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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.




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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.




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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).




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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.




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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.




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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.




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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




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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




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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




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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.




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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] 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.




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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.




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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.




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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

pair potentials.


http://enpub.fulton.asu.edu/cms/potentials/main/main.htm

http://enpub.fulton.asu.edu/cms/potentials/main/main.htm



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Development of an interatomic potential:

Selection of a functional form that can capture the underlying physics of

interatomic interaction in the material of interest (fit specific material properties).

The key issue in developing a good interatomic potential is its transferability, the

ability to accurately describe or predict a material’s property that is not included

in its fitting database.

In the context of dislocation, the physical parameters that matter most are the

relative energies of the most stable crystal structures, elastic constants, point

defect energies and stacking fault energies. An interatomic potential fitted to

these parameters stands a better chance of describing dislocation behaviors

accurately.

Interatomic interactions are usually short range. To improve numerical efficiency,

it is common to truncate interatomic potentials, i.e. by setting the interaction

energy to zero whenever the distance between two atoms exceeds a cut-off

radius rc. In order to smooth out the undesirable artifacts by the truncation

discontinuity of energy at rc, both energy and its derivative should be zero at cut-

off distance.




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For short-range potentials, force on a given atom can be calculated given the

positions of its neighbors only within the cut-off radius. Another useful property of

interatomic potential models is that it is usually straightforward to partition the

potential energy into local energy contributions for each atom. For the EAM-like

potentials, the local energy of atom i can be defined as

It is easy to check that the sum of local energies of all atoms is equal to the total

potential energy as it should be.

Computation efficiency and accuracy

DFT: less efficient, accurate;

TB: mediate, mediate;

Empirical potential: fast and capable of

handling large number of atoms, less

accurate .




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2.2 Equilibrium Distribution

In a N atoms system, the total energy is

H is Hamiltonian. In classical mechanics, the instantaneous state of the system is

a microstate fully specified by the positions and momenta of all atoms {ri, pi}

(phase space).

When the system is in thermal equilibrium temperature T (not too low), the

distribution of the states {ri, pi} obeys Boltzmann law.

Where partition function

The system is said to be canonical ensemble.



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Statistical enemble: Consider a very large number of replicas of the system,

each with N atoms and described by the same Hamiltonian. At any instance of

time, each system’s state is represented by a point {ri , pi} in the 6N-dimensional

phase space. An ensemble is the collection of all these replicas or, equivalently, of

points in the 6N-dimensional phase space. Each replica (a member of the

ensemble) may evolve with time, which is reflected by the motion of a

corresponding point in the phase space. The replicas do not interact. When the

replicas are distributed according to Boltzmann’s distribution, the ensemble is

called canonical.

A macroscopic quantity A at thermal equilibrium can be expressed as a statistical

average of microscopic functions over the canonical ensemble

where

For harmonic system, where the potential V({ri}) is a quadratic function of the

atomic positions, the average potential energy is . And also we know the

average kinetic energy is , the total energy is

2.2 Equilibrium Distribution



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2.3 Energy Minimization

The Steepest-descent Method

Start from an arbitrary initial configuration R. At every iteration, the force vector F

is computed and R is displaced by a small step along F. The iterations continue

until |F| becomes smaller than a prescribed tolerance .

At low temperature, the kinetic energy is zero and the potential energy is

minimized, providing a good description of the atomic structure of the system.

The idea of the steepest-descent algorithm is that, as long as |F| is non-zero,

V (R) can be further reduced by displacing R in the direction of F.

Conjugate Gradient Relaxation

CGR goes through a series of search directions. The (local) minimum energy

point along each search direction is reached before CGR proceeds to the next

search direction. The search sequence is constructed in such a way that

subsequent search directions “avoid” (i.e. are conjugate to) all previously

searched directions.

Inefficient, local minimum



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This algorithm works best when the potential energy is a quadratic function, i.e.

, where G is a 3Nx3N symmetric matrix. The conjugate condition means

that any two search direction must satisfy

Global Minimization

Neither the CGR nor the steepest-descent algorithm guarantees finding the global

minimum of an energy function. If the global minimum is the target, an inefficient

approach is to run minimization many times starting from randomly selected initial

configurations. Then, the lowest local minimum found from a series of relaxations

provides an upper bound on the global minimum.

