Self-assembling fractal particle networks

1. Self-assembling fractal particle networks Joseph Jun and Alfred Hübler Center for Complex Systems Research University of Illinois at Urbana-Champaign Research supported in part by the National Science Foundation (PHY-01-40179 and DMS-03725939 ITR)

2. Growth of a ramified transportation network. random initial distribution compact initial distribution • Experiment: Agglomeration of conducting particles in an electric field • 1) We focus on the dynamics of the system • 2) We explore the topology of the networks using graph theory. • 3) We explore a variety of initial conditions. • Results: • three growth stages: strand formation, boundary connection, and geometric expansion. • networks are open loop • statistically robust features: number of termini, number of branch points, resistance, initial condition matters somewhat • 4) Minimum spanning tree growth model predicts emerging pattern

3. Description of experimental setup source electrode Basic experiment consists of two electrodes, a source electrode and a boundary electrode connected to opposite terminals of a power supply. battery boundary electrode

4. Description of experimental setup source electrode Basic experiment consists of two electrodes, a source electrode and a boundary electrode connected to opposite terminals of a power supply. The boundary electrode lines a dish made of a dielectric material such as glass or acrylic. The dish contains particles and a dielectric medium (oil) battery particle boundary electrode oil

5. Description of experimental setup 20 kV battery maintains a voltage difference of 20 kV between boundary and source electrodes

6. Description of experimental setup source electrode sprays charge over oil surface 20 kV

7. Description of experimental setup source electrode sprays charge over oil surface 20 kV air gap between source electrode and oil surface approx. 5 cm

8. Description of experimental setup source electrode sprays charge over oil surface 20 kV air gap between source electrode and oil surface approx. 5 cm boundary electrode has a diameter of 12 cm

9. Description of experimental setup needle electrode sprays charge over oil surface 20 kV air gap between needle electrode and oil surface approx. 5 cm boundary electrode has a diameter of 12 cm oil height is approximately 3 mm, enough to cover the particles castor oil is used: high viscosity, low ohmic heating, biodegradable

10. Description of experimental setup needle electrode sprays charge over oil surface 20 kV air gap between needle electrode and oil surface approx. 5 cm ring electrode forms boundary of dish has a radius of 12 cm oil height is approximately 3 mm, enough to cover the particles castor oil is used: high viscosity, low ohmic heating, biodegradable particles are non-magnetic stainless steel, diameter D=1.6 mm particles sit on the bottom of the dish

11. Phenomenology The growth of the network proceeds in three stages: I) strand formation II) boundary connection III) geometric expansion

12. Phenomenology Overview { 12 cm stage I: strand formation t=0s 10s 5m 13s 14m 7s

13. Phenomenology Overview { 12 cm stage I: strand formation t=0s 10s 5m 13s 14m 7s 14m 14s stage II: boundary connection

14. Phenomenology Overview { 12 cm stage I: strand formation t=0s 10s 5m 13s 14m 7s { 14m 14s 14m 41s 15m 28s stage II: boundary connection stage III: geometric expansion

15. Phenomenology Overview { 12 cm stage I: strand formation t=0s 10s 5m 13s 14m 7s { 14m 14s 14m 41s 15m 28s 77m 27s stage II: boundary connection stage III: geometric expansion stationary state

16. Motion of the strands The motion of the lead particles of the six largest strands from a single experiment.

17. Motion of the strands The motion of the lead particles of the six largest strands from a single experiment. Distance of lead particle of a strand correlates well with number of particles in strand.

18. N=591 N=784 N=1044 Comparing for different numbers of particles, N. The growth of the strands still tend to correlate for higher N.

19. Phenomenology: stage II (boundary connection) Stage II begins when the “winning” strand connects to the boundary. It is brief in duration, and is best characterized by the particles binding to the boundary.

20. Phenomenology: stage III (geometric expansion) After all the particles bind together, they will now be like charged and spread apart. This expansion into the available space is the main characteristic of stage III.

21. Adjacency defines topological species of each particle Termini = particles touching only one other particle Branching points = particles touching three or more other particles Trunks = particles touching only two other particles Particles become one of the above three types in stage II and III. This occurs over a relatively short period of time.

22. Graph theory measures for trees We allow the physical locations of the particles to define the adjacency. The particles’ positions are digitized. Each particle is considered a node. When the distance between two particles is shorter than a cutoff length, they are considered adjacent; we put a link between them. c=5 c=3 red circles indicate cutoff length yellow lines indicate distance between centers of particles

23. Adjacency (number of neighbors) We can define the average adjacency mathematically as: ci is the adjacency of particle i Θ is the Heaviside step function N is the total number of particles ri & rj are the positions of particles i & j respectively rcut is the cutoff length Ideally, rcut = D, where D is the diameter of a particle. But because of the noise in digitizing the position of the particles, we use a slightly larger value, usually 1.16 ≤ rcut/D ≤ 1.28. Also ideally, 0 ≤ ci ≤ 6; we impose this by hand in the algorithm.

