dierk raabe ab initio simulations in metallurgy

MS&T Oct. 2008 Pittsburgh

Coupling Density Functional Theory with Continuum Mechanics for

Alloy Design

Coupling Density Functional Theory with Continuum Mechanics for

Alloy Design

D. Ma*, M. Friák, W. Counts, D. Raabe, J. Neugebauer

Max Planck Institute for Iron Research, Düsseldorf, Germany

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

MS&T Oct. 2008 Pittsburgh

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

Multi-scale Modeling

Å

nm

μm

mm

EH

bv

C

r

VF

MS&T Oct. 2008 Pittsburgh

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

Multi-scale Modeling

Å

nm

μm

mm

EH

bv

C

r

VF

Å mm

EH C

MS&T Oct. 2008 Pittsburgh

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

Two Examples:(1) β-Ti Alloys for

implants(2) Mg-Li Alloys for

lightweight structures

MS&T Oct. 2008 Pittsburgh

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

β-Ti alloy design1. Motivation2. Phase analysis3. Elastic properties4. Elastic constants

as input for CPFEM5. Summary

MS&T Oct. 2008 Pittsburgh

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

β-Ti alloys design1. Motivation2. Phase analysis3. Elastic properties4. Elastic constants as

Input for CPFEM5. Summary

MS&T Oct. 2008 Pittsburgh

1. Motivation

MS&T Oct. 2008 Pittsburgh

1. Motivation

M. Niinomi, Sci. Tech. Adv. Mater. 2003 M. Niinomi, Mater. Sci. Eng. 1998

Main challenges in designing the bone replacement:(1) Bio-compatibility(2) Reduce the elastic stiffness(3) Stabilize the β-phase

Ti-Nb binary system

~20GPa

~70GPa

>100GPa

MS&T Oct. 2008 Pittsburgh

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

β-Ti alloys design1. Motivation2. Phase analysis3. Elastic properties4. Elastic constants as

Input for CPFEM5. Summary

MS&T Oct. 2008 Pittsburgh

2. Phase Analysis

STHF )()1()()()( 11 XExTixEXTiEXTiH xxxx

vc SSS

))1ln()1()ln((B xxxxkS

DFT

MS&T Oct. 2008 Pittsburgh

wanted bcc-based phase that is softer but metastable

Nb

Tiunwanted hcp-based phase

that is stiffer and stableBCC structure of Ti-Nb alloy HCP structure of Ti-Nb alloy

2. Phase Analysis

MS&T Oct. 2008 Pittsburgh

2. Phase Analysis

MS&T Oct. 2008 Pittsburgh

2. Phase Analysis

MS&T Oct. 2008 Pittsburgh

XRDDFT

2. Phase Analysis

MS&T Oct. 2008 Pittsburgh

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

β-Ti alloys design1. Motivation2. Phase analysis3. Elastic properties4. Elastic constants as

Input for CPFEM5. Summary

MS&T Oct. 2008 Pittsburgh

Lattice constantsMinimum energyBulk modulus

3. Elastic Properties

Ab-initio calculation:Equilibrium lattice constants

MS&T Oct. 2008 Pittsburgh

00

02

10

002

1

)(4

3)( 1211

2 CCU tetr

)(2

312112

2

CCU tetr

3

2 1211 CCB

000

00

00

4422)( CU tri

442

2

4CU tri

Ab-initio calculation: Equilibrium elastic constants

ε, strain tensorδ, strainU, elastic energy densityB, bulk modulus

3. Elastic Properties

MS&T Oct. 2008 Pittsburgh

Ab initio calculation results of the elastic constants:

Composition C11 [GPa] C12 [GPa] C44 [GPa] Az EH [GPa]

Ti-18.75at.%Nb

131.2 114.5 26.8 3.210 49.4

Ti-25at.%Nb 143.6 125.9 21.4 2.418 44.2

Ti-31.25at.%Nb

154.8 118.5 19.2 1.058 54.9

C11, C12, C44: elastic stiffness constantsAZ, Zener‘s ratioEH: homogenized Young‘s modulus by Hershey‘s model