Line search




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A useful method is simulated annealing (SA), which mimics the thermal annealing

of a material. SA simulation starts at a high temperature, making it easier for the

system to escape from the traps of local minima and explore a larger volume in

the configurational space. The SA search then slowly reduces the temperature to

zero. At the end of this procedure, the system converges to a local minimum that

is likely to be less dependent on the initial configuration, and is more likely to be

the global minimum. Tight upper bound on the global minimum can be obtained by

performing SA several times.

SA is usually used together with MC and MD

methods to search for the global minimum of the

potential function.


[1] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi. Optimization by

simulated annealing. Science, 220(4598): 671–680, 1983.

[2] S. Kirkpatrick. Optimization by simulated annealing: quantitative

studies. Journal of Statistical Physics, 34(5/6): 975–986, 1984.

The optimal form of T(t) is system dependent. In practice, is

commonly used.



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2.4 Monte Carlo

Traditional numerical integration (i.e. quadrature) in 6N-dimensional phase space

is prohibitively expensive.

Therefore, Monte Carlo (MC) and Molecular Dynamics (MD) come to the rescue.

Simulate the motion of atoms at a finite temperature in such a way that the fraction

of time the system spends in each state satisfies Boltzmann’s distribution.

If the microscopic function depends only on the atomic configuration R, i.e.

A=A(R), the ensemble average can be written in terms of an integral over the

configurational space (only position, no moment),

where

The idea of the Monte Carlo method is to generate a stochastic process in the

configurational space such that the states visited during the simulation satisfy the

equilibrium distribution ρ(R).



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In the case, the ensemble average A can be replaced by an average of A(R) over

the trajectory of the Monte Carlo simulation. Most Monte Carlo methods simulate a Markov process, meaning that the

probability of moving to a specific state in the next step is only a function of the

current state and is independent of the states visited in the preceding steps. A

Markov process is specified by its transition probability matrix, π(R,R’), which is

the probability density of visiting state R’ in the next step if the system is currently

at R. π satisfies the normalization condition,

If the probability distribution of the system at step n is ρ(n)(R), then the probability

distribution at step n + 1 is

Hence, to sample the equilibrium distribution ρ(R), the Markov process should

satisfy

Here, Metropolis algorithm is used by choosing detailed balance condition

2.4 Monte Carlo



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it states that the net rate of transitions from state R to state R’ is exactly the same

as the net rate of the reverse transitions from R’ to R, when the stationary

distribution ρ is reached. The following restriction on the transition probabilities is

reached

In Metropolis algorithm, π is chosen as

Where acceptance ratio

Two sub-steps of a single Monte Carlo step is included. First, a trial move from

the current configuration R to another configuration R’ is selected with probability

α(R,R’). Usually, the trial moves displace one or several atoms in a random

direction, R’ =R+δR. If the potential energy of the trial state R’ is lower than the

energy of the current state R, the move is accepted. However, if the potential

energy of the new state is higher by dV =V (R’)−V (R), the move is accepted with

probability Pacc = exp(−βdV ). Otherwise, the move is rejected and the system

remains in the old state R for yet another step.

2.4 Monte Carlo



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The algorithm computing the average of function A(R), for a N-atom system with

initial positions ri

It is a common practice to adjust Δ empirically so that the fraction of accepted

moves stays close to 50% during the simulation. The requirement of designing a

MC procedure is (1) trial moves proceed unbiased, i. e. α(R, R’)=α(R’, R); (2) the

simulation trajectory can span the entire configurational space.

2.4 Monte Carlo



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2.5 Molecular Dynamics

Unlike the Monte Carlo method, which generates artificial trajectories spanning

the configurational space and complying with Boltzmann’s distribution, molecular

dynamics (MD) attempts to simulate the “true” dynamics of atoms while also

preserving Boltzmann’s statistics. The classical equations of motion for a system

with Hamiltonian H({ri , pi}) are

If only the potential is the function of atom position while the kinetic energy is the

function of momentum, the equation of motion can be written as

Molecular dynamics, at its heart, is simply the numerical integration of this

Newton’s second law.

