材料科学基础下英文chpt2 11b

Chapter

2材料科学基础（II）Interfaces

in

materials-IIBrief

introduction

of

method

based

onmatrix

and

vector

calculations材料科学基础（II）Dislocation

model

for

semicoherent

interfacesD

b/

for

simple

small

angle

grain

boundariesD

=

b

/

for

Interfaces

with

1D

or

isotropic

misfitHow

to

determine

D

for

a

general

small

angle

grainboundary,

and

for

an

interface

with

anisotropic

misfit?Relations

of

two

lattices:

Rotation:

a

=

Ra

,General:

a

=

Aa

2.

O-lattice

model

for

general

interfaces普适计算工具，知道思路和基本概念即可材料科学基础（II）Definition

of

the

principal

O-lattice

vectorsFrom

张敏SRT

workProgramcan

befoundfrommywebpageO-element:

position

of

best

matching(zero

misfit)Principal

O-latticevectorO-cell

wall:

Positionsof

worst

matchingBollmann,

1970L

+

L

=O-latticeL

=

bcc

{1

1

0}L

=

fcc

{1

1

1}Programs

fromAnisotropic

misfit

in

general

systems

in

3D材料科学基础（II）材料科学基础（II）材料科学基础（II）The

intersections

of

the

O-cell

walls

with

an

interface

arepossible

position

of

misfit

dislocationsSquare

dislocation

network

ofscrew

dislocations

in

a

twist

grainboundary

in

SiDB

Williams

and

CB

Carter1996材料科学基础（II）Dislocations

in

small

anglegrain

boundaries

in

metalsand

in

Al-Al2O3

interfacesNbAl-Al2O3Pt材料科学基础（II）(by

Yang

Xiao-peng,

programavailable

in

my

homepage)Consistence

of

geometryof

O-cell

walls

with

misfitdislocations

in

a

1D

misfitsystemisotropic

dilation2D

O-lattice,

formed

by overlapping

rigidlattices

of

different

lattice

constants材料科学基础（II）Step

1:

Build

a

quantitativeRelationship

between

2

latticesx

=

Ax

x

,

x

:

vectors

in

lattice

and

A:

transformation

matrix,

misfit

distortion

matrixA

=

cos

sin

–

sin

cos

r1

//

x

axisr1

=

[r,

0],

([]

=

column

vector)r2

=

[rcos

,

rsin

]Matrix

calculation

of

O-latticeDetermination

of

misfit

displacement

x

=

x

–

x

=

(I

–

A-1)x

=

Tx

Step

3:

Find

misfit

displacement

associated

with

x

x

m

=

x

x

nwhere

x

n

is

the

nearest

neighbor

from

x

x

can

be

translated

to

x

by

n

x

=

x

+

∑b

n

iL∴

x

m

=

x

x

n

=

x

(x

+

∑b

iL)=

x

∑b

iLiIn

condition of

|

x

m|

<

|

x

m

–

b

L|x

x

n

x

mbax

Step

4:

Find

x

without

misfit

x

m=0

x

=

Tx

=∑b

iLDefinition

of

O-lattice

elements!!Step

2:

Calculate

relative

displacement

at

x

:x

=

Ax

T

=

I

–

A-1Configuration

of

periodic

dislocationsDefine

distribution

of

O-elements

(GMZ)

with

xOTxi

=

b

iO

LGiven

an

interface

normal

to

n

(unit

vector), containing

periodic

dislocations

with

Burgersvector

b

i

(related

to

b

i

)L

LDefineReciprocal

vector

for

O-cell

walls:

ciO

=

T

b

i*Dislocation

direction:Dislocation

spacing:D

=

1/|

i|Special

systems:

ci

=

bi*

=

b

i*

b

i*OBollmann’s

equation,

70

i

=

n

ciObi*

=

biL/|biL|2

=

reciprocal

Burgers

vectorSpecial

boundaries,

i

=

n

bi*,

n

bi*Rotation:

(|bαL|

=

|bβ

|

=

b)LβD

=

1/|

bi*|

=

b2/(bαL

–

b

L)

=

b/2sin(

/2)

b/

α

βIsotropic

deformation:

(b

L

//

b

L, |bL|

=

bL)D

=

1/|

bi*|

=

1/(1/bαL

–

1/bβ

)

