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1 表面・界面のダイナミクスの数理 III

表面・界面ダイナミクスの数理 III May 16～18, 2012 東京大学数理科学研究科

合金材料の結晶粒界と強度特性

北海道大学大学院工学研究院

材料科学部門材料数理学研究室

毛利哲雄


Fe-Pt

L10

L10 L12

L12

Background Phase Diagram


Background

Fe-Pd

Fe-Ni

Fe-Pt Mechanical property

High density magnetic

recording media


B A

Solid Solution

固溶体

Phase separation

相分離

Solubility limit 固溶限

中間相

Intermediate phase

Ordered phase 規則相

Basic concept

金属組織学丸善 (1972)


solid solution type alloy 固溶体型合金

Phase

Diagram

融点melting

point

Cu Ni

Tm

Tem

per

atu

re (

C)

Concentration (Ni)


intermediate phase 中間相型

L10 L12

強度・靭性の付与

機能の付与

Pd Fe

規則相 ordered phase


Theoretical Procedure First-principles calc.

・Cluster Expansion Method (CEM)

・Cluster Variation Method (CVM)

iii STrVErVTg ,,,

・Total energy calculation at the ground state: FLAPW

i lkji

ijkli

ji

ij

BNwNx

NNy

kS

,,,

25

6

,

!!

!!

lnijklijkiji wzyx ),(,, i

i

m

i 21;i . . . ; ) (

m

m

m

ji Ev

1

j

m

jji rVvrVE ,


First-principles calculation of phase stability and

L10-disorder phase equilibria for Fe-Pt system

L10

L10 L12

L12

Background Fe-Pt L10


Fe-Pt

Exp. 1600K

1610K

1900K

2070K

(1st + 2nd n.n.)

Phase Diagram

26, 78

Fe, Pt

L10-disorder

J. Alloys and Compounds (2004)


,)1ln()1(ln

!

!ln xxxxk

Nx

NkS B

i

i

B

Bragg-Williams approx.

xFF

i lkji

ijkli

ji

ij

BNwNx

NNy

kS

,,,

25

6

,

!!

!!

lnCVM

Entropy and Cluster Variation Method (CVM)

50% 2v < 0

5.0 BBAA yy

> 0 2v

0.1ABy

= 0 2v

.ABBBAA yyy

iji yxFF ,

ijkiji zyxFF ,,

Local atomic configuration ?


hi(r)

hi(r) : field variable Microstructure

Free energy of microstructure

dVfFi

iiichem

)(1

2hh

a b

r

hi

interface Free energy of homogeneous system

h1

h2

f [h1, h2]

i : gradient energy

coefficient,

M : mobility

Lij : relaxation constant.

Phase Field Method

Conserved

Non-Conserved

)(i

chemiF

Mt h

h

j j

chem

ij

iF

Lt h

h

Time evolution of hi(r)

C

LRO


ii

i

ii L

t

2

Hybrid Model

CVM f

Multi-scale

1:1

Microstructure

PFM

Atomistic

CVM


Long Range Order parameter

b

a

A

B

bah AA xx LRO

1.0 0.0


Preliminary calculation

2v ; n.n. effective pair interaction energy

1.90(1.893)

T

L10


Microstructure

[100]

[00

1]

Atomistic

T=1.6

Simultaneous calculations for Multi-scales

01.0i

outin

120x120x120 fcc algorithm; Williams

x’=1000

2

50at% L10

1.90(1.89)

M. Ohno Ph.D Thesis Hokkaido Univ. (2004)


Multi-scale Analysis -Hybridized Model-

+

- +

-

+

- +

-

+

-

＝＋

＋

＋

＋

＋・・・・・

Elementary processes

Circle shaped APB

Philosophical Magazine(2003)


