Self-assembling fractal particle networks

$: Self-assembling fractal particle networks$
Self-assembling fractal particle networks

Joseph Jun and Alfred Hübler

Center for Complex Systems ResearchUniversity of Illinois at Urbana-

ChampaignResearch supported in part by the National Science

Foundation

(PHY-01-40179 and DMS-03725939 ITR)

Growth of a ramified transportation network.

Experiment: Agglomeration of conducting particles in an electric field1) We focus on the dynamics of the system2) We explore the topology of the networks using graph theory.3) We explore a variety of initial conditions.

Results:1) three growth stages: strand formation, boundary connection, and

geometric expansion. 2) networks are open loop 3) statistically robust features: number of termini, number of branch points,

resistance, initial condition matters somewhat4) Minimum spanning tree growth model predicts emerging pattern

random initial distribution compact initial distribution

Description of experimental setup

Basic experiment consists of two electrodes, a source electrode and a boundary electrode connected to opposite terminals of a power

supply.

source electrod

e

boundary electrode

battery


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)

source electrod

e

boundary electrode

oil

battery

particle


20 kV

battery maintains a voltage difference of 20 kV between boundary and source

electrodes


20 kV

source electrode sprays charge over oil surface


20 kV


air gap between source electrode and oil surface approx. 5 cm


20 kV


air gap between source electrode and oil surface approx. 5 cm

boundary electrode has a diameter of 12 cm


20 kV

needle electrode sprays charge over oil surface

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


20 kV

needle electrode sprays charge over oil surface

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

Phenomenology

The growth of the network proceeds in three stages: I) strand formation

II) boundary connection

III) geometric expansion

Phenomenology Overview

12 cm

t=0s 10s 5m 13s 14m 7s

stage I:strand

formation


12 cm

t=0s 10s 5m 13s 14m 7s

14m 14s

stage I:strand

formation

stage II:boundary

connection


12 cm

t=0s 10s 5m 13s 14m 7s

14m 14s 14m 41s 15m 28s

stage I:strand

formation

stage II:boundary

connection stage III: geometric expansion


12 cm

t=0s 10s 5m 13s 14m 7s

14m 14s 14m 41s 15m 28s 77m 27s

stage I:strand

formation

stage II:boundary

connection stage III: geometric expansion

stationary state

Motion of the strands

The motion of the lead particles of the six

largest strands from a single experiment.

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.

N=1044

N=591 N=784

Comparing for different numbers of particles, N. The growth of the strands still tend to correlate for higher N.

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.

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.

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.

Graph theory measures for trees

We allow the physical locations of the particles to define the adjacency.

c=5

c=3

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.

red circles indicate cutoff length

yellow lines indicate distance between centers of particles

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.

Adjacency algorithm

Digitize the positions of each particle from the photos.

photos from experiment

Adjacency algorithm


Run the adjacency algorithm on the list of particle positions.


digitization of positions

Adjacency algorithm


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


output from algorithm*

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.



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.



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.


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.

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.

Branching point subspecies

Subspecies b5 and b6 have never been observed in the experiment.

b3 b4

b5 b6

Branching point subspecies

Percentage of branching points that connect to four other particles as a function of

particle number.

Most networks are trees.Only a few rare cases contain loops

(cycles).

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.

Fractal Dimension of Particles

N = 792T = 159B = 153

N = 791T = 170B = 164

N = 794T = 170B = 162

N = 784T = 166B = 161

The mass dimension, dm, is defined by Σρ(r) = N ~ rdm

Fractal Dimension of Particles

N = 792T = 159B = 153

N = 791T = 170B = 164

N = 794T = 170B = 162

N = 784T = 166B = 161

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

Fractal Dimension

Particles arrange themselves similarly in different experiments.

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.

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.

Initial conditions

Qualitative effects of initial distribution


N = 752T = 131B = 85

N = 720T = 122B = 106

N = 785T = 200B = 187

N = 752T = 149B = 146

Initial conditions are a strong constraint on the final form of tree(s).


Will this initial configuration produce a spiral?

?


No, system is unstable to ramified structures.

Perimeter effects (cheat experiments)

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

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.

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.

Overall electrical resistance of system

We estimate the resistance,

as

K = height of oil conductivity of oil

I0= total current

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.

Artificially generated networks

How does the state of the system directly preceding stage II affect the topology of the trees?

Can we predict the final tree at this stage?


Since topology of the networks is established relatively quickly, particles connect to one another before they have moved far.

Thus, we attempt to model the connections formed by the system using only the local information for each particle—it’s neighborhood.



