C C O O R R N N E E L L L L U N I V E R S I T Y MAE 715 – Atomistic Modeling of Materials N. Zabaras (04/02/2012) 1 References Computer Simulations of Dislocation, V. V. Bulatov and W Cai Theory of Dislocations, Hirth and Lothe Introduction to Dislocations, D Hull and D J Bacon Introduction to Solid State Physics, Kittel Computer Simulations of Dislocations
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CCOORRNNEELLLL U N I V E R S I T Y
MAE 715 – Atomistic Modeling of Materials
N. Zabaras (04/02/2012) 1
References
Computer Simulations of Dislocation, V. V. Bulatov and W Cai
Theory of Dislocations, Hirth and Lothe
Introduction to Dislocations, D Hull and D J Bacon
Introduction to Solid State Physics, Kittel
Computer Simulations of Dislocations
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1. Introduction to crystal dislocations
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1. Introduction to Crystal Dislocations
Overview:
The basic elements and common terminology used to describe perfect crystal
structures.
Dislocation as a defect in the crystal lattice and its properties.
Forces on dislocations and atomistic mechanisms for dislocation motion.
It define a great many properties of crystalline materials: crystals’ ability to yield
and flow under stress, creep and fatigue, ductility and brittleness,
indentation hardness and friction, crystal grows, etc.
In material science, a dislocation is a crystallographic defect, or irregularity,
within a crystal structure.
Transmission Electron Micrograph of Dislocations
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1.1 Perfect Crystal Structures
crystal structure=lattice + basis A basis may consist of one or more atoms.
(a) A two-dimensional crystal consisting of two types of atoms (white and gray). (b) The Bravais lattice
is specified by two repeat vectors a and b. (c) The basis contains three atoms.
primitive cell: The smallest parallelepiped with a lattice point at each of its eight
corners (3D).
unit cell: To better reflect the symmetries, certain types of Bravais lattices are
specified by non-primitive lattice vectors a, b and c. The parallelepiped formed
by these vectors is called the unit cell, for example, FCC, BCC.
The Unit cell is larger or equal to the Primitive cell
To avoid confusion, the adopted convention for constructing unit cell is to
associate each crystal structure with the Bravais lattice of highest possible
symmetry and with the basis containing the smallest number of atoms.
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(a) The unit cell of a simple cubic Bravais lattice. (b) The unit cell of a body-centered-cubic Bravais
lattice. (c) The unit cell of a face-centered-cubic Bravais lattice.
the lattice points
positions in the SC
lattice
the lattice points positions
of a BCC lattice
the lattice points positions
of an FCC lattice
The lattice and the basis are not uniquely defined.
1.1 Perfect Crystal Structures
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Miller indices
A vector l that connects one point to another in a Bravais lattice can be written
as a linear combination of repeat vectors, i.e.
where i, j and k are integers. The Miller indices notation for this vector is [i j k].
To specify a crystallographic plane, the Miller indices of the direction normal to
the plane are used. The Miller indices of a lattice plane can also be defined to
be a set of integers with no common factors, inversely proportional to the
intercepts of the crystal plane along the crystal axes, but written between
round brackets, i.e. (i j k).
A family of directions (vectors) is written
between angular brackets <i j k>, while a family
of planes is written in curly brackets {i j k}.
321
1:
1:
1::
xxxkji
Negative component is specified by placing a bar over the corresponding index,
i.e.
1.1 Perfect Crystal Structures
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1.2 The concept of crystal dislocations
They are usually thought of as extra lattice planes (edge) inserted in the
crystal that do not extend through all of the crystal, but end in the dislocation
line.
A dislocation is a defect of crystal lattice topology and can be defined by
specifying which atoms are dislocated or mis-connected with respect to the
perfect, defect free structure of the host crystal.
(a) A perfect simple-cubic crystal. (b) Displacement of two half-crystals along cut plane A by lattice
vector b results in two surface steps but does not alter the atomic structure inside the crystal. (c) The
same “cut-and-slip” procedure limited to a part of cut plane A introduces an edge dislocation ⊥.
