Shear-Induced Homogeneous Deformation Twinning in FCC
Aluminum and Copper via Atomistic Simulation
by
Robert D. Boyer
B.S. Materials Science and Engineering
Case Western Reserve University
SUBMITTED TO THE DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN MATERIALS SCIENCE AND ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
AUGUST 2003
© 2003 Massachusetts Institute of Technology. All rights reserved.
Signature of Author: .........................................................................................................................
Department of Materials Science and Engineering
August 22, 2003
Certified by: ......................................................................................................................................
Sidney Yip
Professor of Nuclear Engineering
Thesis Supervisor
Accepted by: .....................................................................................................................................
Harry L. Tuller
Professor of Ceramics and Electronic Materials
Chair, Departmental Committee on Graduate Students
Shear-Induced Homogeneous Deformation Twinning in FCC
Aluminum and Copper via Atomistic Simulation
by
Robert D. Boyer
Submitted to the Department of Materials Science and Engineering
on August 22, 2003 in Partial Fulfillment of the
Requirements for the Degree of Master of Science in
Materials Science and Engineering
Abstract
The {111}<11 2 > shear stress-displacement behavior for face-centered cubic (fcc) metals,
aluminum and copper, is calculated using empirical potentials proposed by Mishin and by
Ercolessi, based on the embedded atom method (EAM), and compared with published ab initio
calculations. In copper close agreement is observed in the results given by the Mishin potential
for both the ideal shear strength and local atomic relaxation during shear, although the extent of
plastic deformation before failure is over-predicted. In aluminum, both the Mishin and Ercolessi
potentials are used, with only the former able to capture the majority of the behavior exhibited in
first principle calculations. Both potentials are shown to have difficulties modeling the effects of
directional bonding. Calculations of the multiplane generalized stacking fault energy in both
materials reveal that aluminum has a longer range of atomic interaction than copper.
Using molecular dynamics and static energy calculations, deformation twins are shown to
form by homogeneous nucleation, slip and subsequent coalescence of partial dislocations in both
copper and aluminum. The minimum energy path for formation of a two-layer microtwin, and
the energy barriers to its further growth are analyzed for the two metals. The mechanism
observed is interpreted with reference to existing work on the nucleation of microtwins in bodycentered cubic metals.
Thesis Supervisor: Sidney Yip
Title: Professor of Nuclear Engineering
Acknowledgements
I would like to first thank my advisor, Prof. Sidney Yip, for sticking with me and
continually pushing me through a rough start to my graduate career. It is due in no small part to
his tenacity in this arena that this thesis has been produced over the last few months and that I am
well prepared to continue this work.
I would like to acknowledge not just the financial support provided by Lawrence
Livermore National Laboratory from award number B524480 but also technical discussions and
useful advice provided by Dr. Alan Wan, Dr.Geoff Campbell, Dr Vasily Bulatov, and Dr. Wei
Cai during my brief stay at LLNL and beyond.
Prof. Ju Li and Dr. Shige Ogata deserve thanks for providing the first-principles
calculations discussed in this thesis but more importantly for their technical feedback and helpful
suggestions throughout the writing and analysis of this work.
The other members of my research group have taught me much of what I know
concerning the practice of computational materials science and have also supported me
throughout this work by listening to me rant about bugs in my code and lending a hand when
they could. In particular I would like to acknowledge Ting Zhu for his consistent willingness to
answer my questions or discuss results. Elton Chang also deserves a nod for getting me started in
my studies of deformation twinning and providing much of the background to my work.
My friends have kept me afloat through the last few years, and for this I am grateful. In
particular I would like to thank Andy for putting up with me all year in close quarters, and
Megan and Amanda for reading thesis drafts and listening to presentations before they were
bearable.
Lastly, without my family I would not be at MIT, and I would never be the person that I
am becoming. The have instilled so much in me over the years and thankfully continue to do so.
Contents
1 Introduction
12
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Background
16
2.1 Empirical Potentials: The Embedded Atom Method . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Deformation Twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Ideal Shear Strength of FCC Al and Cu
26
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Shear Stress-Displacement Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Local Atomistic Relaxation During Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Multiplane Generalized Stacking Fault Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Deformation Twinning in FCC Metals
48
4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.1 Quasi 2-D Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.2 1-D Chain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Mechanism for Twinning in Bulk Aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Mechanism for Twinning in Bulk Copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68
5 Conclusions
70
5.1 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
A Cohesive Energy for EAM Potentials
76
Bibliography
78
List of Figures
2-1 Schematic of a twinned lattice described by reflection across a mirror plane, which can be
seen by the reversal of the stacking sequence across the twin boundary.
2-2 Illustration of lattice deformation mechanisms. The undeformed lattice is given for reference
in (a) compared to elastic deformation via homogeneous shear strain (b), and two plastic
deformation mechanisms, dislocation slip (c), and deformation twinning (d).
2-3 Schematic of the {111} plane in the fcc structure illustrating the primary slip system
indicated by the Burger’s vector, b, the twinning direction and partial Burger’s vector,
labeled bp, and the stacking sequence in this lattice where successive {111} planes are
centered over C, B, and A.
3-1 Six-atom unit cell where the x, y, and z unit vectors are ao[11 2 ]/2, ao[ 1 10]/2, and ao[111],
respectively, and the atomic positions given as fraction of the unit vectors are (0, 0, 0), (1/2,
1
/2, 0), (1/3, 0, 1/3), (5/6, 1/2, 1/3), (2/3, 0, 2/3), and (1/6, 1/2, 2/3).
3-2 {111} <11 2 > simple (closed symbols) and pure (open symbols) shear stress-displacement
response for copper using the Mishin copper potential. Published DFT curves [Ogata 2002]
are shown for comparison.
3-3 {111} <11 2 > simple (closed symbols) and pure (open symbols) shear stress-displacement
response for aluminum using (a) the Mishin aluminum potential and (b) the Ercolessi
potential. Published DFT curves [Ogata 2002] are presented for comparison in both cases.
3-4 Schematic of the local atomic mechanisms for accommodating shear displacement in copper
(a) and aluminum (b). Black arrows indicate behavior observed with both DFT and the
Mishin copper or Mishin aluminum potential, respectively, while the white arrows indicate
relaxation predicted only by the Mishin aluminum potential. The z direction out of the
plane of the paper is the [111].
3-5 Atomic relaxation patterns presented as the change in interplanar distance as a function of
displacement during pure {111} <11 2 > shear of copper. Calculations performed with the
Mishin copper potential (closed symbols). DFT curves (open symbols) are presented for
comparison [Ogata 2003].
3-6 Atomic relaxation patterns presented as the change in interplanar distance as a function of
displacement during pure {111} <11 2 > shear of aluminum. Calculations performed with
(a) the Mishin aluminum potential and (b) the Ercolessi potential (closed symbols). DFT
curves (open symbols) are presented for comparison in both cases [Ogata 2003].
3-7 Schematic illustrating examples of the three regimes of the multiplane generalized stacking
fault energy: dislocation slip, n = 1, deformation twinning, n = 2, 3, 4..., and affine
shear, n = ∞.
3-8 Unrelaxed generalized stacking fault energy for copper {111}<11 2 > slip calculated with the
Mishin copper (closed symbols) and DFT (open symbols) [Ogata 2002]. The unrelaxed
affine shear stress-displacement response previously discussed is shown for comparison.
3-9 The multiplane generalized stacking fault energy calculated with the Mishin copper
potential. In this figure the plot of the n = 15 case is essentially overlaid on the n = ∞ case.
3-10 Unrelaxed generalized stacking fault energy for aluminum {111}<11 2 > slip calculated
with the Mishin aluminum (closed symbols) and DFT (open symbols) [Ogata 2002]. The
unrelaxed affine shear stress-displacement response previously discussed is shown for
comparison.
3-11 The multiplane generalized stacking fault energy for aluminum calculated with the Mishin
aluminum potential. Aluminum shows significant asymmetry for the n = 1 case and has not
converged to the affine case even at n = 15.
4-1 Undeformed supercell for “quasi-two-dimensional“ molecular dynamics simulations
containing 6912 atoms.
4-2 Diagram of the simplified energetic model used to describe deformation of an fcc lattice by
displacement in the <11 2 > direction. Each atom represents a {111} plane constrained to
move as a rigid unit.
4-3 Energy versus shear displacement for strain controlled molecular dynamics simulation of
aluminum under {111}<11 2 > shear.
4-4 A series of snapshots from the molecular dynamics simulation of {111}<11 2 > shear
deformation in aluminum depicting homogeneously nucleated partial dislocations gliding
past one another to form a twinned structure. Atoms are colored based on the symmetry of
their local environment.
4-5 An analysis of one-layer slip calculated with the 1-D chain model. This analysis corresponds
to the generalized stacking fault energy for aluminum calculated with the Mishin aluminum
potential.
4-6 Excess energy surface for general [11 2 ] displacement on two adjacent (111) planes in
aluminum. The energy minimum at α is the intrinsic stacking fault energy for fcc aluminum,
β is the energy minimum for a two-layer microtwin, and δ is the barrier for displacement in
the anti-twinning direction on two adjacent planes.
4-7 The minimum energy path for shear deformation of aluminum overlaid on the energy
contour for generic displacement of two adjacent (111) planes in the [11 2 ] direction
calculated with the simplified energy model.
4-8 Energy versus number of slipped partial dislocations determined using a simplified energetic
model.
4-9 Schematic illustrating the imposed deformation for the multiple layer growth analysis.
4-10 Representative snapshots from the “quasi-two-dimensional” molecular dynamics simulation
of {111}<11 2 > shear deformation in copper. Atoms are colored based on the symmetry of
their local environment.
4-11 System energy versus shear displacement for {111}<11 2 > shear deformation in copper via
molecular dynamics simulation.
4-12 An analysis of one-layer slip calculated with the 1-D chain model. This analysis
corresponds to the generalized stacking fault energy for copper calculated with the Mishin
copper potential.
4-13 Excess energy surface for general [11 2 ] displacement on two adjacent (111) planes in
copper. The energy minimum at α is the intrinsic stacking fault energy for fcc aluminum, β
is the energy minimum for a two-layer microtwin, and δ is the barrier for displacement in
the anti-twinning direction on two adjacent planes.
