Lecture 2 Thresholds, Islands, and Fractals

NCN@Purdue-Intel Summer School Notes on Percolation Theory

Muhammad A. Alam Electrical and Computer Engineering Purdue University West Lafayette, IN USA NCN www.nanohub.org 1

Outline of lecture 2

1) Basic Concepts of Percolation Theory 2) Percolation Threshold and Excluded Volume 3) Cluster Size Distribution, Cluster Radius 4) Fractal dimension of a random surface 5) Conclusion Application Notes: Nanocrystal Flash

Three topics of random systems Percolation threshold Cluster sizes epidemics, forest fire, Fractal dimension Oil fields, NC Flash telecom grid, www Aerosol, paper, sensors Nanonets, photovoltaics

3 basic concepts: percolation threshold p=0.3 p=0.59 p=0.8

pC p 4

basic concepts: cluster size p=0.3 s=3 p=0.59 s=22 p=0.8 s=infinity

p p C 5

cluster-size distribution and its moments

Number of cluster of size s divided by the number of sites

…plays a role similar to Boltzmann distribution f(E)

6 average cluster vs. infinite cluster

p=0.3 p=0.59 p=0.8

Sav

pC p 7

basic concepts: dimension of a surface

D=0 D=1 D=0

D=?

outline

1) Basic Concepts of Percolation Theory 2) Percolation Threshold and Excluded Volume 3) Cluster Size Distribution, Cluster Radius 4) Fractal dimension of a random structure 5) Conclusion Application Notes: Nanocrystal Flash

9 calculation of percolation threshold

Square Triangular Hexagonal

pc = 0.593 pc = 0.500 pc = 0.697

Percolation threshold (pc=Nc/NT) depends on lattice, there is something wrong here !

area fraction fill-factor

Square Triangular Hexagonal

(V x pc) is universal ….

Square Triangular Hexagonal

pc = 0.593 pc = 0.500 pc = 0.697

Vpc ~ 0.45 Vpc ~ 0.45 Vpc ~ 0.42

12 HW: Percolation in 3D lattices

For simple cubic lattice site percolation,

Vpc ~ 0.16 and pc ~ 0.311.

Use the universality of Vpc to show that the percolation threshold for FCC lattice must be approximately 0.1

percolation involving other shapes

How do I determine the percolation threshold?

Vxpc ~ 0.45 will not work, unfortunately !

excluded area … first an intuitive result

Disk percolation

Stick percolation

15 the concept of excluded area …..

For disks on arbitrary grid …

For arbitrary shape on arbitrary grid … X 4

Percolation begins when excluded volume is routinely breached

excluded volume for a stick ….

percolation threshold correct within a factor of 2 !

18 HW: excluded volume for other shapes ..

curved stick … square …

Ans.

Hint. Compare with circle

Hint. For general shape, use the Use the stick algorithm Monte Carlo code posted 19

outline

cluster-size distribution

p=0.3 p=0.59 p=0.8

ns Sav

p p C pC p

21 small-cluster size distribution

+ + + + 22

features of cluster-size distribution

 ‘Zeros’ at p=0,1 with single peak

 Peak shifts towards pc with s (e.g. s=1 is 0.2, s=2 is 0.25; s=3 is 0.29)

 General form:

: gst increases exponentially.

pC p 23

numerical plots for cluster-size distribution

Analytically … by computer …

n 4 S=45

n6 90 180 n 7 363

24 scaling of cluster sizes

S=45

N S 90 180

363

p z

for large cluster-sizes (s>20)

) c (p s

ns n

pC p s

self-similarity and scale-invariance

irregular regular self-similarity self-similarity self-similarity

ln(ns)

ln(s)

27 outline

1) Basic Concepts of Percolation Theory 2) Percolation Threshold and Excluded Volume D=? 3) Cluster Size Distribution, Cluster Radius 4) Fractal dimension of a random structure 5) Conclusion Application Notes: Nanocrystal Flash

classification of surfaces…

Fractal Dimension (DF)- Box counting technique

h ½ h ½ h ½ h 1/4 2 h 1/4 1 h 1/4 0 N(h)NN ~16 4 h N(h)NN ~ 42 h N(h)N ~ 1 h

Plane DF=2 N(h) Random NW 1< DF<2 Line D =1

Log F

Dot DF=0 Log(1/h) 29

Regular and irregular fractal

h 1/3 h 1/9 h 1/27 h 1/3n N 2 N 4 N 8 N 2n

Bigger than point, But smaller than line

…. keep m piece

of n pieces

30 regular and irregular fractals …. keep m piece

of n pieces

regular

irregular

dimension of quasi-2D cantor stripes

h 1/3

N 6

h 1/9 h 1/3n

n n N 36 N 3 2

h 1/27

N 216

same DF, but different geometry

same dimension, What about this because DF=log(m)/log(n) irregular fractal?

33 dimension of a irregular fractal Fractal Dimension of Composites

h=16h=32h=64h=8h=4h=2 Slope=DF.

ρC = 5.717

cantor transform

Preserve DF during transformation

Fractal dimension at percolation

irregular self-similarity self-similarity

DF~1.85 ln(ns)

ln(s)

36 Summary

 Discussed three key concepts of percolative transport: percolation threshold, island size distribution, and fractal dimension

The concept of excluded volume provides a (nearly) geometry independent way for calculating the percolation threshold for arbitrarily shaped objects on arbitrary grid.

Distribution of island sizes is also described by simple formula with universal constants. At percolation threshold, the island sizes are self-similar and scale invariant.

Fractal dimension provides a generalized technique to describe the dimension of any surfaces, even those defined by randomly oriented sticks.

References

Oil well: www.ambercore.com/Petroleum_iQ.php Aerosol and paper: “ . Brain H Kaye, A Random walk in Fractal dimensions, VCH Publishers; 2nd edition (April 1994).

D. Stauffer and A. Ahrony, Introduction to Percolation Theory, Revised 2nd Edition, 2003. This is an excellent introductory text book. My discussion of percolation clusters follows this book closely.

The discussion on excluded volume is based on I. Balberg, C. Anderson, S