MD-111

Prof. Dr. Siegfried SchmauderIMWF, Universität Stuttgart

• Theory

• EAM Potentials

• Frenkel Defects

• Interaction between dislocations and phase boundaries

Molecular Dynamics (Part I)

MD-112

Macro(Mechanics)

Electrons(Bonding)

Atoms(Cohesion)

Microstructure(Micro Cracking)

Specimen(Controlled Failure)

Component(Integrity)

Micro(FEM)

Nano(MD)

Femto(ab initio)

Macro(FEM)

Materials Science(bottom-up-approach)

Theory

MD-113

Molecular Dynamics (MD) Simulations

• Crystal is considered as a system of classical point particles.

• Numerical integration of Newton‘s equationsof motion

ii x

EF

Interatomic Forces:

MD-114

EAM Potentials

)(2

1)(

,

jiijiji

ii rFE

)( ijij

jii r

j

i

ijr

Embedding part Pair potential part

Local electron density

Embedded Atom Method Potentials

MD-115

Ideal LatticeOne Fe atom

replaced by a void

One Fe atom

replaced by a H atomFe-Lattice

(1 1 0)

(1 -1 0)

Not deformed

deformed

(36%)

Hydrogen Embrittlement, Mechanism 1: Weakening of Bonds

MD-116

Edge Dislocation Movement

(I Ī 0) slip plane, Burgers Vector ½ [ I I I ]

MD-117

Dislocation Movement

MD-118

Time [ps]

w/o H

1 H-atom (substitutional)

1 H-atom (interstitial)

4 H-atoms (along dislocation line)

4 H-atoms (distributed at the dislocation core)

H-atoms

Hydrogen Embrittlement: Mechanism 2: Dislocation Pinning

MD-119

H-atoms

4H-atoms 0 K4H-atoms 300 K4H-atoms 600 K

Time [ps]

X-p

ositi

on o

f the

dis

loca

tion

[0,1

nm

]

Time [ps]

w/o H

1 H-atom (substitutional)

1 H-atom (interstitial)

4 H-atoms (along dislocation line)

4 H-atoms (distributed at the dislocation core)

H-atoms

Hydrogen Embrittlement: Mechanism 2: Dislocation Pinning

MD-120

Simulation of Internal Stresses

• Purpose: Simulation of internal stresses based on atomic structural models and interatomic potentials

• Study of influence of atomic defects (voids, dislocations, lattice defects, dopant atoms) on internal stresses

• Chance to predict failure under external load

• Practical application: Ni/Ni3Al-superalloys

MD-121

Internal Stress in a Carbide MultilayerSystem during Indentation

Simulation of Internal Stresses

0 5 10 15 20 250

2

4

6

8

10

12

14

16

18

20

X-Achse [nm]

z-A

chse

[nm

]

-2E6-1,7E6-1,4E6-1,1E6-8E5-5E5-2E51E54E57E51E61,3E61,6E61,9E62,2E62,5E62,8E63,1E63,4E63,7E64E6

0 5 10 15 20 250

2

4

6

8

10

12

14

16

18

20

x-Achse [nm]

z-A

chse

[nm

]

-2E6-1,7E6-1,4E6-1,1E6-8E5-5E5-2E51E54E57E51E61,3E61,6E61,9E62,2E62,5E62,8E63,1E63,4E63,7E64E6

MD-122

Internal Stresses in Ni / Ni3AlSuperalloys with Atomic Defects

Simulation of Internal Stresses

Bright gray: Ni

Dark gray: Al

MD-123

Molecular Dynamics: Simulation ofInternal Stresses

MD-124

Nanosimulation of the Interaction between

Edge Dislocations and Obstacles (Precipitates)

Nanosimulation

MD-125

EAM Potentials

Fe: G. Simonelli, R. Pasianot, E. Savino: Mat. Res. Soc. Symp. Proc., 291 (1993) 567Cu: A. F. Voter: Los Alamos Unclassified Technical Report #93-3901, 1993Fe-Cu: M. Ludwig, D. Farkas, D. Pedraza, S. Schmauder: Modelling and Simulation in

Material Science and Engineering, 6 (1998) 19Fe-Ni: C. Vailhe, D. Farkas: Mat. Sci. Eng. A 258 (1998) 26Cu-Ni: D. Farkas, J. Clinedist: Mat. Res. Soc. Symp. Proc. 457 (1997) 315

Molecular Dynamics Software

Program „IMD“, developed at the „Institut für Theoretische und Angewandte Physik“ (ITAP) at the University of Stuttgart, Germany.