4-14 The minimum energy path for shear deformation in copper determined by the simplified
energy model overlaid on the energy contour for generic displacement of two adjacent (111)
planes in the [11 2 ] direction.
4-15 Energy versus number of slipped partial dislocations determined using a simplified
energetic model.
A-1 Cohesive energy as a function of lattice parameter for the Mishin copper (a), Mishin
aluminum (b), and Ercolessi aluminum (c) potentials.
List of Tables
3-1 Material properties determined experimentally and with the EAM potentials used in the
current work. The table includes equilibrium lattice constants, ao, cohesive energies, Ec, bulk
moduli, B, elastic constants, Cij, vacancy formation energies, Efvac, stacking fault energies,
ESFE, and phonon frequencies, ν.
3-2 Ideal pure shear and ideal simple shear strengths (σ r and σ u respectively) for aluminum and
copper calculated with empirical potentials and DFT [Ogata 2002].
Chapter 1
Introduction
1.1 Motivation
Ab initio electronic structure calculations such as those provided by density functional
theory (DFT) are powerful tools for the study of mechanical behavior of materials because of
their ability to accurately model the process of bonds breaking and reforming [Kioussis 2002].
The strength of these methods is derived from tracking valence electron interactions, which
requires significant computational work. The length and timescales currently accessible to ab
initio methods are limited by the computationally intensive nature of the calculations. Empirical
potentials allow for faster calculation by following only atomic interactions and ignoring
electronic degrees of freedom. As a result larger and more complex systems can be treated with
certain reduction in accuracy. By fitting to a wide array of ab initio data as well as
experimentally derived properties recent potentials are able to capture increasingly more
complex behavior relating to the nature of atomic bonding and its consequences.
Ab initio calculations are ideally suited to studying the properties of homogeneous
systems, especially when the effects of atomic bonding are of interest. The ideal shear strength,
which has important implications for understanding the deformation and failure of materials, is
such a property. Ab initio results are also useful for benchmarking empirical potentials which
then can be used for deformation problems that require larger systems, such as those containing
extended defects. The present work is an attempt to determine where empirical potentials are a
suitable replacement for first principle methods by comparing calculations using recently
developed embedded atom potentials with DFT results already in the literature.
Specifically, the study of homogeneous deformation twin nucleation in face-centered
cubic (fcc) metals will be addressed with copper and aluminum as test cases. Twinning has
recently been shown to be an important deformation mechanism in nanocrystalline fcc metals
[Yamakov 2002, Chen 2003]. In addition, twinning has long been observed in high strain rate
deformation of these metals [Blewitt 1957, Meyers 2001]. Heterogeneous nucleation of twinned
structures has been shown computationally to occur at crack tips [Farkas 2001, Hai 2003] and
grain boundaries [Yamakov 2002].
In body-centered cubic (bcc) metals, homogeneous
deformation twin nucleation has been observed with atomistic simulations and described in terms
of energetic models [Chang 2003]. The aim of this work is to carefully consider the viability of
homogeneous deformation twin nucleation in fcc metals, following a recent study of bcc
molybdenum by Chang, in the context of low temperature, high strain rate, and an orientation
favorable for twin formation.
1.2 Problem Statement
The deformation responses of fcc aluminum and copper under {111}<11 2 > shear
loading are studied using empirical interatomic potentials developed by several authors [Mishin
2001, Ercolessi 1994, Mishin 1999]. First, the ideal shear stress-displacement curves and
multiplane generalized stacking fault energy of the materials are calculated to determine
differences in both the bond localization and atomic relaxation patterns in these two metals.
Molecular dynamics simulations and simple energetic models are then used to explore the
mechanism for twin formation in defect-free fcc lattices.
The ideal shear strength, the upper limit to material strength under shear [Roundy 1999],
and the shear strain at this maximum stress are obtained from {111}<11 2 > shear stressdisplacement curves calculated using empirical potentials. The multiplane generalized stacking
fault energy in this shear system is calculated to capture the energy cost for shearing adjacent
atomic planes. This quantity is normalized by the number of pairs of adjacent planes, n, being
sheared, and is analyzed here in terms of three distinct regimes, affine shear strain, n = ∞, single
plane slip corresponding to partial dislocation motion, n = 1, and multiple plane slip associated
with deformation twinning, n = 2, 3, 4, etc. The shear-displacement response and both the n = ∞
and n = 1 case of the multiplane generalized stacking fault energy are directly compared with
published DFT calculations [Ogata 2002] to show the extent to which the empirical potentials
can capture the essential feature brought out by first-principles calculation. The energy of the
multiplane case, which is relatively difficult to determine with DFT, is calculated using the
empirical potentials. Examining the generalized stacking fault energy for these two systems
illustrates the relative degree of bond localization in aluminum and copper. In addition the local
atomic relaxation patterns obtained with empirical potentials are determined and related to the
patterns given by DFT.
Quasi-two-dimensional molecular dynamics simulation of high strain rate, low
temperature shear applied in the <11 2 > direction on the {111} plane to both aluminum and
copper is used to probe the feasibility of homogeneous deformation twinning in fcc metals.
Using a simplified model for energetic calculations, the energy barriers to deformation twinning
are calculated. In this way a minimum energy path for strain-controlled deformation in these two
metals is determined and informs the analysis of the molecular dynamic simulations. In addition,
the energy required for growth of the twinned structure after nucleation is investigated using the
same energetic model.
Chapter 2
Background
2.1 Empirical Potentials: The Embedded Atom Method
The current work will primarily employ empirical potentials described by the Embedded
Atom Method (EAM). The EAM was originally developed by Daw and Baskes [1983] as an
improvement to the standard pair potential. With typical pair potentials the total energy of the
system is the sum of the energy between each pair of atoms:
E=
1
∑V (rij )
2 i ,≠ j
(2.1)
where the contribution of each atomic pair is given by a function, V(rij), where rij is the distance
between atoms i and j [Lennard-Jones 1924a, Lennard-Jones 1924b]. The function V(rij) is
determined by fitting experimental data for the material of interest, such as the equilibrium
lattice constant and cohesive energy, to standard mathematical forms. With a pair potential, all
atomic bonds are considered completely independent of one another. This is a particularly poor
assumption for metals where bonding is often described by a sea of valence electrons delocalized
from their associated ions [Ashcroft 1976].
The pair potential’s description of completely independent atomic bonds ignores two
major contributors to the nature of atomic bonding, namely local environment and many-body
effects. For example, cohesive energy calculated with a pair potential scales linearly with
coordination number, Z, since the energy of each atomic bond is determined solely by the
separation between the atom pair. In reality the energy contributed by an additional atomic bond
can be shown to scale approximately as -Z-1/2 [Heine 1990, Daw 1993]. Another illustration of
local environment affecting the nature of bonding is the relaxation of a metallic surface.
Experimentally, interplanar spacings near a surface are shown to be contracted relative to the
bulk, which can be related to an increase in bond strength due to lower coordination. A pair
potential cannot capture this environment dependent behavior and the surface contraction is not
observed [Daw 1993].
Just as pair potentials calculate energy based on the interactions between pairs of atoms
potentials can be constructed that include interactions between clusters of three, four, or more
atoms. These many-body interactions are necessary to observe symmetry and angle-dependent
properties. The simplest example of a property dependent on many-body effects is the open
crystal structure. Without some many-body term, all structures become close-packed. It is
therefore impossible to create a pair potential that produces, for example, a relaxed bcc structure.
To address these problems, EAM potentials contain an additional term to account for the
local electron density surrounding each atom. The term “Embedded Atom Method” is actually
derived from the picture of an atom “embedded” in this cloud of electrons. These potentials take
the following form:

 1
E = ∑ Gi  ∑ ρ j (rij ) + ∑ V (rij )
i
 j ≠i
 2 i ,≠ j
(2.2)
where Gi is the embedding function, ρj is the electron density calculated as a sum of
contributions from neighboring atoms, and V(rij) is the energy of two-atom interactions,
essentially a pair potential. With this general form in place, a wide range of fitting schemes have
been employed to develop physically meaningful potentials for a variety of metallic systems. In
general, one or both of the terms in a potential are fit to empirical data such as lattice constants,
cohesive energy, or elastic constants.
A recent advance in potential development is the use of ab initio data as an additional
input parameter for EAM potentials with the goal of obtaining accurate information about
configurations far from equilibrium. These regions of configurations space are often difficult to
probe experimentally and have not traditionally been included in the fitting parameters for EAM
potentials. The three potentials used in this work were developed using both first principles and
experimental information. The first potential employed a force-matching scheme, in which a
first principles method [Sankey 1989] was used to generate a database of atomic forces and their
consequent atomic trajectories for a range of structures representing bulk and defect
configurations for aluminum. A potential with the form of Eq. 2.2 was then fit to match as
accurately as possible the first principles trajectories. Experimental parameters such as the
equilibrium lattice constant, the intrinsic stacking fault energy, and the elastic constants were
used as additional constraints in the fitting. Ercolessi and Adams developed this force-matching
scheme and used it to produce a potential for aluminum [Ercolessi 1994] that will be referred to
hereafter simply as the Ercolessi potential. The present study employs an additional potential for
aluminum that was developed by Mishin et al utilizing a database of ab initio structural energies
instead of the atomic trajectories used in the force-matching scheme [Mishin 1999]. In addition,
Mishin et al introduced an algorithm for rescaling ab initio data to account for some of the
systematic errors produced by ab initio methods. The goal of the rescaling was to improve the
compatibility of parameters calculated with ab initio methods and those determined
experimentally prior to fitting a potential with the two data sets. The potential used to model
copper throughout the current study was developed using the same technique as the aluminum
potential determined by Mishin et al. However, structures were added to the database of
calculated energies in order to more carefully capture the repulsive range of atomic interactions
[Mishin 2001]. These two potentials will be referred to as the Mishin aluminum and Mishin
copper potentials, respectively.
2.2 Deformation Twinning
Deformation twinning is an important mode of plastic deformation for many materials,
especially metals with body-centered cubic, hexagonal close-packed, and other lower symmetry
structures [Murr 1997, Chichili 1998].
