Atomic models of dislocations and their motion in cubic crystals

Abstract

A discrete model describing defects in crystal lattices and having the standard linear anisotropic elasticity as its continuum limit is proposed. The main ingredients entering the model are the elastic stiffness constants of the material and a dimensionless periodic function that restores the translation invariance of the crystal and influences the dislocation size. Explicit expressions are given for crystals with cubic symmetry: sc and fcc. Numerical simulations of this model illustrate static and moving edge and screw dislocations and describe their cores and profiles.

Full text

European Congress on Computational Methods in A pplied Sciences and Engineering
ECCOMAS 2004
P. Neittaanm¨aki, T. Rossi, S. Kor otov, E. O˜nate, J. P´eriaux, and D. Kn¨orzer (eds.)
Jyv¨askyl¨a, 24–28 July 2004
ATOMIC MODELS OF DISLOCATIONS AND THEIR
MOTION IN CUBIC CRYSTALS
A. Carpio
?
, and L. L. Bonilla
†
?
Departamento de Matem´atica Aplicada
Universidad Complutense de Madrid, 28040 Madrid, Spain
e-mail: ana
−
carpio@mat.ucm.es
†
Escuela Polit´ecnica Superior
Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Legan´es, Spain
e-mail: bonilla@ing.uc3m.es
Key words: Atomic models, dislocations, cubic crystals, anisotropic elasticity.
Abstract. A discrete model describing defects in crystal lattices and having the standard
linear anisotropic elasticity as its continuum limit is proposed. The main ingredients
entering the model are the elastic stiffness constants o f the material and a dimensionless
periodic function that restore s the translation invariance of the crystal and influences the
dislocation size. Explicit expressi ons are given for crystals with cubic symmetry: sc and
fcc. Numerical simulations of this model illustrate static and moving edge and sc rew
dislocations and describe their cores and p rofiles.
1

A. Carpio a nd L. L. Bonilla
1 INTRODUCTI ON
The advances of electronic microscopy allow imaging o f atoms and can therefore be used
to visualize the core of dislocations [1, 2], cracks [3] and other defects that control crystal
growth and the mechanical, optical and electronic properties of the resulting materials
[4]. Emerging behavior due to motion a nd interaction of defects might explain common
but poorly understood phenomena such as friction [5]. Defects can be created in a con-
trolled way by ion bombardment on reconstructed surfaces [6], which allows the study of
effectively two dimensional (2D) single dislocations and dislocation dipoles. Other defects
that are very important in multilayer growth are misfit dislocations [7]. At the nanoscale,
many processes (for example, dislocation emission around nanoindentations [8]) involve
the interaction of a few defects so close to each other that their core structure plays
a fundamental role. To understand them, the traditional method of using information
about the far field of the defects (extracted from linear elasticity) to infer properties of far
apart defects reaches its limits. The alternative method of ab initio simulations is very
costly and not very practical at the present time. Thus, it would be interesting to have
systematic models of defect motion in crystals that can be solved cheaply, are compatible
with elasticity and yield useful information about the defect cores and their mobility.
To see what t hese models of defects might be like, it is convenient to recall a few
facts about dislocations. Consider for example an edge dislocation in a simple cubic (sc)
lattice with a Burgers vector equal to one interionic distance in gliding motion, as in
Fig. 1. The atoms a bove the xz plane glide over those below. Let us label the atoms
by their position before the dislocation moves beyond the origin. Consider the atoms
(x
0
, −a/2, 0) and (x
0
, a/2, 0) which ar e nearest neighbors before the dislocation pa sses
them. After the passage of t he dislocation, the nearest neighbor atoms a r e (x
0
, −a/2, 0)
and (x
0
−a, a/2, 0). This large excursion is incompatible with the main assumption under
which linear elasticity is derived for a crystal structure [9]: the deviations of ions in
a crystal lattice from their equilibrium positions are small (compared to the interionic
distance), and therefore the ionic potentials entering the total potential energy of the
crystal are approximately harmonic. One obvious way to describe dislocation motion is t o
simulate the atomic motion with the full ionic potentials. This description is p ossibly too
costly. In fact, we know that the atomic displacements are small far from the dislocation
core and that linear elasticity holds there. Is there an intermediate description that allows
dislocation motion in a crystal structure and is compatible with a far field described by
the corresponding anisotropic linear elasticity?
If we try to harmonize the continuum description of dislocations according to elasticity
with an atomic description which is simply elasticity with finite differences instead of dif-
ferentials, we face a second difficulty. The displacement vector of a static edge dislocation
is multivalued. For example, its first component is u
1
= a(2π)
−1
[tan
−1
(y/x) + xy/(2(1 −
ν)(x
2
+ y
2
))] for the previously described edge dislocation (ν is the Poisson ratio) [1]. In
elasticity, this fact does not cause any trouble because the physically relevant strain ten-
2

