COMPUTATIONAL MODELING OF MATERIALS: THREE HIERARCHICAL LEVELS
10-9
10-6
10-3
NUMERICALMETHODS
Process model forbulk/sheet forming
relations inpoly crystalline materialsFEM
Eshelby methodMonte Carlo method
Molecular staticsMolecular dynamicsMonte Carlo method
FDM
Phase transformation
100
Macroscopic
(Structural)
Mesoscopic
(Grain level)
Microscopic(Nanoscopic)(Atomic level)
Len
gth
scal
es,m
SCALE THEORYCOMPUTATIONALISSUES APPLICATION
Continuum MechanicsThermodynamics(Constitutive equation)
FEM
Representative volumeelement concept
Quantum mechanicsInteratomic potentials
VisualizationLarge scale computing*Adaptive auto-remeshing
Massively parallel computing*Domain decomposition*Data structure for paralleladaptive solution
Integration of various codesComputation time
Limited by number of grains
Limited by time (ps) andspace (104 -106 atoms)Parallel Molecular Dynamics(PMD, code developed atSandia, Los Alams, Ames)
Self-consistent method
Microstructure evolution
Composite mechanics
Defects (e.g. dislocation,grain boundary)
Grain boundary sliding
Crack-tip evolution
Structural design
Structure-property relationsHomogenization methods
Atomistic SimulationMethod
Molecular StaticsT=0K
Molecular DynamicsT>0K, Stress
Interatomic Potentials, Ei(We use EAM potential for Al)
Specify Xio, T, Specify atomic positionsof all atoms Xio
Positions of all atoms(including defects)
Positions of all atomsat T,
Ei(Xi)
Xi
=0Ei(Xi)
XiFi
=
Atomistic Simulation Method
Embedded Atom Method Energy Functions(D.J.Oh and R.A.Johnson, 1989 ,Atomic Simulation of Materials, Edts:V Vitek and D.J.Srolovitz,p 233Edts:V Vitek and D.J.Srolovitz,p 233)
The total internal energy of the crystal
12
1
1
tot ii
i i ijj
i ijj
E E
E F r
f r
where
and
Internal energy associated with atom i
Embedded Energy of atom i. Contribution to electron density of ith atom and jth atom. Two body central potential between ith atom and jth atom.
iF
ijf r
ij
iE
n
e e
me e
m
F a b
f r
Embedded energy is of the form
where
and e cr r
0
/ 1 / 10
/ /
( ) ( ) ( ), and ( ) ( )
where
( ) , and ( )
( ) ( ) ( ) ( ) / ( )
( ) ( ) ( ) ( ) / ( )
1
e e
e c e
old c r ld c
r r r rold e ld e
c old c old c c
c old c old c c
r r r r
f r f r f r r r
f r f e r e
f r f r g r f r g r
r r g r r g r
g r e
are equilibrium and cut off interatomic separations
The constants required in the above equation are listed for Al, Cu and Mg atoms below
Al Cu Mg
5 5 6
10.5 8.5 10.5
20 20 20
rc 1.9 1.9 1.7
e 0.12538 0.36952 0.14720
a -4.8144 -4.0956 -1.1049
b 0.47685 -1.6979 -1.3122
e 12.793 12.793 12.316
GRAIN STRUCTURE AND COMPUTATIONAL CRYSTAL
CONSTRUCTION OF COMPUTATIONAL CRYSTAL
CONSTRUCTION OF COMPUTATIONAL CRYSTALCONSTRUCTION OF COMPUTATIONAL CRYSTAL
BOUNDARY CONDITIONS FOR GB SLIDING
"PURE" GBS PROCESS AND AND ENERGY
COUPLED GBS AND MIGRATION
3 GB Sliding and Migration at 500K (Applied Shear)
Note: At Same loading and simulation time (5ps)
GB SLIDING MEASUREMENT
0
1
2
3
4
5
6
7
8
9
1 0
-6 0
-5 0
-4 0 -30 -2 0
-1 0
0 10 20 30 40 5 0 60
D istance from gra in bo u nd ary , an gstro m
3 (11 1)_
5 p s
4 p s
3 p s
2 p s
1 .5 p s
0 .5 ps
GB ENERGY AND SURFAC ENERGY
GB (CSL) 3
(112)
9
(221)
9
(114)
11
(113)
33
(332)
43
(225)GB plane(hkl)--------
Egb
Egbmax
hkl
(111) (335)
3 11
0.024 2.03 2.69 2.18 0.91 2.59 2.34 2.62
6.88 8.01 7.90 8.58 9.16 7.56 8.21 7.79
2.45 5.99 6.27 5.46 6.03 5.71 6.64 3.66
All energies are in unit of eV/A2 x10-2
Egb : Equilibrium Grain Boundary Energy
Egbmax : Maxium Grain Boundary Energy During 'Pure' GBS
2hkl : Surface Energy of two Grain Boundary Palnes
Conclusion Since maxium grain boundary energy during GBS (Egbmax)
is less than surface energy of the grain boundary palne (2hkl), GBS ismore favorable than the formation of voids.
