Atomistic Simulations of Elastic-Plastic
Deformation of Amorphous Polymers
by
Franco Mario Capaldi
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2001
@Massachusetts Institute of Technology 2001. All rights reserved.
A A
A uthor .....................
Department of Mechanical Engineering
May 11, 2001
Certified by................
Mary C. Boyce
Professor
Thesis Supervisor
.
I1
C ertified by ......................
I
'
K)
Gregory C. Rutledge
Associate Professor
Thesis Supervisor
Accepted by. .
Ain A.- Sonin
Chairman, Department Committee on Grad
P
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t
MASSACHUSETTS INUSTITUTE
OF TECHNOLOGY
JUL 1 6 2001
I LIBRARIES
r%
Atomistic Simulations of Elastic-Plastic Deformation of
Amorphous Polymers
by
Franco Mario Capaldi
Submitted to the Department of Mechanical Engineering
on May 11, 2001, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
As the demand for polymers with superior properties increases, an understanding of
the fundamental connections between the mechanical behavior and underlying chemical structure becomes imperative. In this work, the thermo-mechanical behavior and
the molecular-level origins of plastic deformation of an amorphous glassy polymer
were studied using atomistic simulations. Understanding of the molecular response
will aid the development of physics-based continuum level models for these materials. A polyethylene-like molecular network was numerically constructed using a
Monte Carlo algorithm and then subjected to uniaxial deformation over a wide range
of strain rates and temperatures using Molecular Dynamics. The model exhibits
many experimentally observed characteristics such as an initial elastic response followed by yield then volume preserving plastic deformation. The stress response was
decomposed into intra and inter molecular components and analyzed throughout deformation. In the glassy regime, activation parameters were calculated in the context
of the Eyring Model of flow in a solid. In addition, observations were made of the
evolution of chain configuration and the correlation of transitions between dihedral
angle states. Below the glass transition, dynamic heterogeneity is observed. It was
also observed that mobility, as measured by the transitioning between dihedral angle
states, increases during plastic deformation to levels observed at much higher temperatures under zero stress. At temperatures near the glass transition temperature, the
mobility approaches levels of undeformed samples at the glass transition temperature.
Thesis Supervisor: Mary C. Boyce
Title: Professor
Thesis Supervisor: Gregory C. Rutledge
Title: Associate Professor
2
Acknowledgments
I would like to thank Professor Mary Boyce and Professor Gregory Rutledge for
their guidance and advice. I am grateful to the National Science Foundation for the
fellowship funding they have given me and for their support of this research through
NSF Grant No. DMR-98-08941.
I would like to thank Dr. Jorgen Bergstrom, Dr. Mark Lavine, Numan Waheed,
and Shawn Mowry for valuable discussions concerning various aspects of this work.
I would also like to acknowledge my lab mates with whom I have had the pleasure
of working and whose company I have enjoyed immensely during the past two years:
Mats, Mike, Ethan, Rebecca, Dora, Sauri, Brian, Prakash, Tom, Hang Qi, Jin Yi,
Matt, Rami, and Steve.
I am especially grateful to my parents, my sister, and Irene for their constant
encouragement, support, and their faith in me.
3
Contents
1
2
3
12
Introduction
1.1
The Glass Transition . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.2
Deformation of Amorphous Glassy Polymers . . . . . . . . . . . . . .
14
1.3
Eyring Model of Flow of Solids
. . . . . . . . . . . . . . . . . . . . .
16
Methodology
19
2.1
Molecular Dynamics Algorithm . . . . .
. . . . .
19
2.2
Monte Carlo Algorithm . . . . . . . . . .
. . . . .
24
2.3
Generation of Initial Structures . . . . .
. . . . .
25
2.4
A Lennard-Jones Fluid . . . . . . . . . .
. . . . .
27
2.5
Polyethylene Chain Model . . . . . . . .
. . . . .
28
2.6
Correlation of Torsion Angles
. . . . . .
. . . . .
31
Structure
34
3.1
Initial Network Characterization . . . . .
3.2
The Glass Transition Temperature
. . .
. . .
44
3.3
Energy Barriers for Transitioning . . . .
. . .
46
3.4
Transitions and Free Volume . . . . . . .
. . .
52
3.5
Correlation of Transitions along a Chain
. . .
57
3.6
Tessellated Volume . . . . . . . . . . . .
. . .
60
34
4 Deformation Results
4.1
64
Dynamical Mechanical Analysis . . . . . . . . . . . . . . . . . . . . .
4
64
4.2
5
Uniaxial Deformation . . . . . . . . . . . . .
. . .
65
4.2.1
Compression . . . . . . . . . . . . . .
. . .
68
4.2.2
Tension
. . . . . . . . . . . . . . . .
. . . . . .
82
4.3
Evolution of Tessellated Volume . . . . . . .
. . . . . . 100
4.4
Strain Rate and Temperature Dependence of Yield Stress
. . . . . .
102
4.4.1
Stress Response . . . . . . . . . . . .
. . . . . .
102
4.4.2
Activation Parameters . . . . . . . .
. . . . . .
105
4.5
Transitions and Yield . . . . . . . . . . . . .
. . . . . .
109
4.6
Correlation of Transitions along a Chain
. . . . . .
115
117
Conclusion
5.1
Structure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
5.2
Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
5.3
Configuration and Deformation . . . . . . . . . . . . . . . . . . . . .
119
5.4
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
120
5
List of Figures
2-1
Pressure-density response of a Lennard-Jones fluid at 0 = 0.9, N = 500
28
2-2
Pressure-density response of a Lennard-Jones fluid at 0 = 2.0, N = 500
29
2-3
Representation of dihedral angle rotation, bond angle bending and
bond stretching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
. . . . . . . . . . . . . . . . .
32
2-4
Energy as a function of dihedral angle
3-1
Measures of equilibration for sample system with 4500 mers at 100K.
The dotted line indicates the time at which the ensemble was switched
from NVT to NPT: (a) density as a function of time, the dashed line
indicates the equilibrium density (b) change in energy as a function of
. . . . . . . . . . . . . . . . . .
time (c) stress as a function of time.
3-2
36
Distribution of bond lengths for crosslinked systems, nw-1-8000-160100K, nw-1-8000-160-200K, nw-1-8000-160-450K . . . . . . . . . . . .
37
3-3
Distribution of bond angles for crosslinked system . . . . . . . . . . .
37
3-4
Distribution of dihedral angles for crosslinked system. . . . . . . . . .
38
3-5
Pole Figures for crosslinked system at temperatures of lOOK, 200K,
and 450K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3-6
Radial distribution function for initial structures at various temperatures. 40
3-7
Distribution of densities at 100K for 40000 mers free chains. Number
of Bins/ Length of Side (Angstroms)
/
Average of atoms per bin and
standard deviation of distribution . . . . . . . . . . . . . . . . . . . .
6
42
3-8
Comparison of bond and angle statistics for various systems: (a) bond
length distribution, (b) bond angle distribution (c) dihedral angle distribution (d) radial distribution profile. . . . . . . . . . . . . . . . . .
3-9
43
Simulations of polyethylene model at atmospheric pressure with 4000
united atom beads. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3-10 Diagram of dihedral angle rotation. . . . . . . . . . . . . . . . . . . .
50
3-11 Diagram of transition definition. . . . . . . . . . . . . . . . . . . . . .
51
3-12 Transition rates for a system of 5000 mers arranged in 20 free chains.
51
3-13 Decay of the torsional angle autocorrelation function at various temperatures. .......
.........................
....
...
52
3-14 Spatial distribution of conformational transitions for a system of 5000
united atoms arranged in 20 free PE chains. Dihedral angles are numbered serially along the chain and the number of transitions at each
dihedral angle are plotted against the dihedral angle. The total number
of transitions at each temperature is 4043. . . . . . . . . . . . . . . .
53
3-15 Correlation between transition rates and density at various temperatures for a system of 5000 mers arranged in 20 free chains using a bin
size of 6.31A x 6.31A x 6.31A. Each bin contains an average of 10.6
m ers. . . . .. .
. . . . . . . . . . .. .
. . . . . . . . . . . . . . . . .
55
3-16 Correlation between transition rates and density at various temperatures for a system of 5000 mers arranged in 20 free chains using a bin
size of 5.23A x 5.23A x 5.23A. Each bin contains an average of 5.84
m ers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3-17 Dihedral angle trajectory for a single, active dihedral angle . . . . . .
58
3-18 Neighbor Correlation for system of 5000 united atoms arranged in 20
free chains at various temperatures. . . . . . . . . . . . . . . . . . . .
59
3-19 M ap of dihedral angles . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3-20 Dihedral angle value over time for active angle in networked system at
100K .
. . . ..
. . . . . ....
.. . . . . . . . .
. . . . . . . . . . .
61
3-21 T = lOOK, networked system. . . . . . . . . . . . . . . . . . . . . . .
62
7
3-22 Tessellated volume distribution at various temperatures for system of
5000 monomers arranged in 20 free chains. . . . . . . . . . . . . . . .
62
4-1
Experimental graph of DMA curve for HDPE [15] . . . . . . . . . . .
66
4-2
Estimated modulus for i = 1 x 10' 0 /s and
=
5 x 10
10 /s
for a system
of 7775 united atoms arranged in 160 networked chains. . . . . . . . .
4-3
Estimated modulus for various strain rates at lOOK and 200K for a
. . .
system of 7775 united atoms arranged in 160 networked chains.
4-4
66
Compressive stress versus strain for uniaxial compression for
=
67
1 x
1010/s and T = lOOK for a system of 7775 united atoms arranged in
160 networked chains . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-5
Partitioning of compressive stress for uniaxial compression along load
direction .
4-6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
71
Partitioning of normalized energy change per united atom versus true
strain in com pression . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-8
70
Partitioning of normalized compressive stress as sample is loaded in
compression along the x-axis.
4-7
69
72
Comparison of structural statistics for compressive true strains of 0,
0.2, and 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
Density versus true strain for uniaxial compression
. . . . . . . . . .
74
4-10 Schematic of Lennard Jones potential and gradient. . . . . . . . . . .
75
4-9
4-11 Compressive stress versus true strain for constant volume compressive
deformation with i = 1 x 1010/s and T = 100K
4-12 Chain statistics versus true strain in compression
. . . . . . . . . . . .
76
. . . . . . . . . . .
77
4-13 Transition rate versus true strain for uniaxial compression
4-14 Pole figures for uniaxial compression
. . . . . .
78
. . . . . . . . . . . . . . . . . .
79
4-15 Line profiles of three dimensional pair distribution for monomers along
different chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
4-16 Diagram of three dimensional distribution profile slices. . . . . . . . .
82
8
4-17 Three-dimensional pair distribution for monomers on different chains
w ith c = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
4-18 Three-dimensional pair distribution for monomers on different chains
w ith c = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
4-19 Three-dimensional pair distribution for monomers on different chains
w ith E = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
4-20 Line profile of three dimensional pair distribution for monomers along
the sam e chain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4-21 Three-dimensional pair distribution for monomers on the same chain
w ith c = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
4-22 Three-dimensional pair distribution for monomers on the same chain
w ith c = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
4-23 Three-dimensional pair distribution for monomers on the same chain
w ith c = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-24 Schematic of chain interactions along compressed axis.
. . . . . . . .
89
90
4-25 Tensile stresses versus true strain for uniaxial tension with i = 1 x
1010/s and T = 100K for system of 7775 united atoms arranged in 160
networked chains.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4-26 Partitioning of tensile stresses versus true strain for uniaxial tension
loaded in the x-direction. . . . . . . . . . . . . . . . . . . . . . . . . .
92
4-27 Density versus true strain for uniaxial tension . . . . . . . . . . . . .
93
4-28 Partitioning of energy change per united atom versus true strain for
tension .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
4-29 Comparison of structural statistics for tensile true strains of 0, 0.2, and
1 .0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-30 Chain statistics versus true strain for uniaxial tension.
95
. . . . . . . .
96
4-31 Transition rate versus true strain for uniaxial tension. . . . . . . . . .
97
4-32 Pole figures for uniaxial tension . . . . . . . . . . . . . . . . . . . . .
98
4-33 Tessellated volume distribution at various strains for system of 5000
mers tested in uniaxial tension at lOOK for a strain rate of 1 x 10 10 sec-1. 10 0
9
4-34 Tessellated volume distribution at various strains for system of 40000
mers arranged in 160 free chains tested in uniaxial compression at lOOK
for a strain rate of 1x10' 0 s- 1 . . . . . . . . . . . . . . . . . . . . . . .
101
4-35 Stress versus strain for various temperatures for a system of 7775 united
atoms arranged in 160 networked chains. . . . . . . . . . . . . . . . .
103
4-36 Change in density versus strain for various temperatures for a system
of 7775 united atoms arranged in 160 networked chains. . . . . . . . . 103
4-37 Change in radial distribution function during the application of compressive deformation at 450K. . . . . . . . . . . . . . . . . . . . . . .
104
4-38 Chemical Structure of (a) polystyrene, (b) polyethylene-terephthalate,
(c) polycarbonate . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
106
4-39 Summary of yield stress obtained through simulations of uniaxial compression for various initial networked structures, strain rates, and temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4-40 Transition rates as a function of true strain. . . . . . . . . . . . . . .
112
4-41 Number of transitions per dihedral angle per angstrom of deformation. 113
. . . . . . . . . . .
114
4-43 Neighbor Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
4-42 Stress as a function of strain for various systems.
10
List of Tables
2.1
Potential Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
General Structure of Chains where N is the number of mers in the sim-
32
ulation, M is the number of mers per chain, C is the number of chains,
< dee > is the average end to end distance, P is given by equation 3.1,
% Trans is the percentage of dihedral angles in the trans configuration,
F2 , F3 and F4 are the number of crosslinks with a functionality of 2,
3, and 4 respectively
3.2
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Crosslinked vs Free Chain Systems. The equilibrium bond length and
angle are 1.53 A and 110.01 degrees respectively. . . . . . . . . . . . .
3.3
41
44
Glass transition temperature, T., and linear thermal expansion coefficient, LTEC, for various molecular weights. . . . . . . . . . . . . . . .
46
. . . . . .
49
3.4
Calculated Activation Energy for Dihedral Angle Switches
4.1
Activation parameters for a selection of glassy polymers [1] . . . . . . 105
11
Chapter 1
Introduction
As the demand for polymers with superior properties increases, so does the need for an
understanding of the relationship between mechanical behavior and underlying chemical structure. The challenge lies in the fact that mechanical behavior in polymeric
materials is governed by phenomena occurring over a wide range of length and time
scales. Atomistic simulations may provide a glimpse into detailed material behavior on length and time scales which are not accessible through current experimental
techniques.
In the present work, information obtained from atomistic simulations is used to
develop a better understanding of the molecular response of amorphous polymers to
deformation. Simulations of a polyethylene-like molecule subjected to uniaxial deformation are carried out over a wide range of strain rates and temperatures. From this
data, activation volumes were calculated. This data was augmented by observations
of the evolution of chain configuration, orientation, and molecular packing during
deformation.
1.1
The Glass Transition
The glassy state is a non-equilibrated state arising from the rapid cooling of a polymer
melt. The glassy structure has excess volume and enthalpy with respect to the equilibrium structure at the same temperature. As the melt is cooled, molecular kinetics
12
slow such that the liquid structure is unable to relax to its lower energy crystalline
state. The resulting glass maintains a disordered structure characteristic of a liquid
while it exhibits mechanical properties similar to those of a crystalline solid. The
highly constrained polymer chains are unable to fully explore their configurational
space during cooling below the glass transition temperature resulting in an inhomogeneous microstructure with varying degrees of disorder. There are regions possessing
lower local densities. The chain segments in these regions are packed less tightly and
due to the diminished constraint can explore their configuration space more readily,
i.e. they have higher mobility.
Although the molecular kinetics may be slow, thermal motion results in continued relaxation at temperatures above absolute zero. If given sufficient time, further
relaxation would occur bringing the structure closer to its equilibrium. Therefore, it
is important to characterize the transition temperature and the associated time scale
for a glassy structure.
Attempts to correlate molecular motions with the glass transition through atomistic simulation have shown both the weaknesses and strengths of these methods.
Gee and Boyd [13] have shown that the glass transition is strongly dependent on
the potential used to describe the dihedral angle. In addition, the cooling rates used
in atomistic simulations are much too large to allow for accurate prediction of glass
transition temperatures at experimental cooling rates.
Fortunately, the qualitative chain dynamics seem to be consistent across a wide
range of model force fields [33, 5]. The transition of dihedral angles from trans to
gauche states has been the subject of many studies. Studies have found that although
transitions in the melt are evenly distributed across all dihedral angles, these transitions become heterogeneously distributed below the glass transition temperature [4].
Relatively few dihedral angles undergo the majority of transitions while other dihedral angles remain constant. This illustrates the heterogeneity of segmental mobility
which is "frozen" into the glassy structure. Correlations between transitions in the
glass and the melt which occur on a polymer chain have shown strong correlations
between dihedral angles separated by 2 bonds which suggests a kink type motion is
13
dominant [33]. Also self correlation of transitions has been shown to become more
significant as the temperature falls below the glass transition temperature making it
a good indicator of vitrification.
Many studies have found that as the temperature falls below the glass transition
temperature, the dihedral angle transition rates decrease dramatically. But a study
by Liang et al. [23] shows that conclusions drawn from transition rates may depend
on the specific definition of a transition. Choosing definitions which encompass the
entire range of phi leads to overestimation of transition rates by counting short lived
changes between very high energy states. In the literature one finds a wide variety of
definitions.
Atomistic simulations have also allowed scientists to probe material states which
are not experimentally accessible. For example, Karatasos, Adolf, and Hotston [21]
found increasing density at constant temperature results in increased correlation between neighboring torsion angles above the glass transition temperature while decreasing the overall transition rates. Yang, et al. [46] also showed that the glass
transition temperature increased with the application of pressure or an increase in
density. This phenomena has also been observed experimentally by McKinney et
al. [27]. Bharadwaj et al. [4] found that increasing pressure increased the torsional
self-correlation. In both of these cases the glass transition was determined from the
change in slope of the specific volume versus temperature.
1.2
Deformation of Amorphous Glassy Polymers
Hasan et al. [16] provide a description of a free volume theory for polymer deformation.
In these theories, mobility is related to the free volume, or the volume
surrounding a chain segment which is available for occupation by that segment. If
a segment has low free volume, it is tightly confined by its neighbors and is unable
to explore its configuration space easily. Since the regions of the polymer chain with
higher mobility transform at a lower applied stress, the structural inhomogeneity described in Section 1.1 leads to the localized deformation observed in polymers. Under
14
an applied load, the regions of the chain with highest mobility undergo shear transformation events. For very small strains, the transformation strain energy is then
stored in the surrounding elastic material which then exerts a back stress preventing
further transformations of this region from occurring. Upon unloading following small
strain deformation, the back stress returns these transformed regions to their original
states leading to the experimentally observed recovery. As stress levels increase, back
stress is relieved, regions of the polymer chain undergo multiple transformations, and
the additional deformation is no longer recoverable. A transformation in one region
increases the likelihood that another transition will follow in a neighboring region.
An increasing stress causes regions with lower mobility to transform decreasing the
average spacing between transformed regions and percolation of these transformed
regions leads to yield.