Cut and slip
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Edge dislocation: a defect where an extra half-plane of atoms is introduced mid way
through the crystal, distorting nearby planes of atoms.
Screw dislocation: atoms round the dislocation line are arranged in a spiral.
In pure screw dislocations, the Burgers vector is parallel to the line direction.
In an edge dislocation, the Burgers vector is perpendicular to the line direction.
Two properties: line direction (sense), the direction running along the bottom of
the extra half plane; the Burgers vector, describes the magnitude and direction
of distortion to the lattice.
(a) An edge dislocation created by inserting a half-plane of atoms B. (b) A screw dislocation created
by a “cut-and-slip” procedure in which the burger vector is parallel to the dislocation line. (c) Mixed
dislocation, a curved dislocation line with an edge orientation at one end (on the left) and a screw
orientation at the other end (on the right).
1.2 The concept of crystal dislocations
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In real crystals, dislocations form in many different ways: shearing along crystal
planes, condensation of interstitials (extra atoms in the lattice) or vacancies (empty
atomic sites).
A more precise way to identify the presence of a dislocation is Burgers circuit
test, which consists of a sequence of jumps from atoms to their neighbors. The
Burgers circuit should form a complete loop when it is drawn in a perfect crystal,
and may not end at the starting atom when in a defective crystal.
line sense The start and end points of the
circuits are Si and Ei ,
respectively. Circuit 1 does not
enclose dislocation ⊥ whereas
circuits 2 and 3 do. The sense
vector ξ is defined to point out of
the paper so that all three circuits
flow in the counterclockwise
direction following the right-hand-
rule.
1.2 The concept of crystal dislocations
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In circuit 2 in the above picture, the vector connecting starting point S2 to end
point E2 of test circuit 2 is the Burgers vector b associated with the enclosed
dislocation. It needs to emphasize that the direction (or sign) of the Burgers
vector is meaningful only when the sense of the dislocation line is either
explicitly defined or implied by context. For example, if we reverse the direction
of line sense vector ξ and make it point into the plane of the paper. Our right-
hand-rule convention then dictates that, to obtain the Burgers vector, circuit 2
should now run clockwise, from atom E2 to atom S2, which obviously reverses
the direction of the resulting Burgers vector.
The algorithm of constructing Burgers circuits (r0 and vi are input)
1. Define the starting position r0 of the circuit to be the position of atom nS. Set
i := 1.
2. Find atom ni whose position ri is nearest to point ri−1 +vi .
3. Compute the difference between the actual and perfect relative positions of
the atom pair connected by the current translation, Δui := ri −(ri−1 +vi).
4. Increment the step counter i := i +1. If i ≤N, go to step 2.
5. Define the index of the end atom, nE := nN.
1.2 The concept of crystal dislocations
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The Burgers vector is obtained either as the difference rN −r0 or, equivalently,
as the sum of vectors
The sum in this equation is a discrete analogue of the equation used in
continuum mechanics to define the Burgers vector of a Volterra dislocation,
where C is any contour enclosing the dislocation line and ∂u/∂l is the elastic
displacement gradient along the contour.
The resulting Burgers vector is unaffected by any deformation and/or
translation of the test circuit as long as such deformation and/or translation
does not make the circuit “cut” through a dislocation line. The Burgers
vector is conserved along any given dislocation. It is an intrinsic property of
the dislocation line that can be regarded as the dislocation’s topological
charge.