A wider range of materials including fcc metals,
intermetallic compounds, metal alloys, semiconductors, and even complicated mineral structures
also exhibited deformation twinning under specific conditions [Blewitt 1957, Suzuki 1958,
Haasen 1958, Paxton 1985, Christian 1987, Pirouz 1987, Christian 1988, Androussi 1989, Huang
1996, El-Danaf 1999, Liao 2003, Meyers 2003]. Deformation at low temperature, high-strain
rates and nanocrystalline grain structures promote twinning in fcc metals [Blewitt 1957, Meyers
2003, Chen 2003].
The theory and experimental work concerning deformation twinning
presented here has been reviewed by a number of authors [Reed-Hill 1963, Mahajan 1973, Gray
1990, Christian 1995]
Formally, a twin refers to a region within a lattice that can be described as either a
reflection across the boundary between the parent lattice and the twin or a rotation of 180o about
a specific axis. Often both the reflection and rotation descriptions are simultaneously accurate,
and in either case the bulk structure of the twinned region is equivalent to that of the original
lattice. The twin boundary is therefore critical in describing the energetics of this defect. The
shaded region in Figure 2-1 is a schematic of a twinned structure, which exhibits a reversal of
stacking order that can be described by a mirror plane at the twin boundary.
A
C
B
A
B
C
A
C
B
A
Mirror plane/Twin Boundary
Figure 2-1: Schematic of a twinned lattice described by reflection across a mirror plane, which
can be seen by the reversal of the stacking sequence across the twin boundary.
Both dislocation slip and deformation twinning are illustrated in Figure 2-2. These are
the two primary deformation mechanisms exhibited by a crystal lattice to accommodate large
strains. In Fig. 2-2, each sheet represents a crystallographic plane of atoms. The picture of an
undeformed lattice is given in Fig. 2-2 (a). The elastic regime for shear deformation can be
described by homogeneous shear strain where each plane is displaced relative to the plane below
it by some common distance (Fig. 2-2 (b)). Dislocation slip is illustrated in Fig. 2-2 (c) where a
single pair of planes is displaced relative to one another by a full lattice spacing in the shear
direction and thereby accommodates strain for the entire crystal lattice.
Fig. 2-2 (d)
demonstrates deformation twinning where many adjacent pairs of planes are displaced relative to
one another. This creates the reorientation of the original lattice that can be described by the
reflection across a twin boundary as shown in Fig. 2-1.
a)
b)
c)
d)
Figure 2-2: Illustration of lattice deformation mechanisms. The undeformed lattice is given for
reference in (a) compared to elastic deformation via homogeneous shear strain (b), and two
plastic deformation mechanisms, dislocation slip (c), and deformation twinning (d).
The formal constraints of deformation twinning as a mechanism for plastic deformation
can be exploited to determine the primary twinning systems for a given structure. Figure 2-3 is a
schematic of a {111} plane in the fcc structure where a full Burger’s vector, b, is found along the
<110> directions. The full Burger’s vector can be dissociated in to partial Burger’s vectors, bp, in
the <112> directions. In addition the three positions labeled A, B, and C indicate the stacking
sequence for the {111} close-packed planes in the fcc structure. The favorable twinning system
for the fcc materials that are the focus of this work is the {111}<11 2 > with each atom being
displaced, in a fully formed twin, by the partial Burger’s vector ao[11 2 ]/6 on the {111} plane.
This displacement corresponds to an intrinsic stacking fault in the fcc lattice. However, when a
succession of adjacent planes are displaced relative to each other by a partial Burger’s vector
they form a fcc lattice with a reversed stacking sequence from the parent lattice, which is a
twinned structured.
bp
bp
b
A
B
C
Figure 2-3: Schematic of the {111} plane in the fcc structure illustrating the primary slip system
indicated by the Burger’s vector, b, the twinning direction and partial Burger’s vector, labeled bp,
and the stacking sequence in this lattice where successive {111} planes are centered over C, B,
and A.
In general, the stress required to form deformation twins in fcc metals is larger than the
stress needed for dislocation slip [Huang 1996]. In addition, the contribution of deformation
twinning to the overall strain in these materials has been shown to be sensitive to both strain rate
and temperature. At high strain rates and low temperatures deformation twinning becomes an
increasingly favored method of relieving large shear strain in the lattice.
A brief discussion of representative experimental work exhibiting deformation twinning
in fcc metals is presented with the goal of providing a context for physically relevant modes of
twinning in these systems. Each example illustrates one of the specific conditions favoring
deformation twinning, low temperature, high strain rate, and small grain size.
One of the earliest experimental evidences of twinning in a fcc metal was provided by
Blewitt, Coltman, and Redman in their low temperature study of deformation in copper [1957].
Single crystal specimens were pulled in tension at 4.2 K and 77.3 K. At both temperatures
twinning was observed after extensive plastic deformation via slip. This was in contrast to
earlier experiments at higher temperatures, which exhibited no twinning behavior. A stronger
orientation dependence for twin formation was observed at 77.3 K, which further indicates that
low temperatures in general favor twinning relative to dislocation slip.
Meyers et al have shown deformation twin formation as a result of “ultra-short shock
pulses” in copper [2003]. Laser-induced shock pulses of about 5 nanoseconds were generated in
single-crystal copper and in situ x-ray diffraction techniques were used to measure the strain rate
and local stress at the shock front. Twinning was observed in shocked specimens subjected to a
pulse of 40 GPa or more. The maximum strain rates in this study were determined to be on the
order of 1 × 107 s-1 with expectation that the technique could be used to reach strain rates as high
as 1 × 109 s-1 [Meyers 2001].
Nanocrystalline aluminum films deformed by both microindentation and mechanical
grinding with a mortar and pestle have been shown to exhibit deformation twinning. [Chen 2003].
A critical dependence on grain size is observed with no twinning seen in samples with grain sizes
larger than 40 nm. The occurrence of twinning under the action of only a mortar and pestle
indicates that extremely high stresses are not necessary for twin formation in nanocrystalline
materials.
The work performed by Chen et al has validated recent computer simulations of twin
formation in nanocrystalline aluminum [Yamakov 2002a, Yamakov 2002b Bilde-Sørensen 2003].
Large-scale molecular dynamics simulations of nanocrystalline columnar grains have predicted a
wide array of complex dislocation processes in aluminum with small grain size. Included in these
dislocation processes is the formation of twinned structures from the successive emission of
partial dislocations from grain boundaries onto adjacent planes.
Deformation twins have been observed experimentally at the tips of cracks in thin foil
specimens of copper [Chen 1999] and at crack tips on the edge of TEM specimens in aluminum
[Pond 1981]. The mechanism for twin formation under the influence of an atomically sharp
crack in aluminum has been studied computationally [Farkas 1999, Tadmor 2003]. The later
example uses a quasi-continuum approach coupling continuum bulk behavior to atomistic
modeling near the crack tip to study the effect of crack tip morphology, loading mode, and
crystallographic orientation on deformation mechanisms in aluminum. These simulations found
good agreement with experimental results. Deformation twin formation by emission of
successive partial dislocations on adjacent {111} planes was exhibited for combinations of
crystallographic orientation and loading condition where the critical resolved shear stress lay
along the direction of the partial Burger’s vector in the fcc structure.
The computational work presented so far for fcc metals exhibits heterogeneous
nucleation via partial dislocations emitted from cracks and grain boundaries. Homogeneous
nucleation has been explored in bcc metals with molecular dynamics simulation [Chang 2003].
In this system nucleation of a two-layer microtwin was observed upon shear loading in the [111]
direction on the ( 11 2) plane, which is the twinning system for the bcc structure. Homogeneous
deformation twin nucleation in fcc metals would likely require highly favorable conditions in
terms of strain rate, temperature, and shear deformation direction.
Chapter 3
Ideal Shear Strength of FCC Al and Cu
3.1 Introduction
Ideal shear stress-displacement behavior is fundamental to understanding the onset of
plasticity and eventual fracture in a material. Materials typically exhibit regions of linear and
non-linear elastic behavior before yielding and entering a regime of plastic deformation. The
maximum shear stress achievable is the ideal shear strength, which can be viewed as an upper
limit to the lattice’s resistance to mechanical stress. The ideal shear strength marks the breaking
of bonds and the onset of plasticity. The current work calculates the ideal stress-displacement
behavior for two fcc materials, copper and aluminum, with the goal of probing the local atomic
relaxation patterns and degree of bond localization in these two materials using empirical
potentials. Insight into the nature of atomic bonding in terms of directionality and bond
localization can inform the study of local atomic relaxation patterns and the deformation
mechanisms favored by a material [Ogata 2002]. These fundamental properties and behaviors
are, in general, accessible using highly accurate ab initio techniques such as density functional
theory (DFT) [Hohenberg 1964, Kohn 1965]. However, the limitations of theses methods in
terms of length and time scales limit the general use of ab initio methods for atomistic
simulation. Finding less computationally intensive methods, such as empirical potentials, capable
of capturing the physical behavior predicted by the more rigorous ab initio techniques would
allow for longer simulations on larger systems without losing the essential physics of the
mechanisms of interest. This work aims to determine how well empirical potentials can account
for the essential behavior of shear deformation revealed by published DFT results.
3.2 Shear Stress-Displacement Response
Stress-displacement curves for both pure affine shear (σ=0 except σ13) and simple affine
shear (finite shear displacement with no relaxation) were calculated for copper and aluminum
using the Embedded Atom Method (EAM) type empirical potentials discussed previously
[Mishin 2001, Ercolessi 1994, Mishin 1999].
Material properties and potential parameters
including cohesive energies, potential cutoffs, and equilibrium lattice parameters for each of
these potentials are listed in Table 3-1. For all calculations the initial fcc supercell was built with
the x, y, and z-axes along <11 2 >, < 1 10>, and <111> directions, respectively, using the
repeatable six atom unit cell shown in Figure 3-1 and the appropriate equilibrium lattice constant
for the potential. In each case the value of the equilibrium lattice parameter for the potential was
verified by plotting the cohesive energy as a function of lattice parameter. These curves are
included in Appendix A.