3. Elastic Properties

MS&T Oct. 2008 Pittsburgh

Ti-18.75at.%Nb Ti-25at.%Nb Ti-31.25at.%Nb

Az=3.210 Az=2.418 Az=1.058

[001]

[100] [010]

Young‘s modulus surface plots

The elastic properties of the Ti-Nb binary alloys become isotropic as the Nb content increases

Pure Nb

Az=0.5027

3. Elastic Properties

MS&T Oct. 2008 Pittsburgh

3. Elastic Properties

MS&T Oct. 2008 Pittsburgh

single-crystallineC11, C12, C44, B0

polycrystallineYoung modulus

“scale-jumping”

(across the meso-scale)macro-scale

micro-scale 64μ H4 + 16(4C11 + 5C12)μ H

3 + [3(C11+ 2C12) × (5C11+ 4C12) -8(7C11 – 4C12)C44]μ H

2 -(29C11 – 20C12)(C11+2C12)C44 μ H –3(C11 + 2C12)2(C11 – C12)C44 = 0

3. Elastic Properties

MS&T Oct. 2008 Pittsburgh

MECHANICALINSTABILITY!!

theory: bcc polycrystals

3. Elastic Properties

MS&T Oct. 2008 Pittsburgh

MECHANICALINSTABILITY!!

theory: bcc polycrystals

3. Elastic Properties

MS&T Oct. 2008 Pittsburgh

MECHANICALINSTABILITY!!

theory: bcc polycrystals

Ti-hcp: 117 GPa

3. Elastic Properties

MS&T Oct. 2008 Pittsburgh

MECHANICALINSTABILITY!!

Ultra-sonic measurement

exp. polycrystals !

bcc+hcp phases theory: bcc

polycrystals

Ti-hcp: 117 GPa

3. Elastic Properties

MS&T Oct. 2008 Pittsburgh

β-Ti alloys design1. Motivation2. Phase analysis3. Elastic properties4. Elastic constants as

input for CPFEM5. Summary

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

MS&T Oct. 2008 Pittsburgh

4. Elastic Constants as Input of CPFEM4. Elastic Constants as Input of CPFEM

Required input data of the materials properties in crystal plasticity finite element method

MS&T Oct. 2008 Pittsburgh

4. Elastic Constants as Input of CPFEM4. Elastic Constants as Input of CPFEM

Plane strain compression: (1) Influence of the elastic anistropy (2) predict the texture evolution

Bending test: Homogenized elastic

properties of textured and non-texture materials

MS&T Oct. 2008 Pittsburgh

4. Elastic Constants as Input of CPFEM4. Elastic Constants as Input of CPFEM

Elastic constants of a single crystal

flow curve from the compression test on solution annealed Ti30at.%Nb

Random texture

The plastic property is kept, and only the elastic property is varied!!!

MS&T Oct. 2008 Pittsburgh

4. Elastic Constants as Input of CPFEM4. Elastic Constants as Input of CPFEM

εh=0

εh=30%

εh=60%

εh=90%

φ1 (0°~90°)

Φ(0°~90°)

φ2=45°

γ-fiber

α-fiber

0° 90°

90°

0°

MS&T Oct. 2008 Pittsburgh

4. Elastic Constants as Input of CPFEM4. Elastic Constants as Input of CPFEM

Elastic constants of a single crystal

Textured and non texture

MS&T Oct. 2008 Pittsburgh

4. Elastic Constants as Input of CPFEM4. Elastic Constants as Input of CPFEM

0

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Displacement [a.u.]

Loa

d [a

.u.]

Ti18.75Nb Non-TexturedTi18.75Nb TexturedTi25Nb Non-TexturedTi25Nb TexturedTi31.25Nb Non-TexturedTi31.25Nb Textured

Ti18.75Nb Non-TexturedTi18.75Nb TexturedTi25Nb Non-TexturedTi25Nb TexturedTi31.25Nb Non-TexturedTi31.25Nb Textured

MS&T Oct. 2008 Pittsburgh

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

β-Ti alloys design1. Motivation2. Phase analysis3. Elastic properties4. Elastic Constants

as Input of CPFEM5. Summary

MS&T Oct. 2008 Pittsburgh

1. Thermodynamic stability of hcp- and bcc-Ti was studied2. Configurational entropy at finite temperature stabilizes bcc Ti-

Nb phase3. Volume fractions have been calculated using the Gibbs

construction 4. Polycrystalline two-phase Young’s modulus has been

theoretically predicted employing the Hershey and CPFEM homogenization methods

5. Very good agreement between theoretical prediction and experiment

6. The calculated elastic constants (DFT) can be used as input for CPFEM

5. Summary

Nb SHOULD BE THE PRIMARY ALLOYING ELEMENTS IN Ti FOR HUMAN IMPLANT MATERIALS

MS&T Oct. 2008 Pittsburgh

Mg-Li alloy design1. Motivation2. Elastic properties3. Analysis of the Elastic Properties4. Summary

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

MS&T Oct. 2008 Pittsburgh

Mg-Li alloy design1. Motivation2. Elastic properties3. Analysis of the Elastic Properties4. Summary

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

MS&T Oct. 2008 Pittsburgh

• Magnesium (and its alloys) are light weight and relatively strong

• Ideal lightweight structural material

• Magnesium (and its alloys) are generally hcp

• Not ductile, textures• Problematic for industrial

applications (anisotropy)

Magnesium Good Magnesium Bad

How can hcp magnesium be transformed into bcc/fcc magnesium?

1. Motivation1. Motivation

MS&T Oct. 2008 Pittsburgh

1. Motivation

Ultra light-weight structural material Li = 0.58 g/cm3 Mg = 1.74 g/cm3

hcpbcc

hcp+

bcc

MS&T Oct. 2008 Pittsburgh

1. Motivation

Ordered alloys (periodic structures) Ground state calculations (0 K)

11 different bcc alloys

Physical Limitations

Use DFT to find the bcc MgLi alloy composition with optimal elastic properties

Goal:

Calculate single crystal Cij’s

Homogenize to get isotropic polycrystal elastic constants

Analyze engineering ratio’s

MS&T Oct. 2008 Pittsburgh

Mg-Li alloy design1. Motivation2. Elastic properties3. Analysis of the elastic Properties4. Summary

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

MS&T Oct. 2008 Pittsburgh

2. Elastic Properties: Bulk Modulus

0

5

10

15

20

25

30

35

40

0 20 40 60 80 100

at. % magnesium

Bu

lk M

od

ulu

s (G

Pa)

Simulation: 0 K + Experiment: 298 K

Li

MS&T Oct. 2008 Pittsburgh

-10

-5

0

5

10

15

20

25

30

0 20 40 60 80 100

at. % magnesium

Sh

ear

Mo

du

lus

(G

Pa)

Voight & Reuss Bounds

Self Consistent Homgenization

Experiment

Experiment is reasonably well reproduced

Li dominate alloys are very soft

Optimal G (17 GPa)around bcc phase

boundary (70 at % Mg)bcc

Mg is unstable

HerG

ReussG

VoigtG

2. Elastic Properties: Shear Modulus

Li

MS&T Oct. 2008 Pittsburgh

2. Elastic Properties:Young‘s Modulus

-20

-10

0

10

20

30

40

50

60

0 20 40 60 80 100

at. % magnesium

Yo

un

gs

Mo

du

lus

(GP

a)

