Molecular dynamics (MD),
empirical potentials, and
molecular statics
Kai Nordlund
Professor, Department of Physics
Adjoint Professor, Helsinki Institute of Physics
Vice dean, Faculty of Science
University of Helsinki, Finland
Prof. Kai Nordlund
Principal investigator
Dr Carolina Björkas
Fusion reactor mat'ls
M Sc Laura Bukonte
Fusion reactor mat’ls
M Sc Elnaz Safi
Fusion reactor mat’ls
Doc. Antti Kuronen
Principal investigator
Dr Jussi Polvi
Fusion reactor mat’ls
M Sc Wei Ren
Carbon nanostructures
M Sc Alvaro Lopez
Surface ripples
Doc. Flyura Djurabekova
Principal investigator
Doc. Krister Henriksson
Nuclear Materials
Dr Andrea Sand
Fusion reactor mat’ls
Dr. Andrey Ilinov
Nanomechanics
Dr Vahur Zadin*
Particle physics mat’ls
(Univ. of Tartu)
M Sc Fredric Granberg
Dislocations
M Sc Morten Nagel
Nuclear materials
M Sc Kostya Avchachov
Dr. Aleksi Leino
Irradiation of metals
Nanostructures in silica
M Sc Shuo Zhang
Ion range calculations
Kai Nordlund, Department of Physics, University of Helsinki
Mr Jesper Byggmästar
Nanowires
M Sc Ekaterina Baibuz
Particle physics mat'ls
Dr Stefan Parviainen
Particle physics mat'ls
M Sc Mihkel Veske
Particle physics mat'ls
Dr Ville Jansson
Particle physics mat'ls
Dr Zhenxing Wang
Particle physics mat'ls
M Sc Junlei Zhao
Nanoclusters
M Sc Anders Korsbäck
Particle physics mat’ls
(CERN)
M Sc Mrunal Parekh M Sc Simon Vigonski
Particle physics mat'ls Particle physics mat'ls
2
Contents
 Molecular dynamics method
 General
approach
 Nonequilibrium extensions -- briefly
 Interatomic potentials
 Overview
 And
of EAM-like and Tersoff-like potentials
the Albe philosophy to fit bond-order potentials
 Case study: Nanocluster simulations
 And
introduction to tutorial case
 Molecular statics
 MD
and conjugate gradient relaxation schemes
 The end: some practical advices
 Examples of recent applications
 Time
permitting
Kai Nordlund, Department of Physics, University of Helsinki
3
Introduction:
The multiscale simulation framework
m
Length
Finite Elements
Rate equations
mm
Discrete dislocation dynamics
μm
nm
DFT
ps
Classical
Molecular
dynamics
ns
Kinetic Monte Carlo
μs
ms
s
hours
years
Time
Kai Nordlund, Department of Physics, University of Helsinki
4
MD method
MD = Molecular dynamics
 MD is solving the Newton’s (or Lagrange or Hamilton)
equations of motion to find the motion of a group of atoms
 Originally developed by Alder and Wainwright in 1957 to
simulate atom vibrations in molecules
 Hence
the name “molecular”
 Name unfortunate, as much of MD done nowadays does not
include molecules at all
 First materials science application:
1960, Gibson simulated radiation
effects in solids
[Phys. Rev. 120 (1960) 1229)]
A
few hundred atoms, very
primitive repulsive potentials (cell
was actually under huge pressure)
Kai Nordlund, Department of Physics, University of Helsinki
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MD method
Simple (trivial) example of atom motion by MD:
thermal motion in Cu at 600 K, cross section of
2 atom layers
Kai Nordlund, Department of Physics, University of Helsinki
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MD method
Less trivial example: small atom collision
event
 1 keV Xe irradiation of Cu
Kai Nordlund, Department of Physics, University of Helsinki
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MD method
Nanocluster deposition event
Kai Nordlund, Department of Physics, University of Helsinki
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MD method
How is this achieved:
MD algorithm, the basic idea
Give atoms initial r(t=0) and v(0) , choose short Δt
Get forces F = − ∇ V(r(i)) or F =
F(Ψ) and a = F/m
Move atoms: r(i+1) = r(i) +v(i)Δt + 1/2 a Δt2 + correction terms
Update velocities: v(i+1) = v(i) +aΔt + correction terms
Move time forward: t = t + Δt
Repeat as long as you need
Kai Nordlund, Department of Physics, University of Helsinki
9
MD method
MD algorithm, in greater detail
Give atoms initial r(i=0) and v(i=0) , set a = 0.0, t = 0.0, i = 0, choose short Δt
Predictor stage: predict next atom positions:
