31th Computational Materials Design (CMD) Workshop

先端事例講義Strength modeling of metals based on the atomistic theory

原子論に基づく金属材料の強度モデリングNational Institute for Materials Science (NIMS)
Research Center for Structural Materials（RCSM）

物質・材料研究機構 構造材料研究拠点

Masato Wakeda譯田 真人
1Structural materials, High-temperature materials〇Structural materials ：Materials, which support load to maintain the structure and shape ofmachines and buildings, used in vehicles and bridgee.g.）Steel, Aluminum alloys, Magnesium Alloys, etc.
〇High-temperature materials ：Materials, which can keep their properties under high-temperature environment,
often used in aircraft and industrial gas-turbine enginese.g.）Ni-based superalloys, etc.

 Maintain performance during designed life time Safety and reliability are crucial
 Improvement of the properties has a large impact on the human society Strong industrial field of Japan2

SteelSteel・Iron alloys with various elements
・Huge amount of production, widely used in industry
・Used in vehicles, ships, buildings, and bridge
・Steel industry creates huge employment in Japan
What’s required now①Further strengthening
 contributes to weight saving and fuel efficient（reduction of environmental load, economic efficiency）
 contributes to safety and life-prolonging of buildings②Proper element usage
 limited resources economic contribution by reducing expense reduction of environmental load

Further strengthening based on the appropriate alloying design is required

3

“Strength” of metals

StressMechanical properties× Yield strength Tensile strength Hardness
 Fracture toughness
 Fatigue strength Creep strength ・・・・

Strainelastic deformation→ plastic deformation（reversible）（ irreversible ）Different strengths for different purposes4Dislocation (転位)
Plastic deformation of crystalline metals is mainly caused bymotion and multiplication of dislocation（metals can deform by the stress, which is much smaller than ideal strength）

Edge dislocation

Screw dislocation
1cm3
Dislocation density

 Slip plane and slip direction depend on the crystal structure

5

Strengthening and crystal defects of metals
vacancy
dislocation
Solute atom
Strengthening techniques solid solution strengthening precipitation strengthening
 work hardening grain refinement strengthening
 etc.

Grain boundary

precipitate
A balance btw. strength and toughness
is also important!

6

Solid solution strengthening
Solid solution strengthening ＝ Strengthening by solute atoms
Interstitial solute atom

Substitutional solute atome.g.）H, C, N, etce.g.）Si, Cr, Mn, etc

A few % addition can lead to a large strengthening effect○ easy strengthening technique× hard to predict due to a non-linear effect7

Solid solution strengthening
Conventional alloying design for structural materials
Based on accumulated knowledge of the relationship between
added element and strength requires large time and expense due to many trials and errors
 cannot obtain individual effect of each defect cannot design materials with huge elements

Perdition of the effect of solute species onmechanical properties from the atomistic theory⇒A priori design of steel,which can reduce trials and errors
8Classical molecular dynamics simulation

Initial configuration and velocity（Update neighbor list）
Calculate interatomic interaction

Update configuration and velocity

t=t+DtInteratomic potentialStartABB

Atomic distance (r )F rPair interaction

Calculate properties
End9

Classical MD and First principles calculation〇Classical molecular dynamics (MD)• evaluates the interatomic interaction empirically or semi-empirically• easy to calculate a large scale model• easy to consider temperature effects〇Large scale, finite temperature calculations〇First principles calculation• evaluate the interatomic interaction based on the first principles (non-empirically)• not easy to calculate a large scale model due to huge computational cost
• not easy to consider temperature effects〇High accuracy calculations

10

Prediction of solid solution strengthening based on the atomistic theory
Screw dislocation has a dominate role on yield stress
Part 1
 Analysis of Fe-Si alloys based on the semi-empirical interatomic potentials• construct Fe-Si interatomic potential (based on the first principles calculation)• evaluate the interaction between screw dislocation and Si• evaluate the effect of Si on the energy barrier for screw dislocation motion• predict CRSS based on the atomistic theory
Part 2
 Systematic analyses of binary Fe alloys based on the large scale first principles
calculation• evaluate the interaction between screw dislocation and substitutional solutes• evaluate the effect of solute speices on the energy barrier for screw dislocation
motion and core structure• predict the effect of solute atoms on yield stress11Part 1Fe alloys with dilute substitutional soluteatoms

