1. 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)

2. Outline (I) Tutorial introduction into the percolation concept: model, critical behavior, fractal structures anomalous diffusion (II) Applications in materials science: composite ionic conductors

3. (I) The percolation concept mean length of finite clusters: size of the infinite cluster: p c : critical concentration: spanning (“infinite”) cluster emerges p > p c : infinite cluster + finite clusters p < p c : finite clusters of occupied sites

4. Fractal structures: ● At p c : ● Above p c :

5. Self-similarity at p c :

6. Self-similarity above p c :

7. A B A B Anomalous diffusion Normal lattice Percolation at

8. Diffusion above Percolation system: Relation between and : Proof: Nernst-Einstein:

9. nanocrystalline Li 2 O:B 2 O 3 composite II. Applications of percolation theory: Nano- and microcrystalline Li 2 O:B 2 O 3 composites

10. DC conductivity of nano- and microcrystalline Li 2 O:B 2 O 3 composites Indris et al, 2000

11. Brick-layer model Cluster of conducting Li 2 0 grains Li 2 0 grain: length a, interface Bulk: normal conducting  0 Interface: highly conducting Ulrich et al, 2004

12. Brick-layer model: connections between grains Ulrich et al, 2004

13. 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. Ulrich et al, 2004

14. Voronoi-type model ● log-normal distribution of grain sizes, ● percolation threshold: p c = 0.85 (also too small!) Ulrich et al, 2004

15. Way out: Ionic diffusion via B 2 O 3 : B 2 O 3 interfaces in the nanocrystalline system p c  0.95p c  0.93 Brick-layer model Voronoi model Ulrich et al, 2004