=

bβ

/

L

Lbβ*

b*b

*

=

(bβL

–

bαL)/bαL

bbβLb

LZhang,APL2005D

=

1/|

b*|

=

1/|b

i*

b

i*|bi*=

b

L/|b

L|2i

iCosine

law:

D

=

bαLbβ

/[(bα

) +

(bβ

) –

2bα

bβ

cos(

)]L L

2 L

2

L

L

1/2Unified

expressions

for

formulas

in

textbooksRec.

vector

for

O-cell

wallsTest

of

the

formula

with

simple

casesO-lattice

model

for

general

interfacesO-element:

position

of

best

matching

(zero

misfit)O-cell

wall:

positions

of

worst

matchingFormulas

for

dislocation

geometry

(Unified

expression)3.

Models

for

other

singular

and

vicinal

interfaces(Up

dated

knowledge,

not

in

the

text

book)CSL/DSC

model

(for

large

lattice

misfit

&

Low

Calculation

ModelsEvidence

of

secondary

preferred

state

and

secondarydislocationsObservation

of

secondary

dislocationsVariation

of

dislocation

spacing and

energy

with

3.

Models

for

other

singular

and

vicinal

interfaces1)

CSL/DSC

model

(for

large

lattice

misfit)CSL:Coincidence

Site

Lattice(重合点阵)Variation

of

energy

with

misorientationFig.

7-14

on

P419材料科学基础（II）Key

parameter1/

=

density

of CSL

points

is

an

odd

number

for

cubic

systemsvolume

of CSL

unit

cellvolume

of crystal

lattice

unit

cell

Dense

CSL

points

maybe

favored

by

nature材料科学基础（II）材料科学基础（II）for

n

=

1,

m

=3S

=

(1+

9)

=10area

of

a

unit

CSL

cell

=

(m2

+

n2

)/2kn

=

1,

m

=

3,

=5

=

2tg-1(n/m)

=

2tg-1(1/3)

=

36.87

(3,1)(1,-3)(3,-1)(-1,-3)

5CSLDerivation

of

(forcubic

crystals)area

of

a

CSL

cell面积=斜边长2

=m2+n2Generating

expressionsThen,

for

cubic

systemFor

other

systems

one

need

to

modify

the

derivationE.g.

{hkl},

<uvw>

may

not

be

equivalent

vectors

(of

samelength)材料科学基础（II）Ranganathan

formula(for

cubic

crystals)The

area

of

square

unit

CSL

cell

=

m2

+

n2Generating

function

for

of

a

square

latticeRotation

axis

is

a

rational

vector

of

(h,

k,

l)plane

normalA

∑3

<111>twin

boundaryWhen

two

grainsshare

a

plane,the

plane

is

twinboundary

A

orsimilarly

C,BRotation

of

,

/3

around

as

axis

<111>

passing

a

lattice

point

in

plane

ABCBCCBProduced

by

shearProduced

by

rotationof

atoms

from

A

layerC

B

CBWhy

is∑3?材料科学基础（II）Possible

low

energyinterface

location:topassdense

CSL

pointsStepped

boundary

alongthe

plane

containingdense

CSL

points

11

<110>

CSL

boundary

forunderstanding

Fig

7-2

in

p41350.5

二次位错的台阶结构纯台阶DSCL柏氏矢量位移Dense

CSL

points

are

presenton

planes

of

particular

locationsCSL

pattern

tends

to

bereserved

locallySmall

deviations

will

lead

to

theloss

of

the

coincidenceTranslations

of

DSCL

vectorpreserves

the

patternDSCL:

Displacement

Complete

Pattern

Shift

LatticeSteps

associatedwithsecondarydislocationsDeviation

from

CSL

OR

withsecondary dislocation

modelBrandon

Criterion

max

-1/2

小角度晶界的上限～

AP

Sutton,

RW

Balluffi,

Interfaces

in

Crystalline

Materials,1995Preferred

state:Reference,

deviation

from

which

defines

misfitIn

good

matching

zone

(GMZ)

between

the

dislocationsPrimary

GMZSecondary

GMZBurgers

vector

=

vector

in

the

DSCLBurgers

vector

=

translationvector

in

crystal

latticesPrimary

dislocations

vssecondary

dislocationsGenerating

expressionsThen,

for

cubic

systemFor

other

systems

one

need

to

modify

the

derivationE.g.