Move atoms: rp = r(i) + v(i) Δt + 1/2 a Δ t2 + more accurate terms
Update velocities: vp = v(i) + a Δt + more accurate terms
Get forces F = - ∇ V(rp) or F = F( Ψ(rp) ) and a = F/m
Corrector stage: adjust atom positions based on new a:
Move atoms: r(i+1) = rp + some function of (a, Δt)
Update velocities: v(i+1) = vp + some function of (a, Δt)
Apply boundary conditions, temperature and pressure control as needed
Calculate and output physical quantities of interest
Move time and iteration step forward: t = t + Δt, i = i + 1
Repeat as long as you need
Kai Nordlund, Department of Physics, University of Helsinki
10
MD method
MD – atom representations
 MD naturally needs atom coordinates (and velocities)
 Atom coordinates can simply be read in from an ASCII
text file
 Simple but for atoms good enough format: .XYZ
 Arrays in an MD code, e.g.:
double precision :: x(MAXATOMS),y(MAXATOMS),z(MAXATOMS)
Kai Nordlund, Department of Physics, University of Helsinki
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MD method
Initializing atom positions
 Atom coordinates can (should) be initialized to the
structure of interest
 Simplest: single crystal: just make atom coordinates
according to basis vectors. Simple 4D loop over
(x,y,z,basis vectors)
 Some
codes like LAMMPS have built-in construction
routines
 Other possible cases are numerous
 Biomolecules:
use databases of coordinates
 Amorphous material: very tricky to get something similar to
experiments. Simplest: start from random atom coordinates,
simulate at high T and cool down slowly. But this gives often
too much coordination defects. Other approach: WootenWeaire-Winer Monte Carlo, can generate coordinationdefect free materials
 Quasicrystals: not known how to do it!
Kai Nordlund, Department of Physics, University of Helsinki
12
MD method
Initializing velocities
 How to set initial velocities depends on the physical case
to be simulated
 Simplest case: thermal equilibrium: give initial Gaussian
velocities to the atom (=Maxwell-Boltzmann distribution)
 Code snippet:
! Gaussian distributed velocities
x(i) = vmean*gaussianrand(0)
y(i) = vmean*gaussianrand(0)
z(i) = vmean*gaussianrand(0)
1
3
 Mean velocity obtained from 2 mv = 2 k T
 Basic physics note: if you start from perfect crystal
2
mean
E pot = Ekin =
E pot = 0
3
k BT
2
B
positions, atoms initially have Epot = 0, whereas by the
equipartition theorem, at a given temperature T, each
1
degree of freedom has 2 kBT of energy in both Epot and Ekin
 Hence in this case to get desired temperature T, you need
to give initial velocities for 2T and simulate about 1 ps to
get correct equipartitioning of Epot and Ekin
Kai Nordlund, Department of Physics, University of Helsinki
13
MD method
MD – Solving equations of motion
 The solution step r(i+1) = r(i) +v(i)Δt + 1/2 a Δt2 + correction
terms is crucial
 What are the “correction steps”?
 There is any number of them, but the most used ones are
of the predictor-corrector type way to solve differential
equations numerically:
Predictor stage: predict next atom positions:
Move atoms: rp = r(i) + v(i) Δt + 1/2 a Δ t2 + more accurate terms
Update velocities: vp = v(i) + a Δt + more accurate terms
Get forces F = - ∇ V(rp) or F = F( Ψ(rp) ) and a = F/m
Corrector stage: adjust atom positions based on new a:
Move atoms: r(i+1) = rp + some function of (a, Δt)
Update velocities: v(i+1) = vp + some function of (a, Δt)
Kai Nordlund, Department of Physics, University of Helsinki
14
MD method
MD – Solving equations of motion
 Simplest possible somewhat decent algorithm: velocity
Verlet
[L. Verlet, Phys. Rev. 159 (1967) 98]
 Another, much more accurate: Gear5, Martyna
I
recommend Gear5, Martyna-Tuckerman or other methods
more accurate than Verlet – easier to check energy
conservation
[C. W. Gear, Numerical initial value problems in ordinary differential equations,
Prentice-Hall 1971; Martyna and Tuckerman J. Chem Phys. 102 (1995) 8071]
Kai Nordlund, Department of Physics, University of Helsinki
15
MD method
MD – time step selection
 Time step selection is a crucial part of MD