A. Carpio a nd L. L. Bonilla
sor contains only derivatives of the displacement vector. These derivatives are continuous
even across the positive x axis, where the displacement vector has a jump discontinuity
[u
1
] = a. If we consider an atomic model, and use differences instead of differentials,
the difference of the displacement vector may still have a jump discontinuity across the
positive x axis.
The previous difficulties have been solved in a simple model of edge dislocations and
crowdions called the IAC model (interacting atomic chains model) pro posed and studied
by A.I. Landau and collaborators [10]. In the equations for the IAC model, the differences
of the displacement vector are replaced by t heir sines. Unlike t he finite differences, these
sine functions are continuous across the positive x axis. Moreover, the equations remain
unchanged if a horizontal chain of atoms slides an integer number of lattice periods a over
another chain. Taking advantage of its simplicity, we have recently analyzed pinning and
motion of edge dislocations in the IAC model [11].
In this paper, we propose a top-down approach to atomic models of dislocations in cubic
crystals. Firstly, we discretize space along the primitive vectors defining the unit cell of
the crystal. Secondly, we replace the gradient of the displacement vector in the strain
energy by an appropriate periodic function o f the discrete gradient, g(D
+
j
u
i
). Summing
over all lattice sites, we obtain the potential energy of the crystal. Next, we find the
equations of motion by the usual methods of classical mechanics. Far from the core o f
a defect, the discrete gradient approaches the continuous one. Then, provided the slope
g
0
(0) is one in the appropriate units, the spatially discrete equations of motio n become
those of the anisotropic elasticity. We illustrate our approach by constructing static and
moving edge and screw dislocations in sc and fcc crystals.
The rest of the paper is organized as follows. The discrete models for simple cubic
crystals are proposed in Section 2. Dislocations are then analyzed in Section 3: Static
and moving screw and edge dislocations are numerically studied in Sections 3.1 and 3.2,
respectively. In Section 4, we study dislocations in f cc cubic crystals, which r equires
formulating the equations o f motion of the model in non-orthogonal coordinates. In fact,
primitive vectors for fcc or bcc crystals are non-orthogonal, and the equations of motion
describing these crystals should be invariant with respect to integer translations along
non-orthogonal primitive vectors. To illustrate our ideas, we calculate numerically several
relevant dislocations for gold (a f cc crystal). Section 5 contains our conclusions.
2 DISCRETE MODEL FOR A SIMPLE CUBIC LATTICE
To construct a spatially discrete model of a crystal having dislocations among its
solutions, we shall start from the strain energy density for the appropriate a nisotropic
elasticity. In the case of cubic symmetry a nd considering infinitesimal displacements, the
strain energy density is [9]
W =
1
2
c
ijkl
e
ij
e
kl
, (1)
3

A. Carpio a nd L. L. Bonilla
-15 -10 -5 0 5 10 15
-15
-10
-5
0
5
10
15
x
y
b
F
F
Figure 1 : Deformed cubic lattice in the pre sence of an edge dislocation for the piecewise linear g(x) of
Eq. (8) with α = 0.1.
c
ijkl
= λ δ
ij
δ
kl
+ µ ( δ
ik
δ
jl
+ δ
il
δ
jk
)
+ 2(C
44
− µ)
δ
ik
δ
jl
+ δ
il
δ
jk
2
− δ
1i
δ
1j
δ
1k
δ
1l
− δ
2i
δ
2j
δ
2k
δ
2l
− δ
3i
δ
3j
δ
3k
δ
3l
!
, (2)
provided the infinitesimal strain tensor is defined by the symmetric part of the distortion
tensor w
ij
[12]:
e
ij
=
w
ij
+ w
ji
2
, w
ij
=
∂u
i
∂x
j
. (3)
In all our equations, sum over repeated indices is always intended unless explicitly spec-
ified. u
i
, i = 1, 2, 3, are the components of t he displacement vector. λ and µ are related
to the elastic stiffness constants C
ij
[9] by
λ = C
12
, µ =
C
11
− C
12
2
. (4)
If C
44
= µ = (C
11
− C
12
)/2, the strain energy density is isotropic, and λ and µ are the
usual Lam´e coefficients.
4