The mechanisms involved in the deformation of glassy polymers are difficult to
characterize due to the many length scales over which they occur. Due to its structural
complexity, there are a wide range of possible responses a polymer may have to
deformation. A polymer may respond through intramolecular deformation such as
bond stretching, bond flexing, and bond rotation or intermolecular deformation such
as chain slippage and chain separation. Creating atomistic models which capture all
of these phenomena is difficult.
Atomic/metallic glasses have fewer constraints and therefore are more easily studied in the context of computer simulations. Studies conducted on these glasses have
shown that plastic deformation occurs by inhomogeneous atomic movements confined
to well-defined volumes undergoing relatively large shear strains [9, 38, 25]. These
regions act as nucleation sites for additional transformations. As these regions link,
flow structures form.
The original studies on polymeric glasses by Brown and Clarke [7] and Theodorou
[42] proved the feasibility of the application of computer simulation techniques to
studying the mechanical properties of polymer glasses. Later molecular statics studies
by Mott et al. [29] and by Hutnik et al. [20] postulated that plastic deformation is
the result of shear transformations occurring within a volume element undergoing a
15
transformation shear strain. Unfortunately, the size of the activation volume element
was on the order of the simulation cell used in these experiments forcing the entire cell
to react at each transformation event. Hutnik, for example, calculated an activation
volume which had a diameter of 10.1 nm while the simulation cell size was around 3-5
nm per side. The activation volume was calculated using the transformation strain
obtained through simulation and the experimental activation volume parameter.
Recent experimental evidence from Zhou et al. [47] on the diffusion of species
within a plastically deforming polymer suggests that atomic mobility of the polymeric glass during flow is similar to that at the glass transition temperature. Nuclear
magnetic resonance studies of deforming polymeric structures has also revealed increased mobility during plastic deformation Loo et al. [24].
1.3
Eyring Model of Flow of Solids
At low temperatures or short time scales, material behavior during deformation is
governed by essentially fixed energy barriers and can be described as a thermally
activated process. At high temperature and long time scales, time dependence may
arise due to both the thermally activated nature of the deformation process and the
occurrence of structural relaxation. The Eyring Model describes deformation as a
stress-biased thermally activated process assuming isostructural deformation.
The Eyring model attempts to predict the effects of strain rate and temperature
on yield by assuming that flow occurs through an activated-rate process. This model
does not take into account the specifics of the underlying activated process which
may be the motion of atoms, molecules, or, in polymers, segments of a molecule. If
this transition has an associated energy barrier of height AE, the jump rate will obey
the Arrhenius equation.
The application of stress to the solid results in a distortion of this energy barrier
favoring jumps which occur in the direction of the applied stress. The resulting strain
rate is assumed to be proportional to the jump rate. Near the high yield stress, the
energy barrier is distorted to such an extent that the reverse jumps may be neglected
16
resulting in an equation relating yield stress to strain rate and temperature[26].
(
AH -rYQ)(
where y is the macroscopic shear strain rate, Ay0 is the rate constant, AH is the
enthalpy of activation in the limit of zero applied stress, Ty is the shear yield stress,
Q is the activation volume, kB is the Boltzmann constant, and T is the temperature.
Equation 1.1 can be rewritten as:
rY
1AH
T
Q
T =-
T
±kBln .
70
(1.2)
At constant strain rate, manipulation of Equation 1.2 gives a convenient equation
for determining the activation volume from yield stress versus strain rate data.
d-ry
dTn
The activation volume,
kBT
Q,
kBT(1.3)
, is commonly interpreted as an estimate of the vol-
ume which must be activated for flow to occur,
Va,
multiplied by the shear strain it
undergoes, YT:
Q = _YTVa.
(1.4)
It is common in polymers for the experimentally determined activation volume
to be significantly larger than the segment length. This suggests cooperative motion
during the flow process for polymers.
It is important to realize that this analysis assumes a single activated process.
Since yield in many materials may be governed by differing processes in different
temperature and strain rate regimes, extrapolation of yield stress from one regime
to another may be incorrect. For example, yield data for polycarbonate below -50
degrees Celsius yields different activation parameters than data at higher temperatures [22].
This suggests that yield occurs by activation of a different mechanism
in these various temperature regimes. If one imagines that there are many possible
17
molecular motions in the polymer system which can be activated each with its own
activation parameters. As the temperature or strain rate vary, one or another of these
mechanisms may be dominant over the others. As long as one stays within a regime
where that particular mechanism is dominant, the data will fit well to the Eyring
model. As the active mechanism changes or more than one mode becomes active, a
more complicated model is required. In addition to various modes of deformation,
the barrier heights for a particular mode may vary due to the heterogeneous nature
of the disorder in the system which leads to the nonlinear transition region between
linear elastic behavior and yield.
18
Chapter 2
Methodology
2.1
Molecular Dynamics Algorithm
Molecular Dynamics is a technique for following the progression of a system of atoms
or molecules through phase space. From a given set of initial conditions and interatomic potentials, one can generate the trajectories for particles. Properties can be
calculated by averaging over the trajectory:
AT = -
dtA(t)
(2.1)
where A is the property of interest and r is the time. The key assumption in molecular
dynamics is that the movement of atoms can be treated using classical mechanics.
If this equation of motion holds, then the same equations of motion that govern
macroscopic objects may be used to model the trajectories of atoms or molecules.
In fact, the motion of all but the lightest of atoms can be treated accurately by
classical mechanics even though the interactions between atoms, which is a product
of the motion of electrons, must be calculated using quantum mechanics. Luckily,
the mass of a nuclei is on the order of ten thousand times that of an electron. The
position of the electron is highly non-localized due to its high velocities and low mass
and has a short relaxation time. In contrast, the heavier nucleus has a highly localized position and requires much more time to reach its equilibrium state than the
19
electrons encircling it. The two motions can therefore be decoupled and treated independently. This is known as the Born-Oppenheimer Principle. Quantum mechanical
calculations can be used to determine the energy of a given configuration of atoms
with equilibrated electrons [37]. By varying distances between atoms, a map of the
potential energy as a function of configuration can be developed and the motion of
large systems of atoms can be calculated without explicitly taking into account the
electrons.
By treating the atoms as localized point masses and assuming their interactions
are captured in a potential function, one can use Newton's equations of motion to
solve for the trajectories of these particles. Recalling that the force on a particle can
be written as the negative gradient of the potential, one can then write the equations
of motion in the following form:
d2 r
au (rN)
=i
dt 2
ri
where mi and ri are the mass and position of atom i respectively, N is the number
of atoms in the simulation, rN
=
{ri, r 2 , - - - rN}, and U (rN) is the potential which
can be expressed as a function of all other atomic positions. This gives a set of N 2 "d
order non-linear ordinary differential equations which can only be solved numerically.
The corresponding Hamiltonian for this system is given by:
N
2
H =
+U (rN)
(2.3)
i12mi
where pi is the particle momentum. The Hamiltonian in this case is equivalent to the
total energy of the system. Therefore, solving these equations produces a trajectory
within an ensemble with a constant number of particles, volume, and energy. This
ensemble is termed the microcanonical ensemble (NVE).
If one would like to simulate other ensembles, changes must be made to the equations of motion. To conduct simulations in an ensemble with a constant number
of particles, volume, and temperature (NVT), the Nose-Hoover thermostat was em-
20
ployed [30] [19].
In this formulation, a heat bath is coupled to the equations of
motion allowing the transfer of energy to and from the system. The equations of
motion become:
d2r
DV (rN)
-r N
~
dti
=
=-
[
N
_ vi
(2.4)
fkBT
(2.5)
Q
f
= 3(N - 1)
(2.6)
where q is a frictional coefficient coupling the particles to a heat bath, Q is an arbitrary
constant determining the rate at which energy is transferred between the system and
heat bath, f is the number of degrees of freedom for the system, N is the number of
particles in the system. There are N particles each with 3 degrees of freedom with
three constraint equations forcing the linear momentum of the system to be zero
which reduces the degrees of freedom to 3(N-1).
The Verlet algorithm is used to integrate equations of motion for a NVT system.
This algorithm can be written as
At 2
r(t + At) = r(t) + Atv(t) + 2 a(t)
v(t + At)
ma(t + At)
v(t
(2.7)
At
= v(t) + 2t (a(t) + a(t + At))
=
+ At) =
(2.8)
-VV(rN)(r(t + At)) - 77(t + At)v(t + At)
(v(t) - AtVr(t + At) +
2m
1+ 71(t +At)At
2m
(2.9)
2
(2.10)
At 2
At) = r(t) + At (t) + 2 i(t)
22
i(t + At) = - Zv(t + At)a(t + At)
r7(t +
(2.11)
(2.12)
The equations of motion can be similarly modified to sample an ensemble with
a constant number of particles, pressure, and temperature (NPT) [32]. Instead, the
21
Berendensen algorithm is used [2]. In this formulation, the coupling between motion of
the simulation cell is assumed to be weakly coupled to the motion of the particles. This
allows the two motions to be integrated separately. Therefore the particle motions
are integrated using the Nose-Hoover equations and the volume and positions of each
particle are scaled at each time step in response to stress changes.
zAt
A(t)
=
s(t + At)
=
As(t)
(2.14)
ri(t + At)
=
Ari(t)
(2.15)
V(t + At)
=
(detA) V
(2.16)
I-
3
#3 (-r(t) - r")
(2.13)
Tp
where A (t) is the scaling matrix, Tp is the time constant, r
0
is the desired stress
tensor, r(t) is the current stress tensor, At is the time step, s is the set of vectors
defining the shape of the simulation cell, ri is the vector position of particle i, and V is
the volume of the simulation cell. This algorithm provides an over-damped response
to pressure changes, thereby removing many of the oscillations that would be found
using the more rigorous ensemble techniques. The stress tensor is calculated using the
atomic virial [41]. Since periodic boundary conditions are employed, the expression
for the virial stress tensor is:
=
M
-
2V
Li
r' ri a
F-'")
-
%
)mn,)
+
(2.17)
Ttail
where V is the volume of the cell, pf is the a component of the momentum of the
ith
particle, mi is the mass of particle i, r' is the a component of the position of
the ith particle,
rjmin(i)
is the a component of the position of the image of the
particle which is closest to the ith particle, Fi is the force on i due to
jmin,
and
jth
Ttail
is the correction to the pressure due to the cutoff radius. The force is defined as
Fi = -Vrj-rj U(rN). The gradient is calculated keeping the separation between all
22
particles other then i and
j
constant. This does not presuppose pairwise additivity
and is applicable to many-body forces. The calculation of long range contributions to
the virial stress tensor and the acceleration can be time consuming. Often interactions
beyond a given distance are discarded. When a truncation is used in calculating the
contribution of a set of interactions to the pressure, the following correction must be
added to the calculated stress:
=
7til tal
oo 23
2
(2.18)
g(r) dUNBr3dr
dr
[00
Jrc
where 6oo is the Kronecker delta which is zero if a -$0 and is equal to one if a = 3, p
is the number density, g(r) is the radial distribution function,
UNB
is the long range
potential, and r, is the cutoff radius.
Assuming that the van der Waals interactions are truncated at rc and are described by a 12-6 Lennard-Jones potential and that the radial distribution function
can be sufficiently approximated as 1 in the region beyond r0 , the equation for the
tail correction becomes
ai
167r
6
3
2
fa
2(
(3
9
r,
or
rc
12)
(2.19)
This correction was employed throughout this study. In order to facilitate the
computation of the van der Waals interactions, a list of the potential interacting
pairs was maintained. Termed a neighbor list, this technique was first used by Verlet
[43]. For each atom a list of particles within a distance d = rcutof f +6r was maintained.
When any particle has moved a distance greater than L, the list is updated. Due to
the constrained motion of particles in the solid phase, this list is valid for a number
of time steps.
The link-cell method [18] was used to aid in the creation of neighbor lists. The
simulation cell is divided into rectangular compartments whose dimensions are slightly
larger than the cutoff distance. Then, for a given mer, only 27 adjacent compartments
must be searched to find neighbors.
Since searching for these neighbors requires
O(N 2 ) work, the link-cell method greatly reduces the computational time required to
23
form the neighbor list.
Due to computational constraints, the particle number in atomistic simulations is
severely limited. Surface effects may negatively affect even the largest of simulations.
In order to reduce the surface effects, periodic boundary conditions were employed.
Atoms or bonds which leave one face of the simulation cell enter the opposite side.
2.2
Monte Carlo Algorithm
The Monte Carlo method is a stochastic method which generates configurations of a
given system within a particular ensemble. Producing a path through configuration
space, it provides no information on the trajectories or velocities of particles. The
original formulation is attributed to Metropolis et al. [28].
In atomistic simulations, one is interested in calculating the properties of a given
potential function within a certain ensemble.
It can be shown that for a NVT ensemble, the probability of finding the system
in a particular configuration, pi, is given by:
-,
p = exp
U(r N
(2.20)
r)
where rN is the set of positions of the particles in the system, U is the potential
energy function, and kb is Boltzman's constant.
The average of a property can then be expressed as an integral over degrees of
freedom weighted by the probability of finding that state.
<For
A >=
f drNeXp
p et
(r))
A(rUk1
N)
N)
where A is the property. For static properties, this average is equivalent to the time
average over a single trajectory.
Determining a given property of a system becomes a matter of integrating this
function.
Only in very rare cases can this integral be evaluated analytically. Re-
sorting to numerical methods, one could randomly sample configurations, calculate
24
the weight and average over the weighted samples. This is inefficient since most of
the configurations sampled would be high energy states with low weights. A more
efficient way to calculate this ensemble average would be to generate configurations
according to the weight e-U(rN).
Using this scheme, one samples more frequently
those states that contribute most to the average. The property can then be expressed
as a simple average of the computed property over the generated states.
I
<A >=
-
N,
N
Z A(r
(2.22)
where A is the property of interest, and N, is the number of generated configurations. To sample configurations according to this given weight, Metropolis developed
the following algorithm:
1. given an initial configuration rN
2. calculate the energy of this state Uj = U(rN)
3. choose an atom at random and perturb it
4. calculate the energy of the perturbed configuration Uj = U(r
)
5. if (Uj - Uj) < 0 accept the perturbation otherwise accept configuration with a
probability e-
U---)
6. calculate property A
7. goto 2.
The average of property A can then be evaluated using Equation 2.22.
2.3
Generation of Initial Structures
The procedure for producing a well equilibrated structure is similar to that used in
the study by Chui et al. [8].
A system of mono-dispersed chains of specified lengths is generated according to
a random walk algorithm with fixed bond lengths and angles. The dihedral angle is
randomly assigned as the walk progresses. This structure is relaxed in the NVT en-
25
semble at a temperature of 1300-2300 K using the Monte Carlo Algorithm described
in Section 2.2 with the Lennard-Jones potential turned off. During this stage the
chains are free to pass through each other and adopt an end-to-end distance characteristic of an idealized chain. The LJ potential is then turned on and the LJ radius
is slowly increased to the desired value.
This equilibrated sample of free chains is then crosslinked according to a procedure
similar to that used by Duering et al. [10]. During crosslinking, functional groups
are attached to all chain ends. When these groups come within a specified critical
distance from one another, one functional unit is removed and all atoms bonded to
this group become bonded to the remaining unit. The system configuration is evolved
using the Monte Carlo scheme and periodic checks are made on the distance between
functional groups.
To quicken the crosslinking process, an artificial potential is added between crosslinks.
ECL
1
2
-KCL$
|rijI
{1 :r7 + 771 > rimax
:
(2.23)
(2.24)
Th + 77'<; 7max
where KCL the potential constant, Iri I is the distance between functional unit i and
j,
rTi is the current functionality of group i, and rimax is the maximum functionality.
The sum is carried out over all pairs of functional groups. The added potential
acts to draw groups that can be joined together while repulsing those whose added
functionality exceeds the maximum. The functionality, potential constant, and critical distance can be varied to change the crosslinking dynamics.
The crosslinked structure is then taken as the initial configuration for the molecular dynamics algorithm described in Section 2.1. The Verlet integration algorithm
requires that the position, velocity, and acceleration of each particle be specified as
initial conditions. The Monte Carlo algorithm gives only configurational data. The
velocities and accelerations must be specified at the start. The initial accelerations
26
are computed from the gradient of the potential and the initial velocities are randomly assigned according to a normal distribution. The probability density, p (vix),
for the normal distribution of velocities is given by the following equation:
m
p (vix) =
2rkBT
exp
'-mv*
kB I
(2.25)
where kB is the Boltzman Constant, T is the absolute temperature, m is the mass,
vix is the x-component of the velocity for the
ith
particle.
Velocities are assigned according to the normal distribution at the elevated temperature. The sample is cooled at a constant rate and equilibrated at the desired
temperature. The pressure control algorithm is used to equilibrate the system at
atmospheric pressure.
2.4
A Lennard-Jones Fluid
In order to test the implementation of the algorithms described in Section 2.1 and
2.2, the equation of state for a Lennard-Jones fluid was determined and the results
compared with previously reported data [11]. Data are presented in terms of a reduced
temperature, pressure, and density:
kBT
(2.26)
Pa3
(2.27)
pr
(2.28)
0
P
=
p* =
(2.29)
where kB is the Boltzman constant, T is the absolute temperature, E and a are the
Lennard Jones parameters, P* is the reduced pressure, p* is the reduced density, 6
is the reduced temperature. This is done so that the results are independent of the
choice of a and c.
The cases of 0 = 0.9 and 0 = 2.0 are shown in Figure 2-1 and Figure 2-2 respec-
27
0.8
o Monte Carlo
.1- Molecular Dynamics
-- Data from Frenkel and Smith (1 996)_
0.6-
0.4-
0.2-
-0.2-
-0.4
-0.6
0
0.1
0.2
0.3
065
0.6
0!4
Reduced Density p = No /V
0,7
0.8
0.9
Figure 2-1: Pressure-density response of a Lennard-Jones fluid at 6 = 0.9, N = 500
tively. A 50 ps simulation was run at each pressure and the values for temperature
were averaged over the last 30 ps. The simulations were conducted with the cutoff
radius set to one half of the simulation cell length. Excellent agreement with the
published data was found.
2.5
Polyethylene Chain Model
The quality and accuracy of atomistic simulations depend upon the quality of the force
field used to model a system. The structure and dynamical properties of a simulated
system are dictated by this force field. Therefore it is of utmost importance to find
a force field that accurately produces the properties of the system which are to be
investigated. The amount of detail that must be captured by the model also depends
on the phenomena to be studied.
Polyethylene was chosen as a model system due to its chemical simplicity: It
consists of a backbone of carbon atoms with pendant hydrogen atoms. It has been
shown in numerous studies that the explicit representation of these pendant hydrogen
atoms in atomistic simulations is not necessary to recreate many static and dynamic
28
1
10
11
1
11
1
Monte Carlo
<1 Molecular Dynamics
-0- Data from Frenkel and Smith (1996)
10
9
8 -
7 -I
6-
/
5
4-
3-
2-
0
0.1
0.2
03
0.4
0.5
0.6
3
Reduced Density p = Na N
0.7
0.8
0.9
1
Figure 2-2: Pressure-density response of a Lennard-Jones fluid at 0 = 2.0, N
500
properties [34]. In the united atom approximation, each methyl group is collapsed
into a single effective potential positioned at the carbon site while leaving the essential
skeletal structure of the carbon backbone unaltered. This approximation has been
widely used due to the three fold reduction in the number of explicit particles in
a given simulation which reduces the computational time required by an order of
magnitude.