For the pure and simple affine shear cases, periodic boundary
conditions are employed to model an infinite homogeneous bulk. The conjugate gradient method
[Press 1996] was employed to relax the stress components other than the imposed σ13 by
minimizing the energy with respect to the supercell shape and dimensions.
C
B
<112>
A
<110>
<111>
Figure 3-1: Six-atom unit cell where the x, y, and z unit vectors are ao[11 2 ]/2, ao[ 1 10]/2, and
ao[111], respectively, and the atomic positions given as fraction of the unit vectors are (0, 0, 0),
(1/2, 1/2, 0), (1/3, 0, 1/3), (5/6, 1/2, 1/3), (2/3, 0, 2/3), and (1/6, 1/2, 2/3).
Table 3-1: Material properties determined experimentally and with the EAM potentials used in
the current work. The table includes equilibrium lattice constants, ao, cohesive energies, Ec, bulk
moduli, B, elastic constants, Cij, vacancy formation energies, Efvac, stacking fault energies, ESFE,
and phonon frequencies, ν.
Aluminum
Experimental1
Mishin
1
2
Ercolessi
Copper
Experimental3 Mishin3
Lattice properties
ao (angtroms)
4.05
4.05
4.032*
3.615
3.615
Ec (eV/atom)
-3.36
-3.36
-3.36
3.54
3.54
11
0.79
0.79
0.809*
1.383
1.383
B (10 Pa)
C11 (1011 Pa)
1.14
1.14
1.181*
1.7
1.699
11
0.616
0.616
0.623*
1.225
1.226
11
C44 (10 Pa)
0.316
0.316
0.367*
0.758
0.762
Efvac
0.68
0.68
0.69*
1.27
1.272
ESFE (mJ/m )
166, 120-144
146
104
45
44.4
νL(X) (THz)
9.69
9.31
9.29
7.38
7.82
νT(X) (THz)
νL(L) (THz)
5.8
5.98
5.8
5.16
5.2
9.69
9.64
9.51
7.44
7.78
νT(L) (THz)
4.19
4.3
4.02
3.41
3.32
νL(K) (THz)
νT1(K) (THz)
νT2(K) (THz)
7.59
7.3
8.38*
5.9
6.22
5.64
8.65
5.42
8.28
7.5*
5.34*
4.6
6.7
4.65
7.17
C12 (10 Pa)
(eV)
2
Ref: 1 - [Msihin 1999], 2 – [Ercolessi 1994], 3 – [Mishin 2001]
- * indicates data fit to different experimental values than shown here
The {111}<11 2 > shear stress-displacement response for copper (both pure and simple
shear), calculated with the Mishin copper potential, is shown in Figure 3-2. The calculated
displacements are normalized by the partial Burgers vector, bp = ao[11 2 ]/6, where ao is the
equilibrium lattice parameter for the potential in use. The normalization allows for direct
comparison of the extent of deformation both between copper and aluminum and between the
two aluminum potentials, which have slightly different equilibrium lattice constants. The stress
as calculated by the empirical potential is systematically higher than the DFT curves [Ogata
2002] for both pure and simple shear in copper. The ideal shear strength of copper for the relaxed
case calculated with the Mishin copper potential is 15% higher than the value calculated by DFT.
In addition the extent of deformation at the maximum stress given in terms of displacement is
significantly over-predicted by the empirical potential (x/bp = 0.28 with the Mishin copper
potential versus 0.19 with DFT) (Table 3-2).
4.5
4.0
Stress(Gpa)
3.5
unrelaxed Mishin
relaxed Mishin
unrelaxed DFT (VASP)
relaxed DFT (VASP)
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0
0.1
0.2
0.3
0.4
0.5
x/bp
Figure 3-2: {111} <11 2 > simple (closed symbols) and pure (open symbols) shear stressdisplacement response for copper using the Mishin copper potential. Published DFT curves
[Ogata 2002] are shown for comparison.
4.0
3.5
Stress(Gpa)
3.0
2.5
unrelaxed Mishin
relaxed Mishin
unrelaxed DFT
relaxed DFT
2.0
1.5
1.0
0.5
0.0
-0.5
0
0.1
0.2
0.3
0.4
0.5
x/bp
(a)
4.0
3.5
Stress(Gpa)
3.0
2.5
unrelaxed Ercolessi
relaxed Ercolessi
unrelaxed DFT
relaxed DFT
2.0
1.5
1.0
0.5
0.0
-0.5
0
0.1
0.2
0.3
0.4
0.5
x/bp
(b)
Figure 3-3: {111} <11 2 > simple (closed symbols) and pure (open symbols) shear stressdisplacement response for aluminum using (a) the Mishin aluminum potential and (b) the
Ercolessi potential. Published DFT curves [Ogata 2002] are presented for comparison in both
cases.
The {111}<11 2 > shear stress-displacement curves for aluminum calculated with both
the Ercolessi and Mishin aluminum potentials are shown in Figure 3-3. Both potentials give
simple shear stress values systematically less than DFT results, although the maximum simple
shear stress given by the Mishin aluminum potential is only 3.6% lower than the DFT versus a
value 41% lower given by the Ercolessi potential. The relaxed ideal shear strength given by the
Mishin potential is 10% higher than DFT values while the Ercolessi potential produced a relaxed
ideal shear strength 32% lower than DFT calculation. The extent of deformation at the ideal
shear strength for each is x/bp = 0.21 and 0.33 for the Mishin and Ercolessi potentials
respectively.
Table 3-2: Ideal pure shear and ideal simple shear strengths (σ r and σ u respectively) for
aluminum and copper calculated with empirical potentials and DFT [Ogata 2002].
potential/method
Al
r
u
σ (GPa) σ (GPa)
Cu
r
u
σ (GPa) σ (GPa)
Mishin
Ercolessi
3.12
1.91
3.60
2.19
2.91
3.92
DFT
2.84
3.73
2.16
3.42
The ideal shear strength calculated by each potential as well as published values
calculated with Density Functional Theory (DFT) [Ogata 2002] are presented in Table 3-2.
In comparing the two fcc metals, DFT calculations show aluminum to have a larger range of
non-linear elastic deformation prior to reaching the ideal strength (xmax/bp=0.28 in Al versus 0.19
in Cu for the pure shear case). DFT calculations of the electron density in these two metals
indicate that copper exhibits essentially spherical charge densities while the charge density in
aluminum is localized which causes directional bonding. However, the form of the EAM
potentials is incapable of containing information about bond directionality. Ogata and Li have
attributed the extended range of non-linear elastic deformation and consequently higher ideal
shear strength in aluminum versus copper that is observed with DFT to aluminum’s directional
bonding. Without the effects of aluminum’s directional bonding, the correct ordering of the
extent of elastic deformation between the two metals has not been observed with the EAM
potentials, although agreement between behavior predicted by DFT and empirical potentials for
each metal is reasonable.
3.3 Local Atomistic Relaxation During Shear
A key advantage of atomistic simulation is the ability to track, in situ, local atomic
behavior. In this study of pure {111}<11 2 > shear stress-displacement, this capacity allows for
careful observation of the mechanisms by which the fcc lattice accommodates shear
displacement. These mechanisms can be directly related to the nature of atomic bonding in
aluminum and copper [Ogata 2002] and it is of interest to determine the extent to which
empirical potentials can capture this detailed behavior.
In the pure shear stress-displacement calculations described in the previous section,
supercell shape and dimensions were relaxed such that the components of stress other than the
imposed shear stress, σ13, were equal to zero. No relaxation of the other shear stress components
was necessary because the symmetry of the fcc structure. The x, y, and z-axes are set up along
the <11 2 >, < 1 10>, <111> respectively, and a change in length of these basis vectors normalized
by the number of planes along each direction yields the change in interplanar spacing for the
system. The observed relaxation can be directly related to the preferred local atomic mechanism
for accommodating shear displacement.
To illustrate consider two adjacent (111) planes in copper being sheared relative to one
another in the [11 2 ] direction. An atom on one (111) plane can be said to slide relative to the
(111) plane below it in the [11 2 ] direction. The scenario is presented schematically in Figure 34. In copper, the bottom plane contracts in the x-direction and expands in the y-direction while
the two planes do not move relative to each other in the z-direction. As the white atom in Fig 34(a) moves in the x direction relative to the gray plane of atoms below it the system minimizes
the energy associated with the shearing processes when the atoms directly in the path of the
upper layer move away from the path of shear and the new nearest neighbor atom comes to meet
the atom being sheared in it’s direction. In Fig. 3-4 the white atom is centered over the three
gray atoms on the left which corresponds to x/bp=0. The atom has moved a full partial Burger’s
vector x/bp=1 when it is centered above the three gray atoms on the right. This position
corresponds to an intrinsic stacking fault in the fcc lattice. Figure 3-5 shows quantitatively the
relaxation patterns observed using the Mishin copper potential by plotting interplanar spacing in
the x, y, and z directions as a function of displacement. The displacement is again normalized by
the partial Burger’s vector of the system. At x/bp = 0.5, the Mishin copper potential shows
approximately 7% expansion in the [ 1 10] and a 6% contraction in the [11 2 ] direction.
Calculations using the Mishin copper potential exhibit close agreement with DFT calculations of
atomic relaxation during {111}<11 2 > shear.
[110]
[112]
a) Cu
b) Al
Figure 3-4: Schematic of the local atomic mechanisms for accommodating shear displacement in
copper (a) and aluminum (b). Black arrows indicate behavior observed with both DFT and the
Mishin copper or Mishin aluminum potential, respectively, while the white arrows indicate
relaxation predicted only by the Mishin aluminum potential. The z direction out of the plane of
the paper is the [111].
1.08
1.06
1.04
Mishin <101>
Mishin <112>
Mishin <111>
DFT <101>
DFT <112>
DFT <111>
a/ao
1.02
1
0.98
0.96
0.94
0.92
0
0.1
0.2
0.3
0.4
0.5
x/bp
Figure 3-5: Atomic relaxation patterns presented as the change in interplanar distance as a
function of displacement during pure {111} <11 2 > shear of copper. Calculations performed
with the Mishin copper potential (closed symbols). DFT curves (open symbols) are presented for
comparison [Ogata 2003].
