Electromigration: cross-coupling between electron and

1

Illustration of Cross-Coupling EffectJ1 = ndash L111 ndash L122 ndash L133

J2 = ndash L211 ndash L222 ndash L233

J3 = ndash L311 ndash L322 ndash L333

1

3(left) = 3(right)1(left) gt 1(right)rarr J3 gt 0

Electromigration cross-coupling between electron and atomic currents

I di

Regan Aloni Ritchie Dahmen

Indiumnanocrystalon metallic

carbon nanotube

2

Ritchie Dahmen and ZettlldquoCarbon

nanotubes as nanoscale mass

conveyorsrdquo Nature 428(2004) 924

2

source

i adatomJi = ndashLiii ndash Liee

Je = ndashLeii ndash Leee

J electronsms

3

sink

Jeelectronsm se = ndashe

is the electrostatic potential (volt)that is externally controlled

Cross-Coupling between Electron and Heat CarriersThermoelectric Effect

Potentiometer

JQ = ndashLQQln ndash LQee

Je = ndashLeQln ndash Leee

temperature T+T

o e o e e

A BJQ

A JQB

JQ heat current Jm2s

0 = J A = L Aln L A A

Potentiometer has high electrical impedance

4

temperature T

Seebeck Effect

JQ JQ 0 = Je = ndashLeQ ln ndash Lee e

rarr eA = ndashLeQ

Aln LeeA

0 = JeB = ndashLeQ

Bln ndash LeeBe

B

rarr eB = ndashLeQ

Bln LeeB

3

bull thermocouple thermometer

5

bull low-grade heat scavenging

Peltier Effect Heat Pump

heat out = -heat in

- I e = Je

A = ndashLeQAln ndash Lee

AeA =

J B L Bl L B B

I JQA

I JQB

heat out heat in JeB = ndashLeQ

Bln ndash LeeBe

B

JQA = ndashLQQ

Aln ndash LQeAe

A neJQ

B = ndashLQQBln ndash LQe

BeB

6

Q

heat in = JQB - JQ

AIn ideal isothermal limit ln = 0

IeLeeA =e

A rarr JQA = ndashLQe

A IeLeeA

IeLeeB =e

B rarr JQB = ndashLQe

B IeLeeB

4

7

Peltier-effect Cooled CPU (Ken Peter)

Effect of Observation Frame on Flux

C-frame

JiC = civi

C

8

L-frame

vCL

JiL = ci(vi

C + vCL) = Ji

C + ci vCL

All fluxes in the Onsager equation are crystal-frame fluxeswhere the observer is co-moving with the RVE

5

Gas diffusion lots of free volume

9

Liquid diffusion less free volume and shared between atom clusters

10

6

Diffusion in crystal free volume localized as vacancy

1

2

3V

11


1

2

3V

12

7

Vacancy thermodynamics in Kossel crystal

V(r) = -+k(r-a0)22e1 = -Z2Z = 4 (2D)

13reference state Sconfig=0 E=Emin=N1e1

Z 4 (2D)Z = 6 (3D)

Vacancy creation by sequential atomic attachment to surface ledges

E = Emin + Nv eVf

eVf = Z2 = -e1

for Kossel crystalbut not in general

vacancy ishellip

14

ldquonanoporosityrdquoldquoatomic-scalefree volumerdquo

8

15

transformation strain 11

normaltraction

t11

16

transformation strain 11If there is tensile traction near a surface then moreldquonanoporosityrdquo is favored nearby

9

vacancyldquonanoporosityrdquo

17

∆VV0 = 3∆LL0

= 3∆aa0 + XV

For simplicity assume - Vf = 0

18

Take one beam out of this construction scaffold what will happen Answer Nothinghellip

10

JV

1111

19

d

JV

1111

20

d

11

21William Conyers Herring ldquoDiffusional viscosity of a polycrystalline solidrdquo J Appl Phys 21 (1950) 437

22

In addition to surfacesGBs climbing dislocations are alsointernal sourcessinks of vacancies

parttcV = JV + (parttcV)source

12

Vacancy hops in monatomic crystal may be uncorrelated

up

down

left right

23

r1= up= (0 a0)

vacancy hopsup down left rightwith equal probability

xV(t)-xV(0) = r1 + r2 + hellip + rK K = VtVacancy hops in monatomic crystal may be uncorrelated

up

down

left right

stillup down h d

24

up down left rightwith equal probability

the second mover2 is

independentof r1

13

Mean Squared Displacement =

E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =

E[ (r1 + r2 + hellip + rK)(r1 + r2 + hellip + rK) ] =

E[ r1r1 + r1r2 + + r1rK +E[ r1 r1 + r1 r2 + hellip + r1 rK +

r2r1 + r2r2 + hellip + r2rK +

hellip +

rKr1 + rKr2 + hellip + rKrK ] =

E[ r r ] + E[ r r ] + + E[ r r ] =

25

E[ r1 r1 ] + E[ r2 r2 ] + hellip + E[ rK rK ] =

KE[ r1r1 ] = Ka02 (Kossel crystal)

