Molecular Dynamics Study of
Solidification in the Aluminum-Silicon
System
Supervisor: Dr. Jeffrey J Hoyt
Peyman Saidi
Winter 2013
Motivation (Importance of Aluminum Alloys)
279.5
300
98.50
Wheels
54.50
Heads
225
250
43.45
Suspension Arms and Links
23.0
Pounds per Vehicle
Transmissin Cases
200
150
17.9
Brake Calipers
32.50
Steering Knuckles
135
22.00
Blocks
100
Bumper Beams
13.80
Closures
13.76
8.65
Subframes
Transfer Cases
50
2.0
33.00
3.30
IP Beams
2.00
Front Structures
1.60
Complete BIW 0.10
0
1990
2000
2006
0
Aluminum – Mineral Commodity Summary 2011
20
40
60
80
100
2
Motivation (Quench Modified Aluminum Silicon Alloy)
Al-Si Eutectic 20μm/s
Al-Si Eutectic 250μm/s
V (μm/S)
20
80
250
600
1000
UTS (MPa)
162
179
190
207
222
Al-Si Eutectic 950μm/s
El (%)
8.7
15
17.6
23.8
12.5
T. Hosch, et al. Material science and engineering A 528 (2010) 226–232.
3
Growth of Silicon crystals
Temperature
Melting
Point
Advacancy
Kink
Step
4
Twin Plane Reentrant Edge Mechanism (TPRE)
Twin Planes
Twin Planes
Twin Planes
Twin Planes
D. R. Hamilton and R. G. Seidensticker J. Appl. Phys. 31, 1165 (1960)
 Propagation mechanism in Silicon dendrites is based on interaction of twinning.
 Quench modified fibrous silicon is twin free.
What is the growth mechanism of quench modified silicon in Aluminum-Silicon alloy?
What is the critical condition for the transition from anisotropic to isotropic growth for Silicon dendrites?
What is the magnitude and anisotropy of step kinetic coefficient, free energy and the stiffness of the interface?
What is the effect of twins on kinetic parameters at different undercooling and compositions?
5
Molecular Dynamics Method:
Newton’s equations of motion:
() ()
()
()
-ÑE x = F x = mv t = mx t
Cutoff
distance
Where are we?
Stability of System
What do we need to run a MD simulation?
G=H-TS
H: Heat Content
S: Randomness
A model describing all interactions in the system
H=U+PV
Implementing model in MD code
U: Internal Energy
U=K+E
Potential energies for all interactions
K: Kinetic Energy
E: Potential Energy
Vibration
Interaction of
atoms
Rotation
Translation
Aluminum
Silicon
Al-Si
6
Molecular Dynamics models: Aluminum (Embedding Atom Method)
(
()
)
( )
(
(
)
E 1, . . . , N = åf1 i +åf2 i, j + å f3 i, j,k +. . . + f N 1, . . . , N
i
i<j
)
i< j<k
Aluminum
FCC Crystal structure, metallic bond
( ) ( )
1
åf r +F r
2 j¹i 2 ij a i
( )
ri = åfb rij
j¹i
Embedding Energy
Pair interaction
Partial Density Contribution
Ei =
0
0
1
2
3
4
5
Distance between Neighboring Atoms
6
1
2
3
4
Distance Between Neighboring atoms
5
0
M. S. Daw and M. I. Baskes PHYSICAL REVIEW B 29, 6443 (1984)
Density Contribution of Neighboring Atoms
7
Molecular Dynamics models: Silicon (Stillinger-Weber)
(
()
)
( )
(
(
)
E 1, . . . , N = åf1 i +åf2 i, j + å f3 i, j,k +. . . + f N 1, . . . , N
i
i<j
)
i< j<k
Silicon
Diamond crystal structure, covalent bonds
Ei =
( )
( ) ( )(
1
åf r + å f r f r cosq jik - cosq0
2 j¹i 2 ij j,kÌT j ij k ik
)
2
8
F. H. Stillinger and T. A. Weber PHYSICAL REVIEW B 31, 5262 (1985)
Molecular Dynamics models: Aluminum Silicon Interactions (AEAM)
Idea:
Reformulation of Embedding Atom Method by extracting a three body term from the Embedding
functional in order to make these two methods (EAM and SW) compatible with each other.
