac-

tical algorithms, was further developed by Finney [76,77] for the case of Voronoi

46

tessellation. Voronoi tessellation partitions the space into convex polytopes called

Voronoi polyhedra. For a molecular system, the Voronoi polyhedron is the region

of space around an atom, such that each point of this region is closer to the atom

than to any other atom of the system. A group of four atoms whose Voronoi

polyhedra meet at a common vertex forms another basic topological object called

a Delaunay simplex. The topological difference between these objects is that the

Voronoi polyhedron represents the environment of individual atoms whereas the

Delaunay simplex represents the ensemble of neighboring atoms. Although the

Voronoi polyhedra and the Delaunay simplices are completely determined by each

other, there exists a significant difference. Whereas the Voronoi polyhedra may

differ topologically (i.e., they may have different numbers of faces and edges), the

Delaunay simplices are always topologically equivalent (i.e., in three-dimensional

space they are always tetrahedra). Delaunay tessellation has been used for struc-

tural analysis of various disordered systems. In most such cases, it has served as

a valuable tool for structure description [78,79].

In two dimensions, a triangulation of a set of vertices V is a set of triangles

T whose vertices collectively form V , whose interiors do not intersect each other,

and whose union completely fills the convex hull of V . The Delaunay triangulation

D of V , introduced by Delaunay [64] in 1934, is the graph defined as follows: any

circle in the plane is said to be empty if it contains no vertex of V in its interior.

(Vertices are permitted on the circle.) Let u and v be any two vertices of V . The

edge uv is in D if and only if there exists an empty circle that passes through

u and v. An edge satisfying this property is said to be Delaunay. This defining

characteristic of Delaunay triangles is called the empty circumcircle property.

Among all triangulations of a vertex set, the Delaunay triangulation maximizes

47

the minimum angle in the triangulation, minimizes the largest circumcircle, and

minimizes the largest min-containment circle, where the min-containment circle

of a triangle is the smallest circle that contains it. The property of max-min angle

optimality was first noted by Lawson [80] and helps to account for the popularity

of Delaunay triangulations in mesh generation.

In three dimensions, a tessellation of a set of vertices V is a set of tetrahedron

T whose vertices collectively form V , whose interiors do not intersect each other,

and whose union completely fills the convex hull of V . The Delaunay tessellation

D of V , is the graph defined as follows: any sphere in the space is said to be

empty if it contains no vertex of V in its interior. (Vertices are permitted on the

sphere.) Let u, v and w be any three vertices of V . The triangle uvw is in D if

and only if there exists an empty sphere that passes through u, v and w. This

defining characteristic of Delaunay tessellation is called the empty circumsphere

property.

The Delaunay tessellation of a vertex set is clearly unique except in lattice,

because the definition given above specifies an unambiguous test for the presence

or absence of an edge in the triangulation.

2.5.2 Torsion Angle

In a chain of atoms A-B-C-D, there is a dihedral angle between the plane contain-

ing the atoms A,B,C and that containing B,C,D). In a Newman projection the

torsion angle is the angle (having an absolute value between 0 and 180) between

bonds to two specified (fiducial) groups, one from the atom nearer (proximal) to

the observer and the other from the further (distal) atom. The torsion angle be-

tween groups A and D is then considered to be positive if the bond A-B is rotated

48

Fig. 2.2: Torsion Angle Definition

in a clockwise direction through less than 180 in order that it may eclipse the bond

C-D: a negative torsion angle requires rotation in the opposite sense. Stereochem-

ical arrangements corresponding to torsion angles between 0 and ±90 are called

syn (s), those corresponding to torsion angles between ±90 and 180 anti (a).

Similarly, arrangements corresponding to torsion angles between 30 and 150 or

between −30 and −150 are called clinal (c) and those between 0 and 30 or

150 and 180 are called periplanar (p). The two types of terms can be combined

so as to define four ranges of torsion angle; 0 to 30 synperiplanar (sp); 30 to 90

and −30 to −90 synclinal (sc); 90 to 150, and −90 to −150 anticlinal (ac);

±150 to 180 antiperiplanar (ap). The synperiplanar conformation is also known

as the syn- or cis-conformation; antiperiplanar as anti or trans and synclinal as

gauche or skew. For macromolecular usage the symbols T, C, G+, G-, A+ and

A- are recommended (ap, sp, +sc, -sc, +ac and -ac respectively). The definitions

are illustrated in Fig. 2.2.

49

2.6 Plastic Shear Zones

The plasticity of glassy material can be described as a rate process with a stress-

dependent activation energy [16]. The shear activation volume, the derivative of

the activation energy with respect to stress, is a key material parameter of me-

chanical properties. For instance, polymers with small activation volumes tend

to be brittle, since large stress concentrations can build up around imperfections

without plastic flow setting in. In contrast, ductile solids, like Biphenyl-A polycar-

bonate, are characterized by relatively large activation volumes. Macroscopically,

the plastic response of glassy polymers is well described by the Eying flow expres-

sion [81].

γ = γ0 exp(−Ea

kT) sinh(

τV ∗

kT) (2.97)

where γ is the strain rate:γ = dγdt

, γ0 is a prefatory, τ the shear stress, Ea the

activation energy, kT the thermal energy, and V ∗ denotes the shear activation vol-

ume [36]. Experimental values for V ∗ vary over an order of magnitude for different

glassy polymers. Shear activation volume V ∗ is a framework for the discussion

of the plastic deformation of glassy solids and can be derived by calculating the

elastic strain energy that is generated by a Plastic Shear Zone (PSZ) in its sur-

roundings. Immediately prior to the transition, the material is under constant

elastic shear stress τ . The work of deformation exerted on the PSZ by the elastic

continuum is then given by

∆W = τγT Ωc, (2.98)

50

where γT is the permanent deformation of the PSZ and Ωc is its volume. On

the other hand, the shape change of the PSZ induced stresses in its elastic sur-

roundings, which will be superimposed onto the homogeneous stress field τ . If

the deformation γT is “forward” (i.e.,in the same direction as the applied stress

τ , otherwise is “backward”), these elastic stress will lead to a decrease in the to-

tal elastic energy of the solid. For an incompressible material, the elastic energy

change resulting from the sudden deformation of an ellipsoidal region is given by

[82]

∆E ≈ 7− 5ν

30(1− ν)µ(γT )2Ωc, (2.99)

where µ and ν denote the elastic shear modulus and Poison’s ratio, respectively.

Note that since we apply linear elasticity, ∆E is independent of the externally

applied stress τ . The energy barrier ∆H , which is the energy difference between

the transitions is:

∆H = ∆E + ∆W. (2.100)

Based on the assumption that thermal activation must overcome the energy

barrier, the strain rate becomes [16]

γ = γ0 exp(−∆E

kT) sinh(

τγT Ωc

kT). (2.101)

The shear activation volume V ∗ is then given by [36]

V ∗ = γT Ωc. (2.102)

For temperatures far below the glass transition temperature, where τγT Ωc ≫

51

kT , sinh(x) ≈ 12exp(x), we have

˙γeq =γ0

2exp(−∆E − τγT Ωc

kT). (2.103)

Under the same approximation, V ∗ is given as the derivative of the activation

energy with respect to shear stress,

V ∗ = kT∂ ln γ

∂τ. (2.104)

The processes involved in plastic deformation of glassy solids factorize the

macroscopic, experimentally accessible activation volume V ∗ into two quantities:

PSZ’s volume Ωc and PSZ’s permanent deformation γT , which are parameters of

the molecular mechanism of deformation.

The size of the plastic shear zones, Ωc, is a key parameter for understanding

the molecular mechanism of plasticity in glassy materials. In the case of atomic

glasses, simulation results [83–86,17] have shown highly localized plastic shear

zones (PSZs), each consisting of the order of ten atoms, undergoing significant

deformation. For polymer glasses, on the other hand, the PSZ’s observed are

diffuse, involving cooperative motion of many segments [19,18]. These atomistic

simulation studies, however, have failed to capture an entire PSZ due to size

limitation of the simulation cell. The size of the PSZ seems to be much larger

in polymers than in atomic amorphous solids, and considerably larger than the

volumes of the simulation cells used up to now.

The question of the length scale of the elementary processes of plastic re-

laxation is unsolved, which is the most important questions to be addressed in

order to understand plasticity in amorphous polymers and to establish the rel-

52

evant structure-property relationships. In this research, we overcome the size

limitations of previous simulation studies by using simplified model systems, like

“united atoms” or “big atoms”, which consist essentially of strings of beads (group

atoms) of various lengths, linked by stiff harmonic springs. Systematic study of

correlation between chain length and size scale is carried out.

2.7 Simulation of Plastic Deformation of Glassy Materials

A two-dimensional model intended for a glassy metal that considers the material as

composed of atoms with spherically symmetric interactions through a Lennard-

Jones potential was used by Deng et al. [83,84,87,85] with molecular dynamic

method. This model captures most of the important physics of the plastic re-

sponse, like five-/seven-coordinated structure and shear transformation. 2D model

is good for simple atomic glass but not atomic glass.

The model used to determine the behavior of the glassy polymer should be in

3D. The method of simulation of 3D glassy material was pioneered by Theodorou

and Suter [88,89] and was utilized in plastic deformation simulations by Mott et

al. [19] and Hutnik et al. [18]. For 3D models, pure shear increments are imposed

by introducing small increments of border displacements on the periodic cell itself,

which amounts to displacing the molecule relative to its images. The calculation

is repeated each time to always attain local mechanical equilibrium. In this way

straining proceeds to larger and larger levels while each time imposing only rela-

tively small increments of border distortions. To evaluate the local response and

to identify the origins of strain increments, an atomic-level strain increment tensor

is introduced [63,16]. For this purpose the material is tessellated into space-filling

Delaunay tetrahedra, which is description of interstitial space defined by four cor-

53

ner atoms of a generic interstice. For any strain increment between the initial and

the final state, the displacement gradients for each interstitial space of material

are calculated. This defines the atomic-level strain increments for all interstitial

spaces as internally homogeneous quantities, calculated with equation (2.47).

Mott et al.was not interest the interstitial space, but strains referred to

atomic environments. For this purpose, the space is retessellated into Voronoi

polyhedra again. Then the local atomic site strain increment tensor is obtained as

the volume-averaged strain increments of the Delaunay tetrahedra that contribute

to the Voronoi polyhedra through their contributions to individual displacement

gradients, which involve the terms

Fij =∑

k

ck(Fij)k, (2.105)

where the sum is carried over all k Delaunay tetrahedra, each contributing a spe-

cific fraction ck to a Voronoi polyhedron. There are 25 or so Delauney tetrahedra

(k ≈ 25) on the average that feed information into a typical Voronoi polyhedron.

Recently Shenogin and Ozisik [90] used Voronoi tessellation to obtain the

atomic strain. From the displacement of nearest neighbors in the deformed state,

local strain tensor, ‖ε‖, can be obtained for each cage from the minimization of

the following equation:

R2 =1

N

N∑

1

(~rdef,i − ‖ε‖ · ~r0,i)2, (2.106)

where N is the number of nearest neighbors, ~rdef,i is the actual relative

displacement of each neighbor after deformation, ~r0,i is the relative position of

each neighbor, and the summation is performed over all the nearest neighbors.

54

Parameter R2 is the accuracy of local strain field with respect to the constant

tensor ‖ε‖ with six independent components.

In this thesis, we do interest the interstitial space. The atomic-level strain

increments for all interstitial spaces as internally homogeneous quantities are cal-

culated with equation (2.47). The internal space then are sampled with grid

points. Then the relative atomic strains of these points are determined. The

correlation of these relative atomic strains can give the size of the location.