A. Carpio a nd L. L. Bonilla
a
a
a
a
a
a
1
3
2
1
2
3
x
y
z
(a)
(b)
Figure 2: Reference systems spanned by: (a) three primitive vectors of the simple cubic lattice, (b) three
primitive vectors of the face centered cubic lattice.
To obtain a discrete mo del for a sc crystal, we shall assume that space is discrete,
so that x = x
1
= l a, y = x
2
= ma, z = x
3
= na, where l, m and n are integer
numbers. In Fig . 2(a), a is the side of the cubic cell, and therefore it is the primary unit
of length. We shall measure the displacement vector in units o f a, so that u
i
(x, y, z, t) =
a u
i
(l, m, n; t) and u
i
(l, m, n) is a nondimensional vector. Let D
+
j
and D
−
j
represent the
standard forward and backward difference operato r s, so that D
±
1
u
i
(l, m, n; t) = ±[u
i
(l ±
1, m, n; t) − u
i
(l, m, n; t)], a nd so on. We shall define the discrete distortion tensor as
w
(j)
i
= g(D
+
j
u
i
), (5)
where g(x) is a periodic function of period one satisfying g(x) ∼ x as x → 0. The strain
energy for the discrete model is obtained by substituting the strain tensor:
e
ij
=
1
2
(w
(j)
i
+ w
(i)
j
) =
g(D
+
j
u
i
) + g(D
+
i
u
j
)
2
, (6)
in Eq. (1). Once the displacement vector is known, the discrete strain energy density is
a function of the po int W ({u
i
}) = W (l, m, n), (l, m, n) = (x, y, z)/a, and the potential
energy of the crystal is
V ({u
i
}) = a
3
X
l,m,n
W (l, m, n). (7)
The definition (5) is useful when dislocations are present. In the far field of a disloca-
tion, the differences of the nondimensional displacement vector are small, and therefore
they approximate the components of the continuum distortion tensor in length scales much
larger than a: D
±
1
u
i
(l, m, n; t) = ±a [u
i
(l±1, m, n; t)−u
i
(l, m, n; t)]/a ≈ ∂u
i
/∂x  1, and
5

A. Carpio a nd L. L. Bonilla
so on. In the far field of a dislocation, Eq. (5) yields w
(j)
i
≈ g(∂u
i
/∂x
j
) ≈ ∂u
i
/∂x
j
= w
ij
because g(x) ∼ x as x → 0. Then our proposed atomic model is compatible with con-
tinuum elasticity in the far field of dislocations. Near the dislocation core, our defini-
tion of discrete distortion tensor is free f rom the two problems of the simpler definition
w
(j)
i
= D
+
j
u
i
, as mentioned in the Int r oduction. Firstly, the continuum displacement
vector has a j ump of size a across t he axis directed along the Burgers vector of an edge
dislocation with strength a. Correspondingly, the discrete displacement vector has a jump
of size 1, but the discrete distortion tensor given by Eq. (5 ) is continuous. Secondly, con-
sider a straight edge dislocation directed along the z axis, with Burgers vector of size
a directed along the x axis and moving f rom left to right, as in Fig. 1. Let us label
the atoms by their position before the dislocation moves beyo nd the origin. Consider
the atoms (x
0
, −a/2, 0) and (x
0
, a/2, 0) which are nearest neighbors before the disloca-
tion passes them. After t he passage of the dislocation, the nearest neighbor atoms are
(x
0
, −a/2, 0) and (x
0
− a, a/2, 0). According t o (5), w
(2)
1
= g(D
+
2
u
1
) = g(D
+
2
u
1
+ 1), and
the discrete distortion tensor can be considered as a “difference” between neighboring
atoms even when the passage of an edge dislocation has distorted the lattice.
The size of the dislocation core and its mobility depend on the function g(x). In
practice, the periodic function g(x) should b e fitted to dat a of the material at hand so as
to have a good description of the defect core. However, in order to illustrate the theory
and determine how the results depend on the shape of g(x), we shall use in our calculations
the odd piecewise linear f unction:
g(x) =
(
x, |x| <
1
2
− α,
(1−2α)(1−2x)
4α
,
1
2
− α < x <
1
2
+ α,
(8)
which is periodically extended outside the interval (α − 1/2, α + 1/2) for a given α ∈
(0, 1/2). Numerical simulations of the g overning equations show that the dislocation
core becomes smaller and the dislocation is harder to move if the interval of x in which
g
0
(x) > 0 shrinks with respect to that in which g
0
(x) < 0.
To complete our description of an atomic model for sc crystals, we need to specify
the equations of motion for the displacement vector. In the absence of dissipation and
fluctuation effects the equations are:
ρa
4
¨u
i
(l, m, n; t) = −
1
a
∂V ({u
k
})
∂u
i
(l, m, n; t)
, (9)
in which ¨u
i
≡ ∂
2
u
i
/∂t
2
. Equivalently,
M ¨u
i
(l, m, n; t) = −
∂
∂u
i
(l, m, n; t)
X
l
0
,m
0
,n
0
W (l
0
, m
0
, n
0
), (10)
provided we define M = ρa
2
, which has units of mass per unit length. Here the displace-
ment vector is dimensionless, so that both sides of Eq. (10) have units of force per unit
6

