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Anomalous Transport and Diffusion in
Disordered Materials
Armin BundeJustus-Liebig-Universität Giessen
in cooperation with
Markus Ulrich (Giessen, Stuttgart)Paul Heitjans, Sylvio Indris (Hannover)
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Outline
(I) Tutorial introduction into the percolation concept:
model, critical behavior, fractal structures
anomalous diffusion
(II) Applications in materials science:
composite ionic conductors
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(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
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Fractal structures:
● At pc:
● Above pc:
fd
rM ~
rr
rrMd
df
,
,~
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Self-similarity at pc:
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Self-similarity above pc:
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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
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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:
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• nanocrystalline Li2O:B2O3
composite
II. Applications of percolation theory: Nano- and microcrystalline Li2O:B2O3
composites
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DC conductivity of nano- and microcrystalline Li2O:B2O3 composites
Indris et al, 2000
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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
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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
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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
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Voronoi-type model
● log-normal distribution of grain sizes,
● percolation threshold: pc= 0.85 (also too small!)
Ulrich et al, 2004
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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
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