1.2 The concept of crystal dislocations
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Multiple dislocations: a test circuit drawn around two dislocation lines reveals a
Burgers vector equal to the vector sum of the Burgers vectors of two enclosed
dislocation. Two dislocation lines ξ2 and ξ3 merge into a single line at junction
node P. The Burgers vector of the resulting dislocation ξ1 can be obtained from
Burgers circuit q drawn on cross-section C. Because circuit q can be obtained
from circuit p by deformation and translation without cutting through the
dislocation lines, the Burgers vector revealed by both circuits must be the same,
that is
If we flip the directions of ξ2, ξ3, b2,
b3, the conservation of Burgers
vector can be written as
Analogy to the conservation of
current in an electric circuit: Cross
section B
Cross
section C
1.2 The concept of crystal dislocations
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1.3 Motion of a Crystal Dislocation
This section illustrates how the driving force for dislocation motion is obtained from
continuum elasticity theory, whereas the dislocation’s response to this force is
governed by discrete atomistic mechanisms.
The driving force for dislocation motion is applied stresses
(a) A perfect crystal with simple
cubic structure and dimension
Lx ×Ly ×Lz. (b) The top surface is
subjected to a traction force Tx
while the bottom surface is fixed.
An edge dislocation nucleates from
the left surface. In (c) and (d) the
dislocation moves to the right. In
(e) the dislocation finally exits the
crystal from the right surface. The
net result is that the upper half of
the crystal is displaced by b with
respect to the lower half.
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The work done by the surface traction is
Where fx is the driving force per unit length on the dislocation line
Generalize to 3D force per unit length at an arbitrary point P on a dislocation line,
we can obtain Peach-Koehler formula [1]
where σ is the local stress field and ξ is the local line tangent direction at point
P. The cross product with ξ ensures that the PK force is always perpendicular to
the line itself.
The significance of the PK formula is that the force acting on a dislocation is
fully defined by the local stress σ on the dislocation, regardless of the origin of
this stress.
The source of the local stress includes: surface traction, other dislocation or
other strain-producing defects interactions.
[1] J.P. Hirth and J. Lothe, Theory of Dislocations, Wiley, New York, 1982
1.3 Motion of a Crystal Dislocation
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The motion of dislocation motion is normally known as the sliding along well-
defined crystallographic planes. Sliding by dislocation motion only requires
significant atomic rearrangements near the dislocation core (local), as opposed
to over an entire plane (global). The level of stress required to make a dislocation
move is usually orders of magnitudes lower than the critical stress to break all
bonds on a crystallographic plane.
Theoretical strength: the shear stress needed to displace the upper half of the
crystal relative to the other half.
Small displacement:
In real crystals: 5 3
max10 10
1.3 Motion of a Crystal Dislocation
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Another important property of dislocation motion is that the area swept out by a
moving dislocation is proportional to the plastic strain it introduces to the crystal.
bi is the component of Burgers vector b in i direction, ni is the component of
vector n normal to the glide plane A, which simultaneously contains the
dislocation line and the Burgers vector. ΔA=vLΔt is the total area swept out by
dislocations, with total length L and average velocity v, during a period Δt
The plastic strain rate can be related to dislocation density ρ by calculating the
time derivative of the plastic strain.
Ω=LxLyLz is the volume of the crystal. Generalization:
z
x y z
NL
L L L , N is the number of dislocations.
1.3 Motion of a Crystal Dislocation
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Conservative versus Non-conservative Motion
According to the motion with respect to glide plane, we can assort the dislocation
motion into 2 categories: glide, motion along the glide plane; and climb, motion
perpendicular to the glide plane.
Dislocation glide is often called conservative motion, meaning that the total
number of atoms is conserved, whereas dislocation climb is non-conservative
(exception is found in prismatic loop [1]). A screw dislocation always glides and
never climbs.
The mobility of non-screw dislocations is usually highly anisotropic. At low
temperatures, climb is usually difficult and glide is dominant. However, at high
temperatures or under conditions of vacancy super-saturation, climb can
become dominant instead.
1.3 Motion of a Crystal Dislocation
[1] Introduction to Dislocations, D Hull and D J Bacon
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Dislocation glide mobility may be influenced by both extrinsic factors,
such as impurities acting as obstacles, and intrinsic factors, such as the
interatomic interactions at the dislocation core.
Two fundamental parameters that characerize intrinsic lattice resistance to
dislocation motion are the Peierls barrier and the Peierls stress.