Chapter 3

Athermal Simulation of Plastic Deformation at Constant

Pressure

3.1 Introduction

The glassy state of matter continues to present many scientific challenges, in spite

of a large amount of research devoted to it over the past century [30]. In particular,

the response of amorphous solids to large amounts of deformation is not yet un-

derstood completely [16]. Plastic deformation of glasses is believed to proceed via

nucleation of stress-induced relaxation events [16]. This has been demonstrated

by computer simulations of the deformation process in a number of different sys-

tems [17,91,85,18,19,21,92]. In addition, the stress-strain behavior of glassy solids

depends strongly on their thermal history, suggesting an intimate coupling be-

tween the slow relaxation processes that cause physical aging, and strain-induced

relaxation during plastic deformation [93]. The precise nature of these relaxation

processes seems to depend strongly on the material. While computer simulations

of atomic glasses show a high degree of localization of the relaxation events, to

about one nearest neighbor shell, no such localization has been observed in sim-

ulations of the deformation of glassy polymers. Indeed, experimental evidence

indicates that the strain-induced relaxation in polymers is much more diffuse,

55

56

and that the relevant size scale is set by the entanglement density of the polymer

network [94,95].

The nature of the discrete relaxation events during plastic deformation of

polymer glasses is currently an unsolved problem of polymer physics. An aspect

of particular importance concerns the volume change associated with the plastic

relaxation events. Volume changes associated with shear deformation are com-

monly referred to as dilatancy. Experimental evidence suggests that there is a

link between the availability of free volume in the amorphous polymer packing

and the resistance to plastic yielding [96]. The shear activation volume, a param-

eter which describes the dependence of the yield stress on the applied strain rate

[16,97], is also known empirically to depend weakly on the isotropic component of

the stress tensor in glassy polymers [98,37], in the sense that hydrostatic pressure

makes plastic deformation more difficult.

Apart from the fundamental interest in dilatancy as an aspect of the nature

of plastic relaxation processes, volume changes that accompany shear deformation

directly influence material behavior. Examples include the formation of shear

bands in fibre composites, which depends on the dilatancy of the matrix [99],

and the initiation of micro-buckling defects in fiber composites under multi-axial

loading [100].

Direct volumetric observation of the density changes associated with plastic

deformation of polymer glasses is difficult, due to the interference of the elastic

Poisson effect. Experimental results on glassy polymers have mostly shown a

positive dilatancy (volume increase with shear deformation) [101], although the

literature is not unanimous on this point [102, and references cited therein]. Com-

puter simulations could be very helpful in order to clarify this situation.

57

However, most computer simulations of plastic flow in glassy solids have

been performed at constant volume. In general, this leads to an increase in system

pressure with shear strain due to the dilatancy [16]: the strain-activated localized

relaxation events not only bring about shear, but also an increase in volume. If

the total system volume is constrained during the simulation, a concommitant

increase in system pressure results [19,22]. While a positive correlation between

strain and system pressure has been observed in most simulations, this seems to

depend strongly on the material studied. Indeed, Hutnik et al. have observed a

negative dilatancy in the plastic deformation of polycarbonate [18]. In contrast to

these simulations, experimental studies of plasticity are usually conducted not at

constant volume, but at constant pressure. Since in most amorphous solids, the

elementary processes of plastic relaxation are known to be dilatant, it is possible a

priori that the kinematics of relaxation are affected by the constraint of constant

volume.

In this research, we introduce an algorithm which facilitates athermal de-

formation simulations at constant pressure, rather than at constant volume, and

we apply it to study the plane strain response of a binary Lennard-Jones glass.

This permits study of the strain-induced relaxation at P = 0, i.e., under a purely

deviatoric stress state. As will be shown in the following, constant volume simu-

lations lead to an increase in the shear stress at high strains due to the pressure

build-up. By contrast, at constant pressure, no such increase is observed, while

the density of the system decreases linearly with strain. In contrast to the stress-

strain response, the kinematics of the relaxation processes, as measured by the

decay of the intermediate structure factor, seems to be the same in both cases.

This new algorithm allows the direct simulation of dilatant shear deforma-

58

tion in glassy materials, including the concomitant volume change. The purpose

of the research in present chapter is to introduce the method, and to demonstrate

its principle. In next chapter, we will report its application to simulate the plastic

response of polymer glasses.

Recently, computer simulations have indicated that the deformation-induced

relaxation processes in atomic glasses lead to an exponential decay of the self part

of the intermediate structure factor, similar to thermal relaxation of a liquid at

temperatures sufficiently above the glass transition. This has been verified both in

systems under continuous simple shear flow [103–107] and in transient deformation

simulations [23].

Ref. [23] has investigated the interplay between thermal and deformation-

induced relaxation in a binary Lennard-Jones glass. It was found that thermal

and deformation-induced relaxation processes involve similar particle kinematics,

and that they superimpose linearly when both occur at the same time. Such

simulations, even though they are carried out using a model atomic glass, can give

valuable insight on the microscopic nature of phenomena that are observed in a

much wider range of glassy materials. For instance, simulations of a Lennard-Jones

glass have shown that physical aging of the glass leads to the development of a

maximum in the stress-strain curve at the yield point, and that the effects of aging

are erased by moderate amounts of plastic deformation [22]. This corresponds to

a well-known behavior of polymer glasses [93], indicating that the phenomenon is

general, and can be seen in all glasses if the time scales allow its observation.

59

3.2 Simulation Method

Plastic deformation can be simulated in the absence of thermal relaxation (i.e., at

T = 0) in the following way: In a first step, the extents of the periodic simulation

cell are changed by a small amount corresponding to the desired strain increment.

In a second step, the potential energy of the system is minimized with respect

to the fractional coordinates of all particles using a conjugate-gradient method.

This procedure ensures that the system is maintained at the bottom of a potential

energy well after each step. If the strain increments are chosen sufficiently small,

the system’s trajectory will follow the lowest potential energy path compatible

with the prescribed strain. This method has been applied successfully to a variety

of different systems in the past [17–23].

Deformation at constant volume is easily implemented in the above method

by using a volume-conserving strain increment. In the present contribution, we

introduce an extension of this method that allows for simulations to approximate

conditions of conserved pressure instead. The algorithm is closely related to the

Berendsen barostat [108,109], which is widely used to keep pressure constant in

molecular dynamics simulations [44].

After each deformation step, the stress tensor t is calculated from the gen-

eralized internal virial tensor

Ξnm =∑

i<j

rij,nfij,m, where n, m = x, y, z, (3.1)

where rij,n and fij,n are the components of the displacement and force vectors

between particles i and j, respectively. The components of the stress tensor are

60

given by

tnm = −Ξnm

V, (3.2)

where V is the system volume. Note that at finite temperature, (3.2) would

contain a kinetic energy (ideal gas) term, also. The instantaneous system pressure

is then given by the trace of the stress tensor

P = −1

3tr t. (3.3)

The constant pressure simulation method introduced here aims at an evolution of

the system pressure P with deformation ε that obeys the differential equation

dP

dε= −P − P0

ǫP, (3.4)

where P0 is a given target pressure, and ǫP is the pressure relaxation strain, which

is analogous to the pressure relaxation time introduced by Berendsen et al. [108].

In practice, the choice of ǫP is quite straightforward. On the one hand, a

small value of ǫP is desirable, in order to ensure an accurately isobaric simulation.

On the other hand, values of ǫP that are small compared to the deformation step

size δε lead to numerical instabilities. Thus, reasonable values of ǫP must lie in

the neighborhood of 1 . . . 5 δε.

The rate of change in the pressure can be converted into a rate of change of

the system volume by using the isothermal bulk modulus K:

dP = −KdV

V. (3.5)

61

We obtain

dV

V= (P − P0)

dε

KǫP

. (3.6)

After each deformation/minimization step, the volume of the system is therefore

adjusted by a factor eµ, with

µ = (P − P0)δε/(KǫP ). (3.7)

Such an adjustment is easily achieved by altering the extents of the simulation box.

Even though the bulk modulus K may not be precisely known for a given system,

an estimated value can be used. Deviations from the true bulk modulus will not

alter qualitatively the system’s trajectory; only the effective pressure relaxation

strain ǫP will be slightly off the desired value.

For the present study, we have used a binary Lennard-Jones fluid that closely

resembles the model introduced by Stillinger and Weber [110] for Ni80P20. The

thermal dynamics of this system [111–114] as well as its behavior under shear

[22,103,105,106,23] have been studied in great detail. The model used here con-

sisted of 3200 type A and 800 type B atoms in an orthorhombic simulation box

subject to periodic continuation conditions. The total potential energy was given

by the sum of the pair contributions

Eij = 4ǫij [(σij/rij)12 − (σij/rij)

6] + cijr2ij + dij (3.8)

for rij < rc, where rij is the distance between atoms i and j, and ǫij and σij

are the energy and length parameters, respectively. The constants cij and dij are

introduced to render Eij continuously differentiable at the cut-off radius rc. The

values of ǫAA = 1.0, ǫAB = 1.5 and ǫBB = 0.5 and σAA = 1.0, σAB = 0.8 and

62

σBB = 0.88 were used. The masses of the particles of type A and type B were mA

= 1.0 and mB = 0.53, respectively. The length of the simulation box was 15. A

cut-off radius rc=2.5 was used in all the cases. All quantities in this contribution

are expressed in terms of reduced units, i.e., length in units of σAA, energy in units

of ǫAA, and stress in units of ǫAA/σ3AA.

Deformation simulations departed from a glassy initial configuration, which

had been obtained by equilibrating a system of 4000 atoms by molecular dynam-

ics at a high temperature, and subsequent energy minimization. A plane strain

deformation mode was used with a strain increment [61]

δεεε =

δεxx 0 0

0 0 0

0 0 δεzz

, (3.9)

and a constant deformation step δεzz = −0.00125. This corresponds to a plane

strain compression. Under conditions of conserved sample volume, or for an ide-

ally incompressible material, εxx = −εzz. In general, the change in volume per

deformation step is given by

µ = lnV + δV

V= δεxx + δεzz. (3.10)

In order to ensure conditions of constant pressure, δεxx was calculated prior to

each deformation step according to (3.7):

δεxx = δεzz

(

P − P0

KǫP

− 1

)

. (3.11)

63

3.3 Results and Discussion

3.3.1 Pressure Conservation

In order to carry out constant pressure simulations, a reasonably accurate estimate

for the bulk modulus K is needed. In the present case, this was obtained from

the elastic shear modulus observed in constant volume simulations, assuming a

Poisson’s ratio of 0.3. This yields a value of K ≈ 49.8, in units of ǫAA/σ3AA. This

is admittedly a crude way of estimating K, and a dedicated simulation using an

isotropic deformation would almost certainly have yielded a more reliable value.

However, the precise value of K is of little concern, since it is the product KǫP

that actually enters the simulation.

The system pressure and density as a function of deformation are shown

in Fig. 3.1 and Fig. 3.2, respectively, for values of ǫP = ∞ (corresponding to

constant volume, dashed line), and ǫP = 0.0033 (approximating constant pressure

conditions, solid line). Each of the two data sets shown represents the average of

20 independent simulation runs, departing from different starting configurations.

In both cases, the system starts out at a state of slight isotropic tension,

with a pressure of P = −0.6. This is due to the chosen initial density, which

leads to zero pressure in molecular dynamics simulations at T = 1.0. After energy

minimization, in the absence of thermal motion, the observed tension results.

It is obvious from Fig. 3.1 that at constant volume (ǫP = ∞), the system

pressure increases systematically with deformation after a brief stage of pressure

reduction associated with the elastic response. This initial response is due to the

fact that the Poisson ratio for the material under study is close to ν = 0.3, so that

elastic plane strain compression would be accompanied by a slight reduction in

volume. Since the dashed curve in Fig. 3.1 has been obtained at constant volume,

64

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

−ε

2

zz

P

Fig. 3.1: System pressure P (in units of εAA/σ3AA) as a function of deformation

for ǫP =∞ (dashed line) and ǫP = 0.0033 (solid line). The data shownrepresents the average of 20 independent simulation runs.

a decrease in pressure results. After a few percent of strain, the pressure increases

monotonically with the deformation. This effect has been observed by several

authors working on a number of different systems [19,22]. For a finite value of

ǫP , however, a different behavior results. The initially negative pressure rapidly

approaches the target value of P0 = 0, which is reached at about 1% strain.