{hkl},

<uvw>

may

not

be

equivalent

vectors

(of

samelength)Ranganathan

formula材料科学基础（II）(for

cubic

crystals)The

area

of

square

unit

CSL

cell

=

m2

+

n2Generating

function

for

of

a

square

latticeRotation

axis

is

a

rational

vector

of

(h,

k,

l)plane

normalPrimary

O-lattice

andprimary

dislocationsSecondary

O-lattice

andsecondary

dislocationsPrimary

O-lattice

vs

secondary

O-lattice材料科学基础（II）Dislocation

model

for

semicoherent

interfacesO-lattice

model

for

general

interfacesO-element/O-cell

wallFormulas

for

dislocation

geometryUnified

expression

of

D

for

simple

boundariesModels

for

other

singular

and

vicinal

interfacesCSL/DSC

model

(for

large

lattice

misfit

&

Low

formulas

for

cubit

crystals<uvw>may

not

be

equivalent

vectors

(of

same

length)Structure

unit

model

(atomic

calculation)Secondary

dislocations材料科学基础（II）Arial

13§3.

Structure

of

interfaces

and

geometrical

models§3.

Structure

of

interfaces

and

geometrical

modelsCSL（重位点阵）:定量描述重位点的模型，用于计算可能的重位共格结构，该结构可以由少量几种结构单元重复构成

=CSL点密度：晶体位向关系(取向差)的函数,与界面取向无关半共格界面：界面上存在位错，位错之间是共格区(成片的共格点）重位共格结构：界面上存在间距很小的离散共格的点，可能存在二次位错，之间是周期性分布的重位共格(CSL)区DSCL(位移点阵)：二次位错的柏氏矢量来自该点阵

O点阵：零错配位置的点阵,是计算好区/错配区分布及界面位错的普适工具，可以计算一次(一般柏氏矢量)或二次位错倒易矢量：代表正空间一组面，长度=1/面间距，方向=面法向，倒易点:倒易矢量定义的点,周期分布的倒易点=倒易点阵，倒易点所在空间=倒易空间材料科学基础（II）材料科学基础（II）CSL

may

not

represent

thetrue

atomic

structureunrelaxed,translatedRelaxed,

tends

to

maximize

distancebetween

black

and

white

atoms,different

(higher)

symmetry

formsRelaxation

from

CSL

structureABBABBAABABAABABFCC点阵以[001]轴旋转的对称倾转晶界的结构单元模型

5的CSL，黑点为重位点，虚线平行于面(210)

5晶界的松弛结构，晶界由B单元组成

17晶界的松弛结构，晶界由A和B单元以ABB顺序重复排列，平行于面(530)

37晶界的松弛结构，晶界由AABAB顺序重复排列，晶界面是(750)(e材)

1料(完科整晶学体)基的情础况，（平I行I于）(110)面构成的结构单元，以A表示余永宁,2000Structural

unit

model材料科学基础（II）Early

structural

unit

model以不同边长比(n:1)的镜面菱形构成界面的结构单元的材料科学基础（II）Variation

of

ratio

of

units

with

misorientationSutton

and

Vitek

11:

B

units

27:

A

unitsfcc:

r=

[110]材料科学基础（II）Structures

of

Interfaces

between

Cm

and

AHowe

and

Spanos,Phil.

Mag.,

1999Zhang

et

al.

Acta

mater.,2000Ye

and

Zhang,

Acta

mater.,

2002Small

DSCL

vectors

for

Burgersvectors

do

not

lie

in

the

planecontaining

dense

CSL

points闫佳易，2009基础（II）材料科学

g1

=

g(02-2)

g(062)s

g2

=

g(020)

g(002)sApplication

to

S

phase

(Al2CuMg)in

Al-Cu-Mg

alloys

(Rule

III)Radmilovic

et

al.