For a screw dislocation, the Burgers vector is parallel to the line direction, hence
the glide plane is not uniquely defined. No insertion or removal. All ways glide and
never climbs. But can change slip plane by cross-slip.
Mixed dislocation: the edge component of Burgers vector determines how many
atoms need to be inserted or removed during climb.
1.3 Motion of a Crystal Dislocation
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Peierls valley: the minimum of this
function mark the preferred dislocation
positions.
Consider a straight dislocation moving in its glide plane. The effect of the crystal
lattice on this motion can be represented by an energy function of the dislocation
position, which has the periodicity of the lattice
Peierls barrier: The energy barrier (per
unit length) that a dislocation must
surmount to move from one Peierls valley
to an adjacent one under zero stress.
Peierls stress: In the presence of a non-zero local stress, the force acting
on a dislocation modifies the periodic energy function. As a result, the
actual energy barrier experienced by the dislocation becomes lower than
the Peierls barrier. The critical stress makes the energy barrier vanish
completely, is called Peierls stress. The minimum stress required to make a
straight dislocation move at zero temperature.
1.3 Motion of a Crystal Dislocation
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When local stress is higher than the Peierls stress, dislocation motion can
happen without other assistance, like thermal fluctuations. In this case,
dislocation can evolve easily, Molecular Dynamics is suitable for the simulation.
Usually, as Peierls stress for ordinary dislocations in the FCC metals is low, so
that MD model is used.
When the local stress is lower than the Peierls stress, a dislocation
cannot move at zero temperature, but can move at a finite temperature with the
help of thermal fluctuations. In this case, rather than moving the whole straight
dislocation at once, motion begins by throwing a short dislocation segment into
the next Peierls valley.
In this case, the kink pair nucleation is a rare event. The time step needs to be
in relatively large scale. It is inefficient to use Molecular Dynamics to simulate
the dislocation evolution, because in most time steps, the dislocation evolution
can not be seen. Kinetic Monte Carlo method is adopted. Most dislocation
motion in semiconductors, belongs to this situation.
1.3 Motion of a Crystal Dislocation
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2. Fundamental of Atomistic Simulations
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Interatomic interactions
Boltzmann distribution
Energy minimization
Introduction to Monte Carlo and Molecular Dynaimics
Fundamentally, materials derive their properties from the interaction between their
constituent atoms. To understand the behavior of dislocations, it is necessary and
sufficient to study the collective behavior of atoms in crystals populated by
dislocation.
2. Fundamentals of atomistic simulations
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When put close together, atoms interact by exerting forces on each other. The
variability of interatomic interactions stems from the quantum mechanical motion
and interaction of electrons. Henceforth, rigorous treatment of the interactions
should be based on a solution of Schrödinger’s equation for interacting electrons,
which is usually referred to as the first principles or ab initio theory. Although
accurate, it is a very inefficient method.
The usual way to construct a model of interatomic interactions is to postulate a
relatively simple, analytical functional form, interatomic potential, for the potential
energy of a set of atoms,
where ri is the position vector of atom i and N is the total number of atoms. The
force on an atom is
The hope is that interatomic potential can capture the most essential physical
aspects of atom–atom interaction. The parameters are usually fitted to
experimental or ab initio simulation data
2.1 Interactomic interactions
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The most obvious physical feature of interatomic interactions is that atoms do not
like to get too close to each other. The other important aspect of the interatomic
interaction is that atoms attract each other at longer distances.
where
Here, ε0 is the depth of the energy well and 21/6σ0
is the distance at which the interaction energy
between two atoms reaches the minimum.
Relatively few materials, among them the noble gases (He, Ne, Ar, etc.) and ionic
crystals (e.g. NaCl), can be described by pair potentials with reasonable accuracy.
For most other materials pair potentials do a poor job, especially in the solid state.