Choice of algorithm for solving equations of motion and time step
are related
 Way too long time step: system completely unstable, “explodes”
 Too long time step: total energy in system not conserved
 Too short time step: waste of computer time

Pretty good rule of thumb: the fastest-moving atom in a system
should not be able to move more than 1/20 of the smallest
interatomic distance per time step – about 0.1 Å typically
Kai Nordlund, Department of Physics, University of Helsinki
16
MD method
MD – Periodic boundary conditions
 A real lattice can be extremely big
 E.g.
1 cm of Cu: 2.1x1022 atoms => too much even for
present-day computers
 Hence desirable to have MD cell as segment of bigger real
system
 Standard solution: periodic boundary conditions
 This
approach involves “copying” the simulation cell to each
of the periodic directions (1–3) so that our initial system
“sees” another system, exactly like itself, in each direction
around it. So, we’ve created a virtual infinite crystal.
Kai Nordlund, Department of Physics, University of Helsinki
17
MD method
MD: periodics continued
 This has to also be accounted for in calculating distances
for interactions
 “Minimum image condition”: select the nearest neighbour
of an atom considering all possible 27 nearest cells
 Sounds tedious, but
can in practice be
implemented with
a very simple comparison:
Kai Nordlund, Department of Physics, University of Helsinki
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MD method
MD – Boundary conditions
 There are alternatives, though:
 Open boundaries = no boundary
condition, atoms can flee freely to
vacuum
 Obviously
for surfaces
 Fixed boundaries: atoms fixed at
boundary
 Unphysical,
but sometimes needed for
preventing a cell from moving or
making sure pressure waves are not
reflected over a periodic boundary
 Reflective boundaries: atoms
reflected off boundary, “wall”
 Combinations of these for different
purposes
Kai Nordlund, Department of Physics, University of Helsinki
19
MD method
MD – Temperature and pressure control
 Controlling temperature and pressure is often a crucial part
of MD
 “Plain MD” without any T or P control is same as simulating
NVE thermodynamic ensemble
 NVT ensemble simulation: temperature is controlled
 Many
algorithms exist, Nosé, Berendsen, …
 Berendsen
does not strictly speaking simulate thermodynamic
NVT ensemble! – but is still often good enough
 NPT ensemble simulation: both temperature and pressure is
controlled
 Many
algorithms exist: Andersen, Nosé-Hoover, Berendsen
 Berendsen
does not strictly speaking simulate thermodynamic
NPT ensemble – but is still often good enough
Kai Nordlund, Department of Physics, University of Helsinki
20
MD method
Berendsen method equations
 In the Berendsen methods, the equations of motion are
modified with a frictional term γ i mi vi as
mai = Fi − γ i mi vi
where the velocity scaling is shown [Berendsen, J. Chem.
Phys. 81 (1984) 3685] to be achieved by a simple scaling
of all velocities v → λv every time step with
where T0 is the desired temperature and τT the
temperature scaling time constant.
 In Berendsen pressure control, the system size and all
atom coordinates are scaled with a similar factor
 Here β is the inverse of the bulk modulus of the system
and τP the pressure scaling time constant.
Kai Nordlund, Department of Physics, University of Helsinki
21
MD method
Note on pressure control
 Never use pressure control if there is an open
boundary in the system!!
 Why??
 Think about it...
 Hint: surface tension and Young’s modulus
Kai Nordlund, Department of Physics, University of Helsinki
22
MD method
MD – cellular method and neighbour lists
 To speed up MD for large (> 100 or so) numbers of
atoms, a combination of neighbour list and a cellular
method to find the neighbours is usually crucial
 If one has N atoms, and want to find the neighbours for a
finite-range potential, a direct search requires N2
operations – killing for large N
 Solution: if potential cutoff = rcut,
divide atoms into boxes of
size >= rcut, search for neighbours
only among the neighbouring cells
 Neighbour list: form a list of
neighbours within rcut+ rskin and
update this only when needed
Kai Nordlund, Department of Physics, University of Helsinki
23
MD method
MD – cellular method and neighbour lists
 To speed up MD for large (> 100 or so) numbers of
atoms, a combination of neighbour list and a cellular
method to find the neighbours is usually crucial
 If one has N atoms, and want to find the neighbours for a
finite-range potential, a direct search requires N2
operations – killing for large N
 Solution: if potential cutoff = rcut,
divide atoms into boxes of
size >= rcut, search for neighbours
only among the neighbouring cells
 Neighbour list: form a list of
neighbours within rcut+ rskin and
update this only when needed
Kai Nordlund, Department of Physics, University of Helsinki
24
Nonequilibrium extensions to MD
Nonequilibrium extensions
 The basic MD algorithm is not suitable for many
nonequilibrium simulations
 But over the last ~30 years augmentations of MD for
nonequilibrium simulations have been developed
 Our
group has specialized in irradiation effects and is deeply
involved in this
 Here
just a few aspects also relevant to other non-
equilibrium systems described
Kai Nordlund, Department of Physics, University of Helsinki
25
Nonequilibrium extensions to MD
Variable time step schemes
 In case the velocities of atoms are varying during the
simulation, it is worth using a variable time step
 Example for irradiation simulations
Here Δxmax is the maximum allowed distance moved
during any t (e.g. 0.1 Å), Δ Emax is the maximum allowed
change in energy (e.g. 300 eV), vmax and Fmax are the
highest speed and maximum force acting on any particle
at t, respectively, cΔt prevents sudden large changes (e.g.
1.1), and tmax is the time step for the equilibrated system.
 This relatively simple algorithm has been demonstrated to
be able to handle collisions with energies up to 1 GeV
Kai Nordlund, Department of Physics, University of Helsinki
[K. Nordlund, Comput. Mater. Sci. 3, 448 (1995)].
26
Nonequilibrium extensions to MD
Special boundary conditions
 In a zone which is far from thermodynamic equilibrium,
one should not use any thermodynamic ensemble
simulation method!
 Only use NVE = direct solution of Newton’s equation of
motion, ”Free MD”
 Example for various irradiation cases:
Kai Nordlund, Department of Physics, University of Helsinki
27
Interatomic potentials
Whence the interactions?
 Recall from the MD algorithm:
Get forces F = − ∇ V(r(i)) or F =
F(Ψ) and a = F/m
 This is the crucial physics input of the algorithm!
 In
the standard algorithm all else is numerical mathematics
which can be handled in the standard cases to arbitrary
accuracy with well-established methods (as outlined above)
 Forces can be obtained from many levels of theory:
 Quantum
mechanical: Density-Functional Theory (DFT),
Time-dependent Density Functional theory (TDDFT)
- Limit: ~1000 atoms for DFT, ~100 atoms for TDDFT
 Classically:
various interatomic potentials
- Limit: ~ 100 million atoms!
- Most relevant to irradiation effects
Kai Nordlund, Department of Physics, University of Helsinki
28
Interatomic potentials
Interatomic potentials
 In general, total energy of a system of N atoms can be
written:
 Because the energy of a bond should not depend on the
rij
i
j
explicit atom coordinates, and if there are no external
forces V1, this can be simplified to:
θijk
rik
k
 From these the forces can be obtained in principle simply
using