Capable of Parallel Processing. Set up a world record in 1997 for a model built of1.2*109 atoms, improved to 5.2*109 atoms at a later time

Software and EAM Potentials

MD-126

EAM-Potentials

EAM-Potentials (EAM=Embedded Atom Method)

Fe-Fe EAM-Potential

Cu-Cu EAM-Potential

Fe-Cu EAM-Potential

M. S. Daw, M. I. Baskes, Embedded atom method: Derivation and application to impurities, surfaces and other defects in metals, Phys. Rev. B, Vol. 29, No. 12 (1984), pp. 6443-6453

MD-127

)(:)(

)(:)exp(

)(

212

2102

2

1

4

1

3

21

xxxxhxhhxx

xxx

x

xzHzxax

R

FRE

a

i

i

ii

ij

ij

ai

i

i

ji

ijtot

Fe-Fe EAM-Potenital

Param. Experiment Calculateda0 0.2866 nm 0.2876 nmEcoh 4.28 eV 4.28 eVc11 241 GPa 248 GPac12 143 GPa 152 GPac44 118 GPa 113 GPaEv,for 1.8 eV 1.6 eV

Adjusted to: lattice parameter a0, cohesive energy Ecoh, elastic constants cij, vacancy formation energy EV,for

G. Simonelli, R. Pasianot, E. J. Savino, Mat. Res. Soc. Symp. Proc., 291 (1993) 567

MD-128

rra

MRr

M

ij

ij

ai

i

i

ji

ijtot

eer

DeDr

R

FRE

MM

296

2

21

2

1)(

A. F. Voter, Los Alamos Unclassified Technical Report #93-3901, 1993

Param. Experiment Calculateda0 0.3615 nm 0.3615 nmEcoh 3.54 eV 3.54 eVc11 176 GPa 180 GPac12 125 GPa 122 GPac44 82 GPa 82 GPaEv,for 1.3 eV 1.3 eV

Adjusted to: lattice parameter a0, cohesive energy Ecoh, elastic constants cij, vacancy formation energy EV,for

Cu-Cu EAM-Potential

MD-129

Fe-Cu EAM-Potential

fxedxcbxa

R

FRE

Fwithrandr

CuFeFeCu

ij

ij

ji

i

i

ji

ijtot

lCuFe

21

)(

0)(:

21

0

M. Ludwig, D. Farkas, D. Pedraza, S. Schmauder, Modelling and Simulation in Material Science and Engineering, 6, pp. 19-28 (1998)

Point defects in Fe, periodic boundary conditions, constant volume, 686 atoms

x=0...1. Parameter: a=1.0, b=4.7, c=0.99, d=4.095, e=1.0, f=4.961

Param. Experiment CalculatedEV 1.8 eV 1.6 eVECu 1.233 eV 1.236 eVEV-Cu 0.14 eV 0.18 eVE2Cu 0.19 eVEk(2) 0.05 eV 0.01 eV

b

b

b

MD-130

Small Cu precipitate in Fe matrix,coherent

Large Cu precipitate in Fe matrix,

becoming unstable

Cu Precipitates in Fe Matrix

MD-131High Resolution Electron Microscopy: Cu Precipitate in Fe Matrix

HRTEM image of a twinned 9R copper precipitate in an Fe-Cu specimen. The angle between the (009)9R basal and (114)9R twin planes is 61º ;

R. Monzen, M. Iguchi, M.L. Jenkins, Phil. Mag. Let. 80 (2000) 137.

Cu Precipitate

MD-132

Simulation Model

Simulation model of an Fe single crystal with uniaxial tensile load

MD-133

The Xi - Yi (i=1,2,...,5) planes for five crystal orientations in bcc-Fe

Crystal Orientations

MD-134

Atomic configuration for orientation No. 1 with free boundary conditions at strain=0.24