(1999)Reported

OR

(Type

II)

:[100]s//[100]

(0-21)s~//(014)

Reported

facet(0

4

3)s//(0

2

1)

Gu

and

Zhang,

2007Interfacial

structure

development

tendency材料科学基础（II）To

form

low

energy

structure,

by

establishing

lowenergy

bondsToform

coherentregions

asfar

aspossible

for

metallicsystems

(primary

preferred

state)To

form

low

energy

structural

units

(regularly

coherent)

ifthe

coherentmisfit

strainis

too

large

for

fullycoherent(secondarypreferred

state)Toform

amorphousif

high

energy

bonds

have

to

form,

e.g.

O-O

in

ceramicsIf

misfit

strain

is

too

large

fora

fullycoherent

(orstrictly

regularly

coherent)

one

to

form,misfitdislocations

will

present

in

a

semicoherent

(semi-regularly

coherent)

one§4.

Interfacial

energyPhysical

basisDislocation

modelNearest-neighbor

broken-bond

modelVariation

of

Us

with

interface

orientationSpecific

interfacial

free

energySurface

tension

(F/L)

and

surface

stress

tensor§5.

Equilibrium

boundary

segregation§6.

Equilibrium

shape

of

crystals,

grains

and

particlesThe

Wulff

plot

and

Wulff

ConstructionLocal

equilibrium

of

facetsForce

balance

of

surface

tensionParticles

at

grain

boundariesEmbedded

particles材料科学基础（II）§4.

Interfacial

energy

(

)材料科学基础（II）Physical

basisRise

of

energy

due

to

broken,

distorted,

and/or high

energy

bondsPossible

rearrangement

of

atoms

in

structure

and compositionsApplicationsMany

phenomena(wetting,

segregation,

absorption,

nucleation

insolidification

and

precipitation,

coarsening,

graingrowth,

microstructure

development,

catalysis)and

material

properties(embrittlement,

intergranular

fracture,

grain

slidingin

creep,

and

microstructure-related

properties)材料科学基础（II）§4.

Interfacial

energy

(

)2.

Dislocation

modelp419

Fig

7-14,

p301

eq

4-107,

p418

eq

7-10typicalMainly

for

low-angle

grain

boundaries,misorientation

angle

value

in

cubic

materials

is

15°.Read-Shockley

model:For

an

array

of

dislocations,

the

long-rangestress

field

depends

on

the

spacing.

Giventhe

dislocation

density

and

the

dislocationcore

energy,the

energy

of

thedislocation

wall

can

be

estimated

bysummation

of

the

dislocation

energy0.40.350.30.250.20.150.10.05003010

20Misorientation

Angle(degrees)Relative

Boundary

EnergyYang,

C.-C.,

A.

D.

Rollett,

et

al.(2001).

“Measuring

relative

grainboundary

energies

and

mobilities

in

an

aluminum

foil

fromtriple

junction

geometry.”

Scripta

Materiala:.（II）材料科学基础§4.

Interfacial

energy

(

)Energy

of

a

single

dislocation:0cGb2

RE

ln( )

E4

(1

)

rGb2

D

E

ln( )

c

0

(

A0

ln

)4

D(1

)

b

D0

Gb4

(1

)0cGb2E

4

(1

)A

D

bdislocation

number

=1/Dlength

=

1R

=

D/2or =

b/2Energy

of

dislocation

array

in

unit

area.:

Read-Shockley

1950A.

Otsuki,

Ph.D.thesis,

Tyoto

University,

Japan

(1990)Energy

depends

onrotation

axis&

boundary

planeFig.

7-14

on

P419上面假设同转轴，同柏氏矢量Misorientation

Axis

[uvw]

;

=

5oEnergy

(m

J/m

2),T=240o

C[001][101][111]Tilt

parallel

disl.190170148Twist

network

disl.200205155材料科学基础（II）§4.

Interfacial

energy

(

)Experimental

results

on

copper.Gjostein

&

Rhines,

Acta

metall.

7,

319

(1959)TiltTwistNo

universal

theory

to

describe

the

energy

of

HAGBs.材料科学基础（II）§4.

Interfacial

energy

(

)TiltPorter

and

Easterling,2001For

high

angle

boundaries,

use

a

constant

valueunless

they

contain

dense

CSL

points材料科学基础（II）§4.

Interfacial

energy

(

)TiltCoherent

vs

incoherent

twinboundariesPorter

and

Easterling,2001CalculationsHasson,G.C.an材d

C料.Go科ux(学197基1).