Ka022 (FCC crystal)

3Ka024 (BCC crystal)

Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14

join at Matano plane x=0

c1-infin

c1infin

lL lR

Bake at high-tempfor some time t1

lab frame originduring baking

x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )

D = fXVDV (independent of c1)

parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R

Width of profile (diffusion length) 6

Infinite-space soln OK as long as

l D t

l l l

General Remark about lD

lD waveconvection

diffusiondiffusion is moreeffective meansof matterinfotransport at

small lengthscales convection is more effective means of

matterinfotransport at

28

tHarry and Sally

send pheromones Harry and Sally send

electromagnetic waves

large lengthscales

15

Self diffusion hops

canrsquotcan t moveat all

29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32

total rate = ZXVVrsquo

D = XVVrsquoa0

2

17

Self diffusion hops will be correlated even if vacancy hops are uncorrelated

upup

down

left right

33

r1= right= (a0 0)


up

r2 is more likely to be

left if r1 = right

This ldquobackflowrdquo

up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K

In dilute substitutional

alloys the

intrinsic diffusivities

36

alloys the interdiffusivityis controlled byself-diffusivity of the solute

19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x

This experiment This experiment

37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures

intrinsic diffusivity

( 002)D X

solubility of Cu and Znin Mo is

785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K

Tran Amer Inst Min Met Eng 171 (1947) 130

38

Zn atoms drives a game of tetris

nearly zeroTM(Zn) 693KTM(Cu) = 1356KTM(Mo) = 2890K

20

J1C

J2C

inert markerwire which

does not participatein diffusion

Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39

if marker wires fixed

to benchKirkendall

Effect

50 years ago

homogeneoushomogeneous

lL lR

today

c2(x)~

g2-rich

g1-rich

bake at 1200Kfor some hours

lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane

lab frame origin50 years ago

origin today

How to find the Matano planein todayrsquos observation frame

x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44

Fat droplets suspended in milk (from Dave Walker) The droplets range in size from about 05 to 3 microm

23

viscous oil

v Stokes law F = 6rv = v

mobility v = F N mobility v F ms msN

Einstein relation D = kBT

Stokes Einstein relation

45

Stokes-Einstein relation between viscosity and self-diffusivity

BB 6

k TD Mk T

r

2

source

i adatomJi = ndashLiii ndash Liee

Je = ndashLeii ndash Leee

J electronsms

3

sink

Jeelectronsm se = ndashe

is the electrostatic potential (volt)that is externally controlled

Cross-Coupling between Electron and Heat CarriersThermoelectric Effect

Potentiometer

JQ = ndashLQQln ndash LQee

Je = ndashLeQln ndash Leee

temperature T+T

o e o e e

A BJQ

A JQB

JQ heat current Jm2s

0 = J A = L Aln L A A

Potentiometer has high electrical impedance

4

temperature T

Seebeck Effect

JQ JQ 0 = Je = ndashLeQ ln ndash Lee e

rarr eA = ndashLeQ

Aln LeeA

0 = JeB = ndashLeQ

Bln ndash LeeBe

B

rarr eB = ndashLeQ

Bln LeeB

3


5



heat out = -heat in

- I e = Je


AeA =

J B L Bl L B B

I JQA

I JQB


Bln ndash LeeBe

B

JQA = ndashLQQ

Aln ndash LQeAe

A neJQ


BeB

6

Q

heat in = JQB - JQ


IeLeeA =e


A IeLeeA

IeLeeB =e


B IeLeeB

4

7



C-frame

JiC = civi

C

8

L-frame

vCL

JiL = ci(vi

C + vCL) = Ji

C + ci vCL


5


9


10

6


1

2

3V

11


1

2

3V

12

7


V(r) = -+k(r-a0)22e1 = -Z2Z = 4 (2D)


Z 4 (2D)Z = 6 (3D)