Plastic Deformation during External Straining

MD-135Atomic Arrangement, Orientation 3

Atomic arrangement for crystal orientation No. 3 with free boundary conditions

(a) strain = 0.136; (b) strain = 0.48

Twinning

MD-136Example for Void Formation (Other Direction)

(a) (b)Atomic configurations for orientation No. 5 with periodic boundary conditions:

(a) strain=0.144; (b) strain=0.16

Void Formation

MD-137

Stress - Strain curves for D=20, H=26 and different crystal orientations, periodic boundary conditions

Stress-Strain-Curves

Orientation 1

Orientation 2

Orientation 4

Orientation 5

Orientation 3

MD-138

Strain

Stress-Strain-Diagram

Voided Material

MD-139

Y

O X

X

X

X

XX

X X

X

XX

XXX

XXX

XX

XX

Atomic Structure during Deformation; 20 Frenkel Defects

Frenkel Defects

MD-140

Dependence of () on Frenkel Defects, Orientation 1

Stress-Strain-Curves

ideal lattice

1 Frenkeldefect

20 Frenkeldefects

5 Frenkeldefects

MD-141

Hideyuki INOUE. Yasuhiro Akahoshi and Shoji Harada

Fig. 1: Initial configuration of single crystals with random orientation.

Fig. 2: Energy distribution of analysed body. (The lightand the shaded positions show atoms with low and highpotential energy, respectively. The encircled numberindicates consecutive number of sub-grains)

Fig. 3: Tensile stress versus total strain up to 2%.

Model Fe polycrystal

Temperature RelaxationTensile problem

300K 100K.300K 500K.700K

Number of atoms RelaxationTensile problem

794S7306

Mass of atom 9.273588x10-26kgTime step 1 .0 fsNumerical integrationof equation

Verlet's method

Potential Morse typepotentialStain rate 1.0x l.0-1/ step

Step number forcalculation

Relaxation

Tensile problem

4.0x104

2.0xl02

Total strain 20.0%

Boundary conditions Periodic B. C.

Fixed B. C.

X1 direction

X2 direction

Strength of Nanocrystalline versus Single Crystalline Metal

Nanocrystalline versus Single Crystalline Metal

Analysis conditions

SC

PC

MD-142

Simulation of Cycle- and Temperature-Dependence of Failure

Comparison of deformation stateat several temperatures.

Relation between stress amplitude and number of cycles to failure.

Process of crack initiation and growth at 300 KRelationship between tensile stress and

total strain at several temperatures.

700K

100K

Initial state

2 = 6.2% (defect generation)

2 = 6.4%

100 K

300 K

500 K

700 K

2 = 6.2%

MD-143

Schematic representation of a section through the sample, showing the initial position of the edge dislocations and the Cu atoms (grey).

The interaction between a moving edge dislocation in an Fe crystal and a Cu-precipitate is investigated by molecular dynamics (MD) calculations.

In the absence of external stresses, two edge dislocations with the same slip plane and opposite Burgers vectors within a perfect Fe crystal lattice are investigated.

Initial Positions of the Edge Dislocations

Nanosimulation of the Interaction between Edge Dislocations and Obstacles (Precipitates)

MD-144

Detailed Structure of one ofthe Dislocation Cores

Detailed structure of one of the dislocation cores during dislocation migration through the obstacle.

Fe atoms yellow, Cu atoms grey.

The distance along the z-axis between the upper and the bottom plane is 6 x 0.176 nm = 1.056 nm

MD-145

Interactions between precipitates and dislocations are investigated using atomistic computer simulations. In particular, the effect of Cu-precipitates on the core structures, slipping behaviour, and Critical Resolved Shear Stress (CRSS) of an edge dislocation in a bcc Fe single crystal is considered.

Model of a bcc Fe single crystal with an edge dislocation and a Cu precipitate under shear deformation.