础“Inte（rfacIiaIl

e）nergies

of

tilt

boundaries

in

aluminum.Experimental

and

theoretical

determination.”Scripta

metallurgica

5:889-894<100>Tilts<110>TiltsTwinObservationsEnergy

valleyEnergy

cusp材料科学基础（II）More

“coherent”

the

interface,the

lower

gb.J.

W.

Martin,

et

al,

1996Some

remarks:材料科学基础（II）Homework

7-2:

whether

the

boundary

will

rotate

or

notdepends

if

energy

is

lowered

after

rotation.Refer

to

p420

for

the

rough

data

of

interfacial

energy

forboth

cases.Annealing:

provides

kinetic

energy

for

the

atoms

tomove

towards

lower

energy

state.Most

materials

are

not

in

stable

equilibrium,

but

in

metastable

statep417Nearest-neighbor

broken-bond

modelAssumptions:Neglect

energy

of

secondary

bondsNeglect

difference

in

atoms

(as

in

unary)Neglect

relaxation

&

reconstruction

Us,

is

not

function

of

T (T

=

0K)

Hs

(heat

of

sublimation)

U

ZNa

/2Us

=

(

ni

pi

/2)/As

spi

:

numbers

of

broken

bonds

per

atom

of

type

isni

:

number

of

type

i

of

surface

atomss§4.

Interfacial

energy

(

)i=原子断键类型断键数量因原子位置而不同材料科学基础（II）fcc

(111)

surfaceExample

of

fcc

{1

1

1}:A

=

enclosed

by

<1

1

0>ni

=

3/2

+

3/6

=

2,

i

=

1!spi

=

3sA

=

a2{

(2)

(2)

[

(3)]/2}/2

=

a2[

(3)]/2Us

=

(

ni

pi

/2)/A

=

2

(3)

/as

s

2§4.

Interfacial

energy

(

)材料科学基础（II）fcc

(111)

surface材料科学基础（II）fcc

(100)

surfacefcc

(110)

surfaceBroken

bonds

of

surface

atomsp1s

=

3??Diagramillustrating

bondsacross

different

(hkl)

planesni

and

pi

vary

with

{hkl}s

snote

different

types

of

surface

atomstop

layer,

two

broken

bondsunder

layer,

x

bond(s)材料科学基础（II）料科学基础（II）Simple

cubic，（0

-1

3）P418

(114)

Fig.

7-11Porter

and

Easterling,2001材Terrace-ledge-kink

(TLK)

broken

bond

model

forsurface

energy

(

)I）材料科学基础（IA

=

1

a,

>

0

ni

pi

=

(cos

+

sin

)/as

sUs

=(

n

sp

s

/2)/AFig.

7-12i

i=

(cos

+

sin

)

/(2a2)=

(2)sin(

/4

+

)

/(2a2),Because

of

4

fold

symmetry,

=

±

/4,Us

=

Us

=

/[

(2)a2]max

=

0,

±

/2, Us

=

Us

=

/(2a2),mineqn

(7-8)Variation

of

Us

with

interface

orientation

ksin

4+

)

ksin

ksin

4+

)

极坐标系极坐标系!直角坐标系sin(

/4)

=

cos(

/4)

=

1/

(2)cos

+

sin

=

(2)[sin(

/4)cos

+cos(

/4)sin

]p418containingSpecific

interfacial

free

energy:

(p347)Internal

energy

(total)

in

a

systeminterfaces,

unitary

system

(C

=

1)eqn(5-89)dU

=

TdS

–

PdV

+

dAG

=

U

+

PV

–TSdG

=

dU

+

PdV

+VdP

–

TdS

–

SdT

+

dAdG

=

VdP

–

SdT

+

dAAt

constant

T

and

P:

dG

=

dAIf

is

isotropic,

A

=

Gxs

p347,

eqn(5-91);

p417,

eqn(7-3)§4.

Interfacial

energy

(

)Definition:

(

=Gxs/A)is

the

excess(过剩)Gibbs

freeenergy

forming

per

unit

area

of

the

surface

(interface)

layerGxs

=G(有界面体系)

G(无界面体系)(Cahn

1977)材料科学基础（II）Reversible

work

to

increase

the

interface

area

by

dA

is dWT,P

=

d(

A)

=

dA

+

Ad

dWT,P

=

dA,

when

(

/

A)P,T

=

06.