The Verlet Algorithm

It is based on the following symmetric finite-difference approximation to the

acceleration,



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which leads to

Using this equation repeatedly, atomic positions are computed step by step. The

atomic velocity does not appear explicitly in Verlet algorithm. It is evaluated by

The Velocity Verlet Algorithm

The numerical estimation of the atomic velocities obtained in the Verlet algorithm

is not very accurate. The velocity Verlet algorithm is proposed to address this

problem

The equations of motion of a Hamiltonian system should conserve the system’s

energy.

Energy Conservation




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The Verlet and the velocity Verlet algorithms belong to a class of the symplectic

integrators[1] that preserve the energy particularly well. Selection of a proper time step t is an important issue in MD simulations. The total

energy of the system

may “drift away” or even diverge when the selected time increment is too large.

For MD simulations of solids, Δt around 0.01 of the inverse Debye frequency νD

is usually a safe choice. Δt is commonly of the order of 10-15.


[1] D. Frenkel and B. Smit. Understanding Molecular Simulation: From Algorithms to Applications.

Academic Press, San Diego, CA, 2002



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3. Case Study of Static Simulation



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set up initial positions of atoms based on solutions of the continuum elasticity

theory.

Boundary conditions.

MD simulation of dislocations.

3. Case study of static simulation



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An atomistic structure is specified by the positions xi of all atoms. A good way to

describe the new structure is by specifying the displacement vector

for every atom i. We can use the continuum solution u(xi) to approximate the

displacement of atom x far away from the dislocation center by assuming the

crystal is a continuum linear elastic solid.

3.1 Setting up an Initial Configuration

( , ) , , 0, 02

z x yu x y b u u except x y

The reasons for the continuum linear elasticity theory to break down in the

dislocation core:

(1) In a discrete crystal structure, relative displacements between neighboring

atoms in the dislocation core can be very large and vary rapidly from one atom

to its neighbors.

(2) The interactomic interactions in the core are highly non-linear.

The equilibrium atomic positions should be determined by minimizing the

interactomic potentials energy of the entire system.



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[112]

[110]

BCC Tantalum

Cut and displaced by . Then relax. ( , )2

zu x y b

Free to move fixed

Periodic in z

Software: MD++


Finnis-Sinclair potential for BCC tantalum.



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# Create single screw dislocation in BCC Ta

# under fixed boundary condition

setnolog

setoverwrite dirname = runs/ta-screw

#Read in potential file (Finnis-Sinclair potential for Ta)

potfile = ~/Codes/MD++/potentials/ta_pot readpot

#Create Perfect Lattice Configuration

latticestructure = body-centered-cubic latticeconst = 3.3058 #(A)

makecnspec = [ 1 1 -2 10 1 -1 0 16 0.5 0.5 0.5 5 ]

makecn finalcnfile = perf.cn writecn

#Create a single screw dislocation b = [111]

# remove all atoms beyond radius 32

# z y x0 y0 Rc rem

input = [ 3 2 0.0334 -0.0208 32 0 ] makecylinder

# 3 : specifies the cylinder axis is along z axis

# 2 : specifies the "local" y axis is the same as our y axis

# (x0,y0) : specifies the center of the cylinder in "local"

# coordinate system in scaled coordinates

# Rc : speficies the cut-off radius

# rem : 0: atoms outside Rc will be removed, 1: atoms to be fixed

# introduce the dislocation

input = [ 1 # activate makedislocation

3.3058 # lattice constant (in Angstrom)

0 0 0.8660 # Burgers vector (in unit of a)

0 0 1 # line direction vector

0 1 0 # normal vector of cut plane

2.65 -1.28 0 # a point on dislocation (in Angstrom)

0.339 # Poisson's ratio (for nonscrew dislocation)

0 # 0 commit displacement, 1 store in memory

]

makedislocation eval

# fix all atoms beyond radius 23.6

# z y x0 y0 Rc fix

input = [ 3 2 0.0334 -0.0208 23.6 1 ] makecylinder

#Plot Configuration

atomradius = 1.0 bondradius = 0.3 bondlength = 0

atomcolor = cyan highlightcolor = purple backgroundcolor = gray

bondcolor = red fixatomcolor = yellow

color00 = "orange" color01 = "red" color02 = "green"

color03 = "magenta" color04 = "cyan" color05 = "purple"

color06 = "gray80" color07 = "white" color08 = "blue"

plot_color_windows = [ 2

-10 -8.0 6 #color06 = gray80

-8.0 -6.0 8 #color08 = blue

1 #1: draw fixed atoms

]