For the purposes of this study, the force field optimized by Paul et al. [34] was
used in a modified form. This model consists of an united atom representation of
polyethylene optimized to reproduce experimental data on the properties of melts of
n-alkane chains with varying molecular weights. This appears to capture the local
and large scale dynamics of the chain well.
The van der Waals forces between united atoms are modelled using the LennardJones functional form of the potential.
EU =
4E
("
12
,
6)
rj<ro
(2.30)
-
0
:evl
dr
> reo
where o- is the excluded volume radius, E is the well depth of the potential, rij is the
29
vector distance separating the united atoms i and
j, and
reu is the cutoff radius for
the Lennard-Jones potential.
The rigid bonds of the Paul model are replaced with a harmonic potential:
E
=
Ik (Iri| - bequi)
(2.31)
where kb is the bond potential coefficient, bequil is the equilibrium length of the bond,
and rij is the vector distance separating the bonded united atoms i and
j.
The angle potential is given by:
ko (cos (jk )
Eik =
cos (Oik)
-
=
cos (Oequii)) 2
-ri
rge(233
ri3
rjk
(2.32)
(2.33)
where ko is the angle potential coefficient, Oequil is the equilibrium angle of the potential, and
0
ijk
is the bond angle between consecutive united atoms ij, and k.
The functional form of the torsional energy is given by:
3
km E (cos' (Oihkl))
m=0
(2.34)
rij x
rjkJ
(2.35)
\rij
rik )|
Eijki =
COS (qijk)
where k, k2, k3, k
Paul,
#ijkl
-
1
x
rjk x rkl
|rjk
x
rk1I|)
are constants which were fit to the torsional potential used by
is the dihedral angle formed between the consecutive united atoms i,
j, k,
and 1, and #ijki = ir radians corresponds to the trans torsional state.
The calculation of the van der Waals potential is the most time consuming step in
the simulation of this system. Due to the rapid decay of the Lennard-Jones potential,
a cutoff was applied without greatly affecting the evolution of the system. Corrections
to the calculated pressure for the truncated system were added to compensate for the
interactions that were not included directly. The proper corrections are discussed in
30
ijkl
1
k
.
-
i
j
k
Figure 2-3: Representation of dihedral angle rotation, bond angle bending and bond
stretching.
section 2.1. The van der Waals interactions were not calculated for atoms which were
separated by fewer then 4 bonds along a chain since the corresponding contribution
was accounted for by the bond, angle, and torsional potentials for these groupings.
2.6
Correlation of Torsion Angles
In order to observe the relaxation or change in the torsional angle over time, the
torsional angle autocorrelation was calculated according to the following formula:
PO(t)
W
(< cos((0))cos(#(t)) > - < cos#(O) >2)
(< cos
2
(2.36)
(O) > - < cos#(O) >2)
where 0(t) is the value of the dihedral angle at time t, 0(0) is the initial value of the
dihedral angle, and the brackets denote an ensemble average.
The correlation between transitions of neighboring dihedral angles on a given
chain was also computed. A transition was defined as the movement of a dihedral
31
20
- -
-
--
-
-- -
-
-
-
.-
-
-- -
-
-
18
-~. --...
--....
-...
-.-.--.-----.----..----.-.
16
..-.- . .. ... .. ... .....-. ..-. ... ..- -. . ....-. . .. .-. -..-.14
0
E 12
.... .- ---.
.. ...
--- . -.
- --.
.
-...-..
-.
-.
.. -..
..- .-. -
10
.....
-.
-.
- .....- -.
.-.
. .. . . . .
-.-.
.
...
.
.... .-.-.-.-.-.-.-.-.. -
8
-.--. .. --.
.. --.-.
-.-. -.
- ...
..
..
. .. ..
-.-.-.
.
.-.
-.
6
4
2
2
1
4
3
Dihedral Angle (Radians)
5
Figure 2-4: Energy as a function of dihedral angle
Table 2.1: Potential Parameters
4.o1 A
0.468
rc 2.5
2000 kJ/kmol/A 2
kb
bequji 1.53 A
ko
5.01 kJ/kmol
0
equil 1.92 rad
14.47 kJ/kmol
ki
k2
-37.59 kJ/kmol
k 3
6.490 kJ/kmol
74 58.49 kJ/kmol
o
32
6
angle from within 10 degrees of a potential minima to within 10 degrees of another
potential minima. By maintaining the time and position of each transition on a given
chain, the probability that a transition will be followed by another transition a given
number of segments away from it was calculated. No restrictions were placed on the
time between transitions. Neighboring transitions occuring within the same time step
(each time step was one femptosecond long) were not counted.
33
Chapter 3
Structure
The behavior of the model polymeric system was characterized as a function of temperature. The glass transition temperature was determined as a function of chain
topology from the specific volume versus temperature response. Since rotation of the
dihedral angle is strongly related to the relaxation of the polymeric chain, energy
barrier transitions between dihedral angle state transitioning were calculated. The
correlation of local density and transition rates and the correlation between neighboring dihedral angles were investigated. These will serve for comparison to the system
response during deformation.
3.1
Initial Network Characterization
There are two issues associated with the short time and length scales employed in
these simulations which must be addressed. Firstly, only a limited amount of phase
space is explored during the course of these simulations leading to a large variation
in the behavior of systems with different initial configurations. Conducting the same
deformation experiment with various initial configurations, one can obtain an estimate
of the variation in behavior to be expected. Secondly, relaxation in these samples
occurs over many time scales. Stretched bonds, for example, equilibrate over femptosecond time scales while chain motion requires much more time. The glassy state
inherently is a non-equilibrium state, but one would like to be sure that structural
34
relaxation will not occur on the time scale of the simulation.
There are a number of indicators which were used to monitor the relaxation of the
system. By following measures of density, energy and stress as well as distributions of
bond lengths, angles, and dihedral angles, one can observe the system's progression
towards equilibrium.
The systems are crosslinked and cooled under NVT conditions at densities which
are slightly lower than the equilibrium densities. The sample is then relaxed under
NPT conditions at atmospheric pressure and the convergence of the density and
energy measures are used to gauge the equilibration of the system. Figure 3-1 is a
characteristic relaxation profile. The density fluctuates rapidly for the initial 50 ps as
the system is switched from NVT to NPT. After this period, the density fluctuates
about the equilibrium value. This particular relaxation was conducted at 300K which,
as will be shown later, is above the glass transition temperature for this particular
system. The energy fluctuations and stress in the system both fluctuate about zero
as shown in Figure 3-1.
A compilation of statistics for equilibrated structures which were used as initial
conditions for the ensuing deformation studies is presented in Table 3.1. In addition to
providing densities, chain statistics, and measures of conformation, the functionality
of crosslinks is reported.
Histograms of the bond length, bond angle, and dihedral angle distributions which
are representative of equilibrated structures are shown in Figures 3-2 to 3-4. The
harmonic bond length potential employed in these simulations results in a normal
distribution of bond lengths. On average, the bond angles are slightly flexed closed
with respect to the potential equilibrium value. Mean bond lengths for the distributions shown in Figure 3-2 at 100 K, 200 K, and 450 K were 1.5276, 1.5285, and
1.531 angstroms respectively while the mean bond angles were 109.779, 109.784, and
110.07 degrees respectively. The equilibrium bond length is 1.53 angstroms while the
equilibrium bond angle was 110 degrees. With an increase in temperature the compressed bonds are relieved and then pulled in tension while the compressed angles are
relieved then flexed beyond their equilibrium values.
35
0.92
2
- -
0.91
1
o9
5.
-van
Bond Stretching
Bond Angle Bending
Dihedral Angle Rotation
Der Waals
A
w
1.,
V
E
A
>0.89
V
0
w
0.5
0.88
0
0.87
50
100
200
150
Time (ps)
50
250
150
Time (ps)
100
200
250
(b)
(a)
200-
-150*
x-direction
y-direction
z-direction
100
CL
50
-50
50
100
150
Time (ps)
200
250
(c)
Figure 3-1: Measures of equilibration for sample system with 4500 mers at lOOK.
The dotted line indicates the time at which the ensemble was switched from NVT
to NPT: (a) density as a function of time, the dashed line indicates the equilibrium
density (b) change in energy as a function of time (c) stress as a function of time.
36
-
0.06-
x
100 K
200 K
300K
0.05-
.20.04-
0.03-x
XXX
cc
x
x
x
x
XXx
x
xx
.
x
X
0.02-
X
x
X
X
x.
XX
0.01-
XX
>x
Xxx
x
'
xX
XXX
. '.
x XX ..
0.9
0.92
X
xxxxxx
--.
1
1.02 1.04
0.96 0.98
Normalized Bond Length (1/b )
0.94
1.06
1.08
1.1
Figure 3-2: Distribution of bond lengths for crosslinked systems, nw-1-8000-160-100K,
nw-1-8000-160-200K, nw-1-8000-160-450K
0.06
-
0.06x
OO
100 K
200 K
300K
0.05-
0.04-
(D
0.03
-X
X
x
0.02-
X
XX
Y
XXx
.'
0.01
.
X
.
x "x
xx
-
-
XXX
xX
1.7
xx
x
x.
x
.
1.8
1.9
2
Bond Angle (Radians)
xx
2.1
2.2
Figure 3-3: Distribution of bond angles for crosslinked system.
37
--
0.09-
x
100 K
200 K
300K
0.080.07.20.06-
xx
16
x
CO
IL
,0.05-
C 0.04 0.03 0.02-
x
x-
0.01 -
x
x
x
2
1
0
4
3
Dihedral Angle (Radians)
xx
x
6
5
Figure 3-4: Distribution of dihedral angles for crosslinked system.
To determine whether the initial structure had any initial bias in orientation,
vector correlation functions were computed. Vectors were constructed connecting
atoms separated by two bonds along a chain. The average angle between the unit
vector along this direction, vi, and the load direction, es, was calculated and the value
is reported in table 3.1.
Pi =< Vj - ex >
(3.1)
Assuming that there are an infinite number of vectors, vi, oriented randomly in
space, then the probability of finding a vector at particular angles # and
e is uniformly
distributed. The value of P can then be computed by
P1
=
dE
f2 dqarccos(cosecos#)cose ~-
57.3'
(3.2)
The radial distribution function was calculated to obtain an estimate of the long
range order within the sample.
Thin shells of thickness 6r are constructed each
centered r away from a particle. The number of particles, N(r), within each of these
38
AF
&
ij
X
;41
Xx
Ao-
-W
(b) 200K
(a) 100K
(c) 300K
Figure 3-5: Pole Figures for crosslinked system at temperatures of 100K, 200K, and
450K.
shells is computed. The radial distribution function is then defined as:
N(r)
_V
g(r)
-
-Nwr)
N 4,Fr26r
(3.3)
where r is the distance to the shell, 6r is the thickness of the shell, V is the volume
of the cell, N is the total number of particles within the cell.
In a perfect crystal, this function will have sharp peaks centered at the spacing of
neighboring particles. On the other hand a liquid will have smoothed peaks at short
distances, where molecular spacing is influenced by the Van der Waals interactions,
and decay to a constant value of 1 at larger distances. This decay is indicative of the
lack of ordering at these larger length scales. An amorphous polymer with its short
range order determined by atomic bonding and lack of long range order will consist of
a mixture of these two profiles. Figure 3-6 provides the radial distribution functions
for initial structures at various temperatures. At 450K the radial distribution function
reveals a polymer melt with peaks representing the first, second, and third neighbors
along a chain. At distances beyond this, the distribution is liquid like. Decreasing
the temperature below the glass transition temperature results in the appearance
of fourth neighbor correlations as the dihedral distribution becomes localized at the
39
5-
- 200 K
x
4.5-
450 K
4-
3.53-
'a; 2.5 2-
1.5 -
1-
0.5-
1
2
4
3
Normalized Distance (d/ulb
5
6
7
Figure 3-6: Radial distribution function for initial structures at various temperatures.
minima as shown in figure 3-4. The long range order, however, is frozen in as the
system kinetics slow and it is unable to achieve a crystalline structure.
To obtain an estimate of the length scale of the inhomogeneity captured in the
structure, the simulation cell was divided into subsections. The density within each
of these sections was calculated and the distribution of these densities was examined.
Each atom was approximated by a cube with uniformly distributed mass with a side
length given by L.
L = V7(0.8u-)
(3.4)
If an atom was cut by the boundary of a subsection, the mass of each atom
was then divided accordingly between the neighboring subsections. A representative
histogram of the density distribution is shown in Figure 3-7. As the size of the bin
increases, the distribution of densities becomes narrow indicating the uniformity of
the sample at this scale. As the section length then approaches the Lennard Jones
parameter -= 4.01 Angstroms, the distribution broadens indicating the heterogeneity
at this level.
40
Sample ID
I.
nw-c-1-4500-90-100K
nw-c-1-4500-90-200K
nw-1-8000-160-100K
nw-1-8000-160-200K
nw-1-8000-160-450K
nw-c-1-10000-200-200K
nw-t-1-4000-80-100K
nw-t-1-4000-80-200K
nw-t-1-4000-80-450K
fc-1-4000-83-100K
fc-1-4000-80-300K
fc-1-4000-40-100K
fc-1-40000-160-100K
N
4380
4380
7775
7775
7775
9722
3946
3946
3946
3984
4000
3920
40000
T
(K)
100
48/90
200
48/90
48/160 100
48/160 100
48/160 450
48/200 200
100
98/40
200
98/40
450
98/40
100
48/83
300
50/80
100
98/40
250/160 100
M/C
p
(g/cm3 )
0.942
0.923
0.939
0.916
0.848
0.921
0.935
0.921
0.854
0.901
0.855
0.914
0.916
< dee >
(A)
20.4
20.58
17.81
18.02
18.83
19.09
23.04
23.26
24.48
17.81
16.76
23.72
30.01
P
(Degrees)
54.86
54.706
58.04
57.81
58.21
58.04
54.67
55.63
56.47
57.35
59.88
56.16
57.07
Trans
%
66.9
66.8
63.7
63.9
65.7
65.7
66.8
67.0
64.7
62.2
62.4
63.5
64.3
F2
F3 F 4
16
16
12
12
12
20
6
6
6
0
0
0
0
19
19
36
36
36
33
12
12
12
0
0
0
0
22
22
47
47
47
64
8
8
8
0
0
0
0
Dimensions
(A)
x 29.43
29.01
x
126.6
127.5 x 29.42 x 29.42
57.75 x 57.75 x 57.75
58.22 x 58.22 x 58.22
59.75 x 59.75 x 59.75
176.6 x 37.28 x 37.28
23.51 x 64.20 x 64.98
23.53 x 64.65 x 65.44
23.48 x 69.80 x 65.57
47.06 x 47.06 x 47.06
24.00 x 68.46 x 66.17
46.48 x 46.48 x 46.48
100.5 x 100.5 x 100.5
Table 3.1: General Structure of Chains where N is the number of mers in the simulation, M is the number of mers per chain,
C is the number of chains, < dee > is the average end to end distance, Pi is given by equation 3.1, % Trans is the percentage
of dihedral angles in the trans configuration, F 2 , F3 and F4 are the number of crosslinks with a functionality of 2, 3, and 4
respectively
0.45
-
6859/5.29/5.8 std= 0.09
5832/5.59/6.86 std = 0.115
- - 4096/6.28/9.8 std = 0.133
- 729/11.2/54.9 std = 0.0815
x 216/16.7/185 std = 0.0338
x
0
0.4-
1
-
0.35 -
x
0.3-
0.25LA.
0.20
0.15-
0.1 -
0.05 00
0.4
0.5
0.6
0.7
0.9
0.8
3
Density (g/cm )
1
1.1
1.2
1.3
Figure 3-7: Distribution of densities at 100K for 40000 mers free chains. Number of
Bins/ Length of Side (Angstroms) / Average of atoms per bin and standard deviation
of distribution
The effect of crosslinking on the initial structure network was also investigated.
The density of the crosslinked systems was higher than similar systems of free chains.
Examination of the bond length distribution, bond angle distribution, dihedral angle
distribution, and radial distribution functions reveal no obvious differences between
the crosslinked and free chain systems. These distributions for two network systems
with 48 and 98 mers per chain and a system of free chains with 250 mers per chain are
shown in Figure 3-8. Referring to Table 3.2, one observes that the change in average
bond length and bond angle is small between the free chain and network systems
but that the density changes are significant. The density change is partly due to the
increased density around crosslinks. A crosslink can have up to four mers within one
bond length away as compared to two atoms within one bond length for a normal
atom and one mer for a chain end. The free chain system which has less dense chain
ends instead of the high density crosslinks has a smaller density.
42
0.0 5
x
0
0.
-
-- MM -= 4898 (network)
xo MM ==4898 (network)
(free chains)
0.04 -
M : 48 (network)
M- 98 (network)
M-48 (Iree chains)
M 8 (free hlns)
0,-
0.05
0.045
0.0
0 0"
(free chains),
0.035
0*
0.0 1-
00
0.03
-0
0
0.025
2-
0.02
0.0
0
0oxo
2-
0
x
0.01
0.0
0.96
x
0.005
000
0.95
8
x
0.015
0.87
0.88 0.99
1
1.01
1.02
Normalized Bond Length (554)..
1.03
1.04
1.05
1.70
1.8
1.85
(a)
2
2.05
(b)
-
x
0.12-
1.9
1.90
Bond Angle (Radians)
o
M
M
M
M
- 48
- 98
=48
98
N49
(network)
(network)
(free chains)
(free chains)
4
.5
xFC 260,
-
0.13 -
0.08
!.S 2-
0.
0.04-
0.02-
x
1
1.5
2
2.5
3
3.5
4
Dihedral Angle (Radians)
4.5
5
1
5.5
2
3
Normalized
4
Distance (d/b
5
6
7
(d)
(c)
Figure 3-8: Comparison of bond and angle statistics for various systems: (a) bond
length distribution, (b) bond angle distribution (c) dihedral angle distribution (d)
radial distribution profile.
43
Table 3.2: Crosslinked vs Free Chain Systems. The equilibrium bond length and
angle are 1.53 A and 110.01 degrees respectively.
Density (g/cm 3 )
Sample
0.942
0.935
0.901
Average Bond Length
(angstroms)
1.5273
1.5272
1.5266
Average Bond Angle
(degrees)
109.77
109.78
109.80
free chains, M = 98
0.914
1.5271
109.76
free chains, M = 250
0.916
1.5270
109.80
network, M = 48
network, M = 98
free chains, M = 48
3.2
The Glass Transition Temperature
The glass transition temperature was determined for various network topologies. Simulations were conducted by varying temperature while maintaining the cell boundaries
at atmospheric pressure. Each temperature was maintained for 150 picoseconds and
the average density obtained from an average over the last 50 picoseconds. At lower
temperatures, the density would continue to increase if allowed more time to relax.