Correspondingly, the structure densities (Fig. 3.2) from the initial ρ = 1.200 to

ρ = 1.215.

After this initial phase, the system density continually falls with increasing

65

0 0.2 0.4 0.6 0.8 11.18

1.19

1.2

1.21

1.22

−ε

ρ

zz

Fig. 3.2: Number density of particles ρ (in units of 1/σ2AA) as a function of

deformation for ǫP = ∞ (dashed line) and ǫP = 0.05 (solid line). Thedata shown represents the average of 20 independent simulation runs.

deformation. After the yield point, which occurs at about 8% deformation in this

system [22,23], there is a linear dependence of the density on strain, with a slope

of dρdεzz

= 0.0175± 0.0005. The pressure remains constant, with small fluctuations

about the target value of P0 = 0. These results demonstrate that the method

described above indeed does keep the pressure constant, and they illustrate the

dilatant nature of the plastic deformation process.

Of course, the results are somewhat sensitive to the selection of ǫP . We have

found that similar results can be obtained with a wide range of choices for the

66

0 0.5 1−2

−1

0

1

2

−ε

εεεε PPPP=0.00125=0.0251=0.251=2.51

0 0.5 1−2

−1

0

1

2

−ε0 0.5 1

−2

−1

0

1

2

−ε0 0.5 1

−2

−1

0

1

2

−ε

0 0.5 11.18

1.19

1.2

1.21

1.22

−ε0 0.5 1

1.18

1.19

1.2

1.21

1.22

−ε0 0.5 1

1.18

1.19

1.2

1.21

1.22

−ε0 0.5 1

1.18

1.19

1.2

1.21

1.22

−ε

Pρ

Fig. 3.3: Effect of different choices of the relaxation strain ǫP on the evolutionof pressure (top row) and density (bottom row) over the course of aplane-strain deformation simulation from εzz = 0 to εzz = 1. For eachvalue of ǫP , results from a single simulation run are shown.

relaxation strain ǫP . Fig. 3.3 shows the system pressure for a range of different

values of ǫP . For ǫP > 1, the barostat is not sufficiently effective to compensate for

the dilatancy, and a systematic drift in the system pressure results (first column

in Fig. 3.3). On the other hand, ǫP < 0.002 results in increasing magnitude

of the density fluctuations with strain, and at ǫP = 0.00125, the computation

becomes numerically unstable at deformations greater than about 0.6. This leaves

a window for ǫP of 0.002 < ǫP < 0.05 that offers good pressure conservation

without excessive fluctuations. We have chosen a value of ǫP = 0.0033 as a

67

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

−ε

τ

Fig. 3.4: Von Mises equivalent shear stress τ as a function of strain εzz forrelaxation strain ǫP = ∞ (dashed line) and ǫP = 0.0033 (solid line).Each data set shown is the average of 20 independent simulation runs.

working compromise for the purposes of the present study.

3.3.2 Stress-Strain Behavior

Running the deformation under conditions of conserved system pressure has a

profound effect to the evolution of the shear stress with deformation, in particular

at large deformations. Fig. 3.4 shows the von Mises equivalent shear stress as a

function of εzz for both constant volume (ǫP = ∞) and constant pressure (ǫP =

0.0033). The von Mises equivalent stress, essentially the second invariant of the

stress tensor, is a positive measure of the amount of shear stress present and is

68

defined as

σeq =

√

3

2tr (td · td), (3.12)

where td is the deviatoric part of the stress tensor

td = t + P I, (3.13)

with the identity tensor I.

At small strains, the von Mises stress behaves similarly for both constant

volume and constant pressure. An elastic linear increase in the von Mises stress is

followed by a plastic regime, where σeq is more or less independent of strain. The

yield stress, as measured by the plateau value in σeq, seems to be slightly lower

in the case of constant volume. Marked differences between the two cases become

apparent at high deformations, above about |εzz| = 0.3. Whereas at constant

pressure, the von Mises stress continues to fluctuate slightly around the same

plateau value, it rises increasingly rapidly in the case of constant volume.

This increase can be understood by the buildup of system pressure, which

progressively hinders dilatant shear relaxation events. As a result, only shear

relaxation events with a small amount of local increase in volume are nucleated.

This leads to the observed increase in resistance to shear.

Under constant pressure conditions, no such effect is observed. The von

Mises equivalent stress remains at the same value up to the maximum deformation

probed in the present simulations. Fig. 3.4 shows a small increase in the magnitude

of the stress fluctuations at high strains. The fluctuations are due to the finite

size of the simulation cell. Since the cell goes from a cubic shape at εzz = 0 to an

elongated shape at εzz = −1, its thickness in the z direction continually decreases

69

10−3

10−2

10−1

100

0

0.2

0.4

0.6

0.8

1

−ε

Φs

Fig. 3.5: Self part of the isotropically averaged intermediate structure factorΦs(k, εzz) as a function of strain εzz for constant volume (ǫP = ∞,dashed line), and constant pressure (ǫP = 0.0033, solid line). Thedashed curve has been displaced vertically by 0.1, otherwise the twocurves would coincide to within the line width. k = 7.251 in both cases;corresponding to the location of the maximum in the static structurefactor.

during the course of the simulation. This leads to less efficient averaging of the

normal stress component tzz in the later stages of the deformation. Since tzz

provides a dominant contribution to both P and σeq, the fluctuations in these

values increase over the course of the simulation.

70

0 1 2 30

1

2

3

4

5

6

7

8

r

gB

B(r

)

−ε=0.0

−ε=0.2

−ε=0.5

−ε=0.9

Fig. 3.6: Pair correlation functions gBB(r) for different degrees of deformation (asindicated in the figure). Solid lines: constant pressure (ǫP = 0.0033);Dashed line: constant volume (ǫP = ∞). The curves have been verti-cally displaced in order to avoid coincidence.

3.3.3 Strain-Induced Structural Relaxation

The structural relaxation of amorphous solids can be monitored conveniently by

the decay of the self part of the intermediate structure factor,

Φs(k, t) ∝∫

Gs(r, t)e−ik·rdr, (3.14)

where the van Hove correlation function Gs(r, t)dr is proportional to the proba-

bility of observing a particle within dr of r at time t, given that it was located at

71

the origin at time 0 [115]. Φs(k, t) is normalized such that Φs(k, 0) ≡ 1.

In the present case, where plastic deformation is simulated under exclusion

of thermal relaxation, time has no meaning and must be replaced by the strain that

increases continually during the simulated trajectory. Therefore, in the present

context, the relevant correlation functions are G(r, εzz) and Φs(k, εzz) [23].

The isotropically averaged relaxation functions Φs(k, εzz) are shown in Fig. 3.5

for k = |k| = 7.251 (this value corresponds to the position of the maximum of the

static structure factor). In both cases, the contribution of the affine deformation

to the decay of Φs, which is purely caused by the shape change of the simulation

box and not the rearrangement of the atoms inside it, has been removed according

to a procedure described in detail in Ref. [23].

The curves for constant pressure and constant volume are indistinguishable,

so that they had to be displaced from one another in Fig. 3.5 in order to avoid

coincidence. This result contrasts sharply with the pronounced difference in the

stress-strain behavior of the two cases. It suggests that the kinematics of the

relaxation events nucleated by plastic deformation in the present binary Lennard-

Jones system are the same under conditions of constant pressure and constant

volume.

This seems surprising, given the marked differences in the von Mises equiv-

alent stress between the two cases. The intermediate structure factor may not

be a sensitive measure of the change in kinematics. In order to rule out this

possibility, the structure of the simulation systems was investigated carefully at

different levels of deformation. No differences apart from the change in density in

the case of conserved pressure could be identified. In particular, the pair correla-

tion functions gAA(r), gBB(r), and gAB(r), measuring the distribution of A − A,

72

B−B, and A−B particle pairs, respectively, have been investigated as a function

of deformation. The correlation functions were not affected by the deformation,

and no differences between the cases of constant volume and constant pressure

could be identified. As an example, gBB(r) is shown in Fig. 3.6 for both constant

pressure (solid lines) and constant volume (dashed lines), at −εzz = 0, 0.2, 0.5,

and 0.9, respectively. gBB provides a sensitive measure of the state of the system,

since contacts between B particles are disfavored energetically by the force field

[22]. The fact that gBB(r) does not differ between systems deformed at constant

pressure and constant volume therefore suggests that they are indeed identical

structurally, up to a small difference in density.

The evolution of the potential energy landscape of atomic glasses upon

changes in density has been investigated in detail by Malandro and Lacks [116].

They found that reducing the density leads first to a lowering in the energy bar-

rier between adjacent minima (“inherent structures” [117,118]), and then to their

complete disappearance. Malandro and Lacks reported that a 3% decrease in vol-

ume leads to the loss of about one quarter of the possible inherent structures [116].

Their study used a single-component atomic glass; however, a similar behavior of

the binary Lennard-Jones fluid studied here must be expected. In the present

case, shear deformation at constant volume seems to lead to an increase of the

height of energy barriers between adjacent minima, as manifested by the increase

in the von Mises equivalent stress. Under conditions of constant pressure, this

increase is apparently perfectly compensated by the decrease in density, leading

to a constant von Mises stress.

73

3.4 Conclusions

In summary, an algorithm similar to Berendsen’s barostat has been introduced

for the simulation of plastic deformation of glassy solids at constant pressure,

and it has been applied successfully to the plane strain compression of a binary

Lennard-Jones fluid. The dilatant nature of the elementary plastic relaxation

processes was demonstrated by a linear decrease of density with strain with a

slope of dρdεzz

= 0.0175± 0.0005.

While no difference in the strain-induced structural relaxation of the system

could be identified between the cases of constant pressure and constant volume,

the response of the von Mises equivalent shear stress to deformation is strongly

affected. Conditions of constant volume lead to a build up of system pressure with

deformation, which in turn leads to an increase of resistance to further deforma-

tion, i.e., an increase in the von Mises equivalent stress. In the case of constant

pressure, no such effect is observed, and the von Mises equivalent stress remains

at the level of the yield point over the entire range of deformation.

The results presented in this contribution give rise to a number of questions.

On the one hand, it is unclear what determines the specific value of the observed

dilatancy. Simulations on different molecular and atomic glasses will be discussed

in the next chapter, in order to obtain information on the variability of the dila-

tancy in different systems. On the other hand, the fact that no change in the pair

correlation functions was found as a function of deformation in the case of constant

volume is intriguing. This means that glasses with the same pair correlation func-

tions can exhibit widely different pressures. However, at zero temperature, the

arrangement of the atoms is the only factor that influences the system pressure.

The systems therefore must be structurally different before and after shear defor-

74

mation at constant volume. We are currently exploring higher-order correlation

functions as a means to quantify these subtle differences.

Chapter 4

Quantification of Strain Localization Using the Relative

Atomic Strain Correlation Function

4.1 Introduction

The study of localization processes of plastic deformation is of great importance

for applied problems of fracture mechanics and plasticity [119] because it gives the

length scale of the elementary processes of plastic deformation in glassy materials.

Plastic deformation is an irreversible process. Once the system undergoes plastic

deformation, its shape can not recovered completely by relaxation if the applied

stress is removed.

Plastic deformation are caused by the mobility of the dislocations [28,29] in

crystalline solids and the nucleation of repeated and spontaneous stress-relaxation

events [19] in glassy solids. The localization of plastic shear events is found in com-

puter studies [17,83,84,87,85], and it was incorporated by the Plastic Shear Zones

(PSZ) [16] theory, which has been discussed in Chapter 2. PSZ theory was de-

veloped into the Shear Transition Zones (STZ) theory by Falk and Langer [120]

and deals with plasticity of amorphous materials at low temperatures. It is based

on the previous work of Cohen, Turnbull [121], Spaepen [122], Argon [123], which

argued that in non-crystalline solids, plasticity is due to atomic rearrangements

75

76

Str

es

s

Strain

Plastic relaxation events

Str

es

s

Strain

Plastic relaxation events

Fig. 4.1: Commonly observed stress-strain response in computer simulation ofglassy materials under plastic deformation (Ref [16]).

at localized sites. This picture has also been confirmed by a number of compu-

tational studies [17,83,84,87,85] However, unlike the earlier theories like PSZ, the

STZ theory focuses in detail on how rearrangements at the localized sites (shear-

transformation-zones) occur, and identifies as important dynamical variables not

only the concentration of the STZs, but also their orientations.