A. Carpio a nd L. L. Bonilla
area. Eq. (10) is equivalent to
M ¨u
i
=
X
j,k,l
D
−
j
[c
ijkl
g
0
(D
+
j
u
i
) g(D
+
l
u
k
)]. (11)
Let us now restore dimensional units to this equation, so that u
i
(x, y, z) = a u
i
(x/a, y/a, z/a),
then let a → 0, use ρ = M/a
2
and that g(x) ∼ x as x → 0. Then we obtain the equations
of linear elasticity [12],
ρ ¨u
i
=
X
j,k,l
∂
∂x
j
c
ijkl
∂u
k
∂x
l
!
. (12)
Thus the atomic model with conservative dynamics yields the Cauchy equations for elastic
constants with cubic symmetry provided the components of the distortion tensor are very
small (which holds in the dislocation far field).
3 Dislocation motion in sc crystals
In this Section, we shall find numerically pure screw and edge dislocations of our atomic
model fo r sc symmetry and discuss their motio n under appropriate applied stresses. The
influence of the periodic function g(x) on the size of the dislocation core and its mobility
will also be considered. In all cases, the procedure to obtain numerically the dislocation
from the atomic model is the same. We first solve the stationary equations of elasticity
with appropriate singular source terms to obtain the dimensional displacement vector
u(x, y, z) = (u
1
(x, y, z), u
2
(x, y, z), u
3
(x, y, z)) of the static dislocation under zero applied
stress. This displacement vector yields the far field of the corresponding dislocation
for the discrete model, which has the nondimensional displacement vector U(l, m, n) =
u(x/a, y/a, z/a)/a. We use U(l, m, n) in the boundary and initial conditions for the
discrete equations of motion of the atomic model.
3.1 Screw dislocations
The continuum displacement field of a dislocation, u = (u
1
, u
2
, u
3
), can be calculated
as a stationary solution of the anisotropic Navier equations with a singularity ∝ r
−1
at
the dislocation core and such that
R
C
(dx ·∇)u = −b, where b is the Burgers vector and C
is any closed curve encircling the dislocation line [12]. A pure screw dislocation along the
z axis with Burgers vector b = (0, 0, b) has a displacement vector u = (0, 0, u
3
(x, y)) [1].
Then the strain energy density becomes W = C
44
|∇u
3
|
2
/2, and the stationary equation
of motion is ∆u
3
= 0. Its solution corresponding to a screw dislocation is u
3
(x, y) =
b (2π)
−1
tan
−1
(y/x) [1]. The same symmetry considerations for Eq. (11) (conservative
dynamics) yield the following discrete equation for the z component of the nondimensional
displacement u
3
(l, m; t):
M ¨u
3
= C
44
{D
−
1
[g(D
+
1
u
3
) g
0
(D
+
1
u
3
)] + D
−
2
[g(D
+
2
u
3
) g
0
(D
+
2
u
3
)]}. (13)
7

A. Carpio a nd L. L. Bonilla
-15
-10
-5
0
5
10
15
-10
0
10
0
0.2
0.4
0.6
0.8
1
x
z+w
i,j
z+w
i,j
z+w
i,j
y
wzz+ u (l,m)
3
Figure 3: Screw dislocation for the piecewise linear g (x) of Eq. (8) with α = 0.1.
Numerical solutions of Eq. (13) show that a static screw dislocation moves if an applied
shear stress surpasses the static Peierls stress, |F | < F
cs
, but t hat a moving dislocations
continues doing so until the applied shear stress falls below a lower threshold F
cd
(dynamic
Peierls stress); see Ref. [1 1] for a similar situation for edge dislocations. To find the static
solution of this equation corresponding to a screw dislocation, we could minimize an energy
functional. However, it is more efficient t o solve the following overdamped equation:
β ˙u
3
= C
44
{D
−
1
[g(D
+
1
u
3
) g
0
(D
+
1
u
3
)] + D
−
2
[g(D
+
2
u
3
) g
0
(D
+
2
u
3
)]}. (14)
The stationary solutions of Eqs. (13) and (14) are the same, but the solutions of (14) relax
rapidly to the stationary solutions if we choose appropriately the damping coefficient β.
We solve Eq. (14) with initial condition u
3
(l, m; 0) = U
3
(l, m) ≡ b (2πa)
−1
tan
−1
(m/l), and
with boundary conditions u
3
(l, m; t) = U
3
(l, m) + F m at the upper and lower boundaries
of our lattice (F is an applied dimensionless stress with |F | < F
cs
). Then the solution of
Eq. (14) relaxes to a static screw dislocation u
3
(l, m) with the desired far field. If F = 0,
Figure 3 shows the helical structure adopted by the deformed lattice (l, m, n + u
3
(l, m))
for for the asymmetric piecewise linear g(x) of Eq. (8) . The width of the dislocation core
depends on C
44
and on g(x). Numerical solutions show that there may be coefficients with
8