The times used simulate a cooling rate of 1.25 x 10- 10 K/sec. Below the glass transition temperature, one cannot speak of equilibrated density. If the simulations were
performed for longer times, one would observe higher densities at these low temperatures. The glass transition is approximated from the point at which the slope changes
in the graph of the specific volume versus temperature. In Figure 3-9(a), results for
varying chain lengths and crosslinking densities are presented. The glass transition
temperature as well as the linear thermal expansion coefficient measured before and
after the glass transition temperature are reported in Table 3.3.
The glass transition temperature seems to respond sensitively to changes in the
chain molecular weight and to the addition of crosslinks. An increasing molecular
weight or crosslink density is found to increase the glass transition temperature. This
result is in agreement with previous work done by Rigby and Roe on a polyethylene
like molecule and suggests that beyond a certain chain length, the glass transition
temperature will be constant with further increase in chain length [35]. Extrapolation
44
of the data from Rigby and Roe suggests that for their model, chains with more
than approximately 200 units have a common glass transition temperature. These
phenomena are also observed experimentally [45]. The chain ends being more mobile
and having higher free volume than mers along the backbone will lower the glass
transition temperature. For low molecular weight chains, chain ends can increase the
free volume significantly but at higher molecular weights, the additional free volume
from chain ends is an insignificant fraction of the total and the glass transition is
constant with molecular weight. Crosslinks are less mobile and have less free volume
than a mer along the backbone. Therefore the addition of crosslinks stifles chain
mobility and acts to increase the glass transition temperature.
It has been shown both in experiment and through simulation that the glass
transition is dependent on cooling rates and pressure[17]. Experimentally the temperature may be varied on the order of degrees per second or degrees per hour while
this molecular simulation experiment was limited to cooling rates on the order 101
K per second. The simulated thermal expansion coefficient was found to be between
175 and 194 x10- 6/C.
The experimentally observed glass transition temperature
for polyethylene is approximately 250 K while the thermal expansion coefficient is
reported to be between 110-200 xlO-6 /K [6]. Noting that the much faster cooling
rate is expected to elevate the glass transition significantly and that the experimental
values are for semicrystalline high density polyethylene, the simulated results compare
favorably with experimental results. The glass transition temperature of polyethylene determined through atomistic studies varies between 200 and 300K depending
on both the cooling rate and the specific potentials employed [46, 13, 40]. Yang et
al. [46] find a glass transition temperature of 306K for a system of 5 chains with
300 united atom beads each at a pressure of 0 MPa. Takeuchi et al. [40] simulate
a chain of 600 mers with a cooling rate of 1.66 x 10" K per second and find a glass
transition at 201K . Gee et al. find that by varying the torsional potential from that
of a freely rotating chain to barriers twice those employed in this study and a chain
length of 768 united atoms cooled at rates on the order of 10" K per second, the
glass transition varies between 100K and 400K [13].
45
Table 3.3: Glass transition temperature, Tg, and linear thermal expansion coefficient,
LTEC, for various molecular weights.
Topology
50 mers per chain
100 mers per chain
1000 mers per chain
Network
Tg
(K)
252
268
280
305
LTEC x10 6 /CO
T = Tg -6T
71
66
63
60
LTEC x10 6 /C
T = T +6T
194
188
175
175
One universal feature observed in all of these systems is that the fraction of dihedral angles in their lowest energy state, i.e. trans state, does not obtain its equilibrium
value. Instead as the temperature falls below the glass transition temperature, the
chains which are no longer able to fully explore phase space become "frozen" into their
current states. The resulting plateau of the fraction of dihedral angles in the trans
configuration is shown in Figure 3-9(b). As shown in the next section, the transition
rate decreases with temperature. A longer amount of time is therefore required to
allow all of the dihedral angles to explore various states and obtain their equilibrium
distribution.
3.3
Energy Barriers for Transitioning
As mentioned earlier, the torsion potential has three potential minima. Gauche plus,
trans, and gauche minus, refer to torsion angles inhabiting these minima as shown in
Figure 3-10 and Figure 3-11. The torsion angle is significantly more compliant than
bond stretch and bond bending and therefore is the degree of freedom utilized for
chain motion. Understanding this motion and the barriers to it is critical to understanding polymer response. Though a specific torsional potential has been specified
for the rotation of dihedral angles, this potential will be modified by intermolecular
interactions.
There are two methods for determining the modified or "effective" barrier potential
for the dihedral angles. The first method results in a full potential and involves the
46
1.3 -
o
x
50 mers
100 mers
o 1000 mers
1.25
>
Network
1.2>
E
0
1.15 0
1.1-
a)
CO,
1.05
1'
0
100
300
200
400
500
Temperature [K]
(a) Specific volume versus temperature.
0.72-
50 mers
x
0.7
- 100mers
1000 mers
Network
0.680.66 0.640.620.6
0
100
300
200
Temperature [K]
400
500
(b) Fraction of dihedral angles in the trans configuration
as a function of temperature.
Figure 3-9: Simulations of polyethylene model at atmospheric pressure with 4000
united atom beads.
47
calculation of the probability that a dihedral angle will assume a given configuration.
The effective potential can then be calculated from this potential. The second method
provides information on the barrier height between the minima. Transition rates can
be calculated for the various states.
The assumption that the rate of transition
between state 1 and state 2, q 12 , will be related to the activation energy, AE
12 ,
by
the Arrhenius relation leads to the following equation:
412 =
K 12 exp
(kBT
(3.5)
where T is the temperature and K 12 is the rate constant. The definition of a transition is ambiguous. For the purposes of this study, gauche plus, trans, and gauche
minus states were assigned to dihedral angles passing within ±10 degrees from a potential minima which they maintain until they pass within this prescribed distance
of another minima. Transitions are counted when the state assigned to a dihedral
angles changes. Bearing in mind that the activation energy between a change in angle
from one configuration to another depends on both the initial dihedral angle and the
final dihedral angle due to the curvature of the potential wells, calculating activation
energies between transitions thus defined gives only an average quantity.
The transitions between the various states were observed for a system of 20 free
chains of 250 mers at various temperatures as shown in Figure 3-12. The temperature
was decreased in 25 degree increments. The sample was equilibrated for 10 ps at
each temperature and was followed by a 200 ps sampling run. Transition rates were
calculated as the total number of transitions divided by the total sampling time and
by the total number of dihedral angles.
The gauche plus and gauche minus states are indistinguishable. Therefore transitions between gauche plus and gauche minus are equivalent to transitions from gauche
minus to gauche plus. The calculated activation energy for transitions reported in
Table 3.4 are very close to the barrier energies in the explicit torsional potential. This
suggests that the torsional potential is not altered significantly by the neighboring
molecules. This is supported by similar results by Boyd et al [5].
48
Table 3.4: Calculated Activation Energy for Dihedral Angle Switches
Transition Type
Gauche to Trans
Gauche to Gauche
Trans to Gauche
AE (kJ/mol)
12.3
15.6
12.4
Also, referring to Figure 3-12, one notes that as the temperature decreases, the
transition rate deviates from a strictly Arrhenius dependence. In fact, the transition
rate is larger than that predicted by such a dependence. At 100K, the predicted log
of the transition rate per bond is -2.5 while the actual value is -1.
observed in a previous study by [5].
This has been
They claim that the transition rates remain
higher than predicted by the Arrhenius model because of excess free volume trapped
in the system below the glass transition temperature. Analysis in Section 3.4 shows
this is not the case.
The decay of the torsional autocorrelation function is closely tied to transition
rates. For the torsional autocorrelation function to approach zero, all angles must
sample other states. The decay function was calculated according to Equation 2.36.
Results at various temperatures are shown in Figure 3-13. At 1OOK one notes that
there is no appreciable drop in the decay function even after a few hundred picoseconds. This indicates that the dihedral angles retain their initial state over these long
time periods. This correlates with the low transition rates observed at these temperatures. The rapid decay of the torsional autocorrelation function at 450K indicates
that structural relaxation is occurring rapidly while at lower temperatures structural
relaxation occurs on much longer time scales.
Agreement between the decay of dihedral angles at 450K and previously published
data from Pant et al. [31] was found. Though Pant et al. incorporated an anisotropic
united atom model verses the simple united atom model employed in this study, the
torsional potential was similar to that employed in this study. The striking similarity
in decay time suggests that the dihedral angle decay is relatively insensitive to this
difference in force field at high temperatures. Since it was shown earlier that the
49
I
'
-
~
%I
-
-
T7
Figure 3-10: Diagram of dihedral angle rotation.
torsional potential is not modified significantly by the surrounding molecules, it is
expected that the decay times should be similar for similar torsional potentials at
high temperatures.
It is also important to note that transitions are not distributed evenly amongst
the dihedral angles.
There are some angles which undergo numerous transitions
while others undergo none. At a temperature of 350K, 40% of the dihedral angles
are involved in the first 4043 transitions, 3-14(a). The remaining 60% undergo no
transitions during this period. The standard deviation about the mean number of
transitions per dihedral angle is 1.65. At a temperature of 250K 23.2% of the dihedral
angles are involved in the first 4043 transitions 3-14(b). In this case the standard
deviation about the mean number of transitions per dihedral angle is 2.57. This
heterogeneity becomes even more exaggerated as the temperature decreases further.
At 1OOK 4.4% of the angles participate in the first 1941 transitions while at 225K
12.3% participate in as many transitions.
50
22
t
g
20
g+
18
16
:14
0
E
12
210
W 8
6
4
K
2
I
100
50
300
250
200
150
Dihedral Angle (Degrees)
350
Figure 3-11: Diagram of transition definition.
. -..
.-.-.-. - -----.-.
.-.--.-.
+ g--t
- t ->g2.5 -
-
0
-
t
c'
2-
1.5
S9-->9+
->g+
-- Total Rate
Total Rate (Gee
---
-
t
g+
a.)
1-
-0.5
~~~~~~~~~~.
-1
............
. .......
. .
.
.....
. .
..
.
..
.
04-0.4
--
- --
- 1- .5 -
2
2.5
3
-
3.5
4
4.5
5
1000/(Temperature (K))
Figure 3-12: Transition rates for a system of 5000 mers arranged in 20 free chains.
51
0.9
2O.8-
=0.5 C3
I0.4 -
0.4
-
0.2
_
0.1
-- --
0
10-1
b
network, T = 100K
network, T = 200K
- -network, T = 450K
0 Pant et al.
r.
.. I
100
.
101
102
103
Time (ps)
Figure 3-13: Decay of the torsional angle autocorrelation function at various temperatures.
3.4
Transitions and Free Volume
An attempt was made to find whether the dihedral angles with the highest transition rates had higher larger local free volume. Two methods were employed in this
attempt. In the first method, the system was split into bins. The density in each bin
was calculated according to the method outlined in Section 3.1. The transitions were
divided into the same bins weighted by the volume of the mer falling into each of the
bins. The density and transition rate were recorded for each bin. When averaged
over multiple samples, the distribution of densities for bins with highest transition
rates was identical to the distribution of densities for bins with the lowest transition
rates. In the same manner, the distribution of transition rates for the bins with the
lowest density was found to be identical to the distribution for bins with the highest
densities. Plots of transition rate versus density are shown in Figure 3-15 and Figure
3-16 for a bin size of 6.31A x 6.31A x 6.31A and 5.23A x 5.23A x 5.23A respectively.
One would expect that the bins with lower density would exhibit more activity but
these figures show no correlation between density and transition rate over the length
52
22
20
18F
16
14
E12
10
10
8
6 -11
Il
4
I I . il
11,|1 111. ill
JIJI
II
2
0
500
1000
1500
2000 2500 3000
Dihedral Angle Number
(a) T
3500
4000
4500
5000
3500
4000
4500
5000
350K
40353025 CD
-
S20--
E
m 15 z
Ii
10 5
500
1000
Dihedral
15000
g
b
DierlAngle Number
(b) T = 250K
Figure 3-14: Spatial distribution of conformational transitions for a system of 5000
united atoms arranged in 20 free PE chains. Dihedral angles are numbered serially
along the chain and the number of transitions at each dihedral angle are plotted
against the dihedral angle. The total number of transitions at each temperature is
4043.
53
scales investigated.
The second method involved correlating transition rates with the tessellated volume for the atoms along the chain making up the dihedral angle which was consistently transitioning. The tessellated volume gives an indication of the local volume
surrounding a single mer as opposed to the first method which averages the density
over regions on the order of 5-10 atoms. One would expect that at least one of the
mers comparing the active dihedral angle has a larger than average tessellated volume
which then gives it more mobility allowing transitions to occur. This was not found
to be the case. In fact, the distribution of tesselated volumes was identical to those
obtained by random sampling.
Though these methods should be able to resolve the correlation between excess
free volume and high transition rates, they were unsuccessful when applied to this
model. As shown earlier the energy barrier for the dihedral angle transition is on
the order of the torsional barrier. This suggests that the transition is unaffected by
the surrounding mers at these temperatures. One notes that the united atom model
employed in this study reduces many of the interactions between neighboring chains
by condensing the methyl groups.
At higher temperatures, one expects very little interaction between chains due to
the low density and increased spacing between neighboring chains. If the transition
rates at these higher temperatures are extrapolated to temperatures below the glass
transition temperature, the predicted rate is lower than that observed. Though it had
been postulated that this was caused by excess free volume trapped in the system due
to cooling, the most active regions appear to be those which are most constrained,
crosslinks in the case of the network and interior mers in the case of free chains.
The methods employed require the positions of each atom at the time of a given
transition. Due to memory constraints, positions may be maintained only at selected
times. These may not correspond to the exact time of a transition. It may be possible
that a given dihedral angle may have additional free volume prior to the transition
but that the free volume collapses by the time the next position file is obtained. For
this reason, transitions were correlated with position files saved within 1 pico-second
54
14
35
a12-
30x
10-
25 -
x
-x2
G20 -
x
ax
8x
-
x .x
x
x
6
x
4-
10 -%
xxd-
x
5-
x
x
x
-
2
"f
x
0 0 x
0
0.2
0.4
0.8
0.6
3
Density (g/cm )
0.2
1.4
1.2
1
0.4
0.6
x
0
0.8
1
1.2
1.4
Density (g/cm)
(b) T = 300K
(a) T = 400K
8
x x
0x
7-
r.6
6-
x
Ii
Cr
3
-
4
-
x
K
xx
x
4-
-
x
x
x
x
x
x
2
x
2
.
1-
x x,
x
x.
x
0
0.5D
-
xx
1I
1.5
Density (g/cm3)
(c) T = 250K
Figure 3-15: Correlation between transition rates and density at various temperatures
for a system of 5000 mers arranged in 20 free chains using a bin size of 6.31A x 6.31A x
6.31A. Each bin contains an average of 10.6 mers.
55
60
100
90
Zx
--
.i
50 -
80 70-
840-
1'
x
60
-
x
40
x
xx
1xxx
20
x
30-
10
20
0.2
0.4
x
.**
xx1
x
x
x
0
%20-
-
10
xt""
x
x
0.8
0.6
Density (g/cm)
1
XX
0.2
0
1.4
1.2
0.4
0.6
0.8
3
1
1.2
1.4
Density (g/cm )
(b) T = 300K
(a) T = 400K
20
18-
16
x
14
12
10-
x xx
x
a
.
42-x
0.2
0.4
0.6
0.8
3
Density (g/cm )
x
-
1
1.2
1.4
(c) T = 250K
Figure 3-16: Correlation between transition rates and density at various temperatures
for a system of 5000 mers arranged in 20 free chains using a bin size of 5.23A x 5.23A x
5.23A. Each bin contains an average of 5.84 mers.
56
of the transition. This may be too large a time slice for the given studies. Reducing
the cutoff time beyond this resulted in too few data points for meaningful analysis.
Upon inspection of the transition angle histories for the most active angles, Figure 3-17, it was found that in the free chain system these angles oscillated about
high energy states whereas the less active angles oscillate about the energy minima.
Figure 3-17 depicts the dihedral angle trajectory for a single, representative, very
active dihedral angle. As shown in Figure 3-17, the active angles are found to frequently transition between two high energy states and are unlikely to oscillate about
the potential minima. This behavior was also observed by Boyd et al. [5]. In the
networked samples, a large fraction of the segments with the highest transition rates
tended to be adjacent to crosslinks. In the case of crosslinks, the potential employed
does not allow for Van der Waals interactions between the monomers adjacent to the
crosslink. Atoms which are separated by fewer than 4 bonds do not include Van der
Waals interactions. For this reason, the crosslink has reduced mobility but the four
mers bonded to the crosslink are less constrained than their neighbors. Movement
of the chain produces transitions in the dihedral angle which contains this mer with
additional mobility.
This may also explain why at low temperatures, the transition rates remain elevated above that predicted by an Arrhenius dependence. At lower temperatures, the
transition rates are dominated by relatively few dihedral angles which are trapped in
high energy states. The effective energy barrier to transitioning is therefore reduced
and rates are increased. These regions of high activity persist in the system even after
nano-seconds of relaxation below the glass transition temperature. Whether they are
artifacts of the extraordinarily high cooling rates employed is not known.
3.5
Correlation of Transitions along a Chain
The correlation between neighboring transitions was calculated according to the methods outlined in Section 2.6 as a function of temperature. Referring to Figure 3-14, one
finds that as the system temperature is lowered below the glass transition tempera57
7
C
-0 4
C
< <
< 3
2:3
AI
3080
3120
3200
3160
Time (ps)
3240
Figure 3-17: Dihedral angle trajectory for a single, active dihedral angle.
ture, dihedral angle transitions become heterogeneously distributed along the chain
and self correlation of transitions increases dramatically. This is consistent with the
notion that chains below the glass transition are "caged in" with heterogeneous distributions of surrounding volume, Figure 3-22. As the temperature is raised, self
correlation decreases, but there are still significant correlations between dihedral angles separated by even numbers of bonds, Figure 3-18. This behavior has been seen
in previous studies [5].
The correlation between changes in dihedral angles is not completely captured by
the distributions shown in Figure 3-18. Motion of one angle does not always cause the
second neighbor to transition but may cause significant changes in the neighboring
dihedral angle. It seems that motion of a single dihedral angle almost invariably
causes the second neighbor angle to move also. Shown in Figure 3-20 are the dihedral
angle histories for neighboring dihedral angles as enumerated in Figure 3-19 at 100K.
One finds this behavior is prevalent in the samples. The dihedral angle oscillates
between two high energy states and each transition is accompanied by motion of
58
0.9500K
400K
x 350K
-300K
-
-
0.8
.-
-
.200K
- OOK
0.50.4 0.37
0.4 LI
0.2 0
0
1
2
3
6
5
4
Correlation position (i±j)
7
8
9
10
Figure 3-18: Neighbor Correlation for system of 5000 united atoms arranged in 20
free chains at various temperatures.
the second neighbor transition. Figure 3-20(a) and 3-20(c) show that rotation of a
dihedral angle is accompanied by movement of the ±2 neighbors while Figure 3-20(b)
and 3-20(d) show that the ±1 neighbor does not seem to be affected by changes in a
given dihedral angle.
In the case of crosslinked systems, the situation is slightly different. The dihedral
angles which are adjacent to crosslinks may oscillates about the dihedral energy minima but undergo many more short lived transitions, Figure 3-21. There seems to be
no correlation between dihedral angle switching on one chain attached to a crosslink
and switches on other chains attached to the same crosslink. This suggests that
communication of dihedral transitions across crosslinks is hampered by the limited
mobility of the crosslink.