The plastic relaxation events are commonly observed in stress-strain re-

sponse by computer simulation of glassy materials under plastic deformation, as

shown in Fig. 4.1 [19,16]. We are interested in the average relative atomic strain

drops instead of the system’s stress drops, which is given in [19,16]. The relative

atomic strains localize in plastic relaxation events, and the localization of these

events gives the length scale of these events.

These plastic relaxation events can be only observed in a small size sample

( several nano-meter long in simulation box size or several thousand particles).

When the size of sample is macroscopic, such spontaneous localized plastic relax-

ation events will be averaged out and the stress-strain response is smooth.

Spontaneous stress relaxation events during plastic deformation are gen-

77

Fig. 4.2: Von Mises equivalent atomic strain observed in a computer model ofpolypropylene undergoing a plastic relaxation event. Scale is indicatedby the size of circles around atoms (Ref [19]).

erally accepted, as showed in figure Fig. 4.1 (Taken from Ref. [16]. The stress

increases with respect to strain, followed by a drop, and increase again. The

discontinuities represent plastic relaxation events, in the schematic plot of stress-

strain response. The stress relaxation is highly localized by computer simulation

of atomic glasses [83]. Many atomic glass simulations [83–86,17] have shown highly

localized plastic shear zones, each consisting of the order of ten atoms, undergo-

ing significant deformation. For polymer glasses, on the other hand, the PSZ’s

observed are diffuse, involving cooperative motion of many segments [19,18], as

shown in figure Fig. 4.2 (Taken from Ref. [19]. These atomistic simulation stud-

ies, however, have failed to capture an entire PSZ due to size limitation of the

simulation cell [19,18]. The size of the PSZ seem to be much larger in polymers

than in atomic amorphous solids, and considerably larger than the volumes of the

simulation cells used up to now.

Even though the plastic relaxation events are observed in simulations of

polymer glasses, their length scale is not clear, due to size limitations. Here we

78

are interested how chain molecules affect the relevant length scale. We will study

the localization of plastic relaxation events [19] in molecular amorphous solids by

atomic strain [16] correlation function to find the length scale of a single plastic

relaxation event.

Instead of showing the localization of atomic strain phenomilogically, in

our research, a new approach to indicate the localization of stress relaxation is

invented. This approach is the correlation function of atomic strain. The process

is following.

Step 1: computer simulation of plastic deformation of amorphous solids, which is

studied in details in chapter 3.

Step 2: Find out the plastic relaxation events during the plastic deformation.

Step 3: Tessellate the system steps in which plastic relaxation event happens. The

Delaunay Tessellation method and its algorithm are introduced in section 2.

Step 4: Once we tessellate the system in Step 3, we can calculate the atomic strain

for each tetrahedron.

Step 5: Calculate the correlation function of the relative atomic strain with Fast

Fourier Transform.

The process of the method is outlined in figure Fig. 4.3 and the result shown

in section 3 and section 4 is the conclusion.

79

plastic deformation

find plastic relaxation events

Tessellation

find relative atomic strain

correlation function

Fig. 4.3: The process of the method to study the localization of plastic relaxationevents

4.2 Model and Simulation Method

The present simulations are based on the united atom potentials for aliphatic

hydrocarbons developed by Smit et al. [124,125] and adapted by Utz et al. [126].

The parameters of this potential have been tuned to represent the phase behavior

of linear and branched alkane in Gibbs-ensemble Monte Carlo simulations. For non

bonded atoms, or atoms separated by more than 3 bonds, the force field is based

on two different pseudoatom types, representing methyl (CH3) and methylene

(CH2) groups, which interact through a two-body potential of the form

Eij =

4ǫij[(σij/rij)12 − (σij/rij)

6] + cijr2ij + dij, r < rc;

0, r ≥ rc.(4.1)

The Lennard-Jones interaction parameters σ and ǫ are given in Table 4.1.

80

ǫ [J/mol] σ [A]CH2 CH3 CH2 CH3

CH2 414.2 3.93CH3 645.1 1004.4 3.93 3.93

Table 4.1: Parameters of the Lennard-Jones interaction between CH3 and CH2

pseudoatoms used in the present models

The cutoff length rc was set to 9.75 A, and the parameters c and d were chosen

to make the potential continuously differentiable at the cutoff distance.

Interactions between bonded atoms include the following terms: the bond-

angle potential is given by

uang(θ) = kθ(θ − θ0)2 (4.2)

where kθ = 158.29 J/mol deg [118] is the bond-angle-potential coefficient, θ is the

bond angle between adjacent atoms, and θ0 = 114 is the equilibrium bond angle.

The torsion angle potential has the form

utors(φ) =∑

ck cosk(φ) (4.3)

where φ represents the dihedral angle and ck are coefficients for the potential.

The constants ck are [124]: c1 = 8.39 kJ/mol, c2 = 0.57 kJ/mol, and c3 = −13.16

kJ/mol. In contrast to the work by Smit et al., who used a fixed bond length in

their Monte Carlo simulations, a flexible C-C bond with a harmonic potential

ubond(l) = kl(l − l)2 (4.4)

was applied in the present work. An equilibrium bond length of l = 1.53 A was

81

used in conjunction with a spring constant kl = 345.8 kJ/mol A [118]. This value

is lower by about a factor of 5 compared to a realistic spring constant tuned to give

correct IR and Raman frequencies [127]. At the low temperatures of interest in

this study, only very small deviations from the equilibrium bond length occur even

with this softened bond potential, and the softening is therefore inconsequential

for the results derived here. However, it allows for a significantly larger time step

in the molecular dynamics integration [127].

The procedure to study the localization of plastic shear events is outlined

in figure Fig. 4.3. First of all, the simulations of plastic deformation are carried

out by the method introduced in chapter 3. Second we tessellate the system after

each deformation step. Third, the atomic strain is calculated. Forth, plotting the

average atomic strain, we can find the steps at which plastic relaxation events

occur. At last, we calculate the correlation function of atomic strain.

4.2.1 Delaunay Tessellation

We need to tessellate the system after each deformation step to calculate the

atomic strain. We use Bowyer-Watson algorithm [128,129] for tessellation, which

is divided into five steps as follows:

Step 1: Create a BIG tetrahedron, which encloses all of the given points.

If the points are contained in a cube, the length of the tetrahedra side should at

least three times bigger than the cube side.

Step 2: Add a point to the triangulation. Construct the delaunay tetrahe-

dron by connecting the added point to the vertex of BIG tetrahedron.

Step 3: Insert the other given points to the system one at a time by fol-

lowing process:

82

(a) Determine the existing tetrahedra whose circumsphere contains the given

point.

(b) The union of these tetrahedra is a polyhedron.

(c) Create a list of boundary triangles of the polyhedron in Step 3(b).

(d) Create new tetrahedra by connecting the new point to the vertices of the tri-

angles in Step 3(c).

(e) Delete the tetrahedra determined in Step 3(a)

(f) Add the tetrahedra created in Step 3(d)

Step 4: Delete all the tetrahedra which share the vertex of the enclosing

BIG tetrahedron. The remaining tetrahedra form the Delaunay tessellation of the

given points.

A system of Ni80P20 containing 4000 atoms with box size 14.76 σNi−Ni was

tessellated by Delaunay Tessellation, as seen in Fig. 4.4.

4.2.2 Location Length

Location length is the correlation length of the correlation function of relative

atomic strain in a glassy material. Location length is introduced to quantify the

localization phenomenon.

To study the localization, it is often necessary to consider the spatial dis-

tribution of some quality or phenomenon in an area consisting of several distinct

regions. One question that arises then is whether the presence of that quality

in one region makes its presence in a neighboring region more or less likely. If

there is such an interdependence, the data exhibit spatial autocorrelation. [130]

So the spatial autocorrelation is a correlation of a variable with itself through

space. The concept of spatial autocorrelation is mainly applied to problems of

83

Fig. 4.4: A system of Ni80P20 containing 4000 atoms with box size 14.76 σNi−Ni

was tessellated by Delaunay Tessellation.

84

geography, economics, ecology, or meteorology. If there is any systematic pattern

in the spatial distribution of a variable, it is said to be spatially autocorrelated. If

nearby or neighboring areas are more alike, this is positive spatial autocorrelation.

Negative autocorrelation describes patterns in which neighboring areas are unlike.

Random patterns exhibit no spatial autocorrelation.

Statisticians have developed a number of measures for quantifying spatial

autocorrelation. [130] Moran’s coefficient I is one of the oldest indicators of spatial

autocorrelation and still a defacto standard for determining spatial autocorrela-

tion. Moran’s coefficient I is defined as:

I =n

2L

∑

ij δij(yi − y)(yj − y)∑

i(yi − y)2, (4.5)

where n is the total number of data points. δij is a connection matrix with

δij = 1 for neighboring points i, j and otherwise δij = 0. y is the property value of

point i. y is the mean property value. L is the total number of connections given

by δij.

Spatial autocorrelation can be applied in study of localization of plastic

shear events. The process is following:

Step 1: Calculate the relative atomic strain of each Delaunay Tetrahedron,

as described in previous section.

Step 2: Sample the system with M × M × M points, where M is the

number of points on one axis. For ith point Pi, if Pi is inside of jth Delaunay

Tetrahedron with the relative atomic strain value of ϑ, the relative atomic strain

of Pi is assigned to the value ϑ.

Step 3: Calculate the autocorrelation function of the relative atomic strain

of the points from Step 2.

85

Here the autocorrelation function of the relative atomic strain C(r) is defined

as:

C(r) =

∑

ij(yi − y)(yj − y)δ(rij − r)∑

ij(yi − y)2 δ(rij − r), (4.6)

where yi, yj are the relative atomic strains of point Pi and Pj located in ri

and rj with the separation of rij = |rj − |ri. y stands for the average value of the

relative atomic strains of all sampled points. y can be the relative atomic strain

tensor, components of the the relative atomic strain tensor and the invariance. The

invariance of Von Mises equivalent relative atomic strain εeq, defined in equation

(2.79) are used in present study.

The deviations of the relative atomic strains are studied by covariance func-

tion. The covariance function is given as:

Cov(r) =

∑

ij(yi − y)(yj − y)δ(rij − r)∑

ij δ(rij − r), (4.7)

which gives the deviations at r = 0 as:

δC = Cov(0) (4.8)

The integration of autocorrelation function of relative atomic strain can be

used to quantify the degree of location of plastic shear events. Location length Lc

is defined as the integration of autocorrelation function of relative atomic strains,

analogous to the Kuhn length defined in The Physics of Polymers [11].

Lc =

∫

C(r)dr. (4.9)

Then we can express the degree of location of plastic shear events as location

86

Name Chain length Molecules Total atoms Box size Number density Density

(A) (1/A3) (g/cm3)

A80B20 1 4000 4000 33.14 0.10993 10.223C5 5 1000 5000 52.909 0.033758 0.80750C10 10 1000 10000 64.502 0.037263 0.87896C20 20 600 12000 67.259 0.039439 0.92374C50 50 300 15000 71.580 0.040899 0.95386C100 100 200 20000 78.559 0.041252 0.96072C200 200 150 30000 89.748 0.041500 0.96584C500 500 100 50000 106.408 0.041500 0.96538

Table 4.2: Materials simulated in the present work with the simulation box size,number density and density

length Lc.

4.2.3 Working Materials and Units

For the present study, we deal with the eight materials, seven molecular glasses

and one atomic glass, as showed in Table 4.2:

where A80B20 is atomic glass as described in chapter 3, which is the model of

Ni80P20, studied by Weber and Stillinger [131]. It was used to compare with

molecular glasses.