24. Adjacency algorithm photos from experiment Digitize the positions of each particle from the photos.

25. Adjacency algorithm photos from experiment digitization of positions Digitize the positions of each particle from the photos. Run the adjacency algorithm on the list of particle positions.

26. Adjacency algorithm photos from experiment output from algorithm* Digitize the positions of each particle from the photos. Run the adjacency algorithm on the list of particle positions. The algorithm picks up how particles are connected. It identifies holes and grain boundaries. *Graphs from algorithm were visualized using the Combinatorica package in Mathematica. rcut = 1.25•D

27. Visualizing the stages with the adjacency By looking at <c> as a function of time from the digitization of the photos, we can see this measure naturally segregates the stages. The average adjacency versus time.

28. Visualizing the stages with the adjacency By looking at <c> as a function of time from the digitization of the photos, we can see this measure naturally segregates the stages. The top dashed lines is an estimate of <c> at t=0 s, given by (circle): The bottom dotted line is the value of <c> in the steady-state (single strand): The average adjacency converges rapidly.

29. Visualizing the stages with the adjacency By looking at <c> as a function of time from the digitization of the photos, we can see this measure naturally segregates the stages. The top dashed lines is an estimate of <c> at t=0 s, given by (circle): The bottom dotted line is the value of <c> in the steady-state (single strand): The inset shows the same plot for several values of the cutoff length. The average adjacency converges rapidly.

30. Visualizing the stages with the adjacency A look at the differences in stages between different particle numbers. The average adjacency converges rapidly for all cases. We conclude that the topology of the network establishes in a relatively short amount of time following stage II.

31. Relative number of each species is robust Graphs show how the number of termini, T, and branching points, B, scale with the total number of particles in the tree.

32. Branching point subspecies b3 b4 b6 b5 Subspecies b5 and b6 have never been observed in the experiment.

33. Branching point subspecies Percentage of branching points that connect to four other particles as a function of particle number.

34. Most networks are trees.Only a few rare cases contain loops (cycles).

35. Loops (cycles) are unstable Insets on the left show two particles artificially placed into a loop separate from one another. The graph on the right shows the separation between the two particles as a function of time.

36. Fractal Dimension of Particles N = 784 T = 166 B = 161 N = 794 T = 170 B = 162 N = 791 T = 170 B = 164 N = 792 T = 159 B = 153 The mass dimension, dm, is defined by Σρ(r) = N ~ rdm

37. Fractal Dimension of Particles N = 784 T = 166 B = 161 N = 794 T = 170 B = 162 N = 791 T = 170 B = 164 N = 792 T = 159 B = 153 dm ~ 1.74─1.83 dm ~ 1.76─1.82 dm ~ 1.75─1.91 dm ~ 1.79─1.90 The mass dimension, dm, is defined by Σρ(r) = N ~ rdm

38. Fractal Dimension Particles arrange themselves similarly in different experiments.

39. Spatial distribution in time The radial distribution of particles for different times in the experiment. The system entered stage II after t=847s. The fractal dimension decreases from Dm=2 to Dm=1.8.

40. Spatial distribution of termini is almost homogeneous, except for small particle numbers The radial distribution of termini for similar number of particles and different number of particles.

41. Initial conditions

42. Qualitative effects of initial distribution

43. Qualitative effects of initial distribution N = 752 T = 149 B = 146 N = 785 T = 200 B = 187 N = 720 T = 122 B = 106 N = 752 T = 131 B = 85 Initial conditions are a strong constraint on the final form of tree(s).

44. Qualitative effects of initial distribution ? Will this initial configuration produce a spiral?

45. Qualitative effects of initial distribution No, system is unstable to ramified structures.

46. Perimeter effects (cheat experiments) Eliminating stage I by artificially placing a connecting strand to the boundary; we call these “cheat” experiments.

47. Perimeter effects (cheat experiments) Eliminating stage I by artificially placing a connecting strand to the boundary; we call these “cheat” experiments. In this case, there are no losing strands that become long termini at the perimeter.

48. Perimeter effects Consequently, there are more termini and branching points for the cheat cases. Initial conditions directly preceding stage II are important to determining the relative number of topological species.

49. Overall electrical resistance of system We estimate the resistance, as K = height of oil  conductivity of oil I0= total current

50. Review of experimental results Growth of trees occurs in three stages.Average adjacency captures the three stages.Topology of network forms relatively quickly.Particles become one of three species.The relative abundance of each species is statistically reproducible.Initial conditions are a strong constraint to formation of networks.