ìï
1
Ei = å f 2 rij + Fi í 1- di
2 j¹i
ïî
( )
(
) åéëf ( r )ùû + 2 å f ( r ) f ( r ) (cosq
2
ji
j¹i
ij
ji
ij
ki
ik
j,kÌTi
jik
- cosq0
)
üï
ý
ïþ
1/2
ci
A. M. Dongare et al. PHYSICAL REVIEW B 80, 184106, 2009.
Implementing A-EAM
Aluminum Potential
Silicon Potential
Al-Si Potential
9
Silicon Potential (Stillinger Weber)
Table 3: Comparing number and distance to the neighbours in wurtzite and diamond crystal structures for
a = 5.431( Å)
1st Neigh.
N um D i s( Å)
W ur tzi te
4
2.351
D i amond
4
2.351
st ruct ure
2nd Neigh.
N um D i s( Å)
12
3.839
12
3.839
3r d Neigh.
N um D i s( Å)
1
3.918
−
−
4th Neigh.
N um D i s( Å)
9
4.501
12
4.501
" #$%
' $()
* !
* is
! an example
+, of
-'a
.%
/hexagonal
' 01!
T his cryst! al st ruct ure, named aft
er &
t%
he
mineral
wurt zit)e,
cryst al
2345!
! "#$%
! "#$%
! "#$%
syst em. More6specifically,
wurtzite
a cryst
for %binary compounds.
&' ( ' % For t he case
&' (is
' ±&)
% al st ruct
&'ure
* &±&)
/ ' 71%
0839 5!
"$$! %
) "$+! %
) "! +%
! : /sit
' 71%
083' ; <21#/ 5!
t hat all at om
es
are occupied by) one
at om type, wurtzite
is called Lonsdaleite
or hexagonal
) ") &++%
)%
) ") &++%
! +* =. (> ? 3' ; <21#/ 5!
?
3@
A
$B0<C/
5
!
&'
"#,
%
&#"*
(
%
&' ",primary
%
@@
diamond. Fig.10
compares diamond and wurt zit e cryst al st ruct ures. T he
block of
? @D3@A $B0<C/ 5!
, "$* %
, "#, %
' "#%
bot h st ruct ures
t et rahedra
t hat form
may form any of t(hese
? EE3@Ais$B0<C/
5!
! "+%in t he melt and
! "+$%
"+% two st ruct ures
%
@@
D
@@
D
@@
D
[37].
%
325!
%
3C5!
3J 5!
Structure
1 crystal
Neighb.
2 Neighb.
3 Neighb.
4 cells.
Neighb.
Figure
10: a) Diamond
structure b) Wurtzite
unit cell. c) Arrangement
of three wurtzite unit
%
st
Num
For any pot ent4ial in
Diamond
4
%
%
Wurtzite
nd
rd
th
Dis(Å)
Num Dis(Å)
Num
Dis(Å)
Num
Dis(Å)
2.351 dynamics,
12
9
4.0501
molecular
st3.839
ability of t he1cryst al 3.918
st ruct ure belonging
2.351
12
3.839
12
4.0501
t o t he component is t he first condit ion t hat should be sat isfied. Regarding t o St illinger%
10
Table 4: Comparing constants of original and modified Stillinger-Weber potentials
Silicon Potential (Modified Stillinger-Weber)
pot ent ial
A
B
p
q
a
d
c0
SW
2.1672 7.0496 0.6022
4
0 1.80
−
−
M SW
7.3835
5
The difference between cohesive energy of diamond and
wurtzite2.1428
structures
should0.6140
be the 3.5496
same as 0the1.80
value 0.0081
calculated

c1
c2
D
−
−
−
-1 DFT.
2.4259 4.5014
with
 Modification should not affect the characteristics of the Stillinger-Weber potential which accurately predicts silicon properties.
number density (in reduced unit s) is compared for several cryst al st ruct ures, including
als.
wurt zit e, in Fig.11.
0
−
1
.8
5
−
1
.9
−
0
.5
L
a
t
t
ic
e
E
n
e
r
g
y
(
# /N
)
2
1
0
−
1
−
3
x
1
0
−
3
1
.5
2
2
.5
−
1
.9
5
−
2
−
2
.0
5
−
2
.1
−
2
.1
5
−
2
.2
2
.2
B
C
C
−
1
−
1
.5
−
2
−
2
.5
−
3
F
C
C
S
C
−
3
.5
−
4
−
2
2
.3
2
.4
r ( Å)
T
w
o
B
o
d
y
in
te
r
a
c
tio
n
(
e
v
)
−
1
.8
M
S
W
S
W
5 3
3
.5
r ( Å)
−
4
.5
−
4
.2
9
4
4
.5
5
0
L
a
t
t
ic
e
E
n
e
r
g
y
(
# /N
)
T
w
o
B
o
d
y
in
te
r
a
c
tio
n
(
e
v
)
3
0
.0
0
8
1
3
e
v
−
5
−
5
0
.3 0
.3
50
.4 0
.4
50
.5 0
.5
50
.6 0
.6
50
.7
D
e
n
s
ity
(
! ²3
)
−
4
.3
!