Pair potential
A well-known pair potential model that describes both long-range attraction and
short-range repulsion between atoms is the Lennard-
Jones (LJ) potential,
2.1 Interatomic interactions
Hard sphere model
L-J model
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many-body potential energy
Example: Stillinger–Weber (SW) potential for silicon, containing two-body and
three-body terms. Embedded-atom model (EAM), decomposing potential into a
pairwise interaction accounting for the effect of core electrons and a field term
defining the energy required to embed atom into an environment with electron
density.
where
is the local density of bonding electrons supplied by the atoms neighboring with
atom i. Because the embedding function is non-linear, the EAM-like potentials
include many-body effects that cannot be expressed by a superposition of pair-
wise interactions. As a result, EAM potentials can be made more realistic than
Free surface: no constraint on the motion of any atom on boundaries. Completely
ignores the effects of atoms outside the simulation volume and introduces
unnecessary surfaces.
Fixed boundary: fix atoms on the periphery of the simulation volume in equilibrium
positions that they would occupy in an infinite solid.
Flexible boundary: allows the atoms in the boundary layer to adjust their positions
in response to the motion of inner atoms.
Periodic boundary: embeds the simulation volume into an infinite, periodic array of
replicas or images. It completely eliminates surface effects and maintains
translational invariance of the simulation volume. No point in space is treated any
more specially than others.
Primary cell
Image cell
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Minimum image convention: the relative displacement vector between atoms i and
j is taken to be the shortest of all vectors that connect atom i to all periodic replicas
of atom j. The potential cut-off distance is sufficiently small so that no more than
one replica of atom j falls within the cut-off radius of atom i.
Scaled coordinates:
Consists of repeat vectors of the simulation cell
Whenever there is an atom at position s=(sx, sy, sz), there are also atoms at
positions s=(sx+n1, sy+n2, sz+n3).
The distance between two atoms with scaled coordinates is
3.2 Boundary Conditions
x
y
(x,y) c2
c1
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The separation vector in the real space is
PBC cannot eliminate the artifacts due to inevitably small number of atoms.
It can also introduce its own artifacts, i.e. in the existence of defect. The remedy
will be discussed later.
3.2 Boundary Conditions
Other types of boundary conditions can be constructed “inside” PBC.
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3.3 Data Analysis and Visualization
It is necessary to perform data filtering: Save memory, extract important information.
Identify crystal defects:
1. Locate higher energy spot.
Low signal-to-noise ratio.
Suppress unwanted noise by partial steepest descent.
The snapshots of MD simulation are used to initiate
steepest descent paths towards underlying local
minima. Limit the steepest descent relaxation to small
number of iterations to preserve system’s
configuration.
2. Centro-symmetry deviation (CSD) for center symmetric crystals (FCC, BCC, but
not HCP, diamond cubic)
Np=4 Np=6
CSD parameter is 0 in perfect crystal.
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Partial dislocations
Stacking fault
Remove
plane and relax
(110)
3.3 Data Analysis and Visualization
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Example: Frank-Read source
The problem of dislocation multiplication is important in the theory of crystal
deformations. Frank-Read source is such a multiplication mechanism of
dislocation.
Schematic representation of the operation of a Frank-Read source. A straight dislocation segment is bowed out by the driving shear stress with two pinning points. After a loop forms, a new dislocation segment is born.
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Frank-Read source, Shockley partials and stacking faults
Perfect crystal Dislocation loop
Remove a plane of
atoms on [-1 1 0]
x[-1 1 0] z[1 1 -2]
y[1 1 1]
relax
Apply shear
stress σxy
Stacking faults
Plotted according to
centro-symmetry
deviation parameters
Example: Frank-Read source
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Frank-
Read
source
Shockley partials
The reason for the
dissociation of the perfect
dislocation is that the
motion of the atom along
the path a to c involves a
larger dilatation normal to
the slip plane, and hence
a larger misfit energy
than does motion along
the path a to b to c.
Example: Frank-Read source
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4 Case Study of Dynamic Simulation
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Build a simulation cell consisting of two rectangular slabs with dimensions: