Anomalous Transport and Diffusion in

Disordered Materials

Armin BundeJustus-Liebig-Universität Giessen

in cooperation with

Markus Ulrich (Giessen, Stuttgart)Paul Heitjans, Sylvio Indris (Hannover)

Outline

(I) Tutorial introduction into the percolation concept:

model, critical behavior, fractal structures

anomalous diffusion

(II) Applications in materials science:

composite ionic conductors

(I) The percolation concept

mean length of finite clusters: size of the infinite cluster:

pc: critical

concentration: spanning (“infinite”) cluster emerges

p > pc: infinite cluster + finite clusters

p < pc: finite clusters of

occupied sites

P

)(~c

pp

2.0p 59.0p 8.0p

)(~c

ppP

Fractal structures:

● At pc:

● Above pc:

fd

rM ~

rr

rrMd

df

,

,~

Self-similarity at pc:

Self-similarity above pc:

A

B

A

B

Anomalous diffusion

wd

ABABRt ~

2~ABAB

Rt

ttr ~2

Normal lattice

Percolation at cp

wd

ttr/22 ~

)3(7.3),2(9.2 ddddww

Diffusion above cp

w

ww

d

dd

ttt

ttttr

~,

~,~)(

/2

2

,/2 TkDneB

dttrD 22

Percolation system:

c

cc

pp

pppp

,0

,)(~

Relation between and :wd /)(2

wd

Proof: )2(2)1/2(1/22 )(~~~~/)(~ wwwwwd

c

dddd pptttrD

)(~c

ppPn

)(~c

pp

Nernst-Einstein:

• nanocrystalline Li2O:B2O3

composite

II. Applications of percolation theory: Nano- and microcrystalline Li2O:B2O3

composites

DC conductivity of nano- and microcrystalline Li2O:B2O3 composites

Indris et al, 2000

Brick-layer model

Cluster of conducting Li20 grains

Li20 grain: length a, interface

Bulk: normal conducting 0

Interface: highly conducting

)(10

)(1024

2

micro

nano

a

Ulrich et al, 2004

01200

Brick-layer model: connections between grains

69.0

~

)1(

10

cp

aaa

86.0

~

)2(

1

cp

aa

90.0

~

)3(

2

1

cp

aa

Ulrich et al, 2004

Brick-layer model: Results

DC conductivity for different grain sizes a and ratios = 1/0 between interface and bulk conductivities, 1 nm.

Nanocrystalline grains: a = 10 nm, = 200; a = 10 nm, = 100; a = 20 nm, = 200; a = 20 nm, = 100.

Microcrystalline grains: a = 10 , = 200; a = 10 , = 100; a = 20 , = 200; a = 20 , = 100.

Comparison of the experimentally observed normalized dc conductivity (p)/(0) with the simulation results for = 1 nm, = 200; a = 10 nm and a = 10 , respectively.m

mmm

m

Ulrich et al, 2004

Voronoi-type model

● log-normal distribution of grain sizes,

● percolation threshold: pc= 0.85 (also too small!)

Ulrich et al, 2004

Way out: Ionic diffusion via B2O3: B2O3 interfacesin the nanocrystalline system

pc 0.95 pc 0.93

Brick-layer model Voronoi model

Ulrich et al, 2004