Using the Mishin aluminum potential, two {111} planes sheared relative to one another
in the <11 2 > direction exhibit expansion in the z-direction while the lower plane expands in the
x-direction and contracts in the y-direction (Fig.3-4b). Figure 3-6 contains the relaxation patterns
given by both aluminum potentials and each is compared to DFT calculation. The Mishin
aluminum potential exhibits an 8% increase in the (111) interplanar distance versus a 6%
increase observed with DFT. Contraction in the y –direction again shows the correct trend and is
less than 1% higher than the DFT value. However, a 2% increase in the (11 2 ) interplanar
spacing seen with the Mishin aluminum potential is not observed with DFT.
The Mishin aluminum potential exhibits an artifact near the theoretical energy
peak which corresponds to σ13 = 0 at x/bp = 0.5 (Fig. 3-3a). The artifact is likely related to the
EAM potential’s inability to model directional bonding. Relaxation in the <11 2 > direction is
not seen with DFT but is observed with the Mishin aluminum potential (Fig. 3-6a). This
relaxation corresponds to a decrease in energy near x/bp = 0.5 that creates a depression in the
energy peak forming a local energy maximum at x/bp = 0.45 and a local energy minimum at x/bp
= 0.5. A direct correlation between expansion in the <11 2 > direction and the energy decrease
has been shown by constraining the relaxation in that direction. With no relaxation in the <11 2 >
direction, the energy depression and the corresponding local maximum at x/bp = 0.45 does not
occur.
The Ercolessi potential has a more serious discontinuity that can be traced to nonphysical atomic relaxation. The change in interplanar spacings calculated with the Ercolessi
potential is within 1% of DFT values for relaxation in all three directions near x/bp = 0.5.
However, the agreement near this displacement is brought upon by a discontinuous change in the
relaxation pattern corresponding to the stress discontinuity observed in Fig 3-3b. This behavior
implies that the potential was strongly fit to some parameter accounting for the configuration
near this energy saddle point. The contribution due to this fitting snaps into place as the
displacement approaches x/bp = 0.5, but the relaxation patterns (Figure3-6b) do not follow the
final trend until a displacement of approximately x/bp = 0.35 where the interplanar spacings
change abruptly. Up until this displacement, contraction in the x-direction and nominal
expansion in the y, and z-directions are observed. This overall relaxation behavior is neither
physically intuitive nor in good agreement with DFT results and should be regarded as an artifact
of the potential’s fitting scheme.
1.1
1.08
1.06
Mishin <101>
Mishin <112>
Mishin <111>
DFT <101>
DFT<112>
DFT<111>
a/ao
1.04
1.02
1
0.98
0.96
0.94
0
0.1
0.2
0.3
0.4
0.5
x/bp
a)
1.08
1.06
1.04
Ercolessi <101>
Ercolessi <112>
Ercolessi <111>
DFT <101>
DFT<112>
DFT<111>
a/ao
1.02
1
0.98
0.96
0.94
0
0.1
0.2
0.3
0.4
0.5
x/bp
b)
Figure 3-6: Atomic relaxation patterns presented as the change in interplanar distance as a
function of displacement during pure {111} <11 2 > shear of aluminum. Calculations performed
with (a) the Mishin aluminum potential and (b) the Ercolessi potential (closed symbols). DFT
curves (open symbols) are presented for comparison in both cases [Ogata 2003].
The goals of the current work are to determine the shear stress-displacement behavior
accessible to empirical potentials and to the study plastic deformation mechanisms under shear
conditions favorable to deformation twinning. The Ercolessi potential has been shown to
underestimate the ideal shear strength, as calculated by DFT, by 32%. Furthermore, a nonphysical relaxation pattern has been observed with the Ercolessi potential. Although the potential
seems well fit near the energy saddle point at x/bp = 0.5, the validity of the potential for the
majority of configurations beyond the ideal shear strength is in doubt. In light of these doubts
and to simplify the discussions to follow, the multiplane study in the next section as well as the
simulations in Chapter Four will be performed using only the Mishin aluminum and Mishin
copper potentials.
3.4 Multiplane Generalized Stacking Fault Energy
The generalized stacking fault energy has long been used to describe material
deformation in terms of the energy penalty at the partial Burger’s vector for shearing two
adjacent planes [Vitek 1968]. More recently the multiplane generalized stacking fault energy has
been introduced to describe the energy penalty incurred when an arbitrary number of planes, n +
1, are sheared relative to one another by a common displacement, x, (Figure 3-7). This quantity is
given by:
γ n ( x) =
E n ( x)
, n = 1, 2, ...
nS o
(3.1)
where En(x) is the total energy penalty compared to the energy at x = 0 and it is normalized in
these functions by both the cross sectional area at x = 0, So, and the number of pairs of adjacent
planes being sheared, n [Ogata 2002]. In this series of functions, the n = 1 case, γ1(x),
corresponds exactly to the conventional generalized stacking fault energy, and the affine shear
strain energy is given by γ∞(x).
n=1
[111]
n=4
n= ∞
[112]
Figure 3-7: Schematic illustrating examples of the three regimes of the multiplane generalized
stacking fault energy: dislocation slip, n = 1, deformation twinning, n = 2, 3, 4..., and affine
shear, n = ∞.
In Chapter Two deformation twins were described as regions within a lattice misoriented
relative to the parent lattice by some reflection or rotation about a common axis. Since the lattice
within the bulk of a fully formed twin is indistinguishable from the parent lattice, the additional
energy associated with a twinned region is the energy of the twin boundary. The expression
γ n ( x) = γ ∞ ( x) +
2γ twin ( x)
+ O (n-2)
n
(3.2)
where γtwin(x) is the unrelaxed twin boundary energy should hold for values of n large enough
that the twin boundary does not interact with itself through the bulk of the twinned region. In
this expression γ∞(x) = 0 for a fully formed twin, i.e. when x = bp, because the bulk of the
twinned region has reformed into a fcc lattice with the reverse stacking order as previously
described.
The generalized stacking fault energy for {111}<11 2 > slip, γ1(x), was calculated with
the Mishin copper potential and is presented in Figure 3-8. Both in Figure 3-8 and the rest of this
analysis the multiplane generalized stacking fault energy will be normalized by the shear
displacement and is therefore presented in terms of stress given by dγn(x)/dx. In addition the
shear displacement, x, is normalized by the partial Burger’s vector, bp=ao[11 2 ]/6, to provide
continuity with the previous sections and published DFT work [Ogata 2002]. The generalized
stacking fault energy produced by the empirical copper potential almost exactly matches the
values calculated with DFT. Also included in Figure 3-8 are the simple (unrelaxed) affine shear
stress-displacement curves calculated with both the Mishin copper potential and DFT, which
were discussed in the previous sections. The output from the Mishin copper potential for these
two cases (n=1 and n=∞) is qualitatively similar. Since these two curves represent the energy
penalty for shear displacement normalized by the number of planes being sheared, their
similarity indicates that the atoms in the pair of planes being sheared in the n=1 case experience
atomic interactions fairly similar to those experienced by a pair of planes sheared in the affine
case, n=∞. The implication is that the interaction range for atomic bonding in copper is on the
order of the nearest neighbor separation. Figure 3-9 shows the extension to the multiplane case,
which would be relatively difficult to calculate with ab initio techniques. This result strengthens
the claim of localized bonding in copper showing that the low additional penalty due to the
boundary between sheared and unsheared lattice is quickly divided among atomic planes. The
n=15 case is indistinguishable from normalized penalty for affine deformation.
5.0
4.0
Mishin
DFT n=1
Mishin n=1
3.0
dγn/dx(Gpa)
2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0
-5.0
0
0.2
0.4
0.6
0.8
1
x/bp
Figure 3-8: Unrelaxed generalized stacking fault energy for copper {111}<11 2 > slip calculated
with the Mishin copper (closed symbols) and DFT (open symbols) [Ogata 2002]. The unrelaxed
affine shear stress-displacement response previously discussed is shown for comparison.
5.0
4.0
n=infinity
n=1
n=2
n=3
n=15
3.0
dγn/dx(Gpa)
2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0
-5.0
0
0.2
0.4
0.6
0.8
1
x/bp
Figure 3-9: The multiplane generalized stacking fault energy calculated with the Mishin copper
potential. In this figure the plot of the n = 15 case is essentially overlaid on the n = ∞ case.
The generalized stacking fault energy for aluminum calculated with the Mishin aluminum
potential shows significant deviation from affine shear-displacement behavior (Figure 3-10). As
the displacement, x, approaches bp aluminum does not recover the energy penalty incurred
during shear deformation which results in a high intrinsic stacking fault energy relative to copper
(Table 3-1). Aluminum’s difficulty in recovering the energy associated with shear deformation
can be seen in the asymmetry of dγ1(x)/dx in Fig. 3-10. The energy maximum calculated with
the Mishin aluminum potential, for the n = 1 case occurs at x/bp = 0.70, which corresponds to
dγ1(x)/dx = 0. This can be contrasted to the behavior observed for copper (Fig. 3-8) where the
maximum energy penalty for shearing a single pair of planes , n = 1, is incurred at x/bp = 0.53.
Aluminum has been shown to exhibit anisotropic electron density, which is associated
with directional bonding [Feibelman 1990, Robertson 1993]. Because of the directional nature
of bonding its, charge redistribution associated with breaking and reforming bonds is more
difficult in aluminum than in copper [Kioussis 2002]. Bonding in copper can be described as
isotropic because of spherically symmetric charge density. Copper’s uniform charge density is
able to adapt more quickly to the changing local environment associated with shear deformation
and is less sensitive to the local structure (fcc versus hcp) than it is to coordination. Therefore,
when an intrinsic stacking fault is formed copper is able to recover most of the energy penalty to
shear [Ogata 2002].
4.0
Mishin
DFT n=1
Mishin n=1
3.0
dγn/dx(Gpa)
2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0
0
0.2
0.4
0.6
0.8
1
x/bp
Figure 3-10: Unrelaxed generalized stacking fault energy for aluminum {111}<11 2 > slip
calculated with the Mishin aluminum (closed symbols) and DFT (open symbols) [Ogata 2002].
The unrelaxed affine shear stress-displacement response previously discussed is shown for
comparison.