Interstitial solute atom

Substitutional solute atome.g.）H，C，N，etce.g.）Si，Cr，Mn，etc

12Stress

Fe-Si alloys
 Si is a well-known solute element of steel（substitutional solute）
 Si increases yield stress at room temperature(solute strengthening) Si induces softening at low temperature Si increases work hardening coefficient(Si changes dislocation substructure)

work hardening
yield stress

Strain118

1H22131415

1617He345678910Li

Be

BC

NO

F

Ne

1112

13

1415

16

1718NaMg

3456789101112Al Si P

S

Cl

Ar

1920

21

22

23

24

25

26

27

28

29

30

33

34

35

36

KCa

Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge AsSeBrKr

37383940

RbSrYZr

5556

LanthanoidCsBa8788FrRa

Actinoid

4142

Nb Mo

3132

43

44

45

46

47

48

49

50

51

52

53

54

Tc

Ru

Rh

Pd

Ag

Cd

InSn

Sb

TeI

Xe

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

Tl

Pb

Bi

Po

At

Rn

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

Rf

Db

Sg

Bh

Hs

Mt

Ds

Rg

Cn

Uut

Fl

Uup

Lv

Uus

Uuo

13

Interatomic potentials for Fe-Si alloys〇Fe-FeG. J. Ackland, M. I. Mendelev, D. J. Srolovitz, S. Han and A. V. Barashev, J. Phys:Condens. Matter, 16 (2004) pp.S2629-2642.
Properties to determine potential parameters〇Fe-Si

Properties of Fe-Si-Vacancy system
N1 N NE    ij  rij    Fi   i 2 i 1 j 1 j i i 1

Fe-Siij

8r    a r  r  H r  r ijk 1

3kkij

kij

F Si      N

1H  x  0 x  0
 x  0

 i    j  rij j i

 Si  SSi  Fe

〇Vacancy formation in a-Fe・Vacancy formation energy: EfVa
・Volume difference (Fe53Va) : DVfVa〇Si solution in a-Fe
・Volume difference (Fe53Si) : DVsSi〇Si-vacancy binding energy
・First nearest neighbor (1nn): EbVa-Si (1nn)・Second nearest neighbor (2nn): EbVa-Si (2nn)〇Distance betw. vacancy and nearest Si atom・First nearest neighbor (1nn): dVa,Si (1nn)・Second nearest neighbor (2nn): dVa,Si (2nn)

M. Wakeda, H. Kimizuka, and S. Ogata, Journal of the Japan Institute of Metals, 77-10(2013) 409-414.

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Definition of the Interaction energySiⅰ_++iiiii

Siiv

Edisl-Si= Ei + Eii – Eiii -Eiv When the distance is infinity, Edisl-Si ＝0
 Negative value→attractive, positive value→repulsive
Edisl-Si =0 (r=∞)

Edisl-Si
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Interaction energy between screw dislocation and Si
Structural relaxation by Conjugate Gradient（＝0 K）

y [112]21az [111]x [110]Screw dislocation

thickness 10b
radius 21a

The interaction is attractive!
16

Calculation of MEP and the energy barrier

initialfinal

Nudged Elastic Band method⇒ finding Minimum Energy Path (MEP)

Activation energy DE17Mechanism of screw dislocation motion（Kink mechanism）

0.69 eV

Pure iron
Kink mechanism

Solute atom18

Effect of Si on kink nucleation energy

Ebarrier

EzEminSolute atom
19

Effect of Si on kink nucleation migration
0.05 0.15 0.25

Si
0.10 0.20l
Si position

0Potential energy [eV]

0.8

0.6

0.4

d = 0.05l
d = 0.10l
d = 0.15l
d = 0.20l
d = 0.25l

0.2

0Si disturbs kink migration−0.2
0510152025

Normalized reaction coordinate 20

Prediction of yield stress based on the atomistic theoryCRSS

Critical Resolved Shear Stress （CRSS）

t =t*+tath
thermalathermaltemperature

t* : thermal CRSS

tath : athermal CRSS

Kink nucleation
SiKink migration
SiSiSiSi
21

Interaction energy [eV]Prediction of athermal CRSS

F ≈ -0.2/b (ev/Å)y coordinate [Å]S. Shinzato, M. Wakeda and S. Ogata, Journal of the Japan Institute of Metals,
80-3 (2016) 197-205.

22

Atomistically-informed predictive model of CRSS

Plastic strain rate   b mv t b: Burgers vector
m: mobile dislocation densityv t  : dislocation velocityt : CRSSv t t23

Atomistically-informed predictive model of CRSS

Plastic strain rate   b mv t b: Burgers vector
m: mobile dislocation densityv t  : dislocation velocityt : CRSS
Peierls-valley{110} plane

111v t  t t 

Peierls-valley

2 112
a3a(=lattice constant)

111

Distance between Peierls valleys

22 4a
b = 0.94b
33 3

  b m kink  nucleation time    kink  migration time 24Formulation of kink nucleation and migration rateenergy

Kink nucleationp
 
(i )
0   t  t   DGkn
DGkn
1   t  kn  qp 
0   t  DEsol. t   DEsol.
1   t  kn  qstress
Kink migration
kink widthm (Si cluster size)interval between two superjogs

The largest cluster m0
statistically predicted from thesolute concentration25

Formulation of kink nucleation and migration rate
Kink-nucleation rate(1)
(2)
 ΔGkn
 ΔGkn ΔGkn(3) t   ΔESi(3) t   t ct   ΔESi(2) t   c  kn 1  c  exp    exp    exp  kT
2kT
2kT

BBB

 

Energy barrier ↓Energy barrier ↑Si increases kink nucleation rate （softening）

 ΔGkm t , c  
exp
Kink-migration ratekm

kT
B


Si decreases kink migration rate （hardening）
26

Predicted CRSS
Calc.

