1. © Imperial College LondonPage 1 Topological Analysis of Packings Gady Frenkel, R. Blumenfeld, M. Blunt, P. King Application to flow properties in granular porous media

2. © Imperial College LondonPage 2 Overview Separation: Topology - Geometry Analyzing networks Obtaining grain and pore networks connectivity and proximity Statistical description (Entropic formalism) Combining back with shape Network flow simulations Combine back with shape Flow & Electrical properties as expectation values over partition function Network flow simulations

3. © Imperial College LondonPage 3 Overview Separation: Topology - Geometry Analyzing networks Obtaining grain and pore networks connectivity and proximity Statistical description (Entropic formalism) Combining back with shape + Shape degrees of freedom Flow & Electrical properties as expectation values over partition function

4. © Imperial College LondonPage 4 Topological Representation of Granular Packing Grains: (transformed) –Polygons (2D) or Polyhedrons (3D) –Corners are the contact points between grains –Assumption: transformed grain is convex Pores: –“Convex” “empty” volumes that are surrounded by transformed grains. Throats: (3D) –Surfaces. the openings that connect two pores:

5. © Imperial College LondonPage 5 2D Packing Example: GRAINS: –Straight lines and planes that connect contacts instead of real boundaries

6. © Imperial College LondonPage 6 2D Packing Example: GRAINS: –Straight lines and planes that connect contacts instead of real boundaries Contact Points

7. © Imperial College LondonPage 7 2D Packing Example: GRAINS: –Straight lines and planes that connect contacts instead of real boundaries Contact Points

8. © Imperial College LondonPage 8 2D Packing Example: GRAINS: –Straight lines and planes that connect contacts instead of real boundaries PORES: –empty” volumes that are surrounded by transformed grains. CONTACT POINT:

9. © Imperial College LondonPage 9 Obtaining pores 2D Grain Pore Directed Grain-edge vectors: Grain: Anti-Clockwise Pore: Clockwise

10. © Imperial College LondonPage 10 3D Pål-Eric Øren Grains  polyhedra Pores  bounded by facets Throat  missing faces

11. © Imperial College LondonPage 11 Finding Throats: Facets of the pore are known Use the 2D algorithm where the radial vector sets the positive edge direction

12. © Imperial College LondonPage 12 Finding Throats:

13. © Imperial College LondonPage 13 Finding Throats: Need to know which faces of grains belong to the pore

14. © Imperial College LondonPage 14 –Growing a deformable object : Inflating a balloon inside the pore until it is filled. –Controlling the inflation by curvature. –Convexity prevents the balloon from exiting the pore. Obtaining Pores 3D: Positively Curved Inflating Balloon X

15. © Imperial College LondonPage 15 Example: Beads in 2D 1.Grains → Polygons 2.Balloons are inflated from each facet

16. © Imperial College LondonPage 16 2D Large Sample with friction Transformed system – polygons Balloon inflated in the pores – perfect recognition of pores and pore grain relations

17. © Imperial College LondonPage 17 Large 3D Sphere Packing Created By: Zdenek Grof

18. © Imperial College LondonPage 18 Obtaining The transformed Grains

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20. © Imperial College LondonPage 20 Foam like structure

21. © Imperial College LondonPage 21 Grain 418 face 2 Grain 418 face 3 Grain 995 face 3 Grain 501 face 0 Sample cells of foam like structure

22. © Imperial College LondonPage 22 Overview Separation: Topology - Geometry Analyzing networks Obtaining grain and pore networks connectivity and proximity Statistical description (Entropic formalism) + Shape degrees of freedom Flow & Electrical properties as expectation values over partition function

23. © Imperial College LondonPage 23 Entropic formalism Statistical Mechanics approach –Averages, Fluctuations, Scaling, … Configurational entropy S Edwards Conjecture of compactivity: –(energy) H  W (volume) –(temperature) T  X (compactivity) Partition Function:

24. © Imperial College LondonPage 24 Entropic formalism Degrees of freedom -Correct number -Independent -Parameterize the volume Density of states Simplifying Assumptions: - Degrees of freedom are uncorrelated - All q’s are of the same kind  Partition function is obtained by a single quadrilateral distribution

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26. © Imperial College LondonPage 26 2D Tessellation: quadrilaterals Quadrilaterals tessellate space Number of degrees of freedom for the isostatic case = number of quadrilaterals The areas of quadrilaterals serve as the degrees of freedom loop i’ g’ r pg RpgRpg loop i g

27. © Imperial College LondonPage 27 Distribution of quadron Area For 5000 spheres with Radii uniformly distributed in (rmin,3rmin) and friction. The distribution is fitted with a Gamma distribution to obtain: - Mean volume per quadron: =1.44/(2.13+1/X). - n-th moment of the volume per quadron: =Γ(1.44+n)/[Γ(1.44)*(2.13+1/X)n]. - Volume fluctuation per quadron: =1.44/(2.13+1/X)2. 2D quadrilaterals area distribution

28. © Imperial College LondonPage 28 Summary Separation: Topology - Geometry Analyzing networks Obtaining grain and pore networks connectivity and proximity Statistical description (Entropic formalism) Combining back with shape Network flow simulations Combine back with shape Flow & Electrical properties as expectation values over partition function Network flow simulations

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30. © Imperial College LondonPage 30 Mean “porosity” Mean “porosity” fluctuations Entropy “Free porosity” We can predict Macroscopic properties of the system using entropic formalism.