T=1.75 1.8

T=1.8

+

-

Kinetics of a Circle Shaped APB



1.90

t’=0

x’=200

t’=400 t’=1400 t’=700

Kinetics of a circle shaped APB

T=1.8

+ -

T=1.75 1.8

100u

T=1.8



Elementary process

<

1. inside/outside APB

homogeneous

+

-

<

+ -

2. at APB

line-shaped APB

+

- +

-

3. movement of APB

100u31u ～

diffusion



T=1.8

e

u

tat h

h

exp

ii

i

iCVM

i

if

Lt

2

eh

kinetics of a Uniform System

31u ～



Elementary process

<

1. inside/outside APB

homogeneous

+

-

<

+ -

2. at APB

line-shaped APB

+

- +

-

3. movement of APB

100u31u ～

diffusion



Wetting T=1.7 1.8

a12

2 Nxx

e

s

s

f

f

ta

tat h

h

expexp

uniform

0.0

0.2

0.4

0.6

0.8

1.0

85 95 105 115

t'=0

t'=1

t'=40

x

t’=40 t’=1

t’=0

+ -

Kinetics of a line-shaped APB

31f ～ 3010s ～

2h



+ -

Hybrid model and multi scale calc.

T=1.8

eAPB

s

s

f

f tt

at

at hh

h

)(expexp

wetting

s =10~30

atomistic

f =1~3

motion

>100



t’=0 500 1000

x’ = 400

Coarsening Process

T=1.8

Curvature driven growth

HLVAPB 2

tbRtR 22

0

H; local mean curvature t’=400 500 600

Coalescence

ii

i

iCVM

i

if

Lt

2

R(t’)

2

t’ 500 1000



Scale?

a12

2 Nxx

Atomistic model of APB

Spatial scale

x’=1000 ?

i : gradient energy coeff.

2v : Pair interaction energy

Crystallographic orientation



rm; Local coordinate

Rn; Global coordinate

mn

mnsCVM ahrRf,

l

lCVM pfF

Inhomogeneous CVM

discrete lattice system

Rn Rn-1 [100]

[01

0]

rm coarse grained region

Global

coordinate

Local

coordinate

mnl rRp ;atomic plane

ahrR mns ;cluster probability

cells

Coarse Graining

)(ahrR mns nsmnsmns RahrRahrR 22

2

1))((

mn

nsnsnsmCVM RRRahrfF,

2,,),(



n sm

s

s

CVM

sm

s

s

CVM

m

nsCVM

ffRfF

,

2

0

2, 0

ssm

ss

ss

CVMf

,, 0

2

2

1

n sm

sms

ms

sms

m

nsCVM JarCJarCRf,

2

,2

,

,1 ,,,,

sm

sms JarC,

2

,3 ,,

sm

sms

ms

sms

m

lsCVM JarCJarCRfL

F,

2

,2

,

,1 ,,,,1

dxJarCsm

sms

,

2

,3 ,,

mn

nsnsnsmCVM RRRahrfF,

2,,),(

homogeneous

state

0,0,),( nsmCVM Rahrf

Coarse Graining [2]



J

mssBss JarVTkN ,,',,

dxfNL

Fss

sssssCVM

,

,'1

Ginzburg-Landau type

Coarse Graining [3]



lattice constant is fixed at adis

lattice constant is optimized

fdis = 2.687 (mRy/atom)

ford = 2.756 (mRy/atom)

f(fdisford) = 0.069 (mRy/atom)

Free energy -LRO profile

free

en

ergy,

fC

VM

(Ry/a

tom

)

LRO, h100

880K

Fe-Pd

1080K

880K

1200K

J. Phase Equil. and Diffusion, (2006)


tfLNt

initial size of ordered particle is about 4nm

First principles Microstructural Calculation

01.0t[100]

[01

0]