A. Carpio a nd L. L. Bonilla
g
0
(x) < 0 in (13) or ( 14) at the core of a dislocation [11]. Thus the size of t he dislocation
core and t he dislocation mobility depend on the shape of g(x), precisely on the width of
the interval of x for which g
0
(x) < 0. As this interval shrinks, the size of the core region
decreases and the dislocation becomes harder to move. For practical use, the function
g(x) should be chosen so as to fit the chara cteristics of the material at hand.
The motion of a pure screw dislo catio n is somewhat special because its Burgers vector
is parallel to the dislocation line. Any plane containing the Burgers vector can be a glide
plane. Under a shear stress F > F
cs
directed along the y direction, a screw dislocation
moves on the glide plane xz. A moving screw dislocation has the structure of a discrete
traveling wave in the direction x, with f ar field u
3
(l − ct, m) + F m; c = c(F ) is the
dislocation speed. This is similar to the case of edge dislocations in the simple IAC model
[10], as discussed in [11] where the details of the analysis can be looked up.
3.2 Edge dislocations
To analyze edge dislocations in the simplest case, we consider a cubic crystal with
planar discrete symmetry, so that u(l, m; t) = (u
1
(l, m; t), u
2
(l, m; t), 0) is independent
of z = na, and ignore fluctuations. We assume that C
44
= (C
11
− C
12
)/2, so that the
material is isotropic.
To find the stationary edge dislocation of the discrete model, we first have to write
the corresponding stationary edge dislocation of isotropic elasticity. An edge dislocation
directed along the z axis (dislocation line), and having Burgers vector (b, 0, 0) has a
displacement vector u = (u
1
(x, y), v
2
(x, y), 0) with a singularity ∝ r
−1
at the core and
satisfying
R
C
(dx ·∇)u = −(b, 0, 0), for any closed curve C encircling the z axis. It satisfies
the planar stationary Navier equations (12) with a singular source term:
∆u +
1
1 − 2ν
∇(∇ · u) = −(0, b, 0) δ(r). (15)
Here r =
√
x
2
+ y
2
and ν = λ/ [2(λ + µ)] is the Poisson ratio; cf. page 114 o f R ef. [12].
The appropriate solution is (cf. Ref. [1], pag. 57)
u
1
=
b
2π
"
tan
−1

y
x

+
xy
2(1 − ν)(x
2
+ y
2
)
#
,
u
2
=
b
2π
"
−
1 − 2ν
4(1 − ν)
ln
x
2
+ y
2
b
2
!
+
y
2
2(1 − ν)(x
2
+ y
2
)
#
. (16)
Eqs. (16) yield the nondimensional displacement vector U(l, m) = (u
1
(la, ma)/a, u
2
(la, ma)/a, 0),
which will be used to find the stationary edge dislocation of the discrete equatio ns of mo-
tion. For this planar configuration, the conservative equations of motion (11) become
M ¨u
1
= C
11
D
−
1
[g(D
+
1
u
1
) g
0
(D
+
1
u
1
)] + C
12
D
−
1
[g(D
+
2
u
2
) g
0
(D
+
1
u
1
)]
+C
44
D
−
2
{[g(D
+
2
u
1
) + g(D
+
1
u
2
)] g
0
(D
+
2
u
1
)}, (17)
M ¨u
2
= C
11
D
−
2
[g(D
+
2
u
2
) g
0
(D
+
2
u
2
)] + C
12
D
−
2
[g(D
+
1
u
1
) g
0
(D
+
2
u
2
)]
+C
44
D
−
1
{[g(D
+
1
u
2
) + g(D
+
2
u
1
)] g
0
(D
+
1
u
2
)}. (18)
9