59
-k2
Figure 3-19: Map of dihedral angles
3.6
Tessellated Volume
Voronoi tessellation was used to distribute the total system volume to each of the
united atoms in the system [44].
In this method the united atoms are treated as
reference points and regions are formed about these reference points such that each
point within this region is closer to the enclosed reference point than to any other
reference point. The Voronoi volume distribution gives details of how the mers are
packed locally. The code used was developed by Richards [14].
With cooling, the distribution of Voronoi volumes becomes more sharply peaked
as the mean value decreases Figure 3-22. As thermal motion decreases, atoms pack
more tightly decreasing the volume per mer. It is also noted that heating above the
previously determined glass transition temperature of 300K results in an increase in
mers with the largest volume. Similar behavior has been observed by Sylvester et
al. [39]. Though the maximum Voronoi volume measured was in the range of 80-100
A
3
,
fewer than one half of a percent of the mers had volumes larger than 60 A 3 . As
is evident from these broad distributions, the local environment varies significantly
from mer to mer. The mean value is simply proportional to the total volume and
therefore exhibits the characteristic change in rate of change at the glass transition
temperature.
The volume reported includes the volume of the mer itself. In order to obtain an
estimate of the free volume surrounding each mer, the Van der Waals volume may be
subtracted from the Voronoi volume. In essence, one assumes that the mers occupy a
60
Dihedral Angle Energy (kJ/mole)
25
20
15
10
5
5
30
-o g-
350
Dihedral Angle Energy (kJ/mole)
15
20
25
10
30
350
01
300
300
250
250
o
o
a200
a200
-@
0150
150
0
-TI
0
100
100"
0
-sJ
50
50
20
60
40
Time (ps)
80
n,
100
(b)
(a) #-2
5
Dihedral Angle Energy (kJ/mole)
25
20
15
10
5
30
80
60
40
Time (ps)
20
100
q1
Dihedral Angle Energy (kJ/mole)
25
10
15
20
30
350
350
02
300
300
250
250
200
-~
o
$
0200
-6
[150
*150
*
*
8
100
1001
50
50
20
60
40
Time (ps)
80
ci
)c~cic
"
ci
ci
20
20
100
~
40
40
60
60
Time (ps)
80
80
100
100
(d) q-1
(c) 02
Figure 3-20: Dihedral angle value over time for active angle in networked system at
100K.
61
I A
Dihedral Angle Energy (kJ/mole)
11
IA
A
in
1n
1a
9n
350-
25
12
0-
00 -
0
oa
--+
.
5
10
20
30
40
50
60
70
80
90
100
Figure 3-21: T = 100K, networked system.
0.2
-
0.18-
- -
0.16-
- -
200K
300K
400K
500K
0.14 -
o0.120.1
-
EI
z 0.08 -
0.06 0.04 0.02 0
10
15
20
25
40
35
30
Volume (Angstromse)
45
50
55
60
Figure 3-22: Tessellated volume distribution at various temperatures for system of
5000 monomers arranged in 20 free chains.
62
region of space given by the Van der Waals radius which no other atom can occupy.
In reality, the mers are not hard spheres. Hence neighboring mers may penetrate this
radius. Additional complexity arises from overlap in the Van der Waals volumes due
to the connectivity of the mers. The bond length is less than half of the Van der
Waals Volume. Overlap of the Van der Waals Volume occurs between adjacent mers
and even second and third neighbors depending on the configuration of the chain.
63
Chapter 4
Deformation Results
The material response was characterized as a function of deformation rate.
The
glass transition temperature is located using DMA type simulations. Both uniaxial
compression and tension experiments are performed at a variety of temperatures and
strain rates. The evolution of stress, internal energy, and chain conformation are
observed with deformation. The Eyring model is applied to determine the activation
volume for the system. Observations of the configurational transitions of the chain
backbone are studied as indicators of mobility and the correlation between these
transitions is investigated during deformation.
4.1
Dynamical Mechanical Analysis
As stated previously, the state of a material is determined not only by a temperature
but also a time scale. For this reason, additional specification was necessary to ensure
that the sample was indeed glassy for the deformation rates employed. In Figure 39(a), the specific volume versus temperature indicated that the Tg for a network
with 50 mers between crosslinks was 305K. Here, in order to verify that this T. is
valid for the time scales of the deformation to be imposed, dynamical mechanical
tests were first simulated. Experimental data obtained from dynamical mechanical
analysis (DMA), Figure 4-1, shows that polyethylene undergoes a decrease of the
storage (real) modulus in the vicinity of the glass transition temperature. During a
64
DMA experiment, a sample is subjected to small amplitude strain at high frequency
and the generated stress is measured as the temperature is varied. The modulus is
sensitive to both the temperature and the frequency of oscillation. A rapid change
in the modulus over a given temperature range is an indicator that a transition in
material behavior will occur for the given frequency in that range.
The Young's modulus was obtained by estimating the slope of the stress strain
curve over the first 3% of true strain for the simulation data at various temperatures
and strain rates in order to characterize the behavior of the material. For temperatures above the glass transition, this becomes increasingly prone to statistical error
as the sample displays a non-linear response even at these small strains. The results
presented in Figure 4-2 agree well with the data presented in Section 3.2 suggesting
that the glass transition is in the range of 275-300 K for a strain rate of 1010/s. At
a strain rate of 5 x 1010 the glass transition temperature is significantly elevated to
330-370K. Data obtained at constant temperature and varying strain rate are plotted
in Figure 4-3. This data indicates that the sample is glassy throughout the sampled
regime at lOOK but that a transition may be in the vicinity of a strain rate of 108/s
for a temperature of 200K.
4.2
Uniaxial Deformation
Having established the glass transition temperature for this material model, the material behavior during deformation in the various regimes may be explored. Samples
were loaded in the x-direction while the stresses in the y and z direction were maintained at atmospheric pressure. Stress components, energies, and a variety of chain
statistics were monitored throughout deformation. Figures 4-4 through 4-12 were
obtained from a constant strain rate uniaxial compression simulation conducted at
100K containing 7775 united atom beads arranged in a network with 160 chains.
Figures 4-25 through 4-30 were obtained from a constant strain rate uniaxial tension
simulation conducted at 100K containing 7775 united atom beads. These results are
illustrative of the qualitative material response observed at all of the simulated strain
65
HDPE
Dynamic Machanikal Analysis
.2
-
i,000,000
+2
-1uu
-as.
40.000
100 E
E-T
Mari,
Law,
"F
Tan DdIta
+
.. .L.....
0
-300
-200
0
-100
j
0
Z00
20X)
'00
-.
Temnperatw% 'IF
Figure 4-1: Experimental graph of DMA curve for HDPE [15].
2.6-
0
2.4
0
2.2C_
2 x
Ci)
_ 1.8 -
0
2
.0
1.6-
U)
i
1.4 1.2 -
0.8
100
'
150
'
200
'
300
250
Temperature
E=
E=5*1010
66
400
350
Figure 4-2: Estimated modulus for = 1 x 10 10 /s and i
7775 united atoms arranged in 160 networked chains.
1*1010
x
o
1 -
=
450
5 x 10 10 /s for a system of
2.'.A
0
x
2.5-
X
xX
0
0
-
2.42.3 CO
S2.2
0
Cn2. 1
2-
0
1.9 x
1.818
10
o
10
1
10
101
100OK
200K
-
10
11
Strain Rate (/s)
Figure 4-3: Estimated modulus for various strain rates at lOOK and 200K for a system
of 7775 united atoms arranged in 160 networked chains.
rates below the glass transition temperature.
Prior to deformation, the sample was relaxed at atmospheric pressure, but the
internal stresses within the network remain quite high. There exist highly tensile
Van der Waals forces and highly compressive bond stresses with negligible stresses
due to bond angle flexure and dihedral angle rotation. Previous studies note this
competition as the source of high internal stresses, but there are conflicting results
in the literature as to the expected nature of these stresses within the dense network
structure [12].
Studies by Chui et al. [8] and Sylvester et al. [39] find compressive Van der Waals
stresses and tensile bonded stresses whereas bonds are compressed in this study and
in studies by Roe et al [36].
This difference in initial stress state is attributable
directly to the force fields employed. Chui and Sylvester employ a torsion potential
which does not energetically differentiate between trans and gauche states. Biasing
was accomplished by adding a repulsive Van der Waals interaction between atoms
separated by three bonds along the chain. The equilibrium bond lengths in the Chui
67
and the Sylvester studies were shorter than in other studies. This was done so that the
stretching of the bonds by the repulsive Van der Waals interactions resulted in a bond
length of approximately 1.53 angstroms. In this study and that by Roe, the torsional
potential incorporates a biasing between states and does not require additional biasing
from the Van der Waals interactions. For this reason Van der Waals interactions are
computed for atoms separated by 4 or more bonds. Atoms separated by 4 or more
bonds will have an attractive Van der Waals interactions which leads to compressed
bonds.
4.2.1
Compression
Results from simulated compression experiments on a network of 7775 mers arranged
in 160 chains is shown in Figure 4-4. The network response to the first few percent
of strain is elastic with a calculated Young's modulus of 2.4 GPa and a Poisson
ratio of 0.4. This agrees well with the results of Brown and Clarke [7] who find the
modulus of a similar polyethylene model to be 2.4-2.5 GPa. Yield initiates at a strain
between 10 and 12 percent and is followed by softening and then by deformation at
a relatively constant stress and then strain hardening. The stress perpendicular to
the load direction fluctuates about atmospheric pressure throughout the simulation.
Qualitatively the network response is similar to that obtained in experimental studies
of glassy polymers, but the quantitative results cannot be compared to experiment
due to the high strain rates employed. The yield stress measured in this simulation
was 260 MPa for a strain rate of 1.0 x 10
10 sec-1.
The stress response was decomposed into its various contributions, thermal stresses,
Van der Waals induced stresses, stresses due to bond streching, bond angle bending
stresses, and stresses due to dihedral angle rotation, Figure 4-5. This provides the
detailed stress response of the network to the imposed strain. In this figure, one notes
the compressive bonded stresses and the tensile Van der Waals stresses at 6 = 0. In
the load direction, the x-direction, the elastic response is governed by the linear relief
of the tensile Van der Waals stresses. Though in this case the Van der Waals stresses
go from tensile to compressive at the yield point, this was not characteristic of all
68
300-
250-
200-
w150a)
100-
50-
0
0.1
0.2
0.3
True Strain
0.4
0.5
0.6
Figure 4-4: Compressive stress versus strain for uniaxial compression for = 1 x 1010/s
and T = 100K for a system of 7775 united atoms arranged in 160 networked chains.
systems tested. Some networks yielded before the Van der Waals stresses reached
zero. The change in slope of the Van der Waals stresses and the bonded stresses,
on the other hand, were characteristic of the network response. During the elastic
portion, the stresses due to bond streching and bond angle bending both become
increasingly compressive while the stress due to the rotation of dihedral angles is
negligible throughout the deformation.
For clarity, the initial stress was subtracted from the curves and the results of
the stress partitioning in the x, y, and z directions are shown in Figure 4-6. Nonlinear behavior of the macroscopic stress strain curve, Figure 4-4, begins at strains of
approximately 6 percent. Prior to this the Van der Waals stress, Figure 4-6, increases
linearly in the load direction. At 10-12 percent strain, the Van der Waals stress is
negligible in the load direction and with further deformation becomes compressive.
The rate of change of the Van der Waals stress also changes at this point. In the y and
z directions, the Van der Waals stresses become increasingly tensile until yield occurs
and this tensile stress begins to be relieved. In the y and z directions the stresses due
69
250
200%
150
100-
Thermal Stress
---
-200
-
a Bond Angle Bending
2-
-200
0.1
0.2
0.3
J Interactions
Bond Length
0.4
0.5
0.6
0.7
Bond Torsion
0.8
0.9
True Strain
Figure 4-5: Partitioning of compressive stress for uniaxial compression along load
direction.
to bond angle streching become increasingly tensile throughout deformation. The
contribution to the stress from dihedral angle rotation is negligible in all directions
throughout deformation.
Energy changes during the first 10-20% of strain are dominated by increases in
Van der Waals energy and an increasing torsion energy, Figure 4-7. Due to the stiff
nature of the bond stretching and bond angle bending potentials, very little energy
is absorbed by these degrees of freedom during elastic deformation.
There is an
increase in the dihedral angle energy, Figure 4-7, as the shoulders of the torsion angle
distribution, Figure 4-8, become broadened. High energy angles are adopted making
transitions between states increasingly favorable.
The bonded stresses pass through an inflection point at approximately 10 percent
strain and yield follows in the range of 10 to 15 percent strain, Figure 4-6. During the
elastic portion, the bonded stresses become more compressive in all directions as the
average bond length decreases slightly. The mean bond length decreases from 1.52765
Angstroms, 1.5275 Angstroms , and 1.52713 Angstroms at 0%, 2%, and 10% strain
70
Thermal Stress
.."."'Van Der Wasa
- - Bond Streching
350
a
300
250
.....
......
-
200
Interactions
Bond Angle Bending
Dihedral Angle Rotation
150
0
100
5
0
0
[
-S
-100'0
0.1
0.2
0.3
0.4
0.5
0.6
True Strain
0.8
0.7
0.9
1
(a) x-direction
100
80
60
Thermal Stress
....Van Der Waals Interactions
- - Bond Streching
0 Bond Angle Bending
'-.Dihedral Angle Rotation
..
,,
'
0.7
0.8
0.9
1
0.8
0.9
1
40
20
CL
S-
0
-20
-40
-60
-80
0.1
0.2
0.3
Strain
0.4 True0.5
0.6
(b) y-direction
100
-
80
- -
60
-
a
Thermal Stress
- Van Der Weals Interactions
Bond Streching
Bond Angle Bending
Dihedral Angle Rotation
,.."'
40-
0-
I.2-
20-
-20
-40
-60
-80
0
0.1
0.2
0.3
0.4
0.5
0.6
True Strain
0.7
(c) z-direction
Figure 4-6: Partitioning of normalized compressive stress as sample is loaded in com-
pression along the x-axis.
71
0.5-
4 Bond Stretching
Bond Angle Bending
- - Dihedral Angle Rotation
0.4 - a Van der Waals Interactions
0.30.2
#
#
X #
a
a#a###aa #
-
-0.1
-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
True Strain
0.7
0.8
0.9
1
Figure 4-7: Partitioning of normalized energy change per united atom versus true
strain in compression.
respectively. This change in mean bond length and a slight redistribution of bond
lengths from 0% to 10% strain results in a 22% change in the bonded stresses, Figure
4-6, while only changing the bonded energy by 1%, Figure 4-7. Following yield, the
mean bond length increases approaching the equilibrium length of 1.53 Angstroms,
but remains slightly compressed throughout deformation. The distribution of bond
lengths also becomes increasingly peaked about the mean value, Figure 4-8.
The density, Figure 4-9, increases during elastic deformation with a characteristic
rapid dilatation occurring just after yield. The decrease in density persists through
softening then the density becomes constant beyond a strain of 25%. The average
Poisson ratio in flow is 0.5. As the activation barrier for structural rearrangement
is crossed, the material softens and the density decreases. The Van der Waals interactions, which are in effect holding the material together in the lateral direction, is
repulsive at distances smaller than 4.50 Angstroms, has a maximum attraction at a
distance of 4.99 Angstroms, and is weakly attractive at large distances, Figure 4-10.
The re-shuffling of neighboring mers in a polymer as simple as polyethylene becomes
72
--
0.0
E-0
-0.2
x E=1.0
0.08 0.07-
-E0.2
0.04
0.0
x
x
5-
,
0.03
0.06-
x
,
x
0.024-.
0.05-
.x
5 0.04x
0.0
0.030.01
x
0.021
0.01
0.0
0.92
0.94
"
xx
x
0.9
2-
-x
0.00 5 -
x
1.02
0.98
1
0.96
Normalized Bond Length (/b q.
1.04
1.06
~
.5
18
18
x
.
.9
Rdas
xx
.5
21
21
.
An
Bon
1.08
(b) Bond angle distribution
(a) Bond length distribution
Sr
0.08-
j=0.2
x
-..
1x
E= 1.0
0.07-
4-.
0.06-
3.5-
0.2
E.1.0
3-
T 0.05 0.04 0.03-
1.5-
x
0.020.01 0
0.5-
0.5
1
1.5
2
2.5
3
3.5
4
Dihedral Angle (Radians)
4.5
5
1
5.5
2
5
3
4
Normalized Distance (d/b,,)
66
7
7
(d) Radial distribution function
(c) Dihedral angle distribution
Figure 4-8: Comparison of structural statistics for compressive true strains of 0, 0.2,
and 1.0.
73
0.95
0.945-
E
.940--
6 0.935-
0.93-
0.925
0
0.1
0.2
I
0.3
0.4
0.5
0.6
True Strain
0.7
0.8
0.9
1
Figure 4-9: Density versus true strain for uniaxial compression
possible when this maximum is crossed. This dilatation continues to a strain of approximately 20-25% strain. The energy stored in Van der Waals interactions and
in dihedral angle rotations increase during this dilatation while the energy stored in
bond stretching decreases slightly, Figure 4-4.
In order to address concerns that the dilatation following yield was not a result of
the barostat, the same initial structure was subjected to deformation with a prescribed
Poisson ratio of 0.5, Figure 4-11 (i.e. under NVT conditions). The lateral pressure was
negative throughout the elastic regime indicating that the material would compact if
the sides were released. The pressure then increases to a positive value just after yield
reproducing the tendency of the system to expand just beyond yield. Perhaps due to
the suppression of the dilational response, the strain softening observed in the uniaxial
simulations was not observed in the NVT case. A constant pressure regime follows
suggesting the deformation is naturally volume preserving at this point.
Similar
observations were made in molecular mechanics experiments on atactic polypropylene
by Mott et al. [29]. By observing system pressure in a model polypropylene system,
they found that there is a tendancy to dilate in the early stages of plastic deformation.
74
0.5
0.5
Uci
0.5 O
----
-0.5
W
-1
4
5
6
7
Distance (Angstroms)
8
9
-1
10
Figure 4-10: Schematic of Lennard Jones potential and gradient.
In studies by Hutnik et al. [20] on a polycarbonate model system, a tendancy for the
system to contract upon yield is found.
The measures of rotation and elongation of bonds and chains for the uniaxial
simulation are given in Figure 4-12. Compressive deformation results in an elongation
of the average end to end distance of the chain. The rate at which the chain elongates
is very small during the elastic portion of deformation and quickly increases. At a
true strain of 1, the chain is approximately 60% of its maximum elongation. The
chains rotate into the plane perpendicular to the load direction as is evident from
the < P 3 > values in Figure 4-12. The percentage of dihedral angles in the trans
configuration changes very little during the course of the simulation, Figure 4-8. It
is notable that during the application of 100% true strain, the fraction of dihedral
angles in the trans configuration changes by only 3%. This suggests that the 27
% change in chain length can be accomplished with the conversion of relatively few
gauche configurations to trans configurations.