All quantities in atomic glass Ni80P20 simulations are expressed in terms of

reduced units, as shown in Table 4.3:

The parameters of the Lennard-Jones interaction between CH3 and CH2

pseudoatoms used in the present models are given in Table 4.1. All quantities in

molecular glass simulations are shown in Table 4.4:

87

Quantities reduced units SI units

Length σNi−Ni 2.2183AEnergy εNi−Ni 7.76kJ/molMass mNi 1.0284× 10−25 kgTime σNi−Ni(mNi/εNi−Ni)

1/2 6.2721× 10−13sDensity mNi/σ

3Ni−Ni 10.223 g/cm3

Pressure εNi−Ni/σ3Ni−Ni

Table 4.3: Units in simulation of atomic glass Ni80P20

Quantities units

Length AEnergy 8.31 kJ/molMass 1.66× 10−27 kgTime 1.1× 10−12s

Density g/cm3

Pressure MPa

Table 4.4: Units in simulations of molecular glass

The molecular structures of the seven molecular glasses were created with a

method first proposed by Muller et al. [69]. The software code is named PolyPack,

including PolyGrow, which deal with linear chain of alkanes. Muller’s method

consists of a heuristic search algorithm in the space of torsion angles, which au-

tomatically delivers the correct conformational statistics of the chains, maintain-

ing rotational-isomeric-state probabilities. The performance and efficiency of this

method have been previously verified for polyethylene and polystyrene. [69]

After the creation of the initial structure, each system was equilibrated with

MC and annealed with MD simulations. The densities are carefully selected so

that the system shear stress are almost zero after equilibrated and annealed. This

can be found that the shear stress-strain curve start from the origin (0, 0).

88

This athermal simulation of plastic deformation at constant pressure was

carried out as described in chapter 3. The system steps in which plastic relaxation

event occurs were determined and tessellated. The local strain of each tetrahedron

was calculated. Then we calculate the correlation function of the atomic strain of

these tetrahedron. From the correlation function, we can study the localization of

the plastic relaxation events, the relationship between the chain length of n-alkane

and the degree of localizations of atomic strains.

89

Name C5 C10 C20 C50 C100 C200 C500

Shear Modulus (GPa) 2.76 3.30 3.59 3.88 3.65 3.80 3.68Bulk Modulus (GPa) 5.98 7.14 7.78 8.41 7.91 8.23 7.97

Table 4.5: Shear modulus of seven simulated molecular glasses and their bulkmodulus, assuming a Poisson’s ratio of 0.3

4.3 Results and Discussion

4.3.1 Plastic Deformation of Molecular Glasses at Constant Pressure

Simulation of plastic deformation of molecular glasses under constant pressure is

necessary because the dilatancy property of molecular glasses [132,133]. In order

to carry out constant pressure simulations, a reasonably accurate estimate for the

bulk modulus K is needed. In the present case, this was obtained from the elastic

shear modulus observed in constant volume simulations, assuming a Poisson’s

ratio of 0.3. The shear modulus of seven simulated molecular glasses and their

bulk modulus is listed in Table 4.5 in units of GPa.

The average value of bulk modulus of these working materials is K ≈ 7.63GPa.

This value is higher than the experiment data of Polyethylene (high density) at

room temperature, which is 0.333−1.083GPa [134]. The reason maybe is the low

temperature in this study. Because in general a rise in temperature decreases the

bulk modulus [135].

The system pressure of different glassy materials as a function of deformation

is shown in Fig. 4.5 for values of ǫP = 0.0706. The lines from bottom to top stand

for C5, C10, C20, C50, C100, C200 and C500 respectively. Values of ǫP = 0.0706

is approximating constant pressure conditions. Values of ǫP = ∞ correspond to

constant volume conditions.

90

0 0.05 0.1 0.15 0.2 0.25 0.3−50

0

50

100

150

200

250

300

350

C5

C10

C20

C50

C100

C200

C500

−εzz

P[M

Pa]

Fig. 4.5: System pressure P of different materials as a function of deformationfor ǫP = 0.0706. The lines from bottom to top stand for C5, C10, C20,C50, C100, C200 and C500 respectively. The curves have been verticallydisplaced in order to avoid coincidence.

The system starts out in a state of slight isotropic tension. This is due

to the chosen initial density, which leads to zero pressure in molecular dynamics

simulations at T = 1000K. After energy minimization, in the absence of thermal

motion, the observed tension results.

The Fig. 4.5 shows that at the condition of constant (zero) pressure was

well kept with the value of ǫP = 0.0706 for the seven molecular glasses. The

initial pressure rapidly approaches the target value of Pt = 0, which is reached

at about 0.1% strain. The system pressures have small fluctuations around the

91

target value. The pressure fluctuation is discussed in the following.

Pressure Fluctuation

The system pressure of the seven glassy materials remains constant, with small

fluctuations about the target value of Pt = 0. We find that the fluctuation of

pentane is the biggest and the C100 is the smallest. The fluctuation can be

quantified with σP , the standard deviation of system pressure P from the target

value Pt, which is defined as

σP =

√

√

√

√

1

N

N∑

1

(Pi − Pt)2. (4.10)

σP of the seven glassy materials can be plotted respect to their system size,

the total number of united atoms N , in Fig. 4.6 with the target value Pt = 0.

We can find that the standard deviation σP decreases from 2.38 MPa to

1.09 MPa when system size N varies from 5000 to 12000. It reaches the smallest

valve of 0.7887 MPa at N = 20000 after a small jump at N = 15000. It increases

slowly when N is greater than 20000. The data presented in Fig. 4.6 is a little

irregular, with a mean value 1.22 MPa.

These seven materials have different chain length L and different system size

N . The system size effect on the pressure shown in Fig. 4.5 need to be studied.

System Size Effect on the Pressure

How does standard deviation σP vary with the system size N? A series simula-

tion of plastic deformation under the constant pressure with the same material

(Pentane) but different system size, N = 625, 1250, 2500, 5000, 10000 were carried

92

0 1 2 3 4 5

x 104

0

0.5

1

1.5

2

2.5

3

C5

C10

C20

C50

C100 C200C500

N

σ p [MP

a]

Fig. 4.6: The standard deviation σP of the system pressure of the seven glassymaterials, with the target value Pt = 0 with respect to their systemsize, the total number of united atoms N .

out. The system pressure of different system size but the same glassy materials

as a function of deformation is shown in Fig. 4.7.

σP of the pentane can be plotted respect to their system size N in Fig. 4.8

with the target value Pt = 0.

We can find that the standard deviation σP decreases from 2.9 MPa to 2.4

MPa when system size N varies from 625 to 5000. It fluctuates around a mean

value of 2.64 MPa when N is greater than 2000. To quantify such fluctuation, we

93

0 0.1 0.2 0.3 0.4 0.5−10

0

10

20

30

40

50

N = 625

N = 1250

N = 2500

N = 5000

N = 10000

−εzz

P [

MP

a]

Fig. 4.7: System pressure P of Pentane at different system size as a functionof deformation. The curves have been vertically displaced in order toavoid coincidence.

define a quantity dP as

dP =

√

√

√

√

1

N

N∑

1

(Pi − Pt

dmax

)2. (4.11)

where dmax is the maximum of the |Pi − Pt|. The quantity of dP has a value

between 0 and 1. The bigger value of dP indicate the distribution of Pi − Pt are

more homogenous or less fluctuation. The plot of dP is shown in Fig. 4.9 with

respect to the system size N . Fig. 4.9 shows that when the system size N > 1000,

dP increases linearly with the system size.

94

0 2000 4000 6000 8000 100002

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

N

σ p [MP

a]

Fig. 4.8: The standard deviation σP of the system pressure of Pentane withrespect to their system size, the total number of united atoms N .

The simulations of pentane with different system size indicate that when

the system size greater than a certain value, about 1000, the standard deviation

of system pressure will be constant, with some fluctuation, which decreases with

the system size.

Consider two systems undergo plastic deformation at constant pressure at

Pt = 0. System S1 has volume V1 = V, N1 = 5000 united atoms. System S2 has

volume V2 = 2V, N2 = 10000 united atoms. Suppose that each shear relaxation

event will increase the volume as v. The relaxation events will cause a deviation

95

0 2000 4000 6000 8000 100000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

N

d p

625

12502500

5000

10000

Fig. 4.9: dP of the system pressure of Pentane with respect to their system size,the total number of united atoms N .

of system’s pressure P1 from 0 as

P1 = c K n1v

V, (4.12)

where c is a constant, K is bulk modulus, n1 is the number of shear events in

system S1. When the system S2 deformed same amount, the number of relaxation

events will doubled, as n2 = 2n1. The amount of deviation of system S2’s pressure

P2 = P1 by equation (4.12) since V2 = 2V. This can explain why the standard

deviation of the system’s pressure will be independent of the system size. The

96

number of relaxation events of system S2, n2 = 2n1, will not happen at the

same time as system S1. This cause the P2 are more evenly distributed and dP

increases.

We can conclude that the system pressure of the seven glassy materials

remains constant, with small fluctuations about the target value of Pt = 0. The

standard deviation of fluctuation almost independent of the system size. Such

fluctuation is caused by the plastic relaxation events and related to the chain

length.

4.3.2 Dilatancy of Molecular Glasses

The dilatancy nature of molecular glasses [132,133] indicates that density of the

system will decrease systematically with deformation. Different glassy materials

have different rate to decease the density. To make it easier in comparison, we

define a normalized number density ρn as

ρn(ε) =ρn(ε)

ρn(0), (4.13)

where ρn(ε) is the number density of the system at deformation state −ε, ρn(0)

is the number density of un-deformed system.

The normalized number density of the system of different glassy materials

as a function of deformation is shown in Fig. 4.10 for values of ǫP = 0.0706. In the

Fig. 4.10, C5 is blue circle line; C10 is green x-mark line; C20 is red plus line; C50

is cyan star line; C100 is magenta square line; C200 is yellow diamond line; C500 is

black down-triangle line. Such marks will keep the same in the following of this

thesis.

The system density falls continuously with increasing deformation. After

97

0 0.05 0.1 0.15 0.2 0.25 0.30.85

0.9

0.95

1

−εzz

ρ C5C10C20C50C100C200C500

Fig. 4.10: Normalized number density ρ of particles of different glassy materialsas a function of deformation at constant pressure.

the yield point, there is a linear dependence of the density on strain. For different

glassy materials, the yield point and the slope are different. But for all these seven

materials, linearity is hold from strain of 10% to strain of 30%. The slopes k is

defined as

k = | ∂ρn

∂εzz

|. (4.14)

The normalized number density ρn of the system linearly decrease with the

strain, as shown in Fig. 4.10. We found that the decay of the number density of

C500 is much faster than the pentane. Since the system size of C500 is bigger

98

0 0.1 0.2 0.3 0.4 0.50.95

1

1.05

1.1

1.15

1.2

1.25

N = 625

N = 1250

N = 2500

N = 5000

N = 10000

−εzz

ρ

Fig. 4.11: Normalized number density ρ of particles of pentane with differentsystem size as a function of deformation at constant pressure. Thecurves have been vertically displaced in order to avoid coincidence.

than pentane, we need to consider the system size effect on the dilatancy.

How does the system size affect the dilatancy of glassy materials? Consider

pentane undergo constant pressure deformation with different system size. The

normalized number density linearly decrease with the deformation. The system

size effect on the dilatancy is shown in the Fig. 4.11. All the normalized number

densities of pentane in different system size linearly decrease with the deformation

at same rate. The fluctuation of larger system size, like N = 10000 is smaller than

that of the smaller system size, like N = 625.

99

0 100 200 300 400 5000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Chain Length [l°]

k

k = 0.073 log(l)−0.02

Fig. 4.12: The slopes k of system normalized number density linearly decreasingwith the strain are plotted for chain length l (solid line) with its fitting(dashed line).

The dilatancy is independent of the system size N . It depends on its molec-

ular structure. The dilatancy can be quantified by the slope k of the normalized

number density to the strain. The slopes k of ρn chain length l are plotted in

Fig. 4.12. Chain length l is in unit of bond length l.