A. Carpio a nd L. L. Bonilla
To find the stationary edge dislocation corresponding to these equations, we set C
44
=
(C
11
− C
12
)/2 (isotropic case), and replace the inertial terms M ¨u
1
and M ¨u
2
by β ˙u
1
and
β ˙u
2
, respectively. The resulting overdamped equations,
β ˙u
1
= C
11
D
−
1
[g(D
+
1
u
1
) g
0
(D
+
1
u
1
)] + C
12
D
−
1
[g(D
+
2
u
2
) g
0
(D
+
1
u
1
)]
+C
44
D
−
2
{[g(D
+
2
u
1
) + g(D
+
1
u
2
)] g
0
(D
+
2
u
1
)}, (19)
β ˙u
2
= C
11
D
−
2
[g(D
+
2
u
2
) g
0
(D
+
2
u
2
)] + C
12
D
−
2
[g(D
+
1
u
1
) g
0
(D
+
2
u
2
)]
+C
44
D
−
1
{[g(D
+
1
u
2
) + g(D
+
2
u
1
)] g
0
(D
+
1
u
2
)}, (20)
have the same stationary solutions as Eqs. (17) and (18). We solve Eqs. (19) and (20) with
initial condition u(l , m; 0) = U(l, m) g iven by Eqs. (16), and with boundary conditions
u(l, m; t) = U(l, m) + F (m, 0, 0) at the upper and lower boundaries of the lattice (F is
a dimensionless applied shear stress). If |F | < F
cs
(F
cs
is the static Peierls stress for
edge dislocations), the solution of Eqs. (19) and (20) relaxes to a static edge dislocation
(u
1
(l, m), u
2
(l, m), 0) with the appropriate continuum far field.
-10
-5
0
5
10
-10
0
10
0
0.5
1
x
y
u
i,j
(a)
1
u(l,m)
-10
-5
0
5
10
-10
0
10
-0.2
0
0.2
x
y
v
i,j
v
i,j
(b)
u(l,m)
2
Figure 4: Pro file of an edge dislocation for the piecew ise linear g(x) of Eq. (8) with α = 0.1: (a) u
1
(l, m),
(b) u
2
(l, m).
In our numerical calculations of the static edge dislocation, we use the elastic constants
of tungsten (which is an isotropic bcc crystal), C
11
= 521.0 GPa, C
12
= 201.0 GPa,
C
44
= 160.0 GPa (C
11
= C
12
+ 2C
44
) [2]. This yields ν = 0.278. Figure 1 shows the
structure adopted by the deformed lattice (l + u
1
(l, m), m + u
2
(l, m)) when ν = 0.278.
The width of the dislocation core depends on the Po isson ratio ν and on the shape of
10

A. Carpio a nd L. L. Bonilla
g(x). The profiles of the displacement vector are shown in Figure 4 for the asymmetric
piecewise linear function g(x). Numerical solutions show that the size o f the dislocation
core and the dislocation mobility depend on the shape of g(x) in the same manner as they
did for screw dislocations. As the interval of x for which g
0
(x) < 0 shrinks, the size of the
core region decreases and the dislocation becomes harder to move.
The glide motion of edge dislocations occurs on the glide plane defined by their Burgers
vector and the dislocation line, and in the direction of the Burgers vector. In our case, a
shear stress in the direction y, will move the dislocation in the direction x. For conservative
or damped dynamics, the applied shear stress has to surpass the static Peierls stress to
depin a static dislocation, and a moving dislocation propagates provided the applied stress
is larger than the dynamic Peierls stress (smaller than the static one) [11]. As α < 1/4 in
Eq. (8) decreases, the static Peierls stress increases, thereby rendering the dislocation less
mobile. A moving dislocation is a discrete traveling wave advancing along the x axis, and
having far field (u
1
(l −ct, m) + F m, u
2
(l −ct, m)). The analysis of depinning and motion
of planar edge dislocations follows that explained in Ref. [11] with technical complications
due to our more complex discrete model.
4 ELASTICITY IN A NON-ORTHOGONA L BA SIS: THE CASE OF FCC
META LS
As shown in Fig. 2(b), the primitive vectors of the unit cell are not orthogonal for a fcc
crystal. To find a discrete model for such a crystal, we should start by writing the strain
energy density in a non-orthogonal vector basis, a
1
, a
2
, a
3
, defined by
a
1
=
a
2
(1, 1, 0), a
2
=
a
2
(1, 0, 1), a
3
=
a
2
(0, 1, 1), (21)
in terms of the usual orthonormal vector basis e
1
, e
2
, e
3
determined by the cube sides
of length a. Let x
i
denote coordinates in the basis e
i
, and let x
0
i
denote coordinates in
the basis a
i
. Notice that the x
i
have dimensions of length while the x
0
i
are dimensionless.
The matrix T = (a
1
, a
2
, a
3
) whose columns are the coordinates of the new basis vectors
in terms of the old orthonor mal basis can be used to change coordinates as fo llows:
x
0
i
= T
−1
ij
x
j
, x
i
= T
ij
x
0
j
. (22)
Similarly, the displacement vectors in both basis are related by
u
0
i
= T
−1
ij
u
j
, u
i
= T
ij
u
0
j
, (23)
and partial derivatives obey
∂
∂x
0
i
= T
ji
∂
∂x
j
,
∂
∂x
i
= T
−1
ji
∂
∂x
0
j
. (24)
11