4.0
n=infinity
n=1
n=2
n=3
n=15
3.0
dγn/dx(Gpa)
2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0
0
0.2
0.4
0.6
0.8
1
x/bp
Figure 3-11: The multiplane generalized stacking fault energy for aluminum calculated with the
Mishin aluminum potential. Aluminum shows significant asymmetry for the n = 1 case and has
not converged to the affine case even at n = 15.
Figure 3-11 illustrates, with several cases of the multiplance generalized stacking fault
energy, the larger interaction range exhibited by aluminum relative to the behavior shown
previously for copper. Even for the n=15 case the behavior of the affine deformation has not
been recovered.
3.5 Discussion
The Mishin copper potential overestimates the ideal shear strength and the extent of
elastic deformation for affine shear stress-displacement calculations. However, it successfully
produces the qualitative behavior for shear deformation including the detailed relaxation
mechanism in affine shear and the generalized stacking fault energy. The form of the EAM is
ideally suited to modeling the behavior of metals with uniform charge densities such as copper.
The strong agreement between the Mishin copper potential and ab initio calculations, when
studying local atomic properties, is, therefore, not surprising.
The Mishin aluminum potential has been shown to reasonably exhibit a variety of
behavior for aluminum despite overlooking the directional nature of its bonding. The ideal shear
strength calculated with the Mishin aluminum potential shows close agreement to DFT values
although the extent of non-linear elastic deformation is underestimated. The relaxation pattern
described by this potential displays some of the behavior observed with DFT although the
accuracy is not as strong as in the copper case. The multiplane generalized stacking fault energy
of aluminum calculated with the Mishin potential also qualitatively captures the asymmetry
observed with DFT that is associated with difficulty in redistributing the localized charge density
in directional bonding.
The Ercolessi potential is consistently inferior to the Mishin aluminum potential. This
result is consistent with a similar study benchmarking empirical EAM potentials to DFT
calculations that used only the generalized stacking fault energy for comparison [Zimmerman
2000]. Zimmerman et al concluded that EAM potentials were not suited to modelling the
behavior of aluminum; however, the present work has shown that the Mishin potential does
capture much of the qualitative behavior observed with DFT.
When comparing the copper and aluminum, DFT shows aluminum to have a higher ideal
shear strength and a longer range of elastic deformation before the onset of plasticity. The
Mishin potentials for these two metals correctly predicts the order of ideal shear strength
between the metals but shows copper to have the longer range of elastic deformation. This is
again related to the EAM’s inability to model the directional nature of aluminum’s bonding
which has been correlated to aluminum’s long range of non-linear elastic behavior [Ogata 2002].
Chapter 4
Deformation Twinning in FCC Metals
4.1 Methods
The present work considers the viability of and potential mechanism for homogeneous
deformation twin nucleation in fcc metals. Aluminum and copper are interesting candidates for
this study because of the differences they exhibit in atomic bonding and intrinsic stacking fault
energy (ESFE = 45 mJ/m2 for copper versus approximately 145 mJ/m2 for aluminum) [Mishin
2001, Mishin 1999].
4.1.1 Quasi-2-D Molecular Dynamics
Quasi-two-dimensional molecular dynamics simulations of fcc aluminum and copper
sheared in the <11 2 > direction on the {111} plane were performed. The initial supercell was
constructed using the same basis set and six-atom repeatable cell as the static calculations
presented in Chapter Three with 16, 6, and 6 unit cells in the <11 2 >, < 1 10>, and <111>
directions, respectively. The undeformed supercell is shown in Figure 4-1. The simulations are
termed “quasi-two-dimensional” because the cell is relatively short in the <110> direction which
to some extent constrains the behavior of the system to the plane containing the <111> and
<11 2 > directions.
[111]
[110]
[112]
Figure 4-1: Undeformed supercell for “quasi-two-dimensional“ molecular dynamics simulations
containing 6912 atoms.
Periodic boundary conditions were applied during strain-controlled {111}<11 2 > shear
deformation. After an initial equilibration stage, the supercell was deformed at each timestep in
order to maintain a constant shear strain rate of 1 × 109 s-1. The system temperature was
maintained at 10K by velocity rescaling every ten timesteps. The goal of these constrained
simulations was to determine potential nucleation mechanisms for twin formation, and as such,
the simulations were designed to favor plastic deformation via twinning with low temperature,
high strain rate, and deformation in the twinning direction on the close-packed plane.
4.1.2 Energy Model
In order to elucidate further the behavior observed in the molecular dynamics simulations
performed for aluminum and copper, a simplified model [Chang 2003] has also been used to
drastically reduce the system’s degrees of freedom. In this model each {111} plane of atoms is
required to move as a single unit and only the displacement of the planes relative to one another
is tracked. Each plane can then be described by a chain of atoms each representing a single
atomic plane (in this case a{111} plane) and the relative displacement of adjacent planes can be
used to described the deformation of the system (Figure 4-2).
xn
[111]
.
.
.
[112]
xi
.
.
.
x2
x1
Figure 4-2: Diagram of the simplified energetic model used to describe deformation of an fcc
lattice by displacement in the <11 2 > direction. Each atom represents a {111} plane constrained
to move as a rigid unit.
For the fcc structure, the relative displacement in the <11 2 > direction on the
{111} plane is the most important parameter for describing deformation twinning. The
deformation in this model is given by a series of displacements in the [11 2 ] direction given by
∆x = (∆x1, ∆x2,
... ,∆xi, ...,
∆xn) where n is the number of planes in the system. With this
description, one-layer displacement describing partial dislocation slip is given by ∆x = (..... 0, 0,
∆xi, 0, 0, ....), where ∆xi is the finite displacement between two (111) planes while there is no
displacement between any other pair of planes. Fig. 4-2 illustrates a three-layer displacement
where three pairs of adjacent planes are sheared relative to one another. The displacement in the
three-layer case is given by ∆x = (.....0, 0, ∆xi, ∆xi+1, ∆xi+2, 0, 0,....). In Fig.4-2, ∆xi = ∆xi+1 =
∆xi+2, and each is the relative displacement illustrated by the white arrows.
The highly
constrained nature of this model has two major consequences. The significant decrease in
degrees of freedom (from 3 N, where N is the number of particles in the system to n, the number
of planes in the system) creates a tractable parameter space that allows for visualization of the
most important parameters describing shear deformation. However, the constrained system is
not able to relax and which comes at a loss of some physical relevance. For the current study,
the approximations inherent in this model are accepted since the goal of this analysis is to inform
molecular dynamics simulations, which will track all 3N degrees of freedom. Using both
methods, a clearer description of the plastic deformation mechanisms of interest is possible than
either technique could provide on its own.
4.2 Mechanism for Twinning in Bulk Aluminum
Molecular dynamics simulation of strain-controlled {111}<11 2 > shear deformation of
aluminum were performed as described in Section 4.1.1 using the Mishin aluminum potential.
Figure 4-3 shows the energy of the system as a function of shear displacement. Snapshots from
the molecular dynamic simulation where atoms are colored based on the symmetry of their local
environment are presented in Figure 4-4. The energy drop at x = 5.36 Å in Fig. 4-3 corresponds
to homogeneous nucleation of a Shockley partial dislocation dipoles and subsequent nucleation
of dislocations separated from the initial dislocation dipole by a single {111} plane (Fig 4-4a and
b). As these partial dislocations glide past one another they first form local hexagonal closepacked structure and upon further strain a twinned structure (Fig. 4-4c and d). Eventually,
because of the cell’s periodic boundary condition, the two halves of the dislocation dipole run
into each other and annihilate creating a fully twinned structure (Fig. 4-4d). This final step in the
simulation’s behavior where the dislocation dipoles annihilate themselves is an artifact of the
simulations boundary condition. However, the simulation indicates a partial dislocation
coalescence mechanism for formation of twinned structures in aluminum.
-3.69E-15
Esys (Joules)
-3.70E-15
-3.71E-15
-3.72E-15
5.26
5.46
5.66
5.86
6.06
x (angstroms)
Figure 4-3: Energy versus shear displacement for strain controlled molecular dynamics
simulation of aluminum under {111}<11 2 > shear.
(a)
(b)
(c)
(d)
(e)
Figure 4-4: A series of snapshots from the molecular dynamics simulation of {111}<11 2 > shear
deformation in aluminum depicting homogeneously nucleated partial dislocations gliding past
one another to form a twinned structure. Atoms are colored based on the symmetry of their local
environment.
For an even simpler analysis of the mechanism for twin formation in aluminum the
energetic model described in Section 4.1.2 was used to examine the minimum energy path for
nucleation of a two-layer twin. For context, the energy versus one layer displacement, ∆x = (.....
0, 0, 0, ∆xi, 0, 0, 0,....), is shown in Figure 4-5. The local energy minimum at ao[11 2 ]/6 is the
intrinsic stacking fault energy for the aluminum. This curve also demonstrates the twinning/antitwinning asymmetry of the fcc structure with the large barrier to reverse shear compared to
forward shear in the {111}<11 2 > system,
700
2
Excess Energy (mJ/m )
600
500
400
300
200
100
0
0
0.2
0.4
0.6
0.8
1
x/(ao[112]/2)
Figure 4-5. An analysis of one-layer slip calculated with the 1-D chain model. This analysis
corresponds to the generalized stacking fault energy for aluminum calculated with the Mishin
aluminum potential.
Figure 4-6 is the energy surface for displacement along two adjacent {111} planes, ∆x =
(..... 0, 0, 0, ∆xi, ∆xi+1, 0, 0, 0,....). The surface essentially constitutes the energy penalty for all
combinations of displacement on two adjacent pairs of {111} planes ranging from ∆xi or ∆xi+1 =
0 to ao[11 2 ]/2. This range samples all possible configurations for displacement in the <11 2 >
direction because of the symmetry of the {111} plane in the fcc structure. In this two-layer
analysis, the displacements will be reported as ∆xi /(ao[11 2 ]/2) in order to easily identify
features of this model inherent to the fcc structure.
Figure 4-6. Excess energy surface for general [11 2 ] displacement on two adjacent (111) planes
in aluminum. The energy minimum at α is the intrinsic stacking fault energy for fcc aluminum, β
is the energy minimum for a two-layer microtwin, and δ is the barrier for displacement in the
anti-twinning direction on two adjacent planes.
