PowerPoint-Pr

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Anomalous Transport and Diffusion in Disordered
Materials Armin Bunde Justus-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
pc critical concentration spanning
(infinite) cluster emerges
p gt pc infinite cluster finite
clusters
p lt pc finite clusters of occupied
sites
mean length of finite clusters
size of the infinite cluster
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Fractal structures

• At pc
• Above pc

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Self-similarity at pc
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Self-similarity above pc
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Anomalous diffusion
Normal lattice
A
B
Percolation at
A
B
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Diffusion above
Nernst-Einstein
Percolation system
Relation between and
Proof
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II. Applications of percolation theory Nano-
and microcrystalline Li2OB2O3 composites

• nanocrystalline Li2OB2O3 composite

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DC conductivity of nano- and microcrystalline
Li2OB2O3 composites
Indris et al, 2000
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Brick-layer model
Li20 grain length a, interface l
Cluster of conducting Li20 grains
Bulk normal conducting s0 Interface highly
conducting
Ulrich et al, 2004
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Brick-layer model connections between grains
Ulrich et al, 2004
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Brick-layer model Results
DC conductivity for different grain sizes a and
ratios t s1/s0 between interface and bulk
conductivities, l 1 nm. Nanocrystalline
grains a 10 nm, t 200 a 10 nm, t
100 a 20 nm, t 200 a 20 nm, t 100.
Microcrystalline grains a 10 , t
200 a 10 , t 100 a 20 ,
t 200 a 20 , t 100.
Comparison of the experimentally observed
normalized dc conductivity s(p)/s(0) with the
simulation results for l 1 nm, t 200 a 10
nm and a 10 , respectively.
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
Voronoi model
Brick-layer model

• pc ? 0.95

pc ? 0.93
Ulrich et al, 2004
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