In reality, doing and coding this derivative for already a 3-body
potentials is incredibly tedious: it has to be perfect for energy
conservation in the MD simulation
Kai Nordlund, Department of Physics, University of Helsinki
29
Interatomic potentials
Interatomic potentials
 Interatomic potentials can in general be divided in 2
classes:
 Molecular mechanics force fields
 Used
in chemistry and biophysics and biology, with a few
exceptions non-reactive, i.e. covalent chemical bonds cannot
break
- Also same element can have different interactions
depending on place in molecule: Carbon-1, Carbon-2, …
 Useless
in most materials science applications, since these
typically involve phase changes, bond breaking etc.
 (Reactive) Interatomic potentials
 Chemical
 Only
bonds can break and reform
one atom type per element
Kai Nordlund, Department of Physics, University of Helsinki
30
Interatomic potentials
Reactive potentials





For classical MD, the often only physical input is the potential
Originally simple 2-body potentials, but by now these are almost
completely out of use (except for noble gases and model studies)
Dominant are 3-body potentials, and increasingly 4-body are used
Two major classes of potentials:
Tersoff-like:
1
Vi =  Vrepulsive (rij ) + bijk (rij , rik ,θijk )Vattractive (rij ); bijk ∝
coordination of i
neighbours

Embedded-atom method-like (EAM)
Vi =



Vrepulsive (rij ) + Fi   ρ (rij ) 
neighbours
 j


Both can be motivated in the second momentum approximation of
tight binding (“extended Hückel approximation” if you are a
chemist)

Related to Pauling’s theory of chemical binding
[K. Albe, K. Nordlund, and R. S. Averback, Phys. Rev. B 65, 195124 (2002)]
Kai Nordlund, Department of Physics, University of Helsinki
31
Interatomic potentials
Embedded-atom method like potentials
 The general “EAM-like” functional form


Vi =  Vrepulsive (rij ) + Fi   ρ (rij ) 
neighbours
 j

is actually shared by a large number of different potentials
 A few descriptions of common forms:
 Finnis-Sinclair (1982) and Rosato group potentials:
F
is simply the square root function
 Motivated
from 2nd moment tight binding
 Rosato group potentials: as above, but electron density just a
simple exponential. Simple but often pretty good.
 Confusingly
called “tight-binding potential” even though fully
classical!
Kai Nordlund, Department of Physics, University of Helsinki
32
Interatomic potentials
Embedded-atom method like potentials
 The original EAM form by Foiles, Daw, Baskes ~1985:


Vi =  Vrepulsive (rij ) + Fi   ρ (rij ) 
neighbours
 j

 Here F is the energy to embed an atom into
an electron gas, and ρ an actual electron
density function for neighbours j of atom i.
 Based
on basic idea of metal as sea of
free electrons with positive ion cores
embedded in it
 EMT potentials: original potentials developed ~1980 from
first principles, underlying the later EAM form. Developed
mostly in Finland (Nieminen, Manninen, Puska)
Kai Nordlund, Department of Physics, University of Helsinki
33
Interatomic potentials
Embedded-atom method like potentials…
 And more potentials within same form
 Glue potentials (Ercolessi): pair part has also an attractive
part
 CEM, Corrected Effective Medium theory potentials:
completely different motivation, but can be squeezed to
same form
 ‘Magnetic’ potentials (Dudarev, Derlet): recent addition of
magnetic energy term into an EAM-like potential
[Dudarev, Derlet, J. Phys.: Condens. Matter 17 (2005) 1]
 Finally, to enable description of covalent materials,
Baskes et al extended the EAM form to include an angular
term, “Modifed EAM” = MEAM potentials
Kai Nordlund, Department of Physics, University of Helsinki
34
Interatomic potentials
Interatomic potential development: general
principles
 The general idea of fitting interatomic potentials is to
adjust the parameters into data obtained experimentally or
from deeper-level calculations (typically DFT)
 In case one gets as an end results a potential that can
describe a wide range of properties outside the original
fitting set, the potential can be considered transferable
 Example:
Stillinger-Weber potential for Si: in spite of being fit
in 1985 to only a small set of properties, it has turned out to
describe well a wide range of other properties
- Even very recently, this potential turned out to be the
best in describing threshold displacement energies
[Holmström et al, Phys. Rev. B 78, 045202 (2008)].
 Set of properties selected for fit depends on field and aims
Kai Nordlund, Department of Physics, University of Helsinki
35
Interatomic potential development
Potentials developed in general
 In general, potentials suitable for irradiation effects exist:

For almost all pure elements (mainly EAM-like for metals)

