Post on 24-Feb-2022
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
Diffusion in crystal free volume localized as vacancy
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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
Diffusion in crystal free volume localized as vacancy
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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
Diffusion in crystal free volume localized as vacancy
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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
Diffusion in crystal free volume localized as vacancy
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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
Diffusion in crystal free volume localized as vacancy
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
6
Diffusion in crystal free volume localized as vacancy
1
2
3V
11
Diffusion in crystal free volume localized as vacancy
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
17
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
upup
down
left right
33
r1= right= (a0 0)
Self diffusion hops will be correlated even if vacancy hops are uncorrelated
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
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
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
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
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
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
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