−
4
.3
1
−
4
.3
2
!
!
W
u
r
tz
ite
−
4
.3
3
−
4
.3
4
1
02
2
.5−
.6
D
ia
m
o
n
d
Diamondvsstructure
stabilized
by taking
−
4
.3
5 third neighbour in
12: Pair potential
distance. was
The value
of potential
aroundinto
distance to
the
Figure
11:
Lattice energy (per atom) vs density base on modified Si potential for different crystal structures.
account
interaction
with
the
third
neighbor.
d structure is not zero. Beside that this function is zero beyond r = 3.77Å
0
.4
4 0
.4
5 0
.4
6 0
.4
7 0
.4
8 0
.4
9
−
1
5
D
e
n
s
ity
(! ² 3
)
.
11
3
.6 3
.8 4 4
.2 4
.4 4
.6 4
.8 5 ! 5
.2
Minimum of
r ( Å)
!
energy curve for diamond lat t ice occurs at ρσ3 = 0.46 where σ, in lengt h
unit , int roduced by St illinger for comput at ional considerat ions [21]. According t o t his density,
Silicon Potential (Modified Stillinger-Weber)
!" #$
Method
MSW
SW
Experiment
!%#$
TM
1686±10oc
1691±10oc
1686oc
!&#$
Figure 13:
! Coexisting of liquid and solid phases in system with 6000 atoms initially in diamond structure
and full periodic condition. Tow S-L interfaces exist in the box. The atoms are colored according the
!
centro-symmetric
parameter. a) Heating whole system to melting point 1686◦ C for 200ps. b) Melting the
middle part. c) Coexistence of two phases at melting temperature at constant enthalpy.
!
4.3.3
.
Elast i c const ant s
¶s ij
T he elast ic propert ies of a syst em det ermine how st ress and st rain t ensors are relat ed in a
cijkl =
mat erial. Accuracy of elast ic const ant s is import ant , because t hese const ant s are direct ly
employed in pract ical uses of mat erials. In addit ion it can serve as a measure of t he reliability
¶e kl
of t he int erat omic pot ent ial in a given t emperat ure. An elast ic const ant t ensor ci j kl is defined
T,e
t o est ablish relat ion between st ress and st rain wit h respectklt o Hook’s law as following:
σi j = ci j kl ε kl
Term
MSW
SW
Experiment
C11
C12
C44
16.47
7.39
5.2
14.98
7.47
5.23
16.7
6.4
8.2
(1011 dyn/cm2)
(12)
Where σi j is t he st ress and ε kl is t he st rain. In t he case of anisot ropic mat erials Hook’s law
is a funct ion of direct ion:
12
Aluminum-Silicon Potential
Target: Aluminum-Silicon Potential should predict phase diagram accurately.
Pure Components: Gibbs-Helmholtz relation
H 0l - Hs0
DG = T0 ò
dT
Tm
T2
T0
Alloy: The alloy’s free energy as a function of composition
¶DG mix
= DmSi/Al
¶x Si
Semi Grand Canonical Monte Carlo (SGCMC) simulation:
DmSi/Al = -k BTln
T, C, P=0
random Si atom is
switched to Al
U1
U2-U1
æ DU
ö
NSi
Si®Al
exp çç ÷÷
N Al +1
k
T
è
ø
B
Atom Switches back
13
P. Sindzingre, D. Frenkel. Chem. Phys. Lett. (1987) 35–41.
Results and Future Work
 So far
 Angular Embedding atom method was implemented on Molecular Dynamics code
 A modified potential for Silicon was proposed
 Current work
 Making an accurate potential for Al-Si interactions
 Studying and modeling nucleation and solidification growth
 Studying the crystalline anisotropies of the interfacial free energy.
 Characterizing the magnitude and anisotropy of the step kinetic coefficient.
 Examining a faceted to non-faceted transition of the interface by changing undercooling.
 Developing a model for the nucleation and growth phenomena.
 Studying the effect of twinning on crystal growth
 Studying growth rate as a function of undercooling on twinned Si crystals in contact with liquid.
 Studying growth rate as a function of composition on twinned Si crystals in contact with liquid.
14
Thank You