#plot_limits = [ 1 -10 10 -0.05 10 -10 10 ]

plot_atom_info = 1 plotfreq = 10

rotateangles = [ 0 0 0 1.5 ]

win_width = 600 win_height = 600

#win_width = 140 win_height = 140

openwin alloccolors rotate saverot refreshnnlist eval plot

#sleep quit

#Conjugate-Gradient relaxation

conj_ftol = 1e-10 conj_itmax = 1000 conj_fevalmax = 10000

conj_fixbox = 1

relax

eval finalcnfile = relaxed.cn writeall = 1 writecn

sleep quit




N. Zabaras (04/02/2012) 45

3.2 Boundary Conditions

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



N. Zabaras (04/02/2012) 46

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


x

y

(x,y) c2

c1



N. Zabaras (04/02/2012) 47

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.


Other types of boundary conditions can be constructed “inside” PBC.



N. Zabaras (04/02/2012) 48

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.



N. Zabaras (04/02/2012) 49

Partial dislocations

Stacking fault

Remove

plane and relax

(110)

3.3 Data Analysis and Visualization



N. Zabaras (04/02/2012) 50

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.



N. Zabaras (04/02/2012) 51

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




N. Zabaras (04/02/2012) 52

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.




N. Zabaras (04/02/2012) 53

4 Case Study of Dynamic Simulation



N. Zabaras (04/02/2012) 54

Build a simulation cell consisting of two rectangular slabs with dimensions:

Upper: Lx1=30/2[1 1 1], Ly1=10[-1 0 1], Lz1=8[1 -2 1]

Lower: Lx2=29/2[1 1 1], Ly2=10[-1 0 1], Lz2=8[1 -2 1]

The lower slab has one atomic plane fewer in x direction than the upper one.

Reshape both slabs to have the same length Lx=29.5[1 1 1]/2 and relax.

y [-1 0 1]

x [1 1 1]

z [1 -2 1]

Periodic boundary

condition is applied

in x and z direction


Create a planar misfit interface between two crystals. Subsequent energy

minimization would automatically lead to dislocation formation.

Vacuum gap,

create free surface

Finnis-Sinclair potential M.W. Finnis and J.E. Sinclair, Philosophical

Magazine A, 50(1): 45-55, 1984

Plot in terms of CSD



N. Zabaras (04/02/2012) 55

4.2 Initializing Atomic Velocities

Thermalization: bring the system to a state of thermal equilibrium at a finite

temperature.

Equilibration run: A preliminary MD simulation starting from an arbitrary initial

configuration can bring the system to a state of thermal equilibrium after some

period of time.

Control temperature by scaling atomic velocities

Instantaneous temperature:

Set initial temperature and atom velocities:



N. Zabaras (04/02/2012) 56

0 200 400 600 800 1000

-200

0

200

400

600

800

1000

1200

1400

1600

1800

2000

En

erg

ies (

eV

)

t (fs)

kinetic

potential

total

0 200 400 600 800 1000

100

200

300

400

500

600

Te

mp

era

ture

(K

)

t (fs)

Initial temperature T*=600K

At the beginning of the equilibrium run, the atomic positions are at a local

potential energy minimum. During equilibration, the average potential energy

increases while the kinetic energy decreases by the same amount so that the

total energy is conserved.

Finnis-Sinclair potential approximated by a harmonic function.

4.2 Initializing Atomic Velocities



N. Zabaras (04/02/2012) 57

4.3 Stress Control

Apply stress to make the dislocation move.

1. Applying forces on surface atoms.

2. Applying stress in simulations with PBC in all

three directions -- Parrinello-Rahman method:

applies stress to the periodic supercell by

dynamically adjusting its shape so that the

virtual stress fluctuates around a specified

value.

In PR approach, the atomistic model acquires

9 additional degrees of freedom, which are

the components of matrix h. For a system with

N atoms, the total number of degrees of

freedom in PR approach is 3N+9.

[1] J. A. Zimmerman, E. B. Webb, J. J. Hoyt, R. E. Jones, P. A. Klein, and D. J. Bammann. Calculation of stress in atomistic

simulation. Modelling and Simulation in Materials Science and Engineering, 12(4): S319–332, 2004.