100nm

16.70 0.83 3.59 7.04

disorder – L10 at c=0.5

T=1200K 880K

nmdin 4

nmdin 4

Fe-Pd

1.01,22,2 vv T-O approximation

PFM+PPM

J. Phase Equil. and Diffusion, (2006)


背景

科学技術基本計画

マテリアル

ナノ材料

機能性材料

磁性、半導体物性、光物性

高比強度軽金属合金

構造用材料

鉄鋼材料

高温耐熱合金 Fe,Ni,Cr,Co,Mo

Ti,Al,Mg

大型建造物：橋梁、超高層ビルディング、原子炉、タンカー、パイプライン；先端輸送機器：航空機、宇宙船、etc etc パソコン筐体

強度

強度の高精度設計

エネルギー問題経済効果

強度の本質強度の科学

ライフサイエンス、情報通信、環境、ナノテク・材料


Micro

マルチスケール性

Meso

Macro

10-8cm - 10-6cm

日常生活

10-1cm - 103cm 大型建造物：橋梁、超高層ビルディング、原子炉、タンカー、パイプライン；先端輸送機器：航空機、宇宙船、etc etc パソコン筐体

内部組織


内部組織

析出物

介在物

転位

結晶粒界


変形過程 (弾性変形)


変形過程

破壊


塑性変形


Slip direction jS

)0(

)0(

T

T

)0(T )(tT

ju

Normal to slip plane jP

塑性変形


合金材料の変形挙動

弾性変形

塑性変形

破断


塑性変形

“しわ“の伝播

金属物理学序論コロナ社（1971）


転位運動と変形

転位線 dislocation

変形＝転位線の運動

地球の赤道周囲の250倍!

106cm/cm3 1012cm/cm3

強度設計

内部組織（障害物ex 析出物）

転位運動の制御



泡モデルと結晶粒界

小傾角粒界

大角粒界



結晶粒界

対称小傾角粒界非対称小傾角粒界

ねじり境界



転位論に基づく結晶塑性の研究

Plasticity within dislocation theory

１．単一転位の挙動

２．単一転位の複数障害物場での挙動

３．多数転位の（複数障害物場での）挙動

Single dislocation behavior

Single dislocation behavior in the obstacle field

Many-body behavior of dislocations


単一転位の挙動 (Elementary Interaction process)[1]

Elastic Interaction

Electric Interaction

Chemical Interaction

Geometrical Interaction


単一転位の複数障害物場での挙動 Single dislocation behavior in the obstacle field

ii fN

iN

if

Number of i-th type obstacle

Elementary interaction force with i-th type obstacle

?

ondistributi,,, Cwf

Line tension

w width of an obstacle

C concentration of an obstacle


x

y

z

100 x 300 x 4


25222),,(

Czzyyxx

zzyyzyxstress

sss

ws


02

2

2

2

b

x

y

t

yB

t

ym eff

;mass of a dislocation m

B ;damping

;line tension

eff ;effective stress on a disloc.

.int. appleff

String model of a dislocation


外部応力と転位の安定平衡位置

y/b

x/b

a

濃度c＝0.2 %

M. Fukushi BE Thesis Hokkaido U. (2007)


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.02

0

0 0.10 0.15 0.20 0.25 0.01 0.03 0.05 0.07 0.09 0.04 0.06 0.08 0.10

0.01

0.03

0

0.04

0.05

0.06

0.07

0.02

CRSS

2/1cCRSS

2/1c

Friedel Regime

01.0c



CRSS

c

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

"contau"

Labusch 統計 Friedel 統計

2

1

C

3

2

C

Other Regimes



Dislocation reactions

Multiplication,

Dynamic recovery,

Immobilization/

mobilization of

mobile/immobile

dislocations

tangling

Heterogeneous distribution of dislocations

Reaction-Diffusion equation

Example：Formation

process of Persistent Slip

Bands(PSBs)

C. Shiller and D. Walgraef:

Acta Metall. 36, 3 (1988)

1st stage 2nd stage 3rd stage

Appli

ed s

tres

s

strain

Prinz and Argon

phys. stat. sol.

(1979)

Mughrabi et al

Phil. Mag.(1986)

U. Essmann,

Acta Met.