There are a few basic features of Fig. 4-6 worth highlighting. First, the energy given
along either axis is the one-layer energy penalty shown in (Fig. 4-5) where the energy minimum
at α is the intrinsic stacking fault energy. The local energy minimum seen at ∆xi = ∆xi+1 =
ao[11 2 ]/6 and labeled β is the local twin minimum corresponding to intrinsic stacking faults on
two adjacent planes. This local minimum represents the formation of a two-layer microtwin. The
energy maximum labeled δ corresponds to the energetic barrier to shear in the anti-twinning
direction on two adjacent planes. The location of these critical points is a general feature of the
fcc structure and will be seen in the analysis of copper as well.
The energy surface in Fig. 4-6 can be used to determine the minimum energy path
separating the global minimum at ∆xi = ∆xi+1 = 0 and the local twin minimum ∆xi = ∆xi+1 =
ao[11 2 ]/6, labeled β. In Figure 4-7, this path is shown overlaid on a contour map of the energy
surface. The minimum energy path is determined by connecting the minimum energy for every
value of the total strain in the system. Initially, for small values of total strain, elastic behavior is
exhibited as the minimum energy path corresponds to identical displacement between both pairs
of planes, ∆xi = ∆xi+1. At point B on this energy path, the strain localizes onto a single pair of
planes with additional strain contributing to the formation of an intrinsic stacking fault between
these two planes at the energy minimum labeled α. Once this stacking fault is fully formed,
further strain activates the adjacent pair of planes. Slip between these two planes moves the
system to β corresponding to the formation of a two-layer microtwin.
Figure 4-7. The minimum energy path for shear deformation of aluminum overlaid on the energy
contour for generic displacement of two adjacent (111) planes in the [11 2 ] direction calculated
with the simplified energy model.
The low degree of freedom calculations imply that the lowest energy path to the
formation of a two-layer twinned structure in aluminum requires first the nucleation of a single
Shockley partial dislocation. Subsequent activation of slip on an adjacent plane could then cause
twin formation. The two-layer analysis indicates that even when loading in the {111}<11 2 >
twinning system for aluminum, partial dislocations are nucleated homogeneously and only upon
dislocation slip and further shear strain do twin structures form. Homogeneous nucleation of a
twinned structure is not predicted by these calculations which matched the behavior observed
with molecular dynamics.
The energetic model can also be used to study the growth of twinned structures by
nucleating partial dislocations on adjacent planes. Figure 4-8 is a plot of the energy penalty as
successive adjacent planes are sheared relative to one another by one partial Burger’s vector to
produce intrinsic stacking faults.
The schematic in Figure 4-9 illustrates the imposed
deformation. A single plane is displaced creating an intrinsic stacking fault represented by the
white arrow. Then, a second plane is displaced creating an intrinsic stacking fault on the
adjacent plane. Subsequent adjacent planes are displaced in the same manner.
Excess Energy (mJ/m2)
250
200
150
100
50
0
0
5
10
15
number of slipped planes
Figure 4-8 Energy versus number of slipped partial dislocations determined using a simplified
energetic model.
[111]
[112]
etc....
Figure 4-9. Schematic illustrating the imposed deformation for the multiple layer growth
analysis.
The picture of twin growth described by Fig. 4-9 corresponds to one proposed mechanism of
twin growth where the energy barrier to additional layers slipping is low compared to the initial
nucleation event. With the Mishin aluminum potentials the unstable stacking energy, which is
the energy barriers to homogeneous partial dislocations nucleation, is 171 mJ/m2. The intrinsic
stacking fault energy, the energy minimum for full displacement of the first pair of planes is
shown in Figure 4-8. In this figure, the barrier to the slip of a second plane is 67 mJ/m2. The
results of this model will be discussed in Section 4.4 in the context of an identical analysis of
copper.
4.3 Mechanisms for Twinning in Bulk Copper
Homogeneous partial dislocation nucleation is also observed in quasi-two-dimensional
molecular dynamics simulations of copper using the Mishin copper potential and the simulation
set up described in Section 4.1.1. Figure 4-10 is a series snapshots from the molecular dynamics
simulation of {111}<11 2 > shear deformation in copper where, again, the atoms are colored
based on the symmetry of their local environment. The energy versus shear displacement for the
system is plotted in Figure 4-11. When the imposed strain reaches x = 5.52Å , a single Shockley
partial dislocation dipole is nucleated (Fig. 4-10a) and there is a corresponding energy decrease
in Fig. 4-11. With further strain, additional Shockley partials are nucleated (Figure 4-10b). These
additional dislocation dipoles are nucleated randomly in the supercell, which is in contrast to the
observed mechanism in aluminum. As the simulation continues, the Shockley partial dislocation
dipoles glide on their respective planes and eventually annihilate themselves because of the
supercell’s periodic boundary condition (Fig. 4-10c and d).
At this point dislocations are
nucleated heterogeneously on the intrinsic stacking faults left by the partial dislocation slip. As
these partials slip, twinned structures are formed (Fig. 4-10d and e) although this behavior is
somewhat artificial because of the periodic boundary condition used in the simulation.
(a)
(b)
(c)
(d)
(e)
Figure 4-10. Representative snapshots from the “quasi-two-dimensional” molecular dynamics
simulation of {111}<11 2 > shear deformation in copper. Atoms are colored based on the
symmetry of their local environment.
-3.88E-15
Esys (Joules)
-3.89E-15
-3.90E-15
-3.91E-15
-3.92E-15
5.44
5.54
5.64
x (angstroms)
5.74
5.84
Figure 4-11. System energy versus shear displacement for {111}<11 2 > shear deformation in
copper via molecular dynamics simulation.
Energy versus one-layer displacement curves for copper, calculated using the model
described in section 4.1.2, are shown in Figure 4-12. The local energy minimum at ao[11 2 ]/6 is
the intrinsic stacking fault energy for the fcc system which is 46 mJ/m2 in the case of copper.
Twinning/anti-twinning behavior is also observed in copper.
900
Excess Energy (mJ/m 2)
800
700
600
500
400
300
200
100
0
0
0.2
0.4
0.6
0.8
1
x/(ao[112]/2)
Figure 4-12. An analysis of one-layer slip calculated with the 1-D chain model. This analysis
corresponds to the generalized stacking fault energy for copper calculated with the Mishin
copper potential.
Figure 4-13 and 4-14 are the energy surface and minimum energy path for two-layer twin
formation in copper derived with the simplified energy model described in Section 4.1.2. The
observed mechanism in copper is similar to the behavior predicted for aluminum. Initially elastic
shear strain localizes onto a single pair of planes at point B and forms an intrinsic stacking fault
at point α. Further shear strain activates slip on the adjacent pair of planes to reach the twin
minimum, labeled β, at ∆xi = ∆xi+1 = ao[11 2 ]/6. The two-layer analysis again indicates that a
partial dislocation would nucleate and form an intrinsic stacking fault prior to further plastic
strain. The two-layer analysis is constrained in that further strain must occur either on the plane
that already contains an intrinsic stacking fault or on the adjacent plane. Displacement beyond α,
the local minimum corresponding to the intrinsic stacking fault, is energetically unfavorable
because of the twinning/anti-twinning asymmetry of the {111} plane in the fcc lattice.
Therefore, this model cannot be used to predict the physical mechanism in an unconstrained
system. However, the model does indicate that formation of a two-layer microtwin occurs
through the local energy minimum for formation of a single intrinsic stacking fault. The initial
plastic deformation event in an infinite bulk of either copper or aluminum, sheared in the
twinning direction, is therefore, predicted to be homogeneous nucleation of a partial dislocation
dipole.
Figure 4-13. Excess energy surface for general [11 2 ] displacement on two adjacent (111) planes
in copper. The energy minimum at α is the intrinsic stacking fault energy for fcc aluminum, β is
the energy minimum for a two-layer microtwin, and δ is the barrier for displacement in the antitwinning direction on two adjacent planes.
δ
∆xi+1 /(ao[112]/2)
β
α
B
∆xi /(ao[112]/2)
Figure 4-14. The minimum energy path for shear deformation in copper determined by the
simplified energy model overlaid on the energy contour for generic displacement of two adjacent
(111) planes in the [11 2 ] direction.
Excess Energy (mJ/m2)
250
200
150
100
50
0
0
2
4
6
8
10
12
14
number of slipped planes
Figure 4-15 Energy versus number of slipped partial dislocations determined using a
simplified energetic model.
The energy barrier to slip on a succession of adjacent planes in copper, calculated with
the energetic model and imposed deformation described by Fig. 4-9, is given in Figure 4-15. The
unstable stacking energy for copper is calculated to be 176 mJ/m2 with the Mishin copper
potential. The energy minimum associated with one fully slipped plane, the intrinsic stacking
fault energy, which is considerably lower for copper than aluminum. Because the intrinsic
stacking fault energy of copper is relatively low, the energy barrier to further slip is significantly
higher in copper than aluminum (156 mJ/m2 in copper versus 67 mJ/m2 in aluminum). In
aluminum, there is a significant energetic advantage to dislocations nucleating on adjacent planes
because the barrier to heterogeneous nucleation is only 40% of the homogeneous nucleation
barrier. In contrast, energetic barrier to heterogeneous nucleation in copper is almost 90% of the
homogeneous nucleation barrier.
4.4 Discussion
Plastic deformation via partial dislocation nucleation is exhibited in quasi-twodimensional molecular dynamics simulations of defect free bulk aluminum and copper. Partial
dislocations gliding past one another on adjacent planes form twinned structures, which is
consistent with the picture of heterogeneous nucleation observed in both simulation and
experiment. Further evidence for a partial dislocation dominated plastic deformation mechanism
for bulk fcc systems has been found using a simple energetic model that also predicts the
formation of an intrinsic stacking fault prior to reaching the energy minimum associated with a
two-layer microtwin.