We can find that the dilatancy slope k increases with the molecule’s chain

length. This means that the plastic shear events will increase with the chain

length when the same amount of plastic deformation is achieved, supposing that

each shear relaxation event will increase the same volume v. Consider the fact

100

that the long chain has bigger number of torsion angle and the transit of these

torsion angles will cause the plastic shear events. The dilatancy slope k have

some relationship with these torsion angles, and then the chain length, since the

number of torsion angle ntors has linear relationship with the chain length l as

ntors = 2(l − 3). (4.15)

This plot Fig. 4.12 is close to an exponential curve. Using exponential fit, we

have:

k = a ln l + b, (4.16)

where coefficient a = 0.073, b = −0.02. And the fitting function was plotted in

dashed line in Fig. 4.12.

The energy of the system energy as a function of deformation is calculated

during the simulations. The dilatant nature of molecular glasses [132,22,133] indi-

cates that the volume of the system will increase systematically with deformation.

The average separation of two atoms is expected to increase, which leads to an

increase in the potential energy of the system. Different glassy materials have

different rate to increase the energy. To make it easier in comparison, we define a

normalized energy En as

En(ε) =E(ε)

|E(0)| , (4.17)

where E(ε) is the energy of the system at deformation state −ε, E(0) is the energy

of un-deformed system. The absolute value was used because the values of the

energy are negative, which is illustrated in Fig. 2.1.

The normalized energy of the system of different glassy materials as a func-

tion of deformation is shown in Fig. 4.13 for values of εP = 0.0706.

101

0 0.05 0.1 0.15 0.2 0.25 0.3−1

−0.95

−0.9

−0.85

−εzz

Ene

rgy

C5C10C20C50C100C200C500

Fig. 4.13: The normalized energy of the system of different glassy materials asa function of deformation for values of εP = 0.0706.

The system energy continually rises with increasing deformation. There is

also a linear dependence of the energy on strain. When we plot the slope of

these linearity, the relationship between the dρn

dεzzand dEn

dεzzare found: they are

almost identical except the sign. This positive correlation between the dρn

dεzzand

dEn

dεzzcan be explained by the fact that system energy depends on the density as

the temperature is constant (T = 0).

102

0 0.05 0.1 0.15 0.2 0.25 0.30

200

400

600

800

1000

1200

C5

C10

C20

C50

C100

C200

C500

−εzz

τ [M

Pa]

Fig. 4.14: The von Mises equivalent stress τ of different materials as a functionof deformation at constant pressure. The curves have been verticallydisplaced in order to avoid coincidence.

4.3.3 Stress-Strain Behavior

Running the deformation under conditions of conserved system pressure was found

to have a profound effect on the evolution of the shear stress with deformation,

in particular at large deformations. The von Mises equivalent stress, essentially

the second invariant of the stress tensor, is a positive measure of amount of shear

stress present, and is defined as equation (3.12).

The von Mises equivalent stress τ of different glassy materials as a function

of deformation εzz are shown in Fig. 4.14 at constant pressure P = 0. The curves

have been vertically displaced in order to avoid coincidence.

103

Fig. 4.14 shows the increment-drop pattern of each τ -ε curve. The von

Mises equivalent shear stress increases linearly with the strain, followed by a

sudden drop and then repeat the increment-drop pattern. Such sudden drops of

von Mises equivalent shear stress destroy the linear relationship of τ -ε and cause

the plastic deformation.

Fig. 4.14 shows the plastic shear events during the plastic deformation of

the seven glassy materials. The plastic shear events are indicated by the sudden

drops of the von Mises equivalent shear stress. The nucleation of plastic shear

events is the mechanism of the plastic deformation of glassy materials.

The Fig. 4.14 shows that for the condition of constant pressure, the von

Mises stress behaves similar for the glassy materials: an elastic linear increase

with strain, followed by a plastic regime. The yield stress, as measured by the

plateau value in τ , seems to be higher in the case of longer chain material. At

high deformations, above about |εzz| = 0.2, the von Mises equivalent shear stress

continues to fluctuate around the same plateau value. The von Mises equivalent

shear stress fluctuates around the same value up to the maximum deformation

probed in the present simulations.

The number of drops of von Mises equivalent stress τ of different materials

as a function of strain increment dεzz was plotted in Fig. 4.15 for seven glassy

materials underwent 30% deformation.

We found that the number of drops increases with the chain length and the

strain increment dεzz decreases with the chain length. This is consisting with the

dilatancy of the glassy materials since during the same amount of deformation,

the system volume of longer chain molecule will increase more than that of shorter

chain molecule.

104

0 0.005 0.01 0.015 0.0210

0

101

102

dεzz

Num

ber

of d

rops

C5C10C20C50C100C200C500

Fig. 4.15: The number of drops of von Mises equivalent stress τ of differentmaterials as a function of strain increment dεzz.

System Size Effect on Shear Stress

How does von Mises equivalent stress vary with the system size N? A series simu-

lation of plastic deformation of pentane under the constant pressure with different

system size were carried out. The system effect on the von Mises equivalent stress

is shown in Fig. 4.7. We found that the von Mises equivalent stress -strain curves

of different system size are almost the same, except the functions. The shear stress

strain curve is independent of the system size. This is verified by the fact that the

shear stress-strain curve is the mechanical property curve of the material [136], in

105

0 0.05 0.1 0.15 0.2 0.25 0.30

10

20

30

40

50

60

70

80

−εzz

τ [M

Pa]

625

1250

2500

5000

10000

Fig. 4.16: Von Mises equivalent stress of Pentane at different system size as afunction of deformation. The curves have been vertically displaced inorder to avoid coincidence.

spite of the system size.

The number of drops of von Mises equivalent stress τ of pentane as a function

of strain increment dεzz was plotted in Fig. 4.17 for different system size underwent

30% deformation.

We found that the number of drops of von Mises equivalent stress τ of

pentane have an peak around the strain increment dεzz = 0.17 averaged for all 5

different system size. This is also verify that the shear stress strain relationship

is almost independent of the system size.

106

0 0.005 0.01 0.015 0.02 0.0250

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

dεzz

Num

ber

of d

rops

N=625N=1250N=2500N=5000N=10000

Fig. 4.17: The number of drops of von Mises equivalent stress τ of pentane withdifferent system size as a function of strain increment dεzz.

4.3.4 Pair Correlation Function and Structure Factor

Pair correlation function g(r), defined in equation 2.95, is plotted in Fig. 4.18

for pentane in un-deformed system. The short-range order is expressed by two

distinct peaks and a broad peak in the short distances, following by a quite flat

tail. The first peak is centered near the bond length lbond = 1.53 Aand the second

peak is centered near 2.57 A, resulting from the bond angle of 114. The third

peak is very broad, about 0.5 Awide, centered near 4.12 A. The flat tail has the

value of 1. The pair distribution function for C10, C20, C50, C100, C200 and C500

are almost the same. This is very typical g(r) plot for C − C chain structure.

107

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

r

g(r)

← (1.5, 16.6353)

← (2.5, 4.5126)

← (4.1, 2.0834)

1.00

Fig. 4.18: Pair correlation function for pentane in un-deformed system. Theshort-range order is expressed by two distinct peaks and a broad peakin the short distances, following by a quite flat tail. The first peakis centered near the bond length lbond = 1.53 Aand the second peakis centered near 2.57 A, resulting from the bond angle of 114. Thethird peak is very broad, about 0.5 Awide, centered near 4.12 A. Theflat tail has the value of 1. This is very typical g(r) plot for C − Cchain structure.

108

The structure factor S(q), where q is the scattering vector defined in equa-

tion 2.96, for pentane in un-deformed system is plotted in Fig. 4.19. The first peak

is artificial because of the limited size of the simulation box. The second peak,

which is centered at q = 1.6197 1/A, corresponds to the position of the maximum

of the static structure factor. This value of q = 1.6197 1/A is quite typical for

C −C chain structure and was used in calculation of self part of the isotropically

averaged intermediate structure factor Φs(k, εzz) for C10, C20, C50, C100, C200 and

C500 in next sub-section .

4.3.5 Strain-Induced Structural Relaxation

The structural relaxation of amorphous solids can be monitored conveniently by

the decay of the self part of the intermediate structure factor Φs(k, t), which is

defined in equation (3.14). Φs(k, t) is normalized such that Φs(k, 0) ≡ 1.

In the present case, where plastic deformation is simulated under exclusion

of thermal relaxation, time has no meaning and must be replaced by the strain that

increases continually during the simulated trajectory. Therefore, in the present

context the relevant correlation functions are G(r, εzz) and Φs(k, εzz) [23], where

k =2π

r. (4.18)

The contribution of the affine deformation to the decay of Φs, which is purely

caused by the shape change of the simulation box and not the rearrangement of

the atoms inside it, has been removed according to a procedure described in detail

in Ref. [23].

Figure Fig. 4.20 is a plot of self part of the isotropically averaged interme-

diate structure factor Φs(k, εzz) for C5, C10, C20, C50, C100, C200 and C500, as a

109

0 0.5 1 1.5 2 2.5−2

−1

0

1

2

3

4

5

6

7

q

S(q

)

← (1.6197, 3.596)

Fig. 4.19: The structure factor S(q) vs the scattering vector for pentane in un-deformed system. The first peak is artificial for the limited size of thesimulation box. The second peak, which is centered at q = 1.6197 1/A,corresponds to the position of the maximum of the static structurefactor. This value of q = 1.6197 1/A is quite typical for C − C chainstructure.

110

10−3

10−2

10−1

100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−εzz

Φs

C5C10C20C50C100C200C500

Fig. 4.20: Self part of the isotropically averaged intermediate structure factorΦs(k, εzz) , for C5, C10, C20, C50, C100, C200 and C500, respectively, asa function of strain εzz undergoing athermal plastic deformation atconstant pressure. k = 1.6197 1/A is corresponding to the locationof the maximum in the static structure factor.

function of strain εzz during athermal plastic deformation at constant pressure.

k = 1.6197 1/A corresponds to the location of the maximum in the static struc-

ture factor. The self part of the isotropically averaged intermediate structure

factor decays to zero after sufficient large deformation.

The anisotropic properties of the molecular glassy materials have been ver-

ified. The self part of the intermediate structure factor in the x,y,z direction are

found to be the same as the isotropically averaged self part intermediate structure

111

Name C5 C10 C20 C50 C100 C200 C500

β 0.98 0.98 0.98 0.98 0.98 0.98 0.99ǫ0 0.18 0.18 0.17 0.17 0.17 0.16 0.15

Table 4.6: The value of its KWW fitting of these self part intermediate structurefactor Φs(k, εzz) for molecular glassy materials.

factor Φs(k, εzz).

The plots were fitted with Kohlrausch-Williams-Watts (KWW) stretched

exponential function, which is defined as:

Φs(k, εzz) = A exp−(εzz

ǫ0)β, (4.19)

where A, ε0, β are coefficients. It was found that Kohlrausch-Williams-Watts

stretched exponential functions can provide very good fits to the simulation data of

the self part of the isotropically averaged intermediate structure factor Φs(k, εzz)

in x, y, z directions and isotropic.

Even though similar studies have been carried out for deforming a glassy

system at finite temperature [103,137] and athermally [23], the self part of the

intermediate structure factor for althermal deformation of molecular glasses has

not been reported. In the absence of deformation, the thermal dynamics of the

binary Lennard-Jones system have been studied previously in terms of Φs by other

authors. [113,114]

The value of the fits with Kohlrausch-Williams-Watts (KWW) stretched ex-

ponential functions for the self part of the intermediate structure factor Φs(k, εzz)

are listed in table 4.6 for molecular glassy materials.