Voight & Reuss Bounds

Self Consistent Homgenization

Experiment

Li dominate alloys are very soft

Optimal E (45 GPa)around bcc phase

boundary (70 at % Mg) bcc Mg is

unstable

VoigtE

ReussE

HerE

Experiment is reasonably well reproduced

Li

MS&T Oct. 2008 Pittsburgh

2. Elastic Properties: Poisson‘s Ratio

0

0,1

0,2

0,3

0,4

0,5

0,6

0 20 40 60 80 100

at. % magnesium

Po

isso

n's

Rat

io

Voight and Ress Bounds

Self Consistent Homogenization

Experiment

Softer alloys have a higher n

Softer alloys have a lower n

Voigtν

Reussν

Herν

Experiment is reasonably well reproduced

Li

MS&T Oct. 2008 Pittsburgh

Mg-Li alloy design1. Motivation2. Elastic properties3. Analysis of the elastic Properties4. Summary

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

MS&T Oct. 2008 Pittsburgh

3. Analysis of the Elastic Properties

Defined as

Based on experimental observationsMeasure of ductile vs. brittle behavior

G

B

ModulusShear

ModulusBulk

1.75 critical value B/G > 1.75 DUCTILE B/G < 1.75 BRITTLE

G Resisting Plastic Flow B Bond Strength

OppositionTo

Fracture

1.75 is more a transition zone

MS&T Oct. 2008 Pittsburgh

3. Analysis of the Elastic Properties

Stiffer bcc Mg-Li alloys Ductile/brittle transition region

1

1,5

2

2,5

3

3,5

4

4,5

0 20 40 60 80 100at. % Solute

B/G

AlLi; Solute:Li; DFT

LiMg; Solute:Mg; DFT

LiMg; DFT Error Est.

LiMg; Solute:Mg; Experiment

B/G = 1.75

ReussG

B

VoigtG

B 0.9

HerG

B

Brittle Region

Ductile Region

MS&T Oct. 2008 Pittsburgh

3. Analysis of the Elastic Properties

Defined as

Design Criteria Maximum stiffness for minimum weight

Typical Values (MPa m3/kg) Graphite Fiber 127.78 Graphite Fiber/epoxy 43.53 Steel 26.41 Aluminum 25.93 PET (polymer) 2.15 Lead 1.41

ρ

E

Density

Modulus sYoung'

MS&T Oct. 2008 Pittsburgh

3. Analysis of the Elastic Properties

Better than Al-Mg. Comparable to Al-Li.

ρ

EReuss

ρ

EVoigt

ρ

EHer

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100at. % Solute

Sp

ecif

ic M

od

ulu

s (M

Pa

m3 /k

g)

AlLi; Solute Li; ExpAlMg; Solute Mg; ExpLiMg; Solute Mg; ExpLiMg; Solute Mg; DFTLiMg; DFT Bounds

MS&T Oct. 2008 Pittsburgh

3. Analysis of the Elastic Properties

MS&T Oct. 2008 Pittsburgh

Mg-Li alloy design1. Motivation2. Elastic properties3. Analysis of the elastic Properties4. Summary

Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf

MS&T Oct. 2008 Pittsburgh

4. Summary

1. DFT and homogenization schemes can be used to predict with reasonable accuracy elastic properties of polycrystalline metals

2. Optimal elastic properties of bcc MgLi alloys are observed around 70 at. % Mg

3. B/G for the optimal bcc Mg-Li alloys is in the brittle/ductile transition region

4. BCC MgLi has a better E/r than AlMg and a comparable E/r to Al-Li

BCC MgLi HAS POTENTIAL AS AN ULTRA-LIGHT WEIGHT STRUCTURAL ALLOY

MS&T Oct. 2008 Pittsburgh

ConclusionsConclusions

+ Understanding trends (thermodynamics, mechanics)

+ Direct use of homogenization theory (elastic)

+ Extract engineering quantities for a rough but quick estimation

+ Get quantities that you cannot get elsewhere

- 0 K

- supercell size

- long calculation times