Surface

tension

(F/L)

and surface

stress

tensor

(fij)For:

A

=

xL,

dA

=

Ldx, d

=

dx/x

W

=

G

W

=

dA

+

Ad

=

(

dx

+

xd

)

L

＝

Fdx

F/L

=

+

d

/d

or f

=

+

/

,

(isotropic)General

case: fij

=

ij

+

/

eij, (eij=strain

tensor)即:因为液相表面和内部的原子可以自由交换，表面结构可随面积改变而基本守恒，

表面张力=(比)表面(自由)能L材料科学基础（II）If

/

A

=

0,

Surface

tension

f

=

f

=

F/L

(N/m)

equal

in

value

to

(J/m2)Average

surfaceenergies

of

selectedsolids

and

liquids.Incoherent

grain

boundary

energy

gb

isabout

1/3

of

the

solid-vaporinterfacial

energy.材料科学基础（II）From:

J

M

HOWE.

Interface

in

Materials,

1997Porter

and

Easterling,2001材料科学基础（II）§5.

Equilibrium

boundary

segregation1.

Equilibrium

solute

concentration

C

p420

(Eq.

7-12)Helmhozltz

free

energy

due

to

solutesand

their

configuration

of

an

alloy:F

=

(PUl

+

QUg)

–

kTln(W)Number

of

states:W

=

N!n!/[P!(N

–

P)!Q!(n

–

Q)!]In

(

F/

Q)P,T

=

0, (note

dP=-dQ)Ug

–

Ul

=

kTln{(n

–

Q)P/[Q(N

–

P)]}Define:

Co

=

P/N

P/(N-P)C

=

Q/n

Q/(n-Q)a

l

g

o

E

=

N

(U –

U

)

=

RTln(C/C

)One

gets

C

=

Coexp(

E/RT)

eqn(7-18)sitesolutelatticeNPgr.

b.nQ

E

>

0,

C

>

CoTwo

sites

areconceivable

toincorporate

a

solute

atominto

the

boundary:

The

atom

may

occupy

a

lattice

site(A)or

a

non-lattice

site

in

the

boundary

core(B)偏聚假设每个晶格位置体积相等,C和Co可理解为体积百分数等体积，等温假设材料科学基础（II）Segregation

enthalpy

of

Pin

-FeH.

Gleiter,

Prog.

Mater.

Sci.

33

(1989)J

M

HOWE.

Interface

in

Materials,

1997材料科学基础（II）Grain

boundary

engineering:

historicalperspective

and

future

prospectsIncreasing

grain

boundary

enrichment

withdecreasingsolidsolubility材料科学基础（II）Porter

and

Easterling,2001材料科学基础（II）保温7s时，试样中硼分布和对应的金相照片Cui

and

He,

2001§6.

Equilibrium

boundary

segregationGibbs

absorption

equation

(Gibbs

Model)Definition

for

system

C>1:dU

=

TdS

–

PdV

+

idNi

+

dABy

increasing

system

at

constant

T

and

P

one

gets

(Cahn

77)U

=

TS

–

PV

+

iNi

A

A

=

U

+

PV

–

TS

–

iNi=

G

of

the

system

–

G

of

the

materials

in

the

systemExcess

quantity

in

a

system

of

+

+

interfaceUs

=

Uxs/ASs

=

Sxs/A

i

=

nxs/AUxs

=

U

–

Uα

–

Uβ,Sxs

=

S

–Sα

–

Sβ,nixs

=

ni

–

nαi

–

nβi,Vxs

=

V–

Vα

–

Vβ

=

0材料科学基础（II）§6.

Equilibrium

boundary

segregationIn

terms

of

excess

quantity,

one

gets: from

dU

=

TdS

–

PdV

+

idNi

+

dAito

dUxs

=

TdSxs

+

idN

xs

+

dAAnd

from

U

=

TS

–

PV

+

iNi

Ai

ito

Uxs

=

TSxs

+

N

xs

AThen

from:dUxs

=

d(TSxs