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Purpose of this talk. Diffusion. Ronald Griessen ACTS workshop May 24, 2007. Water droplet. Waves. Wave equation. Diffusion. Fick’s law Equation of continuity. J. dx. C(x,t). Singularities decay immediately. is a solution of. Diffusion in semi-infinite space. - PowerPoint PPT Presentation
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Purpose of this talk
Ronald GriessenACTS workshopMay 24, 2007
Diffusion
SUSTAINABLEHYDROGEN
SUSTAINABLEHYDROGEN
Water droplet
Wave equation2 2
22 2
u uV
t x
Waves
Diffusion
0c J
t x
Fick’s law Equation of continuity
x
cDJ
dx2
2
c cD
x t
JC(x,t)
-10
12
34
50.0
0.2
0.4
0.60.8
1.0
0.0
0.5
1.0
1.5 D=1
conc
entr
atio
n
time
x
Dt
x
eDt
c 4
2
2
1
2
2
c cD
x t
is a solution of
Singularities decay immediately
01
23
45
0.0
0.20.4
0.60.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
conc
entr
atio
n
Time
x
Dt
xerfc
21
y
p dpeyerf0
22
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
y
erf(y)
Diffusion in semi-infinite space
Diffusion into a membrane of thickness L
2 2
2
2 1
4
0
2 14 1( , ) 1 sin
2 1 2
n Dt
L
n
n xc x t e
n L
c t
c neHR
HL
n n Dt
L
n
14 1
2 1
2 1
4
0
2 2
2
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
c( Dt/L2)
Dt/L2
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
1.2
c( x/L, Dt/L2)
Dt/L2=
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x/L
Hydrogen near a metal surface, for example Pd
H in Nb
Diffusion coefficients of various interstitials
Figs.IV.11 and 12: Temperature dependence of the diffusion coefficient for hydrogen (full line),deuterium (dashed line) and tritium (dotted line) in FCC metals (blue curves) and in BCC metals (redcurves). The host metals are indicated by their symbols. Note the extreme influence of the crystalstructure in the case of the PdCu alloy.
Ni
FCCPd0.47Cu0.53
Pd
Pd0.47Cu0.53 BCC
Fe
V
Nb
Ta
V
V
NbNb
TaTa
Pd
Pd
Ni
CuDiffusion
coefficients of various
interstitials
Pd
Cu
YPd
Pd
Cu
Pd
Cu
Pd
Cu
Pd
Cu
Y
Pd
Cu
Y
Pd
Cu
Y
Pd
Cu
Y
Pd
Cu
Pd
Cu
Pd
Y
Pd
PdCu
Fast H diffusion in bcc Pd-Cu
Y
Fast H diffusion in bcc Pd-Cu
Cu atomic percent palladium
Tem
pera
ture
oC
Figs.IV.11 and 12: Temperature dependence of the diffusion coefficient for hydrogen (full line),deuterium (dashed line) and tritium (dotted line) in FCC metals (blue curves) and in BCC metals (redcurves). The host metals are indicated by their symbols. Note the extreme influence of the crystalstructure in the case of the PdCu alloy.
Ni
FCCPd0.47Cu0.53
Pd
Pd0.47Cu0.53 BCC
Fe
V
Nb
Ta
V
V
NbNb
TaTa
Pd
Pd
Ni
CuDiffusion
coefficients of various
interstitials
Fick’s law Equation of continuity
x
cDJ
0c J
t x
dx?Is this true
?Is this true
P-c isotherms and phase diagram
dG SdT Vdp dN
Fick’s law Equation of continuity
0c J
t x
dx?