For the stoichiometric state of a wide range of ionic materials
- But these do not always treat the constituent elements sensibly,
e.g. in many oxide potentials O-O interactions purely repulsive =>
predicts O2 cannot exist => segregation cannot be modelled
 For
a big range of metal alloys
 Not so many potentials for mixed metal – covalent compounds, e.g.
carbides, nitrides, oxides in non-ionic state
 Extremely few charge transfer potentials
 For organics only ReaxFF for CNOH, extended Brenner for COH
systems
 NIST maintains a potential database, but pretty narrow coverage –
one often really needs to dig deep in literature to find them
Kai Nordlund, Department of Physics, University of Helsinki
36
Interatomic potential development
Potential development aims (materials science
point of view)
 First consider a potential for a pure element A.
 To be able to handle the effects described above, the
potential should give:
 The
correct ground state: cohesive energy, crystal structure etc.
 Describe
all phases which may be relevant
 Describe
melting well
 Describe
defect energetics and structures well
 If we further consider irradiation of a compound AB:
 At high temperatures the compound may segregate, so we
need good models for elements A and B separately!
 Fulfills
all the requirements just given for a pure element
 Describes
well the heat of mixing of the compound
 Describes
defects involving atom types A and B well
Kai Nordlund, Department of Physics, University of Helsinki
37
Interatomic potential development
Potential development approach
 Achieving all this starts to sound prohibitively difficult
 But there is one common factor for the main requirements:
 Melting,
defects and different phases all involve unusual atom
coordination states
 Hence
if we use a framework to fit as many coordination states
of the system as possible, we have some hope of getting many
of the properties right
 A Tersoff (Abell / Brenner)-like formalism can do this!
 This
is our favourite formalism for mixed materials, and hence I
will now describe it in detail
Kai Nordlund, Department of Physics, University of Helsinki
38
Interatomic potential development
Potential development approach
 We start by obtaining information on as many coordination
states as possible:
 Usually
at least:
Z:
1
3
4
6
dimer
graphite diamond SC
8
12
BCC
FCC
 Data from experiments or DFT calculations
 Cohesive energy, lattice constant, bulk modulus for all Z
 Elastic
constants for most important
 Fitting done in systematic approach introduced by Prof.
Karsten Albe (TU Darmstadt)
Kai Nordlund, Department of Physics, University of Helsinki
39
Interatomic potential development
“Albe” fitting formalism
 Use Tersoff potential in Brenner form (unique
mathematical transformation)
 The
3 parameters r0, D0 and β can be set directly from the
experimental dimer interatomic distance, energy and
vibration frequency!
Kai Nordlund, Department of Physics, University of Helsinki
40
Interatomic potential development
“Albe” fitting formalism
 In
nn formulation,
if material follows
Pauling theory of
chemical bonding,
Log Energy/bond
 Key idea:
DFT or expt.
data
dimer
GRA
DIA
BCC
SC
FCC
for all coordinations
Bonding distance
[Albe, Nordlund and Averback, Phys. Rev. B 65 (2002) 195124]
Kai Nordlund, Department of Physics, University of Helsinki
41
Interatomic potential development
“Albe” fitting formalism
 Pair-specific A-B interaction
 Three-body part modified from Tersoff form
Second-moment approximation exponential
No power of 3
ik-dependent angular term
 This form for bij conforms exactly to bijk ∝
1
coordination of i
[Albe, Nordlund and Averback, Phys. Rev. B 65 (2002) 195124]
Kai Nordlund, Department of Physics, University of Helsinki
42
Interatomic potential development
The “blood, sweat and tears” part
 There are all in all 11 parameters that must be specified
 Constructing a good potential means finding suitable values
for these

This is done by fitting to different experimental or densityfunctional theory values of ground state and hypothetical
phases – also for other functional forms than Tersoff