Interaction between Precipitates and Dislocations

MD-146

Profiles of dislocation lines on the slip plane (1 -1 0):(a) equilibrium dislocation(b) slipping dislocation

Dislocation Cutting a Cu-Precipitate

MD-147

Profiles of the dislocation lines on the slip plane (1 -1 0) at different deformation stages (strains: 0.5%, 2.0%, 2.3%, 2.35%)

Profiles of Dislocation Lines

MD-148

Shear stress (a) and normal stress (b) distribution for the pure edge dislocation along Burgers vector (1 1 1) on the slip plane (1 -1 0).

She

ar s

tres

s (G

Pa)

Nor

mal

str

ess

(GP

a)

Initial equilibrium dislocation

----- Slipping dislocation

Dislocation Stress Distribution

MD-149

Stress distribution atthe arm part (z=10 a0 ) (a, b) and at the middlepart (c, d)

She

ar s

tres

s (G

Pa)

Nor

mal

str

ess

(GP

a)

She

ar s

tres

s (G

Pa)

Nor

mal

str

ess

(GP

a)

(a)

(c)

(b)

(d)

Å

ÅÅ

Å

Dislocation Stress Distribution

MD-150

Average shear stress - strain curve for the bcc Fe single crystal with a Cu-precipitate and a single edge dislocation under external shear deformation

(Insert: pure iron).

Shear stress - strain curve with and without a Cu precipitate

Shear Stress-Strain-Curve

She

ar s

tres

s (G

Pa)

Strain

without inclusion(12 MPa)

with inclusion(500 MPa)

MD-151

3-dim. atomistic simulation of dislocation bending and cutting of Cu-cluster in Fe

Scheme of a Dislocation, Blocked byor Cutting a Precipitate

Blocked

Cutting

MD-152

Schematic of a dislocationcutting a precipitate to explainthe definition of the criticalangle.

The angle between the arms of a dislocation, together with the distance between the obstacles, is the key parameter to calculate the increase in matrix strength due to precipitation hardening. The shear stress is given by:

23

2cos

L

GbBrown, Ham

= Shear stress

G = Shear modulus of the matrix

b = Burgers vector in the matrix

L = Obstacle spacing in the slip plane

= Critical angle between the dislocation arms

Shear Stress and Critical Angle

MD-153

23

2cos

L

GbBrown, HamThe shear stress is given by:

Russel&Brown derived the shear stress from a relationship between the energies of thedislocation per unit length inside (E1) and outside (E2) the precipitate as (overaged state):

43

22

211

E

E

L

Gb

Therefore,4

3

22

21

23

12

cos

E

E

where the ratio E1 / E2 depends on the precipitate radius as

002

01

2

1

log

log

log

log

rRrR

rR

E

rr

E

E

E

r = Precipitate radius R = Outer cut off radiusr0 = Inner cut off radius

Shear Stress, Russel&Brown-Theory

MD-154

Russel&Brown adopted the following values for the Fe/Cu system based on experimental strengthening data from literature:

r0 = 2.5 b with b = 0.248 nm (Burgers vector)

R = 1000 r0

Two examples for precipitate diameters d=1.3 nm and d=3.2 nm:

d=1.3 nm : critical angle = 171° (small precipitate)

d=3.2 nm : critical angle = 140° (larger precipitate)

K.C. Russell, L.M. Brown, Acta Met. 20 (1972) 969-974

0.6EE 21 /

Shear Stress, Russel&Brown-Theory

MD-155

Larger Precipitate, Diameter 3.04 nm

Larger precipitate, diameter 3.04 nm

In the case of the 3.04 nm diameter Cu precipitate passing does not happen and the dislocation line is pinned by the precipitate, with free ends oscillating.

The dislocation is not able to cut the obstacle. It can only pass through the precipitate completely as soon as an external shear stress is applied to increase the stress beyond the Peierls stress.

= 140°

MD-156

Small Precipitate, Diameter 1.32 nm

Small precipitate, diameter 1.32 nm

Starting from the initial position, the movement ofthe dislocation line takes place such that it is curvedtowards the precipitate (see Fig. b in comparison toFig. a).

Furtheron, the edge dislocation passes through theprecipitate and after passing, a backward bowing canbe recognized (see Fig. h), indicating the persistingattractive force between the precipitate and thedislocation line.

Altogether, the movement of the dislocation takes placealmost without impedement.

= 170°