COMPUTATIONAL MODELING OF MATERIALS: THREE HIERARCHICAL LEVELS

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)