Not a trivial task!
1-2 years
Kai Nordlund, Department of Physics, University of Helsinki
[Schematic courtesy of Dr. Carolina Björkas]
43
Interatomic potential development
Potentials developed by us
 We, and/or the Albe group, have
so far developed potentials for:
 BN,
PtC, GaAs, GaN, SiC, ZnO,
FePt, BeWCH, FeCrC, FeCH,
+ He with pair potentials
 All
these potentials include all the
pure elements and combinations!
 Fitting
code “pontifix” freely
available, contact Paul Erhart
 Just to give a flavor of complexity
that can be modelled: prolonged
irradiation of WC by H and He
Kai Nordlund, Department of Physics, University of Helsinki
44
Nanocluster simulations
Specific example of MD uses: ways to simulate
nanoclusters and their deposition
 A single nanocluster in vacuum is a prototypical
nanoscience system
…
and not irrelevant experimentally: they can be made and
studied in various high vacuum gas condensation setups’
 Simulation of them is also a good case for MD:
 Limited
number of atoms (a few tens to a few hundred
thousand: easy with modern computers) -- fast
 In
high vacuum, cluster does not interact with anything – just
free NVE simulation describes it perfectly
- Or to be precise, there is always electromagnetic
(blackbody) radiation: but time scale for this is long, at
least tens of microseconds at low temperatures.
 No
boundary conditions needed
Kai Nordlund, Department of Physics, University of Helsinki
45
Nanocluster simulations
Nanocluster basics
 A nanocluster (nanoparticle): an agglomerate of some 10-100
000 atoms or molecules bonded together, where the local
environment of each atom/molecule is similar
 Nanoparticle
= any object in the 1 – 100 nm size range
 Nanocluster: a 1 – 100 nm agglomerate of identical atoms or
molecules
- A single molecule of complex structure is not a nanocluster
 Atom cluster ≈ nanocluster
 Nanocrystal = crystalline or polycrystalline nanocluster
 Several different ‘crystallographic’ shapes exist for clusters of
same material
Wulff polyhedron,
single crystalline
Kai Nordlund, Department of Physics, University of Helsinki
Icosahedron,
polycrystalline x 20
Decahedron,
polycrystalline x 5
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Nanocluster simulations
How to make these nanocrystals in MD?
 Rule of thumb:
 CUT
IT OUT!!
 Also
true for nanowires & surface structures
 It
is almost always much easier to cut out the shape from a
bulk shape than to generate it from scratch
 Basic idea:
- Loop over atom positions
- Create atom i in FCC position
- Check if it belongs inside crystal region
- If it does, add it to array
- End loop over atom positions
Kai Nordlund, Department of Physics, University of Helsinki
47
Nanocluster simulations
Checking of belonging
 Checking whether it belongs to the crystal is a nice small
piece of vector algebra
 E.g. for Wulff polyhedron with 100 and 111 facets:
- Set desired sizes in both directions: s100 and s111 based e.g.
on energy optimization described above
- Loop over atom positions
- Create atom i in FCC position ri
- Outside=0
- Loop over 100 facets: f = 100, 010, 001
- If (|ri· f| > s100 ) outside=1
- End loop over 100 facets
- Loop over 111 facets: f = 111, -111, 1-11,11-1
- If (|ri· f| > s111 ) outside=1
- End loop over 111 facets
- If outside==0, add it to array
- End loop over atom positions
Kai Nordlund, Department of Physics, University of Helsinki
48
Nanocluster simulations
Actual code snippet
 Main loop from code which does this:
Kai Nordlund, Department of Physics, University of Helsinki
49
Nanocluster simulations
Making icosahedra & decahedra
 The icosahedral & multiply twinned decahedral clusters
cannot be made this way, however
 They
are NOT pieces cut out of FCC
 But one can use this approach to first make the
constituent parts, then join them together after suitable
rotation
 For
instance multiply twinned icosahedron:
- Make a perfect tetrahedron with side length
corresponding to the desired size
- Then make 20 copies, rotate them suitably around
20 =
corner closest to origin and join together:
Kai Nordlund, Department of Physics, University of Helsinki
(111)
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Nanocluster simulations
Rotation
 Clusters coming from a cluster ion source or in a liquid
solution will have a random rotation for sure!
 Hence after creation, they should be rotated randomly
 This is a simple piece of geometry, bearing in mind
however how to choose a random angle in 3D with correct
weighting of angles!
 Use e.g. Euler angles, form a rotation matrix and then
rotate
Kai Nordlund, Department of Physics, University of Helsinki
51
Nanocluster simulations
Rotation – actual code for single vector
Kai Nordlund, Department of Physics, University of Helsinki
52
Nanocluster simulations
Thermalizing – or not?
 How to thermalize a nanocluster correctly is a nontrivial
question
 Adding random velocities corresponding to desired
temperature is trivial per se, as usual in MD
 However, what is the distribution of energy to
translational, rotational and vibrational degrees of
freedom??
 This
question does not have a unique answer!!
… but answer will depend on how the nanocluster was
synthesized
 Interesting side aspect: a nanocluster thermalized by
random atom velocities will almost certainly be rotating
 If
no rotations are desired, one should first determine axis of
gyration, then the angular velocity, then subtract velocities
corresponding to this one out of atoms
 But then energy is lost from system – is this correct?
- Again - no unique answer!
Kai Nordlund, Department of Physics, University of Helsinki
53
Nanocluster simulations
Controlling nanocluster temperature and
rotation
 After a cluster has been made, one might want to control
its temperature and rotation
 But also here one has to be careful:
 At
least the very common Berendsen T control has the
artefact that it tends to speed up free nanocluster rotation
- Obviously physically incorrect
 Nose-Hoover does not seem to have the same problem(?)
 Removing angular momentum is possible:
- First one has to calculate the rotational axis, then
subtract out the rotation
- Can be made to work, but also cause artefacts: we
recently got an error in a metal nanocluster melting point
simulation of 100 K due to removing rotation
 On more than one occasion, the only solution that worked
was setting the desired T initially (maybe over a few ps
NVT simulation), then run in pure NVE ensemble with
short enough time step to keep energy very well
conserved
Kai Nordlund, Department of Physics, University of Helsinki
54
Nanocluster simulations
Direct simulation of nanocluster production
 An alternative approach to making clusters: simulate the
formation directly!
 Fairly
(or even fully)
realistic
 Kinetic