Exp.

athermal

In both calculation and experiment, CRSS
 decreases with increasing temperature decreases with increasing Si concentration at low temperature increases with increasing Si concentration at high temperature
27

Yield stress

Softening at low temperature and hardening at high temperaturesofteninghardening

Pure-iron
Fe-Si alloy
Temperature
Sinucleation
SiSi

migration
Si
28

Summary: part 1
Prediction of solid solution strengthening of Fe-Si alloys based on the semi-empiricalpotentials

Ｉ．The interaction between a screw dislocation and Si
 The interaction is attractive

II．Effects of solute Si on the dislocation behaviors
 Si increases kink nucleation rate
 Si decreases kink migration rate
III．Prediction of the solid solution strengthening by Si Si decreases CRSS at low temperature Si increases CRSS at high temperature
We constructed a prediction framework of the yield stress of binary ironalloys with dilute solute atoms based on the atomistic theory29

First-principles calculation of lattice defects
Part 2b

Atomic model of screw dislocationy [ 110 ]cax [ 112 ]z [111 ]few hundred atoms are required for the interaction
huge computational load
analyses between a dislocation and a solute atom ⇒(DFT calculations)

Calculation costsmalllarge

The number of atoms required for calculations
Point defect (vacancy, etc.)
Plane defect (grain boundary)Line defect (dislocation)
Grain boundary-solute interaction
Dislocation-solute interaction
Dislocation-dislocation interaction
Dislocation-grain boundary interaction30

First principles calculation of screw dislocation
First principles calculation software
OpenMX (=Open source package for Material eXplorer)
http://www.openmx-square.org/

Calculated in “K-supercomputer” (RIKEN AICS)
Screw dislocation model・405 atoms
・two screw dislocations with opposite Burgers vector
・dislocation length : 3b・periodic boundary conditions: x, y, z directions

Quadrupolar arrangementby [110 ]cax [ 112 ]z [111 ]31

Dislocation core structure

Differential displacement map

Easy-core

Easy

M-point

Hard

Split

M. Wakeda, T. Tsuru, M. Kohyama, T. Ozaki, H. Sawada, M. Itakura,and S. Ogata, Acta Mater., 131 (2017) 445-456.

32

Peierls potential and core structure
Hard

Easy A(M point)

Easy BSplit

u11243u2u4u333Effects of a solute atom on the energy surface

34

Definition of the interaction energy
Screw dislocation
Substitutional solute

DEint=-E(i)E(iii)++E(ii)

E(iv)

35

Interaction between screw dislocation and Si

The interaction is attractive!
36

Interaction with Easy-core structure37

Dislocation core structure

Differential displacement map

38

Interaction energy map (Si, Cr, Mn, Co, and Ni)(Experiment) Si, Mn, Ni have large strengthening effect Cr and Co have small strengthening effectFundamental data to predict strengthening effect39

Volume change around dislocation core

Size misfit between
a Fe atom and a solute atom
40

Generalized stacking fault energy

generalized stacking fault energy(GSFE)
Chemical misfit between
a Fe atom and a solute atom
41

Effect of the solute species on the Peierls potential

Elementary slip plane
Easy
Hard
Easy{112}

Split

M-point
Easy{110}42

Effect of solute atom on the kink nucleation and migration

43Effect of solute atom on the CRSS

Effect (i)Effect (ii)type (1):－－

type (2):〇－type (3):〇〇44Precited athermal CRSS

45

Summary: part 2
The systematic analyses of the screw dislocation in iron binary alloys based on
the large scale first principles calculations
Ｉ．The interaction between a screw dislocation and solute species
 Quantitative values of the interaction energy
 Quantitative effect on the mobility of screw dislocation
 Whether each solute species change slip plane or not
ＩＩ．Unveil origin of the solute-dislocation interaction, which should be crucial
information for the general alloying design for solute strengthening
ＩＩＩ．Provide the atomic basis to the established meso- and macro-modellingof the strength and deformation of materials.

46

References M. Wakeda, H. Kimizuka, and S. Ogata, Journal of the Japan
Institute of Metals, 77-10 (2013) 409-414.
 S. Shinzato, M. Wakeda and S. Ogata, Journal of the Japan
Institute of Metals, 80-3 (2016) 197-205.
 M. Wakeda, T. Tsuru, M. Kohyama, T. Ozaki, H. Sawada, M.
Itakura, and S. Ogata, Acta Mater., 131 (2017) 445-456.

47