E = Emin + Nv eVf

eVf = Z2 = -e1


vacancy ishellip

14


8

15


normaltraction

t11

16


9


17

∆VV0 = 3∆LL0

= 3∆aa0 + XV


18


10

JV

1111

19

d

JV

1111

20

d

11


22



12


up

down

left right

23

r1= up= (0 a0)



up

down

left right

stillup down h d

24



independentof r1

13


E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =




hellip +


E[ r r ] + E[ r r ] + + E[ r r ] =

25



Ka022 (FCC crystal)


Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

3


5



heat out = -heat in

- I e = Je


AeA =

J B L Bl L B B

I JQA

I JQB


Bln ndash LeeBe

B

JQA = ndashLQQ

Aln ndash LQeAe

A neJQ


BeB

6

Q

heat in = JQB - JQ


IeLeeA =e


A IeLeeA

IeLeeB =e


B IeLeeB

4

7



C-frame

JiC = civi

C

8

L-frame

vCL

JiL = ci(vi

C + vCL) = Ji

C + ci vCL


5


9


10

6


1

2

3V

11


1

2

3V

12

7


V(r) = -+k(r-a0)22e1 = -Z2Z = 4 (2D)


Z 4 (2D)Z = 6 (3D)


E = Emin + Nv eVf

eVf = Z2 = -e1


vacancy ishellip

14


8

15


normaltraction

t11

16


9


17

∆VV0 = 3∆LL0

= 3∆aa0 + XV


18


10

JV

1111

19

d

JV

1111

20

d

11


22



12


up

down

left right

23

r1= up= (0 a0)



up

down

left right

stillup down h d

24



independentof r1

13


E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =




hellip +


E[ r r ] + E[ r r ] + + E[ r r ] =

25



Ka022 (FCC crystal)


Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

4

7



C-frame

JiC = civi

C

8

L-frame

vCL

JiL = ci(vi

C + vCL) = Ji

C + ci vCL


5


9


10

6


1

2

3V

11


1

2

3V

12

7


V(r) = -+k(r-a0)22e1 = -Z2Z = 4 (2D)


Z 4 (2D)Z = 6 (3D)


E = Emin + Nv eVf

eVf = Z2 = -e1


vacancy ishellip

14


8

15


normaltraction

t11

16


9


17

∆VV0 = 3∆LL0

= 3∆aa0 + XV


18


10

JV

1111

19

d

JV

1111

20

d

11


22



12


up

down

left right

23

r1= up= (0 a0)



up

down

left right

stillup down h d

24



independentof r1

13


E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =




hellip +


E[ r r ] + E[ r r ] + + E[ r r ] =

25



Ka022 (FCC crystal)


Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

5


9


10

6


1

2

3V

11


1

2

3V

12

7


V(r) = -+k(r-a0)22e1 = -Z2Z = 4 (2D)


Z 4 (2D)Z = 6 (3D)


E = Emin + Nv eVf

eVf = Z2 = -e1


vacancy ishellip

14


8

15


normaltraction

t11

16


9


17

∆VV0 = 3∆LL0

= 3∆aa0 + XV


18


10

JV

1111

19

d

JV

1111

20

d

11


22



12


up

down

left right

23

r1= up= (0 a0)



up

down

left right

stillup down h d

24



independentof r1

13


E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =




hellip +


E[ r r ] + E[ r r ] + + E[ r r ] =

25



Ka022 (FCC crystal)


Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

6


1

2

3V

11


1

2

3V

12

7


V(r) = -+k(r-a0)22e1 = -Z2Z = 4 (2D)


Z 4 (2D)Z = 6 (3D)


E = Emin + Nv eVf

eVf = Z2 = -e1


vacancy ishellip

14


8

15


normaltraction

t11

16


9


17

∆VV0 = 3∆LL0

= 3∆aa0 + XV


18


10

JV

1111

19

d

JV

1111

20

d

11


22



12


up

down

left right

23

r1= up= (0 a0)



up

down

left right

stillup down h d

24



independentof r1

13


E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =




hellip +


E[ r r ] + E[ r r ] + + E[ r r ] =

25



Ka022 (FCC crystal)


Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

7


V(r) = -+k(r-a0)22e1 = -Z2Z = 4 (2D)


Z 4 (2D)Z = 6 (3D)


E = Emin + Nv eVf

eVf = Z2 = -e1


vacancy ishellip

14


8

15


normaltraction

t11

16


9


17

∆VV0 = 3∆LL0

= 3∆aa0 + XV


18


10

JV

1111

19

d

JV

1111

20

d

11


22



12


up

down

left right

23

r1= up= (0 a0)



up

down

left right

stillup down h d

24



independentof r1

13


E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =




hellip +


E[ r r ] + E[ r r ] + + E[ r r ] =

25



Ka022 (FCC crystal)


Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

8

15


normaltraction

t11

16


9


17

∆VV0 = 3∆LL0

= 3∆aa0 + XV


18


10

JV

1111

19

d

JV

1111

20

d

11


22



12


up

down

left right

23

r1= up= (0 a0)



up

down

left right

stillup down h d

24



independentof r1

13


E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =




hellip +


E[ r r ] + E[ r r ] + + E[ r r ] =

25



Ka022 (FCC crystal)


Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

9


17

∆VV0 = 3∆LL0

= 3∆aa0 + XV


18


10

JV

1111

19

d

JV

1111

20

d

11


22



12


up

down

left right

23

r1= up= (0 a0)



up

down

left right

stillup down h d

24



independentof r1

13


E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =




hellip +


E[ r r ] + E[ r r ] + + E[ r r ] =

25



Ka022 (FCC crystal)


Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

10

JV

1111

19

d

JV

1111

20

d

11


22



12


up

down

left right

23

r1= up= (0 a0)



up

down

left right

stillup down h d

24



independentof r1

13


E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =




hellip +


E[ r r ] + E[ r r ] + + E[ r r ] =

25



Ka022 (FCC crystal)


Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

11


22



12


up

down

left right

23

r1= up= (0 a0)



up

down

left right

stillup down h d

24



independentof r1

13


E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =




hellip +


E[ r r ] + E[ r r ] + + E[ r r ] =

25



Ka022 (FCC crystal)


Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

12


up

down

left right

23

r1= up= (0 a0)



up

down

left right

stillup down h d

24



independentof r1

13


E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =




hellip +


E[ r r ] + E[ r r ] + + E[ r r ] =

25



Ka022 (FCC crystal)


Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

13


E[ (xV(t)-xV(0))(xV(t)-xV(0)) ] =




hellip +


E[ r r ] + E[ r r ] + + E[ r r ] =

25



Ka022 (FCC crystal)


Self diffusion

c1infin

c1-infin

c -infinntra

tion rarr

1 + 1 + V

x=0 infin-infin

c1infin

Con

cen

c1-infinrarr

26

1 + 1 + 3 + V

x=0 infin-infin

c1infin

c1

Con

cent

ratio

n rarr

c3infin

c1-infin

c3-infin

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

14


c1-infin

c1infin

lL lR



x

time t

J1 = -Dc1

parttc1 = (Dc1)

1 1 1 11

2

0

( ) erf2 2 4

2erf( ) exp( )

c c c c xc x t

D t

d

27

t 1 ( 1 )


parttc1 = D2c1

erf(0)=0 erf( )=1 erf(- )=-1

D

D L R



l D t

l l l


lD waveconvection




28

tHarry and Sally



large lengthscales

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

15

Self diffusion hops


29

Self diffusion hops

upup

down

left right

30

rate = XV Vrsquo

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

16

Self diffusion hops

upup

down

left right

31

rate = XV Vrsquo

Self diffusion hops

upup

down

left right

32


D = XVVrsquoa0

2

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

17


upup

down

left right

33

r1= right= (a0 0)


up


left if r1 = right


up

down

left right

34

backflow causes

f lt 1

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

18

35

1 2 2 1

Interdiffusivity

= D X D X D

TM(Cu) = 1356KTM(Ni) = 1726K


alloys the


36


19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

19

XCu=099

XCu=098

XCu=003

XCu=096

XCu=095

XNi=002

XNi=001

Cu

XCu=002

XNi=002

x x


37

Ni

p

measures

interdiffusivity

( 0015)D X Cu Ni

p

measures


( 002)D X


785degC for 1 3 6 13 28 56 days

TM(Zn) = 693K


38



20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

20

J1C

J2C



Initial welded

diffusioncouple

JVC

1-rich 2-rich

if one endis fixed

to bench

game of tetris

39


to benchKirkendall

Effect

50 years ago


lL lR

today

c2(x)~

g2-rich

g1-rich


lL lR gtgt lD

lL lR unknown

arbitrarylab frame

origin today

x~

40

Matanoplane


origin today


x

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

21

Science 304 (2004) 711

What ifnot enough

41

not enoughclimbing

dislocations

voids would form

42

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

22

43

Brownian Motion

44


23

viscous oil





45


BB 6

k TD Mk T

r

23

viscous oil





45


BB 6

k TD Mk T

r

Documents

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