Although the total number of dihedral angles in the trans state changes little,
there is a significant amount of switching between states during deformation. Prior
to and during yield the transition rate between dihedral angle states increases rapidly
75
350 --- -
300 -x
x-direction
y-direction
z-direction
250
200a
U)
0
(D
10-
0
IIPI
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
True Strain
Figure 4-11: Compressive stress versus true strain for constant volume compressive
deformation with e = 1 x 1010/s and T = 100K
indicating that structural rearrangement of the chain is now energetically feasible,
Figure 4-13.
The transition rate plateaus as the dihedral angle energy plateaus.
Flow initiates soon after suggesting that any additional input of energy is dissipated
through rotation.
The Van der Waals stress in the axial direction increases at a slower rate as
it becomes compressive in nature, and the bonded stresses become less compressive.
Figure 4-7 shows that the energy stored in dihedral angle rotations and Van der Waals
interactions plateau as the bonded energy decreases due to the tightening distribution
of bond lengths.
Pole figure diagrams were constructed to obtain information concerning the rotation of bonds away from the load direction, Figure 4-14. The random distribution of
bond directions in the initial state is shown for E = 0. Rotation of bonds away from
the load direction is evident at strains as small as E
=
0.1. The continued deformation
promotes further rotation into the plane perpendicular to the load direction.
A more complete view of the deformation induced orientation can be seen from
76
24
61
2375
70
21
20
6 65
1960
18-
17 0
0.1
0.2
0.3
0.4
0.5
0.6
True Strain
0.7
0.8
0.9
0
0.1
(a) Average end to end distance
0.2
0.3
0.4
0.5
0.6
True Strain
0.7
(b) Average chain angle, <
0.25
63
0.8
0.9
P3 >
-x-direction
y-direction
... z-direction
-x
0.2
0.15
62
0.1
0.05
61
a.j
60
0
-0.05-0.1-
59
-0.15-0.2
58
-0.251
0
0.1
0.2
0.3
0.4
0.5
0.6
True Strain
0.7
0.8
0.9
0.1
(c) Average bond angle with respect to
the load direction, < P 2 >
0.2
0.3
0.4
0.5
0.6
True Strain
0.7
0.8
(d) Bond orientation function, < P >
-50.665
0.66
0.655
0.65
0.645
" 0.64
L 0.635
0
0.1
0.2
0.3
0.9
0.4
0.5
True Strain
0.6
0.7
0.8
0.9
(e) Fraction of Dihedral angles in the
trans configuration
Figure 4-12: Chain statistics versus true strain in compression
77
1
3+ gauche to trans
o trans to gauche
0 gauche to gauche
20
0
CO
-
z
()
0.0
0)0
0
C:
0
C:
--
0 0.1
0)
0
*
.
0
Tru6 Strain
.
0
.
0
.
0
.
.
0.7
0.8
0.
1
0.9
1
2--
-4
0
I
0.1
0.2
0.3
0.4
0.5
0.6
True Strain
Figure 4-13: Transition rate versus true strain for uniaxial compression
constructed three dimensional distribution functions, Figures 4-15 through 4-23. These
were constructed for two cases. The first considering only atoms which are on differing chains, Figure 4-15 through 4-19, and the second considering only atoms along
the same chain, Figure 4-20 through 4-23. Though the full three dimensional profiles
are provided, profiles of the distribution are provided in the x, y, and z directions for
clarity, Figure 4-15 and Figure 4-20.
In the case of atoms along different chains, the compaction of mers occurring
during the initial elastic deformation is evident from the G(x, 0, 0) curve in Figure 415. The steady development of additional packing order in the load direction is evident
from the growth and refinement of the peaks in G(x,0,0). In the plane perpendicular
to the load direction, the distance to the first neighbor grows and the peaks flatten
to a value of 1. This suggests that there is little organization of the mers located on
different chains in this plane.
The three dimensional distribution of mers along the same chain, Figure 4-20,
shows that the chains are rotating out of the load direction even at small strains.
The first and second neighbor peaks decrease in the load direction, i.e. becoming
78
xx
3, 'I.
x
x
x
x
x
x
0.1
0
(a) c
XX
xx
x
x
x
x.
x
-
x XW
x
x
I
WX
.Xx
k I., x x
xX
x
XA
x
x
x
x
x
x
x
it
xxx x.
x ax It
)k.-x
w x
X#x
x
x
xo Xk X
w
xxx AM .
**xx4xx
xx-t
x -k x xx x
xx
x"0J x-Ax 'Am;*xxo
xl-x -kxf : xx x x 9
k I ;,,VIA IA x x
xx x
0
xx x x x Ixx Y'x
x
x
x
x
x
x
xx
9
'no -x.
it r
x xx
xA
3vt
x
x
0.4
(d)
0. 2
(c)
x
x
x
x x
x
x
x
(e) c = 0.6
Figure 4-14: Pole figures for uniaxial compression
79
more peaked in the plane perpendicular to the load direction.
The three dimension radial distribution data suggests that as chain segments rotate out of the load direction, they become on average stacked one on top of the other.
Already directed in the plane perpendicular to the x direction, further compression
can no longer rotate the chains easily. Instead, the Van der Waals interactions become compressive and mers on neighboring chains are compacted together.
This
action tends to stretch the bonds which counteracts its initial compression, Figure
4-24.
This also suggests that the Van der Waals stresses in the axial direction are being
relieved for two reasons. Firstly, the < P2 > correlation function in Figure 4-12, the
pole figure plots in Figure 4-14, and the radial distribution profiles in Figures 4-15
through 4-23, indicate that the highly compressed chains are rotating out of the load
direction. The change in the value of < P 2 > is on the order of 1.5 degrees and
linear in the elastic regime. Slight orientation is also evident in pole figure diagrams
at strains of only 10%, Figure 4-14, and becomes increasingly noticeable at larger
strains. Since these segments have highly tensile Van der Waals interactions aligned
with the chain axis, rotation of the axis out of the load direction results in a relief of
the tensile stresses oriented along the load axis. The Van der Waals stresses become
increasingly tensile in the plane perpendicular to the load direction (i.e. the y and z
directions). These results reinforce the data by Chui et al. [8] and Bergstrom et al.
[3] which suggest that it is rotation of the chain segments which is responsible for the
initial network response.
Secondly, the mer mobility is severely limited by the Van der Waals attractions.
Unable to undergo large structural rearrangements, compression results in a decreased
mean spacing of united atoms along the load direction and relief of the tensile Van der
Waals stresses. Density increases during the elastic portion of deformation necessarily
require a decrease in the mean spacing of mers. This decrease in mean spacing is
seen even in the radial distribution profiles.
Since bond angles and bond length
distributions and mean values change very little, this necessarily translates into a
compaction of neighboring chains. As the mean spacing in the load direction decreases
80
f
,,*0~
/
108
-.
*8
/
0.0
w
I (0.4
- -0G(0.0z)1
0.4
I)
b)2 _4
(a) c
=
(b)
0
010.
=0.
1
1.2 -
0~''*~
0.6
-
0.4
-
0.2
-
.0
--
GO
~*
0
*
/
2
~0,0)
(d) e = 0.4
1.4
--
1.4
/
/
I
(c) E = 0. 2
1.2
I
2
1.2
0
0.8
7
-.
0.
0..4-
o.4.-
*
~
0,0)
0.2 -
0.4
0.2
no.
2
(Ang.t-)
(e) c = 0.6
at ne
(ArNstr-m)
(f) C = 1
Figure 4-15: Line profiles of three dimensional pair distribution for monomers along
different chains.
81
z
Slice 1
Slice 2
X
Slice 3
Slice 4
Slice 5
Figure 4-16: Diagram of three dimensional distribution profile slices.
and bonds are compressed, the bonded stresses in the y and z directions become more
compressive triggering expansion of the chains in plane perpendicular to the load
direction. This results in a decreased mean spacing and increasingly tensile Van der
Waals stresses in this plane. The lack of structural relaxation and dissipation is also
reflected in the rapidly increasing system energy throughout elastic deformation.
4.2.2
Tension
The results for a simulation of 7775 united atoms arranged in a network of 160 chains
which was deformed at a true strain rate of 1 x 1010 per second are presented in
Figures 4-25 through 4-30. The initial elastic response in tension was characterized
by a Young's modulus of 2.3 GPa and a Poisson ratio of 0.35, Figure 4-25. The
Young's modulus was calculated using the stress strain data for the first 3% of strain.
Nonlinear behavior is observed in the stress strain data at 5-7% strain and yield occurs
in the range of 10-15% strain at a stress level of 200 MPa, Figure 4-25. It is well
known that yield in polymers is pressure dependent [26]. The tensile yield stress is
lower than that found in compression due to the difference in the state of hydrostatic
pressure between compression and tension simulations. Note that the macroscopic
82
Slice 1
1.5
20
20
15
G'15 0
N 10
15 52
N
5
20I
a
0
0
5
10
x
15
20
10
20
y (Angstroms)
0 0
1
x (Angstroms)
Slice 3
Slice 2
N
0.5
20
10
20
20
15
15
N
10
10
5
0
0
5
10
y
15
0
20
5
>
1I
5
10
x
15
20
15
20
Slice 5
Slice 4
0
0
10
y
15
10
0
20
5
10
x
Figure 4-17: Three-dimensional pair distribution for monomers on different chains
with c = 0
83
Slice 1
1.5
20
20,
015
E
0
110.10
Cn
..
1
N
0
20
0
10
x
5
15
0.5
E10
20
0 0
y (Angstroms)
x (Angstroms)
Slice 3
Slice 2
20
15
5
N 10
0
5
10
15
10
y
5
0
20
y
15
20
15
20
Slice 5
Slice 4
:
.10
:5 -
:
x' d5
:
,:
:
-. ::
.::o
.
:
v
15
0
5
10
x
15
0
20
5
10
x
Figure 4-18: Three-dimensional pair distribution for monomers on different chains
with c = 0.1
84
Slice 1
1.5
20
20
' 151-
15
E
2
1
10
N 10
<5
N
20jq
5
120
10
5
10
x
15
10
0
y (Angstroms)
20
0.5-
0
x (Angstroms)
Slice 3
Slice 2
20
20
15
15
N
N 10
in.
5
0
0
5
10
y
15
0
20
5
10
y
15
20
15
20
Slice 5
Slice 4
20
15
>'10
5
0
0
5
10
x
15
0
20
5
10
x
Figure 4-19: Three-dimensional pair distribution for monomers on different chains
with c = 1.0
85
-
2 (2,y,2)
I
3-
35-
2.52-
j2.
2 -
.2
0.6
-
12
a
6
4
0421222 (A22.12111)
2
0
,0.
DIM222 (A22212.2)
(b) c = 0.1
(a) c = 0
.-..0(2
y.2)
.2.5 -
22
2--
.2-
1.6
0.-
110.5
-0
2
2
Dist-nc2
(Angst-rn2)
02
4
04
12a
Distwnc (Angst-.m)
a
10
(d) e = 0.4
(c) c = 0.2
4.4.44
3.5I
3.5
32.1
'12.
-
.5-1.2
1s1.
0.5 -
0.5
1
2 4
(Ang2t-.)
042.22. (Angt-.211)
(e) c = 0.6
(f) E = 1
D4.121n
Figure 4-20: Line profile of three dimensional pair distribution for monomers along
the same chain.
86
Slice 1
20.
0.9
0.8
15.
0.7
0.6
0.5
0.4
E
N
5
10
0.3
0.2
201
20
10
0
5
10
x
15
20
0 0
y (Angstroms)
x (Angstroms)
Slice 3
Slice 2
N
0.1
20
20
15
15
10
N10
5
0
0
0U
0
5
10
y
15
20
5
10
y
15
20
15
20
Slice 5
Slice 4
20
20
>,10
5:
00
5
10
x
15
0
0
20
5
10
x
Figure 4-21: Three-dimensional pair distribution for monomers on the same chain
with c = 0
87
Pp
Slice 1
20
0.9
15
20,
0.8
------- ' 15.
0.7
E
0.6
10
N
0.5
0.4
10-
51 -
110.3
0
20
5
0.2
20
---- --0
0
5
10
x
15
20
10 .
0 0
y (Angstroms)
0.1
x (Angstroms)
Slice 3
Slice 2
20
20
15
14
N
10
5
0
5
10
y
15
0
20
5
10
y
15
20
15
20
Slice 5
Slice 4
20
20
15
15
>'10
>'10
5
5
00
0
5
10
x
15
0
20
0
5
10
x
Figure 4-22: Three-dimensional pair distribution for monomers on the same chain
with E = 0.1
88
Slice 1
0.9
......
20 .
0.8
[lilt15
,
0.7
0.6
0.5
E
5
[
0.2
20A
201
10
0
5
20
10
x
15
0 0
y (Angstroms)
20
0.4
I0.3
0.1
x (Angstroms)
Slice 3
Slice 2
20
N
15
15
10
N 10
5
0
0
5
10
y
15
00
0
20
5
15
20
15
20
Slice 5
Slice 4
20
20
15
15
> 10
>110
5
0
0
10
y
5
5
10
x
15
0
0
20
5
10
x
Figure 4-23: Three-dimensional pair distribution for monomers on the same chain
1.0
with c
89
A
-41$
4-
f t
t
ft
ft
Figure 4-24: Schematic of chain interactions along compressed axis.
density during deformation in the system undergoing tensile deformation is lower
than that for a system undergoing compression deformation precisely because of this
hydrostatic pressure difference.
Following yield, the material softens.
The stress
remains relatively constant between 20% and 50% strain. Beyond 60% strain there
are indications of hardening. The stresses in the y and z directions fluctuate about
atmospheric pressure throughout deformation.
The initial structure contains bonds which are compressed and tensile Van der
Waals stresses. The initial stress is subtracted from the response and the results for
stress in the x, y, and z direction are shown in Figure 4-26. As in the case of the
compression tests, the Van der Waals interactions dominate the elastic response in
the load direction. These interactions become increasingly tensile in a linear fashion
in the load direction while the tensile Van der Waals stresses are relieved in the y
and z directions. The stresses due to bond angle bending and dihedral angle rotation
remain negligible throughout deformation in all directions. In the x, y and z direction
the bonded stresses become less compressive.
Bonds are being slightly decompressed as indicated by the relief of compressive
bond stresses in all directions. This decompression is minor and does not affect the
bond energies significantly, Figure 4-28. The mean bond length is 1.52765 Angstroms,
90
300 x-direction
- - y-direction
250-
z-direction
200a
150 -
100 (0
50 -5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
True Strain
Figure 4-25: Tensile stresses versus true strain for uniaxial tension with = 1 x 1010/s
and T = 1OOK for system of 7775 united atoms arranged in 160 networked chains.
1.52766 Angstroms, and 1.5284 at 0%, 2%, and 10% strain respectively. There is an
accompanying redistribution of bond lengths and a 32% change in bonded stresses
and a 0.5% drop in bond energy from 0% to 10% strain.
At yield, the Van der Waals stresses decrease. As the atoms are re-shuffled they
find lower energy positions with an accompanying lower stress. The stresses remain
more tensile during flow than they were prior to deformation. Also the level of the
Van der Waals stress in the axial direction remains relatively constant with further
deformation.
In the plane perpendicular to the load direction, the stress response is characterized by an alleviation of internal stresses throughout deformation. The bonded
stresses become less compressive and the Van der Waals stresses become less tensile,
Figure 4-26. This is attributable to the highly stressed backbone rotating out of the
y-z plane and to the decompression of the bonds.
The structure has an initial Poisson ratio of 0.35 in the elastic regime. The density
decreases by 4% from 0% to 15% strain, Figure 4-27. This large decrease in volume
91
200
-
Thermal Stresa
....Van Der Weals Interactions
Bond Streching
a Bond Angle Bending
16 0Dihedral Angle Rotation
-
0
1
14
0
12 0
0
I'
-
s
agtr
%
.5
-
10
0
*
5
60 4 0.
.
2
0
0.2
0.1
0.3
0.4
0.5
0.6
True Strain
0.7
0.9
0.8
1
(a) x-direction
-
150
.
-
a
--
-
Thermal Stress
Van Der Waals interactions
Bond Streching
Bond Angle Bending
Dihedral Angle Rotation
100
50
'
0
1=
o
-
..
-+-
-50
.
. . . .. - -
-
--------------
-.-
I
.,-,.PIN
-100
0.1
0.2
0.3
0.4
0.7
0.5
0.6
True Strain
0.8
0.9
1
(b) y-direction
Thermal Stress
..... Van Der Weals interactions
- - Bond Streching
a Bond Angle Bending
-Dihedral Angle Rotation
150
,
100
.
Is-
so
CL
--
0
---.-.- --- --- --- -... --..
-50
-100
0
0.1
0.2
0.3
0.4
0.5
0.6
True Strain
0.7
0.8
0.9
1
(c) z-direction
Figure 4-26: Partitioning of tensile stresses versus true strain for uniaxial tension
loaded in the x-direction.
92
0.95
I
0.940.930.92K
0.91
-
0.9 -
0.890.880.870.860.851
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
True Strain
Figure 4-27: Density versus true strain for uniaxial tension
is sensitive to the deformation rates employed. The magnitude of the drop is smaller
for lower strain rates. The same large density drops were observed in work by Brown
and Clarke [7]. The slope of this drop then decreases after yield as the Poisson ratio
approaches 0.5. There is a minor decrease in the density over the remainder of the
deformation. The inter molecular spacing is increased during this dilation since the
bond length and angle changes are too small to account for the density changes.
The increasing mean spacing would result in an increasingly tensile interaction until
neighboring mers cross the maximum in the gradient of the potential, Figure 4-10.
The energy change is dominated by increasingly energetic Van der Waals interactions, Figure 4-28. During elastic deformation, atoms are trapped by their neighbors
unable to undergo large structural rearrangements. The imposed deformation results
in an increased mean spacing between chains and hence an increase in energy as atoms
are displaced from their minimum energy configurations. Energy changes seem to be
confined to the Van der Waals interactions as the energy stored in bond stretching,
bond angle bending, and dihedral angle rotation remain relatively constant during
elastic deformation. Further deformation results in decompressed bonds and hence
93
0.7Bond Stretching
Bond Angle Bending
0.6 - - - Dihedral Angle Rotation
a Van der Waals Interactions
-
4
0.5 -
a33
aaa,
so#3#*###
0.40.23W
0.2 -
-0.1
-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
True Strain
0.7
0.8
0.9
1
Figure 4-28: Partitioning of energy change per united atom versus true strain for
tension.
a decrease in the bond energy. The torsion energy increases then plateaus beyond
yield.
Observations of the structural statistics, Figure 4-29, shows that the bond length
distribution narrows with deformation and the shoulders of the dihedral angle distribution broaden. The radial distribution shows an increasing peak corresponding to
the trans configuration. There is also a slight shift in the mean bond length as the
bonds are decompressed.
During the elastic portion of deformation the chain end-to-end distance does not
change. Chains are elongated by 44% at a true strain of 1, Figure 4-30. The chains
rotate into the load direction as the sample is strained as indicated by the decrease
in < P3 >, which measures the average angle made between the end points of a chain
and the load direction. both segments and bonds are rotated into the load direction
as indicated by < P 2 > and < P >. Though the fraction of dihedral angles occupying
the trans configuration changes twice as much as it does in compression, there is only
a 6% change in the fraction trans as the chain is elongated by 44%.
