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