effects included
 Rotations,
vibrations etc.
will come automatically
 But quite time-consuming;
long MD sim needed
[E. Kesälä, A.Kuronen and K. Nordlund, Phys. Rev. B (2005) ]
Kai Nordlund, Department of Physics, University of Helsinki
55
Nanocluster simulations
Direct simulation of nanocluster production
 A more recent example: formation of mixed Si/Ag clusters
Kai Nordlund, Department of Physics, University of Helsinki
[J. Zhao, F. Djurabekova, K. Nordlund 2014]
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Nanocluster simulations
Direct simulation of nanocluster production
 Final shape may be all but spherical!!
 Example:
Kai Nordlund, Department of Physics, University of Helsinki
SiGe clusters with different Ge compositions
57
Nanocluster simulations
Simulation of nanocluster deposition
 After the nc is created, simulating
deposition/bombardment on a surface is easy
 Place it above and give desired kinetic energy towards it
E=X meV/atom
<hkl>
Kai Nordlund, Department of Physics, University of Helsinki
58
Nanocluster simulations
Example, with rotation and all
 Cu
cluster on
Cu at
very low
energy
Kai Nordlund, Department of Physics, University of Helsinki
[Meinander et al, Thin Solid Films 425, 297 (2002); Appl. Phys. Lett. 89, 253109 (2006)]
59
Nanocluster simulations: introduction to tutorial
Introduction to tutorial simulations
 A classic question in physics is how the melting
temperature of nanoclusters depends on their size
 Theory predicts a smooth behaviour with cluster melting
point going down with diameter due to weakening from
surface [original derivation:
Area
Volume
C
r2
− C2 3 = Tmelt ,bulk − 2
r
r
Tmelt (r ) = Tmelt ,bulk − C1
P. Pawlow Z. Phys. Chem. 1909]:
(C1, C2 are constants determined
from surface/interface energy)
= Tmelt ,bulk
 Experimental confirmation
came in 1976
[Buffat, Borell, Phys. Rev. A 13, 2287]
Kai Nordlund, Department of Physics, University of Helsinki
60
Nanocluster simulations: introduction to tutorial
How to set up a nanocluster melting
simulation?
 Remember the note on possible T control artefacts above!
 Safest approach:
 Create
 Set
nanocluster of desired size (e.g. d=5 nm)
the initial temperature of the atoms using the Maxwell-
Boltzmann initialization (see above) for T = 2xdesired T
 Run
simulation of nanocluster in pure NVE ensemble (no T
or P control) in cell with periodic boundary conditions much
larger than cell diameter
 Determine
actual temperature of cell by averaging over a
large number of time steps
 Monitor
visually or some analysis method whether it melts
 Repeat
using binary search in a wide T range until you find
the melting point with desired accuracy
 Compare
Kai Nordlund, Department of Physics, University of Helsinki
with experiments and literature
61
Nanocluster simulations: introduction to tutorial
Recent example
 Test of melting point of a new
bond-order potential for Au
[M. Backman, N. Juslin, and K. Nordlund, Bond order
potential for gold, Eur. Phys. J. B 85, 317 (2012)]
Kai Nordlund, Department of Physics, University of Helsinki
62
Molecular statics
Molecular statics: definition
 In some applications one wants just to find the ground
state atom coordinates of a system, i.e. minimize the total
energy to the nearest energy minimum. This is called
molecular statics
 Note:
now we do not discuss trying to find the global energy
minimum, which is still another task
 If you have an MD code, you can always do this by
simply scaling the T in the simulation slowly down to
0, and run MD
 However, this is not particularly efficient approach, since
the atoms vibrate due to the T (velocities)
 There are more efficient ways for this, in particular
conjugate gradients (also widely used in DFT)
Kai Nordlund, Department of Physics, University of Helsinki
63
Molecular statics
Steepest descent energy minimization
 Before we describe conjugate gradients, let’s look at the
obvious approach: steepest decent energy minimization
 Simply
always move down the energy gradient = force
 Obviously
will lead to local energy minimization
 Algorithm:
Kai Nordlund, Department of Physics, University of Helsinki
64
Molecular statics
Line minimization
 Step 3 in the algorithm above is a so called “line
minimization”: simply move in the given vector (force)
direction until you find a minimum
 Standard approach: “minimum bracketing”, Brents method
 If
interested in details, see Press et al, “Numerical recipes”
Kai Nordlund, Department of Physics, University of Helsinki
65
Molecular statics
Conjugate gradient energy minimization
 Steepest decent has a problem, however: it tends to lead
to a zig-zag motion, inefficient for energy minimization
 The “Conjugate gradient” approach is designed to
overcome this problem
 Basic idea: use instead of only force, two vectors g and h
constructed in a way to avoid the zig-zag motion, by being
“conjugate” to each other. Can be achieved fairly simply:
Kai Nordlund, Department of Physics, University of Helsinki
66
Molecular statics
Conjugate gradients (CG) algorithm
 In algorithmic form, this becomes only slightly more
complicated than steepest descent:
If interested in details, see Press et al, “Numerical recipes”
Kai Nordlund, Department of Physics, University of Helsinki
67
Molecular statics
Adaptive conjugate gradients (ACG)
– optimized for interatomic potentials…
 However, in MD the force/energy calculation is the most
time-consuming part, and hence the line minimization is
fairly heavy.
 Nordlund et al developed a scheme that avoids the line
bracketing by using the knowledge that for an atomic
system, “the minimum is out there”.
 Step
size selected with an educated guess of typical shapes
of interatomic potentials
 Increase
 If
step size slowly
energy goes up, return to previous position and reduce
step size x 2
 Speedup of 3-4 compared to normal CG
Kai Nordlund, Department of Physics, University of Helsinki
68
Molecular statics
Adaptive conjugate gradients (ACG) algorithm
[Nordlund et al, J. Appl. Phys. 88, 2278 (2000)]
Kai Nordlund, Department of Physics, University of Helsinki
69
Molecular statics
Example: strain field for Co nanocluster in Cu
 Example of results: analysis of strain field for Co
nanoclusters in Cu
 Note:
displacements from perfect positions exaggerated by a
factor of 3
[C. Zimmermann, M. Yeadon, M. Ghaly, K. Nordlund, J. M. Gibson, R.
S. Averback, U. Herr, and K. Samwer, Phys. Rev. Lett. 83, 1163 (1999)]
Kai Nordlund, Department of Physics, University of Helsinki
70
Molecular statics
MD relaxation
 The velocities available in MD can actually be used to
achieve fairly fast relaxation by a simple trick:
 Start at any temperature T you may be. The switch away
from any other temperature control, and instead for every
atom, form the dot product of the velocity and force:
 