Thus, we attempt to model the connections formed by the system using only the local information for each particle—it’s neighborhood.We use data from the experiments: a snapshot of the particles directly preceding stage II.



Thus, we attempt to model the connections formed by the system using only the local information for each particle—it’s neighborhood.We take data from the experiments: a snapshot of the particles directly preceding stage II.

Digitize the positions.

Run the adjacency algorithm to obtain a base neighborhood.

cutoff length = 3 particle diameter


From the base neighborhood, we apply algorithms to generate trees.

In other words, particles can only connect to particles that neighbor it. All the links shown on the left are potential connections for the final tree.

Algorithms run until all available particles connect into a tree.

Some particles will not connect to any others (loners). They commonly appear in experiments.

loner

loner


From the base neighborhood, we apply algorithms to generate trees.

In other words, particles can only connect to particles that neighbor it. All the links shown on the left are potential connections for the final tree.

Algorithms run until all available particles connect into a tree.

Some particles will not connect to any others (loners). They commonly appear in experiments.

We chose three algorithms to implement:

1) random (RAN)

2) minimum spanning tree (MST)

3) propagating front model (PFM)

loner

loner

Random

The random algorithm randomly selects a link from the neighborhood graph and determines whether to connect the two particles based on whether the link maintains or violates a tree structure.

In practice, we do this by tracking a “tree label” for each particle.

If two particles in a potential connection have the same label, the connection would produce a cycle, and consequently it is rejected.

PARTICLE STATE

unconnectedconnected to

tree i

unconnected accept accept

connected to tree j

accept reject if i=j

particle 1

part

icl

e 2

summary of RAN connection rule

RAN

movie of random algorithm

RAN

Typical connection structure from RAN algorithm.

Distribution of termini produced from 105 permutations run on a single

experiment.

Number of termini produced for all experiments, plotted as a function of

N.

Minimum Spanning Tree

Uses the identical acceptance/rejection criterion as RAN.

The difference between the two is in how the potential connections are chosen.

MST picks shortest links first (particles that are closest to one another).

Since there are degeneracies in links, we run the algorithm through 105 permutations of degenerate ordering.

graph (non-tree) tree (non-minimal) tree (minimal)

MST

movie of minimum spanning tree algorithm

MST

Typical connection structure from MST algorithm.


experiment.


N.

Propagating Front ModelSince only one strand reaches the boundary, the connections should propagate from a particular direction.

To capture this, we propose a model where particles link in order by their geographic location.

Particles can connect only when they are adjacent to a particle that already belongs to the boundary.




grey thatched particles are already in the network, connections are shown in black lines

white particles are available to connect

dotted particles are not allowed to connect because they are not yet adjacent to a particle in the network




the grey filled particle was randomly chosen

it must now randomly select one of its neighbors that are already in the network




the chosen particle joined the network

any particles adjacent to it are now added to the list of particles that may connect




the process repeats until all particles join the boundary.

PFM

movie of propagating front model

PFM

Typical connection structure from PFM algorithm.


experiment.


N.

Comparison of all models to experiments

The number of termini and branching points for all three models and the natural experiments.

MST produces the closest match with experiments.

Comparison of all models to experiments

cheat initial condition (without stage I) and natural initial condition

<T>-<B>

<T>-<B>=1+b4+2b5+3b6

<T>-<B> is independent of b3 subspecies

Thus, PFM and RAN are not only generating more branching points, they are generating higher order branching points.

natural cheat

Review of simulations

We applied three algorithms to produce trees using local connection rules.

We found that the algorithm which uses the interparticle spacing but neglects the direction of connection produces the best match to the experiments.

Hebbian Learning in a three-electrode system

M. Sperl, A Chang, N. Weber, A. Hubler, Hebbian Learning in the Agglomeration of Conducting Particles, Phys.Rev.E. 59, 3165 (1999)

Predicting the growth of a fractal network.

Experiment: J. Jun, A. Hubler, PNAS 102, 536 (2005)1) Three growth stages: strand formation, boundary connection, and

geometric expansion;2) Networks are open loop;3) Statistically robust features: number of termini, number of branch

points, resistance, initial condition matters somewhat;4) Minimum spanning tree growth model predicts emerging pattern.5) To do: derive result from first principles, random initial condition, predict

other observables, control network growthApplications: Hardware implementation of neural nets, nano neural nets with

SC particles - M. Sperl, A Chang, N. Weber, A. Hubler, Hebbian Learning in the Agglomeration of Conducting Particles, Phys.Rev.E. 59, 3165 (1999)

random initial distribution compact initial distribution

Self-assembling fractal particle networks

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