Coupling of different length scales in molecular dynamics simulations [Elektronische Ressource] / Anja Streit

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• Physik

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• Publié par: technische_universitat_kaiserslautern
• Publié le: 01 janvier 2007
• Nombre de lectures: 55
• Langue: English
• Poids de l'ouvrage: 6 Mo

Extrait

Coupling of di eren t length scales in
molecular dynamics simulations
Anja Streit
Dissertation
D368
Vom Fachbereich Mathematik der Universit at Kaiserslautern
zur Verleihung des akademischen Grades Doktor der
Naturwissenschaften (Doctor rerum naturalium, Dr. rer. nat.)
genehmigte Dissertation
Gutachter:
Prof. Dr. Axel Klar
Prof. Dr. Karsten Albe
Datum der Disputation: 21.12.20062I would like to thank all the people who supported the work on this thesis in various
ways.
First of all, I thank Prof. Michael Junk for the supervision during the rst one and
a half years of my work on this thesis, Prof. Axel Klar for his supervision afterwards
and Prof. Karsten Albe for being the co-referee.
I thank Prof. Achi Brandt and Dr. Sc. Valery Ilyin for interesting and helpful dis-
cussions and comments and a very interesting visit at the Weizmann Institute of
Science.
Special thanks to Dr. Peter Klein for suggesting this interesting problem, for ex-
plaining me the important aspects in this eld from the point of view of a physicist,
many interesting discussions and his help with various problems.
This work was done with the nancial support of Fraunhofer ITWM Kaiserslautern.
I thank Dr. Franz-Josef Pfreundt for giving me the opportunity to work in the Com-
petence Center High Performance Computing and Visualization and all colleagues
from the department. Especially Dr. Susanne Hahn and Dr. Robert Zillich for the
good cooperation and help in theoretical physics.
I thank Dr. Joachim Wipper for carefully reading the manuscript.
Special thanks to my family and friends for their continuous support.Contents
Symbols 5
Introduction 7
1 Molecular dynamics simulations of a sputtering process 9
1.1 Sputtering process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Approximations in molecular dynamics simulations 17
2.1 Appro in the simulation of the sputtering process . . . . . . 17
2.2 De nition of microscopic stress tensor . . . . . . . . . . . . . . . . . . 19
2.3 Comparing the solutions . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Concurrent coupling of length scales method . . . . . . . . . . . . . . 24
2.5 Bridging scales approximation . . . . . . . . . . . . . . . . . . . . . . 27
3 One dimensional analysis and extension of existing methods 33
3.1 Harmonic chain of atoms . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Concurrent coupling of length scales method . . . . . . . . . . 37
3.1.2 Bridging scales approximation . . . . . . . . . . . . . . . . . . 43
3.2 Longer ranging harmonic and anharmonic potentials . . . . . . . . . 50
3.2.1 Concurrent coupling of length scales . . . . . . . . . . . . . . 52
3.2.2 Bridging scales approximation . . . . . . . . . . . . . . . . . . 55
4 Hamiltonian formulation, boundary conditions and interpolation
functions 61
4.1 Bridging scales approximation not energy conserving . . . . . . . . . 62
4.2 Orthogonal displacement splitting . . . . . . . . . . . . . . . . . . . . 63
4.3 Coupling of atomistic and coarse scale region . . . . . . . . . . . . . . 66
4.4 Calculation of the random force . . . . . . . . . . . . . . . . . . . . . 71
4.5 Mori-Zwanzig projection operator formalism . . . . . . . . . . . . . . 75
4.6 Choosing the interpolation functions . . . . . . . . . . . . . . . . . . 79
34 CONTENTS
5 Approximations and numerical results for one dimensional har-
monic potentials 83
5.1 Computation of better interpolation weights . . . . . . . . . . . . . . 83
5.2 Approximations in the Lagrangian for harmonic potential . . . . . . . 90
5.3 Numerical integration and other numerical errors . . . . . . . . . . . 95
6 Two dimensional examples 103
6.1 Two simulations . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 New interpolation functions in two dimensions . . . . . . . . . . . . . 105
6.3 Re ectionless boundary condition in two dimensions . . . . . . . . . . 110
6.3.1 Small atomistic system . . . . . . . . . . . . . . . . . . . . . . 111
6.3.2 Coupling of atomistic and coarse scale . . . . . . . . . . . . . 112
6.4 Test cases, comparison of interpolation functions . . . . . . . . . . . . 115
Summary 125Symbols
a equilibrium distance of atoms0
a atomistic accelerations
b ne scale displacements