GB Energy vs. Misorientation angle
(
111)
(
112)
(
221)
1
4)
(113
)
(334
)
(331
)(
552)
(225
)
(443
)
(556
)
(551
)
(441
)
(332
)
(335
)
(a)
0
1
2
3
4
0 20 40 60 80 100 120 140 160 180
Our Calculation
Experimental Result
0
1
2
3
0 20 40 60 80 120 140 160 180
100
(b)
(
111)
(113
)
(
112)
Egb
,eV
/A2
Egb
,eV
/A2
MAGNESIUM EFFECT ON GB ENERGY OF Al
0
0.1
0.2
0.3
0.4
0.5
-15 -10 -5 0 5 10 15
with Mg
without Mg
3
Mg location from the grain boundary plane
Egb
,eV
/A2
Egb
,eV
/A2
2.65
2.7
2.75
2.8
2.85
2.9
2.95
3
3.05
-15 -10 -5 0 5 10 15
without Mg
with Mg9
Mg location from the grain boundary plane
y
Mg
Al
GB
Energy distribution at grain boundaries
-3.6
-3.56
-3.52
-3.48
-3.44
-10 -5 0 5 10
-3.6
-3.56
-3.52
-3.48
-3.44
-10 -5 0 5 10
-3.6
-3.56
-3.52
-3.48
-3.44
-10 -5 0 5 10
-3.6
-3.56
-3.52
-3.48
-3.44
-10 -5 0 5 10-3.6
-3.56
-3.52
-3.48
-3.44
-10 -5 0 5 10
-3.6
-3.56
-3.52
-3.48
-3.44
-10 -5 0 5 10
3 5
9 11
27 57
Distance from grain boundary (A)
Ene
rgy
perat
om(e
V)
oDistance from grain boundary (A)
o
Distance from grain boundary (A)o
Distance from grain boundary (A)o
Distance from grain boundary (A)o
Distance from grain boundary (A)o
Ene
rgy
perat
om(e
V)
Ene
rgy
perat
om(e
V)
Ene
rgy
perat
om(e
V)
Ene
rgy
perat
om(e
V)
Ene
rgy
perat
om(e
V)
MAGNESIUM EFFECT ON GB Al-3
0 ps 1.5 ps
2 ps 5 ps
1
2
R
1
2
R
1
2
R
1
2
R
Al
Mg
Applied shear stress: 0.02 eV/A2Note:
Energy distribution in equilibrium structure of tilt grain boundary
-3.7
-3.65
-3.6
-3.55
-3.5
-3.45
-3.4
-3.35
-25 -15 -5 5 15 25
Distance from grain boundary
Egb,
eV
pure Al
Al5%Cu
Al5% Mg
Energy distribution in presence of Mg atoms
-3.6
-3.55
-3.5
-3.45
-3.4
-3.35
-25 -15 -5 5 15 25
Distance from grain boundary
Egb,
eV
pure Al
Al1%Mg
Al2%Mg
Al4%Mg
Al5% Mg
Energy distribution in presence of Cu atoms
-3.66
-3.64
-3.62
-3.6
-3.58
-3.56
-3.54
-25 -15 -5 5 15 25
Distance from grain boundary
Egb,
eV
pure Al
Al1%Cu
Al2%Cu
Al4%Cu
Al5%Cu
Energy variation of a atom whrn Mg atoms distributed with in 10 atomic layers across
GB
-3.59
-3.58
-3.57
-3.56
-3.55
-3.54
-3.53
-3.52
-50 -30 -10 10 30 50
Distance from grain boundary
Egb,
eV Al 1%Mg
pure Al
Radius Distribution Function (RDF)