94
0.07
=0
.x
x
0.06
.X
~
~
0.0
0.1
=
x
X
E
E
E= 0.1
4 -I
=1.0
x
E
=1.0
0.03
x x
5-
0.05-
0.0 '3 -a
0.02
0.04-
0.0 '2
.0.03 -
2-
'5
0.01
xx
x
0.020.0
0.01
0.005-
x
0.94
xx
1.02
1
0.98
Nonnalized Bond Length (/b u)
0.96
1.06
1.04
1.7
1.8
1.85
1.9
Bond Angle
1.95
2
(Radians)
2.05
2.1
2.15
(b) Bond angle distribution
(a) Bond length distribution
0.08 -
5-
_E-0
E=0.1
x E.1.0
0.07-
1.75
4.5-
x
x
E=0.1
E11.0
40.063.50.05 -
3-
0.04-
62.5 -
c0 0.03-
2 1.5 -
0.02
x
-
1-
0.5 -
0.01 -
1
1.5
2
3.5
2.5
3
Dihedral Angle (Radians)
4
4.5
5
5.5
Normalized Distance (d/b,,)
(d) Radial distribution function
(c) Dihedral angle distribution
Figure 4-29: Comparison of structural statistics for tensile true strains of 0, 0.2, and
1.0.
95
27
60
26
I
!55023'U
45-
2221 -
W
40-
2019-
35-
18
17
--
0
0.1
0.2
0.3
0.4
0.5
0.6
True Strain
0.7
0.8
0.9
1
30
0.1
0
0.2
0.3
0.4
0.5
0.6
True Strain
0.7
0.8
0.9
(b) Average chain angle, < P 3 >
(a) Average chain end to end distance
57-
0.3 --
5655-
0.2 --
S54-
o53
0.1
-
-
A
52-
-L0 ,-
51
50
0.1
--
4948
0
0.1
0.2
0.3
0.4
0.5
0.6
True Strain
0.7
0.8
-0.
0.9
-
0.2
0.3
0.4
0.5
0.6
True Strain
0.7
0.8
0.9
(d) Bond order parameter, < P >
(c) Average segment angle with respect
to the load direction, < P 2 >
0.69
0.1
----
0.68
0.67
0.66
0.65
0.64
06
0.1
0.2
0.3
0.4
0.6
0.5
True Strain
0.7
0.8
0.9
(e) Fraction of dihedral angles in the
trans configuration
Figure 4-30: Chain statistics versus true strain for uniaxial tension.
96
3-
CL
cld)
C
0
I-
C-2
-3) -
+
o
o
-4'''''
0
0.2
0.4
0.6
True Strain
gauche to trans
trans to gauche
gauche to gauche
0.8
Figure 4-31: Transition rate versus true strain for uniaxial tension.
As the material flows at constant stress levels, energy changes and transition rates
plateau. At this point any additional input of energy is dissipated through structural
rearrangements. Though the transition rates are significantly higher following yield
than they are in an undeformed sample, the fraction of dihedral angles in the trans
configuration changes by only 6% percent from 0% to 100% true strain, Figure 4-30(e).
In fact, this fraction does not change at all from 0% to 50% true strain. Referring
to Figure 4-31, one finds that the transitions from the Gauche state to the Trans
state are matched on average by the transitions from the Trans to the Gauche state.
Therefore, the average configuration of the system does not change while individual
dihedral angles are in fact sampling multiple states.
Construction of pole figures, Figure 4-32, shows an increasing alignment of chains
with the load direction. This alignment is evident at strains of 0.1 and becomes
increasingly visible in the plots.
From these measures of orientation, one deduces that the change in the Van der
Waals stresses is attributable both to an increase in mean spacing of particles and to
rotation of the chains in the load direction. As there are highly tensile Van der Waals
97
x
X
,I-
K
x
x
T' W.
x,
x
x
x
= 0.1
(b)
(a) f = 0
xm
x
x xx
'k
Ix
x
x
x
Nx
gab
;
A
Wx
fi
1; x
x
_111
v x
x
Vi
x
x
x
4
ex
x
x
x
x
x
x
x 04
x
4x
t
li x
x
x
x
0. 4
0. 2
(d)
x
x I
x# xxxxx xx x rx
IFx
x
xx
waxw
x
x
ON x
x
x
wN 'A
(c) c
xxx x
'k
x
t
Ox
x
x awxx
x
x
x
x
x
17
10
fxf
*
x
x
x
x
; x,
-. x
x
x
x
x x
'kx
x*
x
it
. x
0
(e) c
jF
xmks
*
x
xr
x
x
x
x
i-x
_x
x
il
x
x.0
xIN
x
xx!x
x;wx
Ipx
x
0. 6
Figure 4-32: Pole figures for uniaxial tension,
98
IF x x
x*N j
x
stresses along the chain backbone, rotation of these chains into the load direction
results in the observed increasingly tensile Van der Waals stresses.
99
0.15 -
1=0.1
= 0.2
- F = 0.3
-
--
0.1
E =0.6
S---=0.8
-
.0
E
z
0.05-
10
15
20
25
30
35
40
Volume (Angstrom3)
45
50
55
60
Figure 4-33: Tessellated volume distribution at various strains for system of 5000
mers tested in uniaxial tension at 1OOK for a strain rate of lx 10 10 sec- 1 .
4.3
Evolution of Tessellated Volume
The evolution of the tessellated volume with deformation indicates how the volume
is redistributed locally amongst the atoms. During elastic deformation in both compression and tension, one notes a broadening of the free volume distribution, Figures
4-33 and 4-34. The curve corresponding to those mers with free volumes from 10 to
21
A, i.e.
those which are tightly packed, does not change much. The distribution
then becomes static beyond a strain of 0.3. This corresponds to the initiation and
continuation of flow. This indicates that both the local volume distribution and the
macroscopic volume are static during flow.
In addition, the volume distributions during flow in tension and compression are
similar to those found at the temperatures around the glass transition temperature.
The broadened shoulder and peak heights are similar. This suggests that the volume
distribution in flow evolves to a state near the glass transition temperature after
which molecular mobility can accommodate further deformation without increases in
volume.
100
0.15-- CE=0.1
-
e=0.2
--- e=0.3
-
=0.6
S=0.8
0.10
CO)
.0
E
z
0.05-
0
10
'
15
20
25
30
35
40
Volume (Angstrom3 )
45
50
55
60
Figure 4-34: Tessellated volume distribution at various strains for system of 40000
mers arranged in 160 free chains tested in uniaxial compression at lOOK for a strain
rate of 1x10 10s- 1 .
101
4.4
Strain Rate and Temperature Dependence of
Yield Stress
4.4.1
Stress Response
The response of the network at various temperatures and strain rates was determined.
Presented in Figure 4-35 are stress strain data for two temperatures each tested at
two strain rates. As previously stated, the temperature and strain rate are both
important in determining the material response. If the strain rate is slow enough
or temperature high enough, the response is leathery or rubbery. This difference is
exhibited by the samples shown in Figure 4-35.
At 1OOK the model is well below its glass transition temperature while at 450K
it is well above the glass transition temperature.
At 100K, one notices that the
magnitude of strain softening increases with increasing strain rate but the proportion
of softening to yield stress remains relatively constant. The flow stress as increases
with increasing strain rate. Though as the strain gets larger the response is similar.
At 450K there is no softening as the material exhibits a rubbery behavior.
The density change as a function of strain is provided in Figure 4-36. Note also
that the dilation seen after yield in samples below the glass transition is sensitive to
the strain rate. At slower rates the magnitude of the dilation becomes smaller but is
always present. Though one might expect constant volume deformation for a rubbery
material, the model being tested is known to crystallize easily given enough time and
temperature. At 450K, one must worry about crystallization of the model. This may
explain why the density at 1 x 10
10s- 1
increases more rapidly than at 5 x 10's--1.
Given more time, the sample may crystallize further. This may result in additional
hardening of the material as the strain is increased.
Attempts to characterize amounts of crystallinity through observation of the radial distribution function were unsuccessful. There is little change in the radial distribution function as the sample is deformed.
The thermal fluctuations at higher
temperatures broaden the peaks and make interpretation of the distribution changes
102
4 00
--
- -
- - -- -
-
-
'
----
-A--
F
350
- -
-a~
S300
C
0
0 250
200
--
C
lO-,E-110-
S150
U)
S100
--
100K, E=51X 10
--
450K,sE= 5x101
50
0
0
0.2
0.1
0.4
0.3
0.5
0.6
0.7
True Strain
Figure 4-35: Stress versus strain for various temperatures for a system of 7775 united
atoms arranged in 160 networked chains.
-.
0.05
...- . ......
.
.
- 100K, E = 1Xl
-- - 100K, F =5x10
-- 450K, E = 1X10
- - -450K= 5 10
0.04
0.03
... .. ....
450K~e .........
-
-
-
. ..
......
sx
*.
...
9. 0.02
-.
-.
....
.....
0.01
-
0
..... -.
...
.....
..
.....
-..
>-0.01
. -0.02
-0.03F-00 A
..........
I
0
0.1
0.2
0.3
0.4
True Strain
0.5
0.6
0.7
Figure 4-36: Change in density versus strain for various temperatures for a system of
7775 united atoms arranged in 160 networked chains.
103
2--
1.8
450K, e = 0
450K, e = 0.8
1.6-
1.4 1351.2-
0-
0.2
-
0
0
1
2
3
Normalized Distance M/
4
5
6
Figure 4-37: Change in radial distribution function during the application of compressive deformation at 450K.
difficult.
In addition, there may be crystalline and amorphous regions within the
same sample cell.
From the radial distribution function, Figure 4-37, there is little change in the
short range peaks at a normalized distance of 1 and 1.63. One finds a slight sharpening of the peak at a normalized distance of 3.25. The long range liquid like structure is
compacted as g(r) is shifted to the left at normalized distances above 3.5. It appears
that the long range order is being altered through the application of deformation.
Although changes in the density suggest crystallization, the crystallites may be dispersed in an amorphous structure and so the radial distribution function does not
capture the appearance of crystallization clearly.
104
Table 4.1: Activation parameters for a selection of glassy polymers [1]
Polymer
£ (nm3)
Polystyrene
Polycarbonate
Polyethylene-terephthalate
4.4.2
4.95
18.4
17.2
Activation Parameters
The yield stress was determined for various strain rates, shown in Figure 4-39, in an
attempt to determine the activation volume as described in section 1.3 for the Eyring
model. The close agreement in yield stress measurements for different initial samples
leads to confidence in the results. A least squares fit to the yield stress vs strain rate
data as shown in Figure 4-39 results in an activation volume of Q = 0.18 nm 3 at 100K
and 0.21 nm 3 at 200K. The RVE in our simulations ranged in size from 105 nm 3 for
a cell of 4500 mers to 240 nm 3 for a cell of 10000 mers.
Referring to Table 4.1, the activation volume obtained for this model is over an
order of magnitude smaller than the experimentally determined values for various
other polymers. Noting the simpler chemical structure of polyethylene compared to
polycarbonate, polystyrene, and polyethylene-terephthalate, Figure 4-38, one would
expect it to have a lower activation volume since it does not contain large side groups
or ring structures.
Molecular mechanics studies on atactic PP with an RVE of at most 39 nm 3 by
Mott and Argon [29] suggest the shear transformation strain should be on the order
of 1-5%. Assuming a transformation strain of 1% and 5%, the transformed volume,
Va,
according to Equation 1.3 is then 4 nm 3 and 20nm 3 , respectively. At the given
densities, this represents cooperation by 150-800 mers.
At 100K and strain rates above 2 x 10"/s, there is an obvious deviation in the
behavior of the yield stress. A more rapid increase with strain rate which corresponds
to a smaller activation volume of 0.03 nm 3 . Assuming a strain of 1% and 5%, this
suggests cooperation by 20 mers and 120 mers, respectively. Though this transition
may be gradual and may involve many intermediate modes, the inaccuracy of the
105
a)
I I
-F-C-Cm4n
H
0)
polystyrene
b)
C)
/
--
C)
"7
C--CHsCH4t
4:
terephthalate
group
ethylene
group
carbonate
group
Figure 4-38: Chemical Structure of (a) polystyrene, (b) polyethylene-terephthalate,
(c) polycarbonate
106
measured yield stress and the scatter in data does not allow for the reliable interpretation of these intermediate modes. The structural relaxation of the glassy polymer
may be imagined as being composed of many relaxation modes each of which operates with a given time constant at a given temperature. Deformation processes which
occur on time scales shorter than this relaxation time constant will not sample that
particular relaxation mode. On the other hand, a mode will be extremely active in
deformation processes occurring on time scales much longer than its time constant.
The yield stress at realistic strain rates was determined by extrapolation using
the calculated activation volume at the lower strain rates. The yield stresses at 100K
and 200K at a strain rate of 0.1 per second are -138 MPa and -429 MPa respectively.
These negative values for the yield stress do not come as surprises. As suggested by
the modulus changes observed as a function of strain rate in Figure 4-3, this model
polymer may no longer be glassy at experimental strain rates. The assumption of
isostructural deformation may not hold under these circumstances. In addition, one
expects additional relaxation modes to be activated as the deformation rate decreases.
An analogous error would be made if one were to extrapolate the behavior at 100K
for strain rates above 2 x 10
10 /s
to lower strain rates at that temperature.
107
4-
3.5 -
- o 1OOK > 100K 200K + 200K x
200K -
- 8000 mers
4500 mers
8000 mers
4500 mers
10000 mers
-
-
a.
0.55
9
.
Iog 1o(e)
0
05
1
Figure 4-39: Summary of yield stress obtained through simulations of uniaxial compression for various initial networked structures, strain rates, and temperatures.
108
4.5
Transitions and Yield
By monitoring transitions rates as a function of deformation, it was possible to observe changes in material behavior during yield. Referring to Figure 4.5, one finds
that transition rates quickly increase during yield to relatively constant levels as flow
initiates. Since very few transitions are recorded during deformation at low temperatures, ensemble averaging would be necessary to obtain more accurate profiles
of the transition rates. Figure 3-13 shows that various initial samples produce very
similar transition rate profiles. In fact, the cross-linking of the system seems to make
no difference in the profile obtained. This indicates that at the deformation rates
used in these studies, the rate of configurational transitioning is governed by the local
interactions and not by chain connectivity.
The general features of a significant increase in transition rate leading to and just
after yield and a decrease in the slope of this transition rate as flow initiates is seen at
all strain rates and at all temperatures tested. Increasing the temperature, Figure 440(b) and 4-40(c), or increasing the strain rate, Figure 4-40(d), both serve to increase
the transition rates. The difference in transition rates at a constant strain rate for
various temperatures cannot be explained by taking into account the transition rate
of a sample at the given temperature with no deformation. One might expect that
not all transitions contribute equally to the accommodation of deformation. Some
extraneous transitions occur. In an attempt to correct for this, the transition rate of
a sample with no deformation was subtracted, Figure 3-12, from the transition rate
at that same temperature for the various strain rates but failed to collapse the data,
Figure 4-40(b) and Figure 4-40(c), into a single curve.
The initial drop in transition rate seen in the first few percent of strain is most
probably attributable to the increase in density during compression. The increased
density in the cell stifles transitioning. At low transition rates, the values measured are
statistically uncertain. Interpretation of this initial drop may simply be a consequence
of the small number of transitions occurring within the given time period. One finds
the most variation amongst different samples in this low transition rate regime, 4-
109
40(a).
Before yield, transition rates at a given temperature appear to rise at similar rates
at various strain rates, Figure 4-40(d). Beyond yield, the transition rates increase as
the strain rate is increased. This suggests that as the strain rate increases, accommodation of deformation through the re-shuffling of neighboring mers is decreased.
Instead, the deformation is accommodated through transitional rotations. This is
consistent with the observed decrease in activation volume at the higher strain rates,
Figure 4-39.
It is clear that after yield the transition rates have increased significantly above
the pre deformation values. At 250K the transition rates after yield are very similar to the transition rates at the glass transition temperature for the given strain
rate. For example, at 250 K and a strain rate of 1 x 1010 sec~ 1 the transition rate
is approximately 14-20 transitions per angle per nano-second, Figure 4-40(b). This
transition rate corresponds to a temperature between 285K and 310K, Figure 3-12.
The glass transition temperature at this strain rate as measured by the DMA type
simulation was 275-300K, Figure 4-1. Similarly at a strain rate of 5 x 10
10sec- 1
the
transition rate at 250K is approximately 30-40 transitions per angle per nano-second,
Figure 4-40(c). This corresponds to a temperature between 335K and 360K. The
glass transition temperature was measured to be 330-370K, Figure 4-1.
As the temperature was lowered, the transition rates for a given strain rate decreased. If one compares the transition rates at high strain to the transition rates
for static samples at various temperatures, these decreased transition rates correlate
to a lower temperature. For example, at 1OOK for a strain rate of 1 x 10
5 x 10
10 sec- 1
10 sec- 1
and
the transition rates were 3-5 and 10-20 transitions per angle per nano-
second, respectively, Figure 4-40(b) and 4-40(c). These transition rates correspond
to temperatures 210-235K and 265-310K respectively, i.e. the mobility has increased
to a level observed in the material at much higher temperatures when under zero
stress. Although the mobility is that observed at much higher temperatures, it is
not as enhanced as that observed at Tg. The glass transition temperatures for the
employed strain rates is much higher than this. At 1 x 10 10 sec
110
1
and 5 x 10 10 sec
1
the estimated glass transition temperature obtained from DMA type experiments is
270K and 350K respectively. The dependence of the transition rates on strain rate
and temperature suggest it is a stress-biased thermally activated process.
If one normalizes the number of transitions obtained in a given time unit by the
displacement in that time unit, one obtains a curve which peaks during flow, Figure
4.5. This suggests that the number of transitions required to accommodate a given
unit of deformation becomes constant during flow. The plateau value decreases with
decreasing temperature or increasing strain rate.
This suggests that as the time
scale of molecular relaxation becomes smaller in comparison to the strain rate, the
amount of deformation accommodated by dihedral angle rotation lessens. Instead
the sampling of the dihedral angle configuration is stifled. At slower strain rates and
higher temperatures more of the deformation is being accommodated by rotation.
One expects that there are two sets of transitions occurring in the system. Those
which contribute to accommodation of deformation and those which do not. The
trend of an increasing number of transitions per unit of deformation as the temperature is increased or the strain rate is decreased might be an artifact of the fact that at
higher temperatures there are more transitions even in the static case. To account for
this, the initial transition rates were used to calculate the number of transitions which
would have taken place in the static case during the time required to deform the sample by one Angstrom. This number was subtracted from the curves and the trend of
an increasing number of transitions with increasing temperature or decreasing strain
rate persisted.
Referring to the stress strain plots for these systems, Figure 4-42, one finds that
the plateau in the number of transitions per unit deformation seems to coincide with
the end of softening. The transition rate appears to decrease in the initial elastic
regime while increasing with the application of stress beyond the initial few percent
of strain. The decrease in slope of the transition rate corresponds to a leveling off of
the stress with increasing strain.
111
21
21
1.5
1.5
a)
1
00.