The stretching exponent β obtained from Φs tends to unity at high tem-

112

perature, whereas slow dynamics in the deeply supercooled regime lead to lower

values of β. The values of β for seven molecular glasses are close to unity, in

x, y, z directions and isotropic, which is in quantitative agreement with results

reported previously. [114,23]

The stretching exponent β, in x, y, z directions and isotropic, obtained

from Φs for seven molecular glasses, tends to increase with respect to the molecular

chain length. The following figure (Fig. 4.21) is the plot of the stretching exponent

β, in x, y, z directions and isotropic, obtained from Φs for seven molecular glasses:

C5, C10, C20, C50, C100, C200 and C500, respectively, as a function of molecular chain

length L. Fig. 4.21 shows that the stretching exponent β, in x, y, z directions

and isotropic, obtained from Φs for seven molecular glasses, are close to unity

and tend to increase with respect to the molecular chain length. This trend may

be the result of the increasing flexibility of the molecular chain with respect to

increasing chain length.

The correlation strain ε0, obtained from KWW fitting of Φs, has the opposite

trend, deceasing with respect to the molecular chain length. ε0 is 0.18 at L = 5

bond length, decreasing to 0.15 at L = 500 bond length. Shorter chain has bigger

ε0. But this varies are small as 0.03 decrease when chain length 100 times longer.

The following figure (4.22) is a plot of the correlation strain ε0, in x, y, z

directions and isotropic, obtained from Φs for seven molecular glasses: C5, C10,

C20, C50, C100, C200 and C500, respectively, as a function of molecular chain length

L.

Figure 4.21 shows the correlation strain ε0, obtained from KWW fitting of

Φs, has a value of 16% and tends to decease with respect to the molecular chain

length. This trend may be the result of the decreasing mobility of the molecules

113

0 100 200 300 400 5000.975

0.98

0.985

0.99

0.995

L

β

isoxyz

Fig. 4.21: The stretching exponent β, isotropically and in x, y, z directions,obtained from Φs for seven molecular glasses: C5, C10, C20, C50, C100,C200 and C500, respectively, as a function of molecular chain length L.

with respect to increasing chain length.

Figure 4.21 also shows that the correlation strains, ε0, ε0x, ε0y, ε0z, isotropic

and in x, y, z directions, respectively, are identical for molecular glasses.

114

0 100 200 300 400 5000.15

0.155

0.16

0.165

0.17

0.175

0.18

0.185

L

ε 0

isoxyz

Fig. 4.22: The correlation strain ε0, in x, y, z directions and isotropic, obtainedfrom Phis for seven molecular glasses: C5, C10, C20, C50, C100, C200

and C500, respectively, as a function of molecular chain length L.

4.3.6 Localization of Plastic Shear Events

After simulating the plastic deformation and finding the drops of shear stress

which indicate plastic shear events occur, we calculate the relative atomic strain

to find the localization of these plastic relaxation events by calculating correlation

function of the Von mises equivalent relative atomic strains, which can quantify

the localization.

115

Finding Plastic Shear Events

Plastic relaxation events are observed in computer simulation of glassy materi-

als under plastic deformation as a typical saw shape stress-strain response [19].

Sudden drops of shear stress during the deformation indicate plastic relaxation

events. These events can be observed in a system with sufficient small size and

small increment strain step δεzz, because in such system in deformation, only a

few relaxation events occur and will not average out.

In this study, the systems are small, in order of nanometer in length. The

pressure is kept in constant. The events of drops of shear stress indicate the plastic

relaxation events. Four typical plastic shear events are shown in figure Fig. 4.23

for the stress-strain relationship of pentane, a system of 5000 atoms in a cubic box

with the size of 52.909 A, where −εzz is from 0.15 to 0.2, and increment strain

step δεzz = 0.0001.

The shear stress drops, as indicated in Fig. 4.23, always follow the same

pattern, linear increase followed by sudden drops. But such drops with a certain

amount keeps the pressure at the target value and the von Mises stress continues

to fluctuate around a certain plateau value. These sudden drops are irreversible

in the shear load and cause the plastic deformation of the glassy materials.

Fig. 4.23 shows the sudden drops of von Mises equivalent stress coinci-

dentally happen with drop of system energy, system pressure and volume. The

amount of drops have positive correlation with the plastic shear events, which are

related to the torsion motion of chains.

During the plastic relaxation events, not only the shear stresses are signifi-

cantly changed, but also pressure, energy and density. The von Mises equivalent

relative atomic strain εεεreq, which is defined in equation (2.79), was specially inter-

116

0.2 0.033

0.0332

0.0334

ρ

0.2 −50

0

50

P

0.2 0

100

200

τ

0.15 0.16 0.17 0.18 0.19 0.2−985

−980

−975

E

−εzz

Relaxation events [MP

a][M

Pa]

[kJ/

mol

]

Fig. 4.23: Shear relaxation events in pentane (C5) under althermal plastic defor-mation at constant pressure, a part from −εzz = 0.15 to −εzz = 0.2.

esting. The average εεεreq was calculated by averaging von Mises equivalent relative

atomic strain of each Delaunay Tetrahedron, which tessellates the system of each

deformation step. It was found that the average εεεreq of Delaunay Tetrahedra of

the system jumps when the stress of the system drops. So the plastic shear events

are characterized by the shear stress drops and εεεreq jumps. It is not surprising

that during the plastic shear events, large and irreversible deformation happens

in some regions of the system, which cause the average εεεreq to increase suddenly.

For those deformation steps without plastic shear events, the average εεεreq is the

same as the system’s increment strain, which is 0.0001 here.

117

Phenomenon of Localization of Plastic Shear Events

Jumps in the average relative atomic strains or the drops in the system’s shear

stress indicate the plastic shear events. These plastic shear events are the result of

large and irreversible deformation in specific regions of the system. The relative

atomic strain is the measurement of the deformation of certain location of the

system. So the relative atomic strain relates to a certain area or location of the

system.

The von Mises equivalent relative atomic strain εεεreq is inhomogeneous and

exhibits sharp peaks in areas where the shear relaxation event takes place. The

localization of plastic shear events can be studied by the distribution of relative

atomic strains.

To calculate the relative atomic strain, we need to build the Delaunay Tetra-

hedra by grouping united atoms of the system. The relative atomic strains of the

points inside of the tetrahedron are equal. This make it possible to observe the

phenomenon of localization of plastic shear events by computational geometry.

The system is meshed by these Delaunay Tetrahedra with color which is weighted

by relative atomic strain of the Delaunay Tetrahedra.

This method is plausible because only Delaunay Tetrahedra close to the

surface can be viewed. To observe the locations with high strains, the DT with

high strains were selected. Phenomenon of localization of plastic shear events can

be viewed by plotting these high strain DT only.

The following figure (Fig. 4.24) shows the localization of the plastic shear

events. The atomic glass Ni80P20underwent plastic deformation at constant pres-

sure as described in chapter 3. The reason that this system is selected to illustrate

the phenomenon of localization of plastic shear events is its general. In Fig. 4.24,

118

(a) shows 4000 atoms in a simulation box of Ni80P20at the deformation stage of

elastic increment shown in (e). (b) shows the Delaunay tessellation of the system

of the deformation stage of elastic increment shown in (e-a). The von Mises equiv-

alent relative atomic strain εεεreq is small and not varies much from DT to DT. (c)

shows the Delaunay tessellation of the system of the deformation stage of plastic

shear events shown in (e-b). The average value of εεεreq is big and varies much from

DT to DT. (d) shows the DT with high εεεreq, 30 times greater than the average,

of the system (e-b). (e) shows the location of system a elastic increment and b

plastic shear event in the stress strain curve.

The relative atomic strain distributes non-homogeneously in the system un-

dergoing plastic shear events. The relative atomic strains are localized, as shown

in Fig. 4.24(d). DTs with high value of relative atomic strain in the system under-

going plastic shear events are spotted to be separated. There are several clusters

with 2 or 3 DTs. The relative atomic strain distributes more homogeneously in

the system without plastic shear events.

It was of great interest to find out what went on in the interior of the mate-

rial during plastic shear events. We find that the plastic shear event state is very

different from the elastic increment state. To compare two states, with and with-

out plastic shear events, the distributions of Delaunay Tetrahedra corresponding

to the relative atomic strains and their volumes are studied. The glassy material

used here is C500. The number of united atoms in the system of material of C500

is 50, 000, and Delaunay Tetrahedra is 326, 160. The simulation box with 50, 000

united atoms are shown in Fig. 4.25(A). The elastic increment state is illustrated

in the von Mises equivalent shear stress-strain curve Fig. 4.25(B-a). The plas-

tic shear event is illustrated in the von Mises equivalent shear stress-strain curve

119

(a) (b)

(c) (d)

0.15 0.16 0.17 0.18 0.19 0.2 0.2118

20

22

24

26

28

30

τ [M

Pa]

−εzz

Plastic Shear Events

b

a

Elastic increment

(e)

Fig. 4.24: Localization of the plastic shear events. (a) shows 4000 atoms in asimulation box of Ni80P20. (b) shows the Delaunay tessellation of thesystem of the deformation stage of elastic increment shown in (e-a).(c) shows the Delaunay tessellation of the system of the deformationstage of plastic shear events shown in (e-b). (d) shows the DT withhigh εεεr

eq, 30 times greater than the average, of the system (e-b). (e)shows the location of system a elastic increment and b plastic shearevent in the stress strain curve.

120

0.07 0.0702 0.0704 0.0706 0.0708 0.071 0.0712 0.0714210

220

230

240

250

260

270

280

τ [M

Pa]

−εzz

Plastic Shear Events

b

a

Elastic increment

(A) (B)

Fig. 4.25: (A)The simulation box with 50, 000 united atoms of glassy materialC500. (B) The elastic increment state and plastic shear event state isillustrated in the von Mises equivalent shear stress-strain curve.

Fig. 4.25(B-b). The distribution of distributions of Delaunay Tetrahedra of elas-

tic increment state is plotted in Fig. 4.26, with −εzz = 0.0703. The distribution

of distributions of Delaunay Tetrahedra of plastic shear event state is plotted in

Fig. 4.27, with −εzz = 0.0704.

Fig. 4.26 shows the distribution of distributions of Delaunay Tetrahedra of

elastic increment state at −εzz = 0.0703 of the glass material of C500. (A) shows

the the surface of the 3d distribution of distributions of Delaunay Tetrahedra with

respect to the von Mises equivalent relative atomic strain εreq and their volume.

(B) shows the contour of the distribution. (C) shows the DT distribution respect

to the volume only. (D) shows the DT distribution respect to εreq.

Fig. 4.27 shows the plastic shear event state at −εzz = 0.0704 of the glass

material of C500. Fig. 4.27 shows a big different of the distribution with respect

the εreq, but the same distribution with respect to the volume. The εr

eq are broadly

distributed from 0 to 0.06, 600 times the strain increment δεzz = 0.0001, with the

121

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x 104

0

5

10

15

0

0.02

0.04

0.060

1

2

3

4

5

x 104

Volume

Delaunay Tetrahedron Distribution

DT

s

Strain 0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

−3

Volume [10−30m3]

ε eqr

(A) (B)

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

4

Volume [10−30m3]

Num

ber

of D

T

0 0.01 0.02 0.03 0.04 0.05 0.060

0.5

1

1.5

2

2.5

3

3.5x 10

5

εeqr

Num

ber

of D

T

(C) (D)

Fig. 4.26: The distribution of distributions of Delaunay Tetrahedra of elastic in-crement state at −εzz = 0.0703 of the glass material of C500. (A)shows the the surface of the 3d distribution of distributions of De-launay Tetrahedra with respect to the von Mises equivalent relativeatomic strain εr

eq and their volume. (B) shows the contour of the dis-tribution. (C) shows the DT distribution respect to the volume only.(D) shows the DT distribution respect to εr

eq.

