Diusion on disordered fractals

Transcription

1 Diusion on disordered fractals Janett Prehl D. H. N. Anh K. H. Homann S. Seeger S. Tarafdar Institut für Physik, Computerphysik

2 Anomalous Diusion Anomalous Diusion Diusion of particles in disordered materials inside a bulk on the surface (deposit) Mean square displacement: 0 r(t) r 2 (t) t γ γ > 0 γ < 1 sub-diusion γ = 1 normal diusion γ > 1 super-diusion r(t): Distance a particle has traversed in time t from its origin

3 Anomalous Diusion Anomalous Diusion Anomalous diusion of water in biological tissues. M. Köpf, et al. Biophys. J., 70(6): , Anomalous diusion of hydrogen in amorphous metals. W. Schirmacher, et al.; Europhys. Lett., 13(6):523529, Submonolayer growth with repulsive impurities. S. Liu, et al.; Phys. Rev. Lett., 74(22): , Investigation of diusion in disordered materials by numerical simulation

4 Disordered materials Porous Materials Properties of porous materials: Holes, barriers and connections on all length scales Self-similar over certain length scales At larger length scales rather homogeneous Smallest length scale given by material Aim: Modeling porous materials by fractals

5 Disordered materials Regular Sierpinski Carpets Iteration repeated ad innitum Sierpinski carpet k-fold iteration iterator of depth k Fractal dimension: M L d f d f = ln M ln L Transition from regular to randomized fractals

6 Disordered materials Randomized Sierpinski Carpets Random mixing of dierent generators Combining dierent iterators to one carpet More realistic model: Self-similarity at certain length scales Smallest length scale: Depth of iterators ( ˆ= pore size) Large length scales: Homogeneity

7 Random Walk Random Walk Dimension Random walk dimension: d w = 2 γ r 2 (t) t γ t r dw power law behavior also valid for randomized Sierpinski carpets D. Anh, et al.; Europhys. Lett., 70(1): , 2005 <r 2 (t)> timestep t

8 Random Walk Open Questions What happens by mixing dierent generators with d w? Investigations by Reis with regular random fractals (dierent from our construction procedure) F. Reis; J. Phys.A: Math. Gen.,29(24): , 1996 No signicant changes in exponents But we obtained quite dierent results!

9 Numerical Results Results D. Anh, et al.; Europhys. Lett., 70(1): , 2005 Mixing pairs of dierent generators Two generators of dierent d f and dierent d w A <d w > A x B 100-x 2,5 2,4 2,3 B 2,2 2, x Maximum of d w can be observed

10 Numerical Results Results Two generators with dierent d w but same d f C <d w > C x D 100-x 2,6 2,5 2,4 2,3 D 2,2 2, x Maximum of d w can be observed

11 Numerical Results Results Two generators with same d w and same d f C <d w > 2,25 C x C 100-x 2,2 C 2,15 2, x Variation even for similar d w and d f

12 Numerical Results Results Two generators with same d w and same d f D <d w > D x D 100-x 2,6 2,55 2,5 D 2,45 2, x More disorder can enhance diusion

13 Discussion Question Can we predict dynamical properties just by knowing the structure? If yes: Analysis porous materials for experiments: Mass Connectivity Explanation of dynamics due to structural properties

14 Discussion Connection Points Structural property: Connection points between generators C and C` Slowing down of diusion

15 Discussion Connection Points Structural property: Connection points between generators D and D` Enhancement of diusion

16 Discussion Shortest Path Structural property: Shortest path through generators <d w > C x D 100-x 2,6 2,5 2,4 2,3 2,2 2, x More connection points but longer shortest path Slowing down of diusion

17 Summary Summary From structure to dynamics: Increasing number of connection points enhancement of diusion For longer shortest paths diusion slows down Number of connection point and shortest path length are important for d w

18 Summary Acknowledgment Thank you for your attention!