(1964)

Cu

Actual situations

Fatigue

T. Shoji Ph.D Thesis Hokkaido U. (2003)


U. Essman : phys. stat.sol. (1963)

・Three stages of S-S curve

・dislocation microstructure

・Energetic origin of

dislocation

structural formation

Calculation of a S-S curve

under

Constant Strain Rate Tensile

Test

Objective


),,,(2 k

m

k

i

j

m

j

i

j

i

j

ii

j

i fDdt

d

n± Sign of Burgers vector

immobile(i), mobile(m)：n = (i, m)

j j-th slip system：j = (p, s)

),,,(2 k

m

k

i

j

m

j

i

j

m

j

mx

j

m

j

mxm

j

m fvDdt

d

(111) [101] Primary

100

001

010

Slip systems and evolution equations (R-D)

Conservative motion of dislocation

Glide motion

Reaction

Secondary

(111) [011]



k

kn

jn

jkinter

jn

jintra

kn

jn

jn fff ),()(),(

Intra slip system Inter slip system

R-D equation and Reaction terms

)()()( j

n

j

veconservatinon

j

n

j

veconservati

j

n

j

intra fff

Conservative reaction Non-conservative reaction

Mobile and Immobile


1. Mobilization

2. Immobilization；Dipolization・Multi-polization

ji

ji

jm

jmpvr

jm

jm

jmd vr

Effective stress

Immobile dislocation

jiefffr

Mobile dislocation

Reactions: Intra slip system; conservative

)()()( j

n

j

veconservatinon

j

n

j

veconservati

j

n

j

intra fff



3. Multiplication

4. Recovery

Climb

ji

jiirvr

jm

jmmmvr

ji

jm

jmimvr

Reactions: Intra slip system; non-conservative

)()()( j

n

j

veconservatinon

j

n

j

veconservati

j

n

j

intra fff



Tangling of dislocations

Lomer-Cottrell lock formation ),,( kjspkjvr km

jm

jkmLC

),,(2

kjspkjvr ki

jm

jmt

Reactions: Inter slip system



Mobile dislocation

Plastic strain →

Immobile dislocation

Plastic strain →

Evolution of dislocation structures



U. Essman : phys. stat.sol. (1963)

・Three stages of S-S curve

・dislocation microstructure

・Energetic origin of

dislocation

structural formation

Calculation of a S-S curve

under

Constant Strain Rate Tensile

Test

Objective



immobile

0.0

2.0×10-3

4.0×10-3

6.0×10-3

8.0×10-3

0.0 0.2 0.4 0.6 0.8 1.0

No

rmal

ized

ten

sile

str

ess

(a

pp/

)

Tensile strain

Work hardening rate of 1st stage

Work hardening rate of 2nd stage

mobile

Stress-Strain Curve and Dislocation structure



弾性変形 (Elastic deformation）

外力下での系の静的平衡状態

塑性変形 (Plastic deformation ）

外力下での系の状態の時間的変化

降伏強度 (yielding), クリープ強度 (creep), 超塑性 (super plasticity)

etc.

転位密度(dislocation density), 内部組織 (microstructure) etc

非平衡状態

変形の非平衡熱力学 Non-equilibrium Statistical Thermodynamics

Static equililbrium of a system under the applied stress

Time evolution process of a microstructure under the applied stress

静止転位

原理？


2/1

2 1ln

)1(4

21)(

xx

bbeline

Line energy of dislocation per unit length

xxxxx dedEE line

local

store

hetero

store ))(()(1

))((1

Internal energy due to stored dislocations

)( line

homo

store eE ・Homogeneous distribution

・Heterogeneous distribution

0.0

1.0×102

2.0×102

3.0×102

4.0×102

5.0×102

0 0.2 0.4 0.6 0.8 1

strain

Sto

red

en

erg

y (

J/m

3)

Homogeneous

Heterogeneous



転位運動と変形

Burgers vector

b


2

i

ib

多数転位の挙動 Many-body behavior of dislocations

=minimum 静止転位（static dislocation）


弾性変形 (Elastic deformation）

外力下での系の静的平衡状態

塑性変形 (Plastic deformation ）

外力下での系の状態の時間的変化

降伏強度 (yielding), クリープ強度 (creep), 超塑性 (super plasticity)

etc.