122

0

200

400

600

800

1000

1200

1400

1600

1800

0

5

10

15

0

0.02

0.04

0.060

500

1000

1500

2000

Volume

Delaunay Tetrahedron Distribution

Strain

DT

s

200

400

600

800

1000

1200

1400

1600

1800

0 5 10 150

0.01

0.02

0.03

0.04

0.05

0.06

Volume [10−30m3]

ε eqr

(a) (b)

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

Volume [10−30m3]

Num

ber

of D

T

0 0.01 0.02 0.03 0.04 0.05 0.060

2000

4000

6000

8000

10000

12000

εeqr

Num

ber

of D

T

(c) (d)

Fig. 4.27: distribution of distributions of Delaunay Tetrahedra of elastic incre-ment state at −εzz = 0.0704 of the glass material of C500. (A) showsthe the surface of the 3d distribution of distributions of DelaunayTetrahedra with respect to the von Mises equivalent relative atomicstrain εr

eq and their volume. (B) shows the contour of the distribution.(C) shows the DT distribution respect to the volume only. (D) showsthe DT distribution respect to εr

eq.

123

average value of 0.0196. On the contrary, the εreq are narrowly distributed from 0

to 0.001 with the average value of 0.0001.

There are five picks in the DT-volume curve of Fig. 4.27(C). This is corre-

sponding to the component of the vertices of DT, which lays on the statistics of

the nearest neighbors. So this plot will not give more information than the pair

distribution function plot of Fig. 4.18.

The Strain Weighted DT in a system without plastic shear events are more

homogenously distributed with respect the relative atomic strains, i.e, close to

the mean value. On the contrary, the distribution of Strain Weighted DT in a

system undergoing plastic shear events are heterogenous with respect the relative

atomic strains. The same phenomenon can be observed in other molecular glassy

materials.

Quantify the Localization of Plastic Shear Events

The heterogenous distribution of SWDT in a system undergoing plastic shear

events are demonstrated in Fig. 4.24. The phenomenon of localization of plastic

shear events are observed. Since the plastic shear events are the elementary pro-

cesses of plastic deformation in molecular glasses, the property of localization of

plastic shear events will be the frame work to discuss plastic deformation of molec-

ular glasses and needs to be described precisely. The autocorrelation function of

relative atomic strain (equation (2.77)) can be used to describe the location of

plastic shear events.

The following figure (Fig. 4.28) is a plot of autocorrelation function of the

von Mises equivalent relative atomic strain εreq for glassy materials of Ni80P20,

C5, C10, C20, C50, C100, C200 and C500. The length unit is σAA, the diameter of

124

0 0.5 1 1.5 2 2.5 3 3.510

−2

10−1

100

r[σ]

C(r

)

NiPc5c10c20c50c100c200c500

Fig. 4.28: Autocorrelation function of the von Mises equivalent relative atomicstrain εr

eq for glassy materials.

majority atom in the system. This length unit is selected for easy comparison of

autocorrelation of εreq among different glassy materials.

The downward straight lines in log-plot of figure Fig. 4.28 indicate that the

relative atomic strains autocorrelation function decays exponentially with the sep-

arations. Fig. 4.28 shows that the relative atomic strain autocorrelation function

exponentially decays very fast, 10% after two diameters away. The result also in-

dicates that the decay rate of the relative atomic strain autocorrelation function

for C5, C10, C20, C50, C100, C200 and C500 is almost the same.

125

We expect the decay rate of correlation function of εreq are different for

different glassy materials. The glassy materials with longer chains should be more

entangled and decay rate should be smaller. But we have not observed such slower

decay rate for longer chain molecules.

The location length Lc, the integral of autocorrelation function of relative

atomic strain (equation (4.9)), can be used to quantify the location of plastic shear

events. The location length Lc is almost the same for seven molecule glasses. The

average value is Lc = 1σAA. This indicates that the plastic shear events are highly

localized, as the order of the diameter of the composing atoms. This also indicates

that the plastic shear events occurs in the rearrangement of the atoms. In the

molecular glasses, Lc = 2.6l, where l is the C − C bond length with l = 1.53

A . Because the length of a trans is 2.574l, and the length of a gauche is 2.41l,

the location length Lc = 2.6l indicates that the plastic shear events in molecular

glasses are caused by the rearrangements of torsion angles along the chains.

But when we consider Ni80P20, the location length Lc = 2.6l is not universe,

since there is no chemical bond in atomic glass Ni80P20. So the statement that

the plastic shear events in molecular glasses are caused by the rearrangements of

torsion angles along the chains is not true for atomic glass. Keep in mind that

the autocorrelation of relative atomic strain in Ni80P20decays at the same rate as

molecular glass, like C500, as shown in Fig. 4.28. There are must be something

in common among glassy materials that determine the decay rate and location

length. The average value of Lc = 1σAA from both atomic glass Ni80P20and

molecular glasses suggests that the repulsion core of potential is the length scale

of the plastic relaxation events. The repulsion force may play an important role in

the plastic relaxation events. Further study of the relationship between repulsion

126

0 0.5 1 1.5 2 2.5 3 3.5

10−6

10−4

10−2

100

r[σ]

Cov

(r)

NiPc5c10c20c50c100c200c500

Fig. 4.29: Covariance function of the von Mises equivalent relative atomic strainεr

eq. for glassy materials

force and dilatancy will be helpful to make this clear.

Although the location length Lc = 1σAA are almost the same in the glassy

materials, the deviations of the relative atomic strains expressed by covariance

function (equation (4.7)) are different. The covariance function of the von Mises

equivalent relative atomic strain εreq for different glassy materials, Ni80P20, C5,

C10, C20, C50, C100, C200 and C500 is shown in figure Fig. 4.29.

The covariance function cov(r) is the same as autocorrelation function ex-

cept the normalization. cov(0) gives the square of the standard deviation of the

127

variables. The standard deviation of the εreq not only depends on the molecular

structure, but also the system size N . The effect of the system size N on the

covariance function is predictable by√

NN

. The effect is tested in the following

by pentane (C5) with N = 625, 1250, 2500, 5000, 10000, respectively. The system

effect on the εreq is shown in Fig. 4.30. System size effect on autocorrelation func-

tion of the von Mises equivalent relative atomic strain εreq in plastic shear events of

pentane was shown in the top of Fig. 4.30. The bottom plot is for the system size

effect on covariance function. We found that Both autocorrelation function and

covariance function decay at almost the same rate, independent of the system size.

But cov(0) varies with the system size N . The cov(0) was plotted with respect to

N in the dotted line in Fig. 4.31.

We found that the cov(0) decay with respect to N when N ≥ 1250 as a

power of −0.5. cov(0) of the glassy materials was plotted with respect to N in

line in Fig. 4.31. We found that for shorter chain, such as C5, C10, C20, cov(0)

varies with system size by a power of −0.5. For chain length longer than 20,

cov(0) decay with a power about −0.15.

The relative atomic strains correlation length Lc gives the length scale of

the plastic shear events, the elementary process of plastic deformation, which is

Lc = 1σAA in this study. The system size affects the covariance function by a

power of −0.5, but not correlation function.

128

0 0.5 1 1.5 210

−2

10−1

100

r[σ]

C(r

)

N625N1250N2500N5000N10000

0 0.5 1 1.5 210

−3

10−2

10−1

r[σ]

Cov

(r)

N=625N=1250N=2500N=5000N=10000

Fig. 4.30: System size effect on autocorrelation function (top) and covariancefunction (bottom) of the von Mises equivalent relative atomic strainεr

eq in plastic shear events of pentane.

129

102

103

104

105

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

N

cov(

0)

C5

C10

C20C50

C100C500

Fig. 4.31: cov(0) varies with respect to system size N of pentane (dotted line)and cov(0) of the glassy materials plotted with respect to N (line).

4.4 Conclusion

The algorithm for the simulation of plastic deformation of glassy solids at constant

pressure has been applied successfully to the plane strain compression of molecular

glasses, such as C5, C10, C20, C50, C100, C200, C500. The system pressures of

seven molecular glassy materials are well kept to the target value during the

simulations of plastic deformation for values of ǫP = 0.0706. The densities of

the system decrease with respect to the deformation, which indicates that the

dilatancy nature of glasses was confirmed in molecular glasses. The system energy

rises continually with increasing deformation, negatively correlated with the the

130

system density. The von Mises equivalent equivalent shear stress increases linearly

with respect to deformation, then fluctuates around the same value up to the

maximum deformation.

The stretching exponent obtained from KWW fitting for seven molecular

glasses are close to one and tend to increase with respect to the molecular chain

length and the correlation strain close to 16% and tends to decease with respect

to the molecular chain length.

The jumps of average relative atomic strains or the drops of system’s shear

stresses sign the plastic shear events. The plastic shear events are localized, which

described the the autocorrelation function of the relative atomic strain, corre-

sponding to DTs in the system. The DTs in a system without plastic shear events

are more homogenously distributed with respect the relative atomic strains, i.e,

close to the mean value. On the contrary, the distribution of DTs in a system

undergoing plastic shear events are heterogenous with respect the relative atomic

strains.

The relative atomic strains autocorrelation function exponentially decays

with the separations. The relative atomic strains correlation length Lc gives the

length scale of the plastic shear events, the elementary process of plastic defor-

mation, Lc = 1σAA. This length scale indicates that the plastic shear events are

highly localized, as the order of the diameter of the composing atoms and the

plastic shear events occurs in the rearrangement of the atoms. In the molecular

glasses, Lc = 2.6l, indicates that the plastic shear events in molecular glasses are

caused by the rearrangements of torsion angles along the chains.

Chapter 5

Conclusion

In summary, an algorithm similar to Berendsen’s barostat has been introduced for

the simulation of plastic deformation of glassy solids at constant pressure, and it

has been successfully applied to the plane strain compression of a binary Lennard-

Jones fluid. The dilatant nature of the elementary plastic relaxation processes was

clearly demonstrated by a linear decrease of density with strain with a slope of

dρdεzz

= 0.0175± 0.0005.

While no difference in the strain-induced structural relaxation of the system

could be identified between the cases of constant pressure and constant volume,

the response of the von Mises equivalent shear stress to deformation is strongly

affected. Conditions of constant volume lead to a build up of system pressure with

deformation, which in turn leads to an increase of resistance to further deforma-

tion, i.e., an increase in the von Mises equivalent stress. In the case of constant

pressure, no such effect is observed, and the von Mises equivalent stress remains

at the level of the yield point over the entire range of deformation.

The system pressures of seven molecular glassy materials are well kept

to the target value during the simulations of plastic deformation for values of

ǫP = 0.0706. The densities of the system decrease with respect to the deformation

, which indicates that the dilatancy nature of glasses was confirmed in molecular

131

132

glasses. The system energy continually rises with increasing deformation, nega-

tively correlated with the the system density. The von Mises equivalent equivalent

shear stress increase linearly with respect to deformation, then fluctuate around

the same value up to the maximum deformation.

The stretching exponent obtained from KWW fitting for seven molecular

glasses are close to one and tend to increase with respect to the molecular chain

length and the correlation strain close to 16% and tends to decease with respect

to the molecular chain length.

The jumps of average relative atomic strains or the drops of system’s shear

stresses sign the plastic shear events. The plastic shear events are localized, which

described the the autocorrelation function of the relative atomic strain, corre-

sponding to DTs in the system. The DTs in a system without plastic shear events

are more homogenously distributed with respect the relative atomic strains, i.e,

close to the mean value. On the contrary, the distribution of DTs in a system

undergoing plastic shear events are heterogenous with respect the relative atomic

strains.

The relative atomic strains autocorrelation function exponentially decays

with the separations. The relative atomic strains correlation length Lc gives the

length scale of the plastic shear events, the elementary process of plastic defor-

mation, Lc = 1σAA. This length scale indicates that the plastic shear events are

highly localized, as the order of the diameter of the composing atoms and the

plastic shear events occurs in the rearrangement of the atoms. In the molecular

glasses, Lc = 2.6l, indicates that the plastic shear events in molecular glasses are

caused by the rearrangements of torsion angles along the chains.

The average value of Lc = 1σAA from both atomic glass Ni80P20and molec-

133

ular glasses suggests that the repulsion core of potential is the length scale of

the plastic relaxation events. The repulsion force may play an important role in

the plastic relaxation events. Further study of the relationship between repulsion

force and dilatancy will be helpful to make this clear.

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