J cL
x x x t
c c
Lx c x t
x
cDJ
J Lx
A real diffusion experiment
0.0 0.5 1.0 1.5 2.0 2.5 3.010
-32
10-28
10-24
10-20
10-16
10-12
10-8
10-4
100
PH
2 (1
05 Pa)
x=H/Y Kooij et al. 1999
Pressure-composition isotherm of YHx at T=293 K
Hydrogenography in Yttrium
Den Broeder, van der Molen et al, Nature 394 (1998) 656
Y
Y2O3 Pd
H
1
H/Y
0
2
3
hcp-
insulator
hcp-
metal
1
H/Y
0 2 3
hcp-
insulator
hcp-
metal
1
H/Y
0 2 31
H/Y
0 2 3
hcp-
insulator
hcp-
metal
hcp-
insulator
hcp-
metal
1
H/Y
0
2
3
hcp-
insulator
hcp-
metal
0.0 0.5 1.0 1.5 2.0 2.5 3.010-32
10-28
10-24
10-20
10-16
10-12
10-8
10-4
100
PH
2 (
105 P
a)
H/Y
1
H/Y
0 2 3
hcp-
insulator
hcp-
metal
1
H/Y
0 2 31
H/Y
0 2 3
hcp-
insulator
hcp-
metal
hcp-
insulator
hcp-
metal
This picture demonstrates that
instead of j Lx
c
j Dx
Y2O3 PdY
SiO2
H
V
Switchable mirrors as indicators
SiO2
V V
Sample architecture
Pd 10 nm
Y 50 nm
SiO2
V V
Sample architecture
HPd
Y
SiO2
V V
x
x=0
Hydrogen loading
dV
25 nm
50 nm
75 nm
100 nm
125 nm
10 m
m
Pd
YH2 front
x=0
x
H-loading: 473K, 1mbar, 3h
Diffusion in a multilayer
The chemical potential MUST be continuous
The concentration MAY have discontinuities
x
cDJ
Usual diffusion Real diffusion
J Lx
c
Lc x
?
0
En
erg
y
1
1
kTe
n
n
nkT
1ln
oc
ckT
1ln
)1( cc
kT
c
Particles in a lattice gas
X
x
cDJ
Usual diffusion Real diffusion
J Lx
c
Lc x
o
cJ D
x
(1 )
kT
c c c
1
kT cL
c c x
(1 )
o
c cL D
kT
However, the chemical potential MUST be continuous
Ni
Ti
Ni
Ti
Mg
Mg
Ti-H Mg-H Ni-HΔH < ΔH < ΔH
Ni
Ti
Ni
Ti
Mg
Mg
c = c(x,t) cH = f(t)
Ti-H Mg-H Ni-HΔH < ΔH < ΔH
Ni
Ti
Ni
Ti
c = c(x,t) cH = f(t)
Mg
Mg
Fast loading, Very slow unloading
Relatively slow loading, Very fast unloading
Diffusion and Snell’s law
2
2 a
U UD v U
t x
Random walk of a photon
With v the velocity of lighta the absorption coefficient
So
The aquarium experiment
O’Leary et al., Phys. Rev. Lett. 69 (1992) 2658
SiO2
Pd 10 nmV 50 nm
V 250 nm
Y 50 nm
Sample architecture
YH2
YH3
Pd
32 min1 bar373 K
V 50 nm
V 250 nm
Hydrogen loading
YH2
YH3
Pd
110 min1 bar373 K
YH2
YH3Pd
216 min1 bar373 K
YH2
YH3
Pd
442 min1 bar373 K
2’ 32’ 55’ 110’216’332’442’ 90’
1
2
1
2
sin
sin
Time evolution of contours
1
2
1
2
1
2
sin
sin
D
D
Ray tracing alongthe phase – gradientat small angles
A. Remhof, R. J. Wijngaarden, and B. R. Griessen, Refraction and reflection of
diffusion fronts, C. Phys. Rev. Lett., 90 (2003) 145502
Snell’s law for diffusion !
Electromigration
L eZx
j E
In presence of an electric field
x
cDJ
Usual diffusion Real electro-diffusion
J L eZEx
c
L eZEc x
(1 )o
c c cJ D eZE
x kT
(1 )
kT
c c c
(1 )o
c cL D
kT
Electromigration
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
c( x/L, Dt/L2)
Dt/L2=
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
x/L
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
c( x/L, Dt/L2)
Dt/L2= 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
x/L
Electric field ON Electric field OFF
Electromigration of H in V
Electric field ON Electric field OFF
ElectromigrationDen Broeder, van der Molen et al. Nature 394 (1998) 656
Pd
Y
Y2O3H H
Electro-diffusion of hydrogen in
yttriumDen Broeder, van der Molen et al. Nature 394 (1998) 656
Pd
Y
Y2O3
j=0
j=20 mA
j=40 mA
+_
H behaves like a negative ion
H H
SUSTAINABLEHYDROGEN
SUSTAINABLEHYDROGEN