B matrix of interpolation weights from ne scale values to the
equilibrium positions of the atoms
c() wave speed
C elasticity constant
lC ( ;
) l-times continuous di eren tiable mapping from
to
1 2 1 2
d displacements of coarse scale nodes
E energy
E instantaneous kinetic energykin
E potential energypot
f,F forces
F coarse scale deformation gradient
g(x) weight function
H Hessian
I identity matrix
k spring constant
k Boltzmann’s constantB
L Laplace transformation
^L Liouville operator
m atomic mass
m~ space dimension
m^ number of extra weights for new interpolation
M atomic mass matrix
n number of atoms
n~ ratio of the distance of coarse scale nodes, compared
to equilibrium distance of atoms
N matrix of interpolation weights from coarse scale values
to the equilibrium positions of the atoms
p atomistic momenta
P projection operator
q atomic positions
0q atomistic equilibrium positions
Q orthogonal projection operator
56 SYMBOLS
r interatomic distance in undeformed state0;
r in in deformed state
R(t) random force
T temperature
u atomic displacements
u coarse scale part of atomistic displacementsc
u ne scale part oftsf
w interpolation weight
W energy density
x positions of coarse scale nodes
(x) delta function
Kronecker symbolj;k
t time discretization
T coarse scale time step
x, y atomic discretisation
X, Y coarse scale
relative error in the energyr
wave vector
phase space density
stress tensor
con gurational part of the stress tensorC
kinetic part of the stress tensorK
( t) memory kernel
potential energy
! wave frequency
B ne scale momenta
H Hamiltonian function
L Lagrangian
O Landau symbol
P coarse scale momentaIntroduction
Molecular dynamics, the numerical solution of the Newtonian evolution equations
for particles, rst came up in the 1950th when the rst computer simulations of
liquids were carried out [5]. It is constantly developed further and is nowadays a
widely used method to study diverse e ects in material science of liquids and solids.
A problem of such simulations is, that the system under consideration often
consists of a huge number of particles with complicated interaction and requires a
very small time step for the numerical solution of the system of ordinary di eren tial
equations. The simulation of a huge system with molecular dynamics is therefore
often not directly possible. In many cases, periodic boundary conditions [5] can
be used to avoid this problem, but it depends on the e ects of interest if such an
approximation is applicable. E.g. for the simulation of surface coating by sputtering,
we cannot use periodic boundary conditions, at least not in growth direction, since
a free surface is necessary there. A possible solution is to describe the surface e ects
with an atomistic model and to use a coarser model several atomic layers away from
the boundary.
Such problems do not only occur in molecular dynamics simulations but in dif-
ferent areas in the computer simulation of real materials. To describe an e ectiv e
material behaviour often requires to take into account di eren t physical e ects. The
characteristic length and time scales of these can be very di eren t. Therefore, a sim-
ulation of all relevant e ects on the nest scale is often impossible. The di eren t
e ects can thereby be important in di eren t parts of the domain or in the whole do-
main, and they usually in uence each other. Di eren t mathematical models, valid
on di eren t length and time scales or in di eren t parts of the domain, are then
necessary to describe the complex behaviour. This models can be e.g. quantum-
mechanical, atomistic or continuum descriptions.
The coupling of di eren t models, the transition between them and the e ectiv e
numerical solution are therefore a basic requirement for the study of material be-
haviour by computer simulation. Dependent on the type of interaction between the
di eren t scales, several coupling or approximation methods have been developed.
The rst class are the hierarchical methods. The computations on each scale are
performed separately, and the results determine the parameters or the constitutive
equations on the next coarser scale (e.g. [24, 25, 26, 6]). The ne scale computations
in this upscaling approaches are often performed on a small representative volume
element, to compute the e ectiv e properties for the model on the next coarser level.
An example is the computation of macroscopic properties like the di usion coe cien t
or the viscosity from microscopic simulations. They are then used in macroscopic
78 INTRODUCTION
simulations on the level of continuum mechanics.
Another possibili