0.5
(D
0
.5
0
-L0.5
05
-0.5
1
+
-p- +
-.-
1 0
1
nw-c-1-4500-90-100K, E= 1X1010 ' s1
nw-1-8000-160-100K, E = 1x10 s7
fc-1-40000-160-100K, E= 1Xl0101 0S-1
nw-c-1-4500-90-100K, e = 5x10 s~1
1
nw-1-8000-160-100K, E = 5x10 ' s-'
fc-1-40000-160-100K,
E
0.6
0.4
True Strain
0.2
= 5X1010 S-
2 0
-9-
1
- --
0.2
1
0.8
0
fc-1 -40000-160-100OK, -= 1 x0 aS1
10
tc-1-40000-160-200K, c = 1x10 s-1
fc-1-40000-160-250K, E = 1X100 S
0.4
0.6
True Strain
0.8
(b) Varying the temperature with e
1 x 10 10 /s
(a) Comparison between free chains
and network systems.
2-
1
=
21
1.5
1.5-
Q-
0.50-
,
.5 -
02
CD
0
0-
0-
c
-0.5 -
.5 -
---
fc-1-40000-160-100K,
E = 5xII10 S-1
-40000-160-200K, E = 5xl 10 s-1
fc-1 -40000-160-250K, E = 5xl 10 S-1
+fc-1
-eS
0.2
0.4
0.6
True Strain
0.8
(c) Varying the temperature with
1
i=
-1
-
tc-1-40000-160-100K,
- -
fc-1-40000-160-100K, e = 1Xl
Sfc-1 -40000-160-100K, E = 5x10 s-
-4-
0
0.2
E
0.4
0.6
True Strain
=
5x101 0 s-
0.8
(d) Varying the strain rate with T
100K
5 x 10'1/s
Figure 4-40: Transition rates as a function of true strain.
112
1
1
=
lor
9a
0
a
a
70)
6-'
a
1
5-
0-
4-
CL
05
3-
2-
--
-
0
1
-
fc-1-40000-160-100K, E =
--
_C) r-1
0
0.1
0.2
0.3
0.4
True Strain
a 0=5x1010 s0
0.5
0.6
0.7
- --
0
0.1
0.2
fc-1-40000-160-200K, E=
fc-1-40000-160-250K, E =
0.3
0.4
True Strain
0.5
0x10
0 aI
1x1 0 "s1
' a's'
0.6
0.7
(b) Varying the temperature with i
1 x 1010/8
(a) Ensemble average at 100K.
=
4.5-
42
.5-
E 3.5 E
E
2
2-
3-
EC
2.5-
-1
CL
.50.5 --
-.
-,
00
+
-.-
0.1
0.2
fc-1 -40000-160-1OOK, E = 5x10 S-1
0
fc-1--40000-160-200K, E= Sxl1 s- 1
fc-1-40000-160-250K, E = 5x1 010 S7
0.3
0.4
True Strain
0.5
9
5x10 slx1 010 S~1
0
-4- fc-1-40000-160-100K, E = 5x1O' S-1
0.7
0.6
(c) Varying the temperature with c
5 x 101s
00K,
E=
fc-1 -40000-160-1
fc-1-40000-160-1OOK, E=
0
0.1
0.2
0.3
0.4
True Strain
0.5
0.6
(d) Varying the strain rate with T
lOOK.
=
0.7
=
Figure 4-41: Number of transitions per dihedral angle per angstrom of deformation.
113
---.----.-.-.--.-.-.--
300
500
450
-
250
400f
--
,
-.
-.
- ..
A,
-.
-.
....
Cal
----.
--.
-----..
..
-. .
...... -.. - .... -.-..
lt
a- 350
C
0
C
0
2200
6 300
a
-D
0
-j
50
as 5
00
200
-
-0
(75
65 1 0 0
2 150
2
a
F-
50
100
8000, =1x10
-
-.
----
50
--0
C
0.2
0
secsec~40000, e = 1x10 sec1
40000, E = 5x1 0 sec-
-8000,
E=5x10
0.6
0.4
True Strain
40000 mers, T=100OK, E=1x100 sec-l
10
0.8
10
1
---.--.-.
--.. -.
-.
.....
---
--
30 0
......-..
250 -
-.
---
3 50 -
- 0 0 --.
0.8
- .....
1
e=
.----
-
--
- - --
-
-
300
-.
..-.
C
-
-.
,cs250 -
(2
0.4
0.6
True Strain
sec-
(b) Varying the temperature with
1 x 1010/s.
(a) Varying sample and strain rate.
35 0
0.2
-
0
K
40000 mers, T = 250K ,E= 1 x1 0
-
..-.
.-
0200 ----
0
150-
100
1000
-sec I
- -- 40000 mers, T = 200K, e =5x1 00 sec
-- 40000 mers, T = 250K ,E = 5xI 10 Sec
50 ..--..--.--.
0
-- 40000 mers, T = 100K,
0.2
0.4
0.6
True Strain
50
e =5xI1
0.8
(c) Varying the temperature with
1
e=
-- - -
-- -
-
--
40000 mers, T = 100K, e= 1x1099SeC-- 40000 mers,T=100K,E_=1x10 sec- - - 40000 mers, T = lOOK, E =1 Xl10 sec'
40000 mers, T = 1OOK , = 5x1 010 sec'
0
0.2
0.4
0.6
True Strain
0.8
(d) Varying the strain rate with T
100K.
5 x 10-0/s.
Figure 4-42: Stress as a function of strain for various systems.
114
=
4.6
Correlation of Transitions along a Chain
During deformation at strain rates of less than 1 x 10 10sec- 1 , it was shown in Section 4.5 that the transition rate increases and then plateaus as flow commences. In
addition to this increase in rate, the nature of coopertivity between these transitions
changes. Prior to yield, the transitions are highly self correlated. Following yield and
throughout flow, the transition coopertivity resembles that at the glass transition
temperature. Throughout flow, one finds a decreased self correlation and strong correlations with dihedral angles separated by an even number of bonds. This behavior
seems to be consistent across the investigated temperature and strain rate regimes.
Unfortunately, statistical error in these correlations is much greater than in the determination of transition rates due to the small number of each type of transition.
115
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6-
!40.4
0.4
0.3
0.3
0.2
0.2
-static
-- ____-'-----.
------'--_0.1
0.1
"0
2
4
6
Correlation position (i±j)
(a) fc-40000-160-100K,
100K, i = 5 x 10 9 /s
0.9
8
10
T
0
2
4
6
Correlation position (itj)
_
8
10
T
=
0.7
-- static
--- f
i
static
low
0.8
-- --
(b) fc-40000-160-100K,
100K, = 1 x 1010/s
=
flow
---
0.6
0.7
0.5
0.6
-0.4
0.4
-- - - - - - - - - - - - - -
0.3
0.3
0.2
0.2
0.1
0.1
0
2
4
6
Correlation position (i±j)
(c) fc-40000-160-100K,
100K, E = 5 x 10 10 /s
10
8
T
"0
2
4
6
Correlation position (i±j)
(d) fc-40000-160-200K,
200K, i = 5 x 10 10 /s
=
Figure 4-43: Neighbor Correlation
116
8
T
10
=
Chapter 5
Conclusion
The results of this investigation are divided into three segments, structural behavior
as a function of temperature, stress response to deformation, and chain configuration
as a function of deformation. The chapter closes with remarks concerning future
studies.
5.1
Structure
The initial structure of the network was characterized as a function of temperature.
Networks and free chain systems were created and simulations run at various temperatures. The structural response can be characterized as follows:
" The glass transition temperature was found to decrease with decreasing molecular weight for free chain systems and increase upon the addition of crosslinks.
As the temperature decreases below the glass transition temperature, dihedral
angles no longer explore their configuration space and are frozen into a particular configuration.
" Studies of transition rates as a function of temperature reveal that dihedral
configuration transitions occur as single barrier events. At temperatures above
the glass transition temperature, the activation energy for transitioning is found
to be similar to the explicit dihedral angle potential.
117
"
Deviation from an Arrhenius behavior is detected at temperatures below the
glass transition. This occurred as dihedral angles became trapped by the surrounding matrix and oscillated between two high energy states. The effective
barrier height for these transitions is lower and thus a higher transition rate than
predicted by the Arrhenius extrapolation was observed. Attempts to correlate
transition rate with local density and tessellated volume were unsuccessful using
the techniques described in Chapters 3 and 4.
" The cooperative nature of transitions along a chain was investigated and found
to be significant at all temperatures below the glass transition. A strong correlation between dihedral angles separated by an even number of mers was found.
The especially strong ±2 which is characteristic of kink type motion was observed as temperatures fell. The self correlation increases below the glass transition temperature indicating that many dihedral angles are oscillating between
two or more states while neighboring dihedral angles are not transitioning at
all.
5.2
Deformation
The system response to deformation was determined as a function of strain from
simulations which mimic uniaxial tension and compression experiments. The system
density, energy, and stress response as well as analysis of yield stress as a function of
strain rate and temperature can be characterized as follows:
" Deformation response below the glass transition temperature captured qualitatively the experimentally observed behavior. DMA type experiments and
studies of specific volume as a function of temperature were used to characterize the glassy regime. The system response is marked by a stiff elastic regime,
yield, softening, flow, and hardening.
* Yield in the system is accompanied by dilation which is followed by volume
preserving deformation. The distribution of tessellated volume during flow ap118
proaches the distribution for an undeformed sample at the glass transition temperature.
" Analysis of the yield stress as a function of temperature and strain rate below the
glass transition temperature using the Eyring model reveals a small activation
volume parameter. As the strain rate was increased at 100K, the activation
volume describing the dependence of yield stress on strain rate changed. This
may correspond to the change in activated processes.
" Though the polyethylene chain theoretically provides the opportunity to directly
observe shear events due to the small activation volumes involved, it has not
yet been possible to correlate local motion in the system directly.
5.3
Configuration and Deformation
The changes in configuration of the chains during deformation were observed as measures of mobility in the system. Characterization of the transition rates and correlations between transitions led to the following observations:
" The self correlation decreases drastically during flow as the cages around atoms
are destroyed by the deformation. A sharp increase in the transition rate per
dihedral angle during and following yield occurs at all temperatures and strain
rates. The transition rate depends only on temperature and strain rate while
being insensitive to the initial structure and the chain connectivity.
" It appears that in the elastic regime, the response is dominated by rigid motions
of the chain. The application of stress modifies the dihedral angle transitioning
barriers leading to yield and then flow.
" The number of transitions per unit deformation also levels off during flow suggesting deformation is being accommodated by configurational changes along
the backbone. A decrease in temperature or an increase in strain rate decreases
the number of transitions per angstrom of deformation indicating a competition
119
between accommodation through chain angle rotation and chain slippage. At
high temperatures or low strain rates, the chain is allowed to explore its configurational space and accommodates the deformation through these configurational
changes leading to a higher number of transitions per angstrom of deformation.
At lower temperatures or higher strain rates, this mechanism is being frozen out
and the chains react in a more rigid fashion with fewer configurational changes
per angstrom of deformation.
* The flow state for samples deformed at temperatures near the glass transition
temperature seems to resemble the state of the material at the glass transition
temperature. The transition rates during flow approach those of an undeformed
sample at the glass transition temperature. The neighbor correlation function
calculated during flow also resembles that calculated for an undeformed sample
at the glass transition temperature.
5.4
Future Work
In the future, attempts will be made to correlate the particle trajectories to determine
the activation volume directly. The influence of side groups and ring structures on the
mobility and mechanical properties of the system could also be studied. Observations
of mobility changes during unloading will also be made.
120
Bibliography
[1] A.S. Argon and M.I. Bessonov. Plastic deformation in polyimides, with new
implications on the theory of plastic defromation of glassy polymers. Phil. Mag.,
35(4):917, 1977.
[2] H.J.C. Berendsen, J.P.M. Postma, W.F. van Gunsteren, A. DiNola, and J.R.
Haak. Molecular dynamics with coupling to an external bath. J. Chem. Phys.,
81:3684, 1984.
[3] J. Bergstrom. Large Strain Time-Dependent Behavior of Elastomeric Materials.
PhD thesis, MIT, 1999.
[4] R.K. Bharadwaj and R.H. Boyd. Effect of pressure on conformational dynamics
in polyethylene: A molecular dynamics simulation study. Macro., 33:5897-5905,
2000.
[5] R. H. Boyd, R. H. Gee, J. Han, and Y. Jin. Conformational dynamics in bulk
polyethylene: A molecular dynamics simulation study. J. Chem. Phys., 101:788,
1994.
[6] J. Brandrup and E.H. Immergut. Polymer Handbook, 3rd ed. Wiley-Interscience,
1989.
[7] D. Brown and J.H. Clarke. Molecular dynamics simulation of an amorphous
polymer under tension. 1. phenomenology. Macromolecules, 24:2075, 1991.
[8] C. Chui. Application of Monte Carlo Techniques to the Study of Amorphous
Polymer Deformation. PhD thesis, MIT, 1998.
121
[9] D. Deng, A.S. Argon, and S. Yip. Topological features of structural relaxations
in a two-dimensional model atomic glass ii.
Philos. Trans. R. Soc. London,
A329:613, 1989.
[10] E.R. Duering. Structure and relaxation of end-linked polymer networks.
J.
Chem. Phys., 101(1):8169-8192, 1994.
[11] D. Frenkel and B. Smit. UnderstandingMolecular Simulation. Academic Press,
1996.
[12] J.P. Gao and J.H. Weinger. Nature of stress on the atomic-level in dense polymer
systems. Science, 266(5186):748, 1994.
[13] R.H. Gee and R.H. Boyd. The role of torsional potential in relaxation dynamics:
a molecular dynamics study of polyethylene. Comp. and Theor. Polymer Sience,
8(1):93, 1998.
[14] M. Gerstein, J. Tsai, and M. Levitt. The volume of atoms on the protein surface:
Calculated from simulation, using voronoi polyhedra.
J. Mol. Biol., 249:955,
1995.
[15] E.R. Griffin. Analysis of thermoplastics used in fold-and-form pipe liners. Website, 1997.
[16] O.A. Hasan. An experimental and analytical investigation of the thermomechanical properties of Glassy polymers. PhD thesis, MIT, 1994.
[17] R.N. Haward and R.J. Young, editors. The Physics of Glassy Polymers, chapter
Relaxation processes and physical aging. Chapman and Hall, 1997.
[18] R.W. Hocknet, S.P. Goel, and J.W. Eastwood. Quit high-resolution computer
models of a plasma. J. Comput. Phys., 14:48, 1974.
[19] W. Hoover. Canonical dynamics: Equilibrium phase-space distributions. Physical Review A, 31:1695, 1984.
122
[20] M. Hutnik, A.S. Argon, and U.W. Suter. Conformational characteristics of the
polycarbonate of 4,4' - isopropylidenediphenol.
Macromolecules, 24(22):5956-
5961, 1991.
[21] K. Karatasos, D.B. Adolf, and S. Hotston. Effects of density on the local dynamics and conformational statistics of polyethylene: A molecular dynamics study.
J. Chem. Phys., 112(19):8695, 2000.
[22] A.J. Kinloch and R.J. Young. Fracture behavior of polymers. Applied Science
Publishers, 1983.
[23] T. Liang, Y. Yang, D. Guo, and X. Yang. Conformational transition behavior
around glass transition temperature. J. Chem. Phys., 112(4):2016, 2000.
[24] L.S. Loo, R.E. Cohen, and K.K. Gleason.
Chain mobility in the amorphous
region of nylon 6 observed under active uniaxial deformation. Science, 288:116,
2000.
[25] K. Maeda and S. Takeuchi. Atomistic process of plastic-deformation in a model
amorphous metal. Philos. Mag. A, 44:643, 1981.
[26] N.G. McCrum, C.P. Buckley, and C.B. Bucknall. Principles of Polymer Engineering. Oxford University Press, 1988.
[27] J.E. McKinney and M. Goldstein. J. Res. Nat. Bur. Std, 1974.
[28] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Tellar, and M. Tellar. Equation
of state calculations by fast computing machines. J. Chem. Phys., 21:1087, 1953.
[29] P.H. Mott, A.S. Argon, and U.W. Suter.
Atomistic modeling of plastic-
deformation of glassy-polymers. PhilosophicalMagazine A, 67(4):931-978, 1993.
[30] S. Nose.
A unified formulation of constant temperature molecular dynamics
methods. J. Chem. Phys., 81(1):511, 1984.
[31] P.V. Pant, G. D. Smith, and R.H. Boyd. A molecular dynamics simulation of
polyethylene. J. Chem. Phys., 99(1):597, 1993.
123
[32] M. Parrinello and A. Rahman. Polymorphic transitions in single-crystals - a new
molecular-dyanmics method. J. Appl. Phys., 52(12):7182, 1981.
[33] W. Paul, G. D. Smith, and Y. D. Yoon. Static and dynamic properties of an
n-clooh 202 melt from molecular dynamics simulations. Macromolecules, 30:77727780, 1997.
[34] W. Paul and Y. D. Yoon. An optimized united atom model for simulations of
polymethylene melts. J. Chem. Phys., 103(4):1702, 1995.
[35] D. Rigby and R. Roe. Molecular dynamics simulation of polymerliquid and glass.
i. glass transition. J. Chem. Phys., 87(12):7285, 1987.
[36] D. Rigby and R.J. Roe. Molecular dynamics simulation of polymerliquid and
glass. 3. chain conformation. Macromolecules, 22:2259, 1988.
[37] M. Sprik. Effective pair potentials and beyond. In M.P. Allen and D.J. Tildesley, editors, Computer Simulation in Chemical Physics, volume 397 of Series C:
Mathematical and Physical Sciences, pages 211-259. Kluwer Academic Publishers, 1993.
[38] D. Srolovitz, V. Vitek, and T. Egami. An atomistic study of deformation of
amorphous metals. Acta Metall., 31:335, 1983.
[39] M.F. Sylvester. Molecular Dynamics Studies of the Liquid-Glass Transition in
Atactic Poly(Propylene). PhD thesis, Massachusetts Institute of Technology,
1992.
[40] H. Takeuchi and R. Roe. Molecular dynamics simulation of local chian motion
in bulk amorphous polymers. ii. dynamics at glass transition. J. Chem. Phys.,
94(11):7458, 1991.
[41] D. N. Theodorou, T. D. Boone, L. R. Dodd, and K. F. Mansfield. Stress tensor
in model polymer systems with periodic boundaries. Makromol. Chem., Theory
Simul., 2:191-238, 1993.
124
[42] D.N. Theodorou. Local-structure and the response of a polymeric glass to elasticdeformation. Macro., 19(2):379, 1986.
[43] L. Verlet. Computer experiments on classical fluids. i. thermodynamics properties
of lennard-jones molecules. Phys. Rev., 159:98, 1998.
[44] G.F. Voronoi. Z. Reine Angew. Math., 134:198, 1908.
[45] I.M. Ward. Mechanical Properties of Solid Polymers. Wiley-Interscience, 1971.
[46] L. Yang, D.J. Srolovitz, and A.F. Yee. Molecular dynamics study of isobaric
and isochoric glass transitions in a model amorphous polymer. J. Chem. Phys.,
110(14):7058, April 1999.
[47] Q.Y. Zhou, A.S. Argon, and R.E. Cohen. Enhanced case-ii diffusion of diluents
into glassy polymers undergoing plastic flow. Polymer, 42:613, 2001.
125