Copper has a lower intrinsic stacking fault energy than aluminum, which is thought to
favor twin formation. However, once an intrinsic stacking fault is formed on the aluminum
lattice the barrier to additional dislocation nucleation is smaller than that of copper (156 mJ/m2 in
copper versus 67 mJ/m2 in aluminum). The lower energy barrier favors nucleation of additional
partial dislocations near intrinsic stacking faults, which results in the formation of twinned
structures in aluminum. In copper, the energy barrier to the nucleation of an additional partial
dislocation is similar for both homogeneous nucleation and nucleation near an intrinsic stacking
fault. As a result, the formation of twinned structures via build up of partial dislocation, even in
this constrained case, is not significantly favored over general partial dislocation formation.
Partial dislocations are not predisposed to form on existing stacking faults and the formation of
twinned structures, therefore, requires more extensive plastic deformation for the random
nucleation of two partials on adjacent planes.
Despite the observation of twinned structures in aluminum and copper, homogeneously
nucleated deformation twinning, as observed by Chang in bcc molybdenum, does not seem
viable in fcc metals. The presence of a local energy minimum in the form of the intrinsic
stacking fault in the fcc twinning direction seems to favor partial dislocation nucleation over
homogeneous twinning even in conditions favorable for twin formation. The bcc twinning
direction does not contain an intrinsic stacking fault. As a result, two-layer micro twins are
homogeneously nucleated in the bcc lattice [Chang 2003].
Chapter 5
Conclusions
5.1 Summary
The present work has explored the extent to which empirical EAM potentials can account
for the essential behavior of shear deformation revealed by published DFT calculations [Ogata
2002] in fcc aluminum and copper. The mechanisms for plastic deformation in the {111}<11 2 >
twinning systems for these metals have then been explored using classical molecular dynamics
and a simplified 1-D chain model [Chang 2003].
Both the shear stress-displacement behavior and atomic relaxation patterns observed with
the Mishin copper potential show close agreement to the behavior seen in ab initio calculations.
The extent of deformation before yielding would occur, however, is over predicted with this
potential.
The EAM potentials chosen to model the behavior of aluminum both exhibit artifacts that
can be related to the difficulty of fitting EAM potentials for metals, such as aluminum, with
localized charge density and consequently directional bonding. The behavior past the ideal shear
strength, which is associated with breaking and reforming of bonds show qualitative errors in
both potentials. The Mishin aluminum potential exhibits an expansion in the <11 2 > direction
not observed with DFT although the relaxation in the <111> and <110> directions are captured.
The Ercolessi potential shows a physically unintuitive relaxation behavior that seems to indicate
that the potential is strongly fit to data near the unstable stacking energy but has no information
regarding the configurations between linear elastic behavior and this instability. However, the
Mishin aluminum potential produced the majority of the local atomic behavior observed in first
principles calculations and should be regarded as an advancement over the Ercolessi potential.
The potential validation performed in the present study should be considered in the
context of a similar study comparing the generalized stacking fault energy calculated with
empirical EAM potentials to DFT calculations of the same parameter [Zimmerman 2000].
Zimmerman et al calculated behavior for several copper potentials. However, the Mishin copper
potential was not used in their study. By comparison to the DFT calculations used in the present
work, the Mishin copper potential should be considered the most accurate potential to model the
behavior of copper among those discussed in both works. However, Zimmerman et al present
different DFT calculations for which both the Mishin copper potential and a potential fit by
Voter [1994] show similar agreement. In addition, Zimmerman et al concluded that EAM
potentials were not suited to modelling the behavior of aluminum; however, the present work has
shown that the Mishin aluminum potential does capture much of the qualitative behavior
observed with DFT.
The redistribution of charge density associated with bond breaking and reforming is a
relatively difficult process for directionally bonded materials, such as aluminum, compared to
those with uniform charge density, such as copper. The difficulty in charge redistribution leads
to an extended range of deformation prior to bond breaking in directionally bonded materials.
Ab initio calculations indicate that this extended deformation range leads to a higher ideal shear
strength in aluminum when compared to copper [Ogata 2002]. Without the effects of aluminum’s
directional bonding, the EAM potentials in this study have not produced the correct ordering of
the extent of deformation between the two metals, although the two Mishin potentials do
correctly show aluminum to have a higher ideal shear strength than copper.
Both molecular dynamics simulations and simple energetic models, using the Mishin
copper and Mishin aluminum potentials, predict nucleation of partial dislocations and not
homogeneous deformation twins during {111}<11 2 > shear deformation despite favorable
conditions for twin formation. This behavior is in contrast to the observed mechanism of
homogeneous two-layer twin nucleation in bcc molybdenum [Chang 2003].
In both metals, nucleation of Shockley partial dislocations leads to eventual twin
structure formation. Aluminum exhibits homogeneous nucleation of a partial dislocation dipole
and the nucleation of partial dislocations on nearby planes in the “quasi-two-dimensional”
simulations performed in the current work. When these dislocation dipoles glide past one
another, a region of local hexagonal close-packed structure is formed, which, upon further strain
forms a twinned structure.
In copper homogeneous partial dislocation nucleation is also
observed, although dislocations are not subsequently nucleated on nearby planes. These
dislocations do not necessarily form twinned structures when they glide, and it is only after a
relatively large total strain that twinned structures are eventually formed.
The different mechanisms observed in copper and aluminum can be related to their
intrinsic stacking fault energy.
In general, low stacking fault energy is thought to favor
deformation twinning. However in these simulations studying perfectly homogeneous bulk
crystals, the opposite seems to be true. Aluminum and copper have similar energetic barriers to
initial homogeneous nucleation of partial dislocations. Once an intrinsic stacking fault has been
formed, however, aluminum has a significantly lower barrier to the nucleation of an additional
partial dislocation on an adjacent plane because, unlike copper, it does not recover the energy
penalty incurred during the initial shear deformation. The energetic barriers to heterogeneous
nucleation on an intrinsic stacking fault and homogeneous nucleation are nearly identical in
copper; therefore, dislocations are not preferentially nucleated near intrinsic stacking faults in
this material. With extended deformation, however, twin formation has been observed by a
random nucleation of partial dislocations on adjacent planes. Again this behavior is contrasted
with work performed by Chang, which indicates that the bcc structure has an inflection point and
not an energy minimum for one-layer slipped in the twinning direction [2003]. Because of this
inflection, the first stable structure observed is a two-layer microtwin and not a partial
dislocation as seen in the fcc metals studied in the current work.
5.2 Future Work
All of the results presented in this thesis should be considered work in process, and, as
such, there are many potential directions for continued research.
The 1-D chain model used in the present work to calculate the minimum energy path for
two-layer microtwin formation and the energy for successive slip on adjacent planes can be
further exploited. First, the energy path for the observed mechanism of twin formation in
aluminum, homogeneous partial dislocation nucleation, further dislocation nucleation and glide
to form a local hcp structure, and finally further slip to form a twinned structure, should be
calculated and compared to the energy path for two-layer twin formation. Also, the minimum
energy path for general slip on three, four, or more planes can be calculated.
The model can also be used to analyze the competition between dislocation slip and
twinning by analyzing the energetics of these mechanisms in light of a constitutive relation
proposed by Tadmor and Hai [2003] for deformation at atomically sharp crack tips in fcc metals.
This relation utilizes the unstable stacking energy, the intrinsic stacking fault energy, and the
energetic barrier described in the present work as the energy barrier to slip on a second adjacent
plane to predict the outcome of competition between dislocation motion and twinning under the
influence of crack tips. The application of this model to the defect-free bulk systems studied in
the present work seems straightforward and potentially enlightening.
The quasi-two dimensional molecular dynamics simulations presented in this work
represent a first shot at observing the mechanisms of shear deformation in fcc aluminum and
copper under conditions favorable for twin formation. The geometric constraint of these
simulations, both in shear orientation and relative system dimensions, severely limits the
viability of full dislocation slip. Extension of shear deformation studies to full three-dimensional
systems will reintroduce the competition between twinning and dislocation slip. It is of interest
to determine whether the mechanisms observed in these quasi-two-dimensional simulations are
still favored in a full three-dimensional simulation and to study the effects of shear orientation on
the observed mechanism. The strain rate and temperature dependencies of the dominant
deformation mechanism are also of interest.
Eventually, this type of careful study utilizing both simple static calculations and
controlled molecular dynamics simulations will be extended to more complex systems.
Nanocrystalline materials, for example, exhibit deformation twinning. The effects of grain
boundary sources for partial dislocations in these materials are currently of significant interest,
and simple deformation studies of systems including grain boundaries may shed light on the
processes that have already been observed in experimental work and large-scale simulations.
Appendix A
Cohesive Energy for EAM Potentials
The equilibrium lattice parameter for a potential is the lattice parameter that minimizes
the cohesive energy of the perfect crystal structure. The cohesive energy as a function of lattice
parameter has been plotted in Figure A-1 for each of the potentials used in the present work to
verify the lattice constants cited for each potential. The values obtain are as follows: Mishin
copper, 3.615 Å, Mishin aluminum 4.05 Å, and Ercolessi aluminum 4.032 Å. The values
calculated here match exactly the referenced lattice constants [Mishin 2001, Mishin 1999,
Ercolessi 1994].
-3.5375
Cohesive Energy (eV)
-3.538
-3.5385
-3.539
-3.5395
-3.54
-3.5405
3.585
3.59
3.595
3.6
3.605
3.61
3.615
3.62
Lattice Parameter (angstroms)
(a)
3.625
3.63
3.635
-3.359
-3.3591
Cohesive Energy (eV)
-3.3592
-3.3593
-3.3594
-3.3595
-3.3596
-3.3597
-3.3598
-3.3599
-3.36
-3.3601
4.025
4.03
4.035
4.04
4.045
4.05
4.055
4.06
4.065
4.07
4.075
4.045
4.05
4.055
Lattice Parameter (angstroms)
(b)
-3.3586
Cohesive Energy (eV)
-3.3588
-3.359
-3.3592
-3.3594
-3.3596
-3.3598
-3.36
-3.3602
4.005
4.01
4.015
4.02
4.025
4.03
4.035
4.04
Lattice Parameter (angstroms)
(c)
Figure A-1: Cohesive energy as a function of lattice parameter for the Mishin copper (a), Mishin
aluminum (b), and Ercolessi aluminum (c) potentials.
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