The Effect of Different Geometries on Percolation in

Two DimensionsBy

Allison MorganDr. Alan Feinerman and Jared Weddell

Overview

• Introduction to Percolation• Percolation Theory• Project Goals

• Experimental Design and Methods• Results• Conclusion

Percolation Theory• Percolation: the movement of a mass across a

porous material [1].

– Geologists study percolation to analyze the flow of fluid through micro-fractures in rocks [2].

– Biologists investigate how drugs might diffuse through blood vessels [3].

[1] B. Last and D. Thouless: Physical Review Letters, 1971, 27, 1719 – 1722.[2] B. Berkowitz and R. Ewing, Surveys in Geophysics 19, 23 (1998). [3] J. Baish et. al., Role of Tumor Vascular Architecture in Nutrient and Drug Delivery: An Invasion Percolation-Based Network Model, Microvascular Research 51, 327-346 (1996).

Percolation Theory Continued• Percolation threshold: a critical fraction of

area where a medium can still be conductive or fluid can still diffuse [4].

• To treat solid cancerous tumors, doctors would like to observe how blood carries drugs: this process can be modeled with fractals and percolation clusters.• At the percolation threshold, qualities like the blood

flow rate, oxygenation rate, and drug delivery rate approach zero [3].

[4] A. Hunt, Percolation Theory for Flow in Porous Media, (Springer, Berlin Heidelberg 2005).

Project Goals

• Establish a better experimental set-up based on previous work.

• Determine the percolation threshold for rectangle-shaped holes.

• Examine how experimental rectangle data compares to previous experimental data for ellipses.

MatLab OutputThe shapes are randomly distributed and oriented using MatLab. AutoCAD then visualizes the design.

Percolated region Effective area region

Current Experimental Setup

Acrylic fixtures and nylon screws*

Aluminum baseplate*

Layer of double-sided adhesive

Layer of Mylar with conductive aluminum

Brass rods

* New items

Evaluation of Kerf• The error due to Weddell et. al.’s experimental setup was the

result of underestimating the width of the laser’s cuts (kerf).• Previously, the width of the laser was estimated to be in a

window of values.• We determined the kerf of the laser to be 114 microns.

Kerf

Experimental ProcessAt the same time the laser is cutting, the current across both regions is measured.

As the number of holes increases across the left square, resistance increases.

A portion of the right square is eliminated every time the area on the percolated square is decreased by 1%.

Electrical ModelDefinition of resistance:• ρ = resistivity • L = length over which the current is measured• τ = thickness of the conductive sheet

Initial resistance : Ro = (ρ * L) / (Ho * τ)

– H0 is the initial height of the sheet

Final resistance: Rr = (ρ * L) / (Hr * τ)

– Hr is the height remaining across

effective area square when the other square has been fully percolated.

The percolation threshold is Hr / H0 = R0 / Rr = Ir / I0

H0

Hr

L

Example of Experimental Output

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Time** (s)

Curr

ent (

A)

Percolated square Effective area square

Raw data from 1500 circular cuts. The experimental percolation threshold was 0.3387.

Ir

I0Pc = Ir / I0

** Area removed is a non-linear function of time.

Validation of Testing• Averaged 5 runs of

roughly 1500 circle shaped cuts.• Theoretical percolation

threshold is 0.33 [5].• Previous experimental

results yielded 0.351 [6].• Experimental

percolation threshold was 0.343 ± 0.073.

[5] B. Xia and M. Thorpe, Physical Review A 38, 2650 (1988).[6] J. Weddell, A. Feinerman, Percolation Effects on Electrical Resistivity and Electron Mobility, Journal of Undergraduate Research, 5, 9 (2011).

Experimental Rectangle Results

Aspect ratio

Percolation threshold for rectangles (± STD )

Percolation threshold for ellipses [6]

1.0000 0.496 ± 0.077 0.351

0.7000 0.533 ± 0.046 0.392

0.6000 0.492 ± 0.039 0.414

0.2500 0.563 ± 0.024 0.588

0.1000 0.697 ± 0.113 0.723

0.0125 0.969 ± 0.008 0.923

1.000

0.2500

0.1000

[6] J. Weddell, A. Feinerman, Percolation Effects on Electrical Resistivity and Electron Mobility, Journal of Undergraduate Research, 5, 9 (2011).

The percolation threshold results below are an average for at least 4 runs of data at each aspect ratio. 1500 rectangular cuts were made for each trial.

Rectangles with aspect ratio 1, 0.25, and 0.1

Relationship Between Rectangles and Ellipses

The percolation threshold is affected by geometry at high aspect ratios.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Rectangles Ellipses

Aspect Ratio

Perc

olati

on T

hres

hold

Conclusions• The new experimental set-up for measuring

the percolation threshold is in agreement with prior published results.

• For ellipses and rectangles, both follow the same trend: percolation threshold increases as the aspect ratio decreases

Conclusions• For aspect ratios less than or equal to 0.25,

where both shapes are similarly stick-like, percolation thresholds cannot be differentiated.

• For larger aspect ratios, where the shapes are more defined, the percolation thresholds for rectangles and ellipses begin to deviate.

AcknowledgmentsThe financial support from the National Science Foundation, EEC-NSF Grant # 1062943 is gratefully acknowledged.• Dr. Alan Feinerman• Dr. Gregory Jursich• Dr. Christos Takoudis• Dr. Prateek Gupta• Jared Weddell• Ismail Mithaiwala

Appendix

Relationships Between Rectangles and Ellipses Continued

The best-fit model shows percolation threshold as a function of log of aspect ratio.

-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 00.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f(x) = − 0.135138976838273 x + 0.363845478312865R² = 0.977509063468525f(x) = − 0.11647147379133 x + 0.439988433895346R² = 0.979645357590344

Rectangles Linear (Rectangles)Ellipses Linear (Ellipses)

Log (Aspect Ratio)

Perc

olati

on T

hres

hold