A. Carpio a nd L. L. Bonilla
Then the strain energy density W = (1/2)c
iklm
e
ik
e
lm
can be written as
W =
1
2
c
ijlm
∂u
i
∂x
j
∂u
l
∂x
m
=
1
2
c
0
rspq
∂u
0
r
∂x
0
s
∂u
0
p
∂x
0
q
, (25)
where the new elastic constants are:
c
0
rspq
= c
ijlm
T
ir
T
−1
sj
T
lp
T
−1
qm
. (26)
Notice that the elastic constants have the same dimensions in both the orthogonal and
the non-orthogonal basis. To obta in a discrete model, we shall consider that the dimen-
sionless displacement vector u
0
i
depends on dimensionless coordinates x
0
i
that are integer
numbers u
0
i
= u
0
i
(l, m, n). As in Section 2, we replace the distortion tensor (gradient of
the displacement vector) by a periodic f unction of the corresponding forward difference,
w
(j)
i
= g(D
+
j
u
0
i
). As in Eq. (6), g is a periodic function with g
0
(0) = 1 and period 1. The
discretized strain energy density is
W (l, m, n) =
1
2
c
0
rspq
g(D
+
s
u
0
r
) g(D
+
q
u
0
p
). (27)
The function g may b e fitted to experimental o molecular dynamics data, but, to illustrate
the theory, we have chosen the piecewise linear function (8) in the numerical results
presented in this paper. The elastic constants c
0
rspq
can be calculated in terms of the
Voigt stiffness constants for a cubic crystal, C
11
, C
44
and C
12
. Eq. (5) yields c
ijlm
=
C
44
(δ
il
δ
jm
+ δ
im
δ
lj
) + C
12
δ
ij
δ
lm
− H(δ
1i
δ
1j
δ
1l
δ
1m
+ δ
2i
δ
2j
δ
2l
δ
2m
+ δ
3i
δ
3j
δ
3l
δ
3m
), where H =
2C
44
+ C
12
− C
11
measures the anisotropy of the crystal. Then Eq. (26) provides the
tensor c
0
rspq
. The elastic energy can be obtained from W by means of Eq. (7), and the
conserva tive equations of motion (9) are then
ρa
3
∂
2
u
0
i
∂t
2
= −T
−1
iq
T
−1
pq
∂V
∂u
0
p
.
Together with Eqs. (7) and (2 7), these equations yield
ρ
∂
2
u
0
i
∂t
2
= T
−1
iq
T
−1
pq
D
−
j
[c
0
pjrs
g
0
(D
+
j
u
0
p
) g(D
+
s
u
0
r
)]. (28)
Now we can analyze the motion of dislocations. The initial and boundary data for
the numerical simulations are constructed from the far fields of dislocations in anisotropic
elasticity. To calculate the elastic far field of any straight dislocation, we shall follow the
method explained in Chapter 13 of Hirth and Lothe’s book [2]. Firstly, we determine the
elastic constants in an orthonormal coordinate system e
00
1
, e
00
2
, e
00
3
with e
00
3
= −ξ parallel to
the dislocation line. The result is
c
00
ijkl
= c
ijkl
− H
3
X
n=1
(S
in
S
jn
S
kn
S
ln
− δ
in
δ
jn
δ
kn
δ
ln
). (29)
12