転位密度(dislocation density), 内部組織 (microstructure) etc

非平衡状態

変形の非平衡熱力学 Non-equilibrium Statistical Thermodynamics

Static equililbrium of a system under the applied stress

Time evolution process of a microstructure under the applied stress

静止転位動転位??

原理？転位密度-応力??


保存則 Conservation law

u 内部エネルギー(internal energy)

ut

uJ

st

sJ

s

t

ss J

nt

nJ

nn BBBB 332211

n

反応速度 (reaction rate) j

iiju

x

w

dt

du

J

エントロピー生成速度

(entropy production rate)

n 粒子数 (number of species)

密度(density)

iw 速度(component of the velocity)

ij 応力 (stress)

熱エネルギー (thermal energy)

力学的エネルギー (mechanical energy)


j

iiju

x

w

dt

du

J

Energy conservation

ijij

j

i

x

w

i

j

j

iij

x

w

x

w

2

1

i

j

j

iij

x

w

x

w

2

1

uijijt

vp

dt

duJ

1'

1

密度(density)

iw 速度(component of the velocity)

uJ 熱流束(component of the heat flux)

u

j

iij

x

w

t

vp

dt

duJ

1'

1

zzzyzx

yzyyyx

xzxyxx

ij


Local equilibrium and energy conservation

dvpdsTdu

t

N

t

vp

t

sT

dt

du

zzzyzx

yzyyyx

xzxyxx

ij

dN 転位密度 (dislocation density)

dN

u

uijijt

vp

dt

duJ

1'

1

st

ss J

usT

JJ1

t

N

TTijiju

11J

ddN

K. Nishioka et al., (1978)


Entropy Production Rate

t

NbvN m

mh

hieffapp

32

effkv

mi NC

2132

321

mm NCNbkT

s

mNf

ieffs

N

st

ss J

t

N

TTijiju

11J

t

N

T

1

動転位の相互作用応力 (interaction stress)

app 付加応力(applied stress)

eff 有効応力(effective stress)。

i

h 加工硬化(work hardening)

NN

s

bvNm


v

mN

歪

歪速度

応力

stress strain

Strain rate

引っ張り試験

Tensile test

塑性変形の原理 (principle of deformation)

動転位の挙動原理

ieffs

N

NN

s

ひずみ速度一定条件下では（ある変形速度が与えられたとき）、運動転位の密度は、そのひずみ速度を保つのに必要な変形応力が最小となるように決まる。

変形応力が一定の条件下では変形速度を最大とするように運動転位の密度が決定される。

K. Sumino (1971)


超塑性 (Superplasticity)

Non-equilibrium Statistical Thermodynamics

粒界すべり(超塑性)の非平衡統計熱力学

数ミクロン程度の微細結晶粒

高温変形

異常に大きな破断伸び

Refined grain

High temperature

Abnormal deformation


超塑性 (Super plasticity)


Slip deformation （すべり変形）

bvNm Strain rate(歪速度)

mN

v

b

Mobile dislocation density (可動転位密度)

Burgers vector (バーガースベクトル)

Average dislocation velocity (可動転位の平均速度)

Super plasticity (超塑性)

0 gNStrain rate(歪速度)

gN

0

Rotational grain density (可動結晶粒密度)

回転の素過程の歪への寄与

Average rotational velocity (可動粒の平均回転速度)

超塑性 (Super plasticity)

T. Mohri 未公表


Hall-Petch relation 1

金属組織学丸善 (1972)



h

A

dxx

xDA

0 a

fa

dxx

xDAaf

xh

ndxxD

a

a


2122

1

xa

nxD

21

xa

xanxD


0 fa constant a

転位論入門アグネ (1974)


2122 xa

x

AxD

fa

a

a

adxxa

xDAf

0 cos1

cosa

fad

a

f

afa

a

a

221

ffa

a

2

1a


0

dxxD

a

a



21 dfcfa



Thank you !

Division of Materials Science and Engineering, Graduate School of Engineering, Hokkaido University

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