[2] A. M. Pendas. Stress, virial, and pressure in the theory of atoms in molecules. Journal of Chemical Physics, 117(3): 965–

979, 2002.

[3] J. Gao and J. H. Weiner. Excluded-volume effects in rubber elasticity. 1. Virial stress formulation. Macromolecules, 20(10):

2520–2525, 1987.



N. Zabaras (04/02/2012) 58

The equations of motion

where

is the stress one wishes to maintain during the simulation,

is its hydrostatic pressure component, is the matrix describing the shape of the

periodic cell at the beginning of the simulation– this matrix defines the reference

frame in which the orientation of the applied stress is expressed. W is a “mass”

parameter that determines how fast the box changes its shape in response to the

imbalance between the desired and the actual stress.

Feedback, adjust h when the virial stress

becomes different from the desired value.

4.3 Stress Control



N. Zabaras (04/02/2012) 59

4.4 Temperature control

Maintain the constant temperature during dislocation motion–

1)scale the atomic velocities at periodic intervals during the simulation.

2)Nose-Hoover thermostat, which mimics heat exchange between a simulation

volume and its surroundings.

3)Berendsen.

Ensembles:

Micro-canonical or NVE ensemble: fixed or periodic boundary conditions with

fixed repeat vectors. Constant E.

Canonical or NVT ensemble: a system exchanging heat with an external

thermostat will maintain a constant temperature T.

Isothermal isobaric or NPT ensemble: the system interacts with an external

barostat, its total volume will adjust so as to maintain a constant pressure.

Grand-canonical or μVT ensemble: mass exchange with a large external reservoir

will cause the total number of atoms in the system to fluctuate so that its chemical

potential remains close to that of the massostat.

The Nose-Hoover thermostat is to reproduce the canonical ensemble. The total

system is assumed to consist of two interacting sub-systems, the atomistic

system plus the additional fictitious degrees of freedom. The total system is

closed but the atomic sub-system is open.



N. Zabaras (04/02/2012) 60

Introducing a dynamical variables s that serves as a scaling factor of time. It

distorts the atomic trajectories, but maintains the temperature of the atomistic sub-

system close to a preselected value.

feedback

The Nose-Hoover equations of motion are derived from the standard variational

principle of classical mechanics. The trajectories conserve the following total

energy of the extended system.

Atomistic sub-system Heat bath

4.4 Temperature control



N. Zabaras (04/02/2012) 61

The original Verlet algorithm has to be modified to take Nose-Hoover thermalstat.

where

Solving these two equations for the atomistic positions at time

The equation for the fictitious parameter is

where

in this case study.



N. Zabaras (04/02/2012) 62

4.5 Extracting Dislocation Velocity

A simple definition of the dislocation position would be the center of mass of

the core atoms. Here, the core atoms are defined as atoms whose CSD

parameter is within certain domain.

Drag coefficient

Transient period,

measures

dislocation’s

effective mass



N. Zabaras (04/02/2012) 63

After the system achieves thermal equilibrium, shear stresses σxy = 300Mpa are

applied on both free surfaces. The total force is 9.287 eV/A. As the top surface

contains more atoms than the bottom surface (720 versus 696), the force per atom

fx is difference on the two boundaries: on the top surface fx=0.01290 eV/A; on the

bottom surface, fx=-0.01334 eV/A.

Example



N. Zabaras (04/02/2012) 64

From the linear relationship between dislocation core position and time, a

constant velocity under specific stress is predicted. Finding the slope, for the case

at temperature 300K and shear stress 300 Mpa, the velocity is approximately

1176 m/s.

50 100 150 200 250 300

200

400

600

800

1000

1200

Data: Data24_B

Model: Line

Equation: y = A + B*x

Weighting:

y No weighting

Chi^2/DoF = 2614.81905

R^2 = 0.9829

A 125.13333 ±47.60437

B 3.70686 ±0.24447

stress (Mpa)

ve

locity (

m/s

)

The dislocation velocity increases nearly, but not exactly, linearly with shear

stress. Fitting this curve, we can obtain a viscous drag coefficient B=σb/v. In this

case, B = 7.17e-5 pa.s.

CDS=[1.5, 10]A

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