d = Fi ⋅ vi
 If the dot product is negative, atom is moving up the force.
In this case, set

vi = 0
 This way all atoms are guaranteed to move downwards in
force
 This approach may be good to get close to minimum fast,
but for very accurate minimization, CG or ACG is likely
still more efficient
Kai Nordlund, Department of Physics, University of Helsinki
71
The end: practical advices
“Big” MD codes
 MD codes in wide use:
 LAMMPS
= Sandia national labs parallel MD code. By far
the most used materials science code. Open source, free,
well documented, big user base. Works like a script-like
language. Only possible downside “creeping featurism”
- Used now in tutorial
 DL_POLY
= Popular fairly widely used code from the UK.
Works for both inorganic and organic materials.
 Gromacs
= Dominating MD code for biophysics simulations.
Only supports molecular mechanics force fields, so not
suitable for materials science
 More info on any of these: just google on the name
Kai Nordlund, Department of Physics, University of Helsinki
72
The end: practical advices
Further reading
 General:
 Classic
book: Allen-Tildesley, Molecular dynamics
simulations of liquids, Oxford University Press, 1989.
 Newer
book: Daan Frenkel and Berend Smit. Understanding
molecular simulation: from algoritms to applications.
Academic Press, San Diego, second edition, 2002
 Ion irradiation-specific reviews:

K. Nordlund and F. Djurabekova, Multiscale modelling of irradiation
in nanostructures, J. Comput. Electr. 13, 122 (2014).

K. Nordlund, C. Björkas, T. Ahlgren, A. Lasa, and A. E. Sand,
Multiscale modelling of Irradiation effects in fusion reactor
conditions, J. Phys. D: Appl. Phys. 47, 224018 (2014).
Kai Nordlund, Department of Physics, University of Helsinki
73
Examples of MD modelling results
Research examples
 Time permitting…
Kai Nordlund, Department of Physics, University of Helsinki
74
Examples of MD modelling results
1. Chemical sputtering of metals
 Using classical molecular dynamics and experiments, we
showed that pure metals can sputter chemically
−
Light metals under H bombardment sputter by breaking bonds!
[Björkas et al, New J. Phys. 11, 123017 (2009);
Nordlund et al, NIM B 269, 1257 (2011)]
75
Kai Nordlund, Department of Physics, University of Helsinki
75
Examples of MD modelling results
2. Swift heavy ion effects on materials
 Swift heavy ions (Ekin > 100 keV/amu) can be used to
fabricate and modify nanostructures in materials
 We are using multiscale modelling of electronic energy
transfer into atom dynamics to determine the effects of swift
heavy ions on materials
 We have explained the
mechanism by which swift
heavy ions create
nanostructures in silicon, silica
and germanium and change
nanocrystal shapes
[Ridgway et al, Phys. Rev. Lett. 110, 245502 (2013); Leino et al, Mater. Res. Lett. (2013)]
Kai Nordlund, Department of Physics, University of Helsinki
76
Examples of MD modelling results
3. Surface nanostructuring
 Together with Harvard University, we are examining the
fundamental mechanisms of why prolonged ion irradiation
of surfaces leads to formation of ripples (wave-like
nanostructures)
 We overturned the old paradigm that ripples are produced
by sputtering and showed ab initio that they can in fact be
produced by atom displacements in the sample alone
Kai Nordlund, Department of Physics, University of Helsinki
[Norris et al, Nature commun. 2 (2011) 276]
77
Examples of MD modelling results
4. Modelling of arc cratering
 We have developed new concurrent multiscale modelling
methods for treating very high electric fields at surfaces
 Using it we are examining with a comprehensive
multiscale model the onset of vacuum electric breakdown
 We have shown that the complex crater shapes observed
in experiments can be explained as a plasma ion
irradiation effect – multiple overlapping heat spikes
[H. Timko et al, Phys. Rev. B 81 (2010), 184109]
Kai Nordlund, Department of Physics, University of Helsinki
78
Examples of MD modelling results
5. Cluster cratering over 40 orders of magnitude
 Using classical MD, we demonstrated that at a
size ~ 10000 atoms, cluster bombardment
starts producing craters with the same
mechanism as meteorites on planets
−
100 million atom simulations with statistics
[J. Samela and K. Nordlund, Phys. Rev. Lett. 101 (2008) 27601, and cover of issue 2]
Kai Nordlund, Department of Physics, University of Helsinki
79
Summary
Molecular dynamics is a powerful
tool
– when you know what you are
doing!
And a lot of fun! 
Kai Nordlund, Department of Physics, University of Helsinki
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Kai Nordlund, Department of Physics, University of Helsinki
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