The Effect of Different Geometries on Percolation in Two Dimensions By Allison Morgan Dr. 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 Output The shapes are randomly distributed and oriented using MatLab. AutoCAD then visualizes the design. Explain magenta lines … this design will be cut onto aluminum mylar which we measure current across and will be described with more detail in the next slide Percolated region Effective area region

Current Experimental Setup Aluminum baseplate* Layer of double-sided adhesive Brass rods Acrylic fixtures and nylon screws* Aluminum baseplate can fit two runs & Two sheets of Mylar found way to roll mylar out smoother New nylon screws to avoid shorts Layer of Mylar with conductive aluminum * 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. Placing the rows a specified distance apart Circles were chosen because it was consistent with our testing method Kerf

Experimental Process At 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 Model Definition 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 L H0 Hr

Example of Experimental Output Raw data from 1500 circular cuts. The experimental percolation threshold was 0.3387. I0 Pc = Ir / I0 Ir Percolated square Effective area square ** 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. Because rectangle shaped holes have never been experimentally done before, we ran control tests with circles. [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 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. 1.000 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 0.2500 Here is our data First note the effects of changing aspect ratio Overall trend in comparison to ellipses. 0.6 aspect ratio is low. 0.1000 Rectangles with aspect ratio 1, 0.25, and 0.1 [6] J. Weddell, A. Feinerman, Percolation Effects on Electrical Resistivity and Electron Mobility, Journal of Undergraduate Research, 5, 9 (2011).

Relationship Between Rectangles and Ellipses Discuss overall trend Describe leftmost point in agreement Righter points not as much The inconsistency is mostly due to randomness of these networks? The percolation threshold is affected by geometry at high aspect ratios.

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.

Acknowledgments The 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 Professor Feinerman doesn’t get this slide. Maybe we can’t make this model with so few points between -2.25 and -4.5. The best-fit model shows percolation threshold as a function of log of aspect ratio.