A. Carpio a nd L. L. Bonilla
Here the rows of t he orthogonal matrix S = (e
00
1
, e
00
2
, e
00
3
)
t
are the coordinates of the e
00
i
’s in
the old orthonormal basis e
1
, e
2
, e
3
. In these new coordinates, the elastic displacement
field (u
00
1
, u
00
2
, u
00
3
) depends only on x
00
1
and on x
00
2
. The dislocation line is oriented along
ξ = −e
00
3
. In the new coordinates, the Burgers vector and the elastic displacement field
satisfy b
00
1
= b
00
2
= 0 and u
00
1
= u
00
2
= 0 for a pure screw dislocation in an infinite medium. For
a pure edge dislocation, b
00
3
= 0 and u
00
3
= 0. Secondly, the displacement vector (u
00
1
, u
00
2
, u
00
3
)
is calculated as follows:
• Select three roots p
1
, p
2
, p
3
with positive imaginary part out of each pair of complex
conjugate roots of the polynomial det[a
ik
(p)] = 0, a
ik
(p) = c
00
i1k1
+ (c
00
i1k2
+ c
00
i2k1
)p +
c
00
i2k2
p
2
.
• For each n = 1, 2, 3 find an eigenvector A
k
(n) associated to the zero eigenvalue for
the matrix a
ik
(p
n
).
• Solve Re
P
3
n=1
A
k
(n)D(n) = b
00
k
, k = 1, 2, 3 and R e
P
3
n=1
P
3
k=1
(c
00
i2k1
+c
00
i2k2
p
n
)A
k
(n)D(n) =
0, i = 1, 2 , 3 for the imaginary and real part s of D(1),D(2),D(3).
• For k = 1, 2, 3, u
00
k
= Re[−
1
2πi
P
3
n=1
A
k
(n)D(n) ln(x
00
1
+ p
n
x
00
2
)].
Lastly, we can calculate the displacement vector u
0
k
in the non-orthogonal basis a
i
from
u
00
k
.
We use this strategy to calculate the elastic displacements of the perfect edge dislo-
cation and pure screw dislocation in the case o f gold. We have considered two straight
dislocations: the perfect edge dislocation directed along ξ = (1, −1, 2)/
√
6 (with a Burg-
ers vector which is one of the translation vectors of the lattice, and therefore glide of
the dislocation leaves behind a perfect crystal [7]; and the pure screw dislocation along
ξ = (−1, −1, 0)/
√
2. For the perfect edge dislocation, we select:
e
00
1
= (−1, −1, 0)/
√
2, e
00
2
= (1, −1, −1)/
√
3, e
00
3
= (1, −1, 2)/
√
6, (30)
which are unit vectors parallel to the Burgers vector b, the normal to the glide plane
n, and minus the tangent to the dislocation line −ξ, respectively. For the pure screw
dislocation, we have:
e
00
1
= (1, −1, 2)/
√
6, e
00
2
= (−1, 1, 1)/
√
3, e
00
3
= (−1, −1, 0)/
√
2, (31)
where e
00
2
is a unit vector normal to the glide plane and e
00
3
is a unit vector parallel to the
dislocation line and to the Burgers vector (but directed in the opposite sense).
For gold, C
11
= 18.6, C
44
= 4 .20, C
12
= 15.7 and H = 5.5 in units of 10
10
GPa. The
lattice constant is a = 4.08
˚
A and the density is ρ = 1.74 g/cm
3
. Figures 5 a nd 6 show
the perfect edge dislocation and the screw dislocation obtained as stationary solutions of
model (28 ) . Their far fields match the corresp onding elastic far fields of the dislocations
(written in the non-orthogonal coordinates corresponding to the primitive cell vectors a
1
,
13

A. Carpio a nd L. L. Bonilla
a
2
, a
3
). Dark and light colors are used to trace points placed in different planes in the
original lattice. Note that the planes perpendicular t o the Burgers vector in Fig. 5 have
a two-fold stacking sequence ‘dark-light-dark-light . . . ’ The extra half-plane of the edge
dislocation consists of two half planes (one dark and one light) in the dark-light-dark-light
. . . sequence. Movement of this unit dislocation by glide retains continuity of the dark
planes and the light planes across the glide plane, except at the dislocation core where
the extra half planes terminate [7].
Figure 5: Perfect edg e dislocation in a gold lattice displaying a two-fo ld stacking sequence of planes
containing dark and light atoms. The dislocation line is perperdicular to the paper.
5 CONCLUSIONS
We have proposed discrete models describing defects in crystal structures whose con-
tinuum limit is the standard linear anisotropic elasticity. The main ingredients entering
the models are the elastic stiffness constants of the material and a dimensionless periodic
function that restores the translation invariance of the crystal (and together with the elas-
tic constants determines the dislocation size). For simple cubic crystals, their equations
14

A. Carpio a nd L. L. Bonilla
Figure 6: Screw dislocatio n in a gold lattice. The dislocation line is parallel to the z axis.
of motion with conservative dynamics are derived. Numerical solutions of these equations
illustrate simple screw and edge dislocations. For fcc metals, t he primitive vectors along
which the crystal is tra nslationally invar ia nt are not orthogonal. Similar discrete models
and equations of motion are found by writing the strain energy density and the equations
of motion in non-orthogonal coordinates. In these later cases, we determine numerically
stationary edge and screw dislocations.
This work has been supported by the MCyT gr ant BFM2002-04127-C02, and by the
European Union under grant HPRN-CT-2002-00282.
15

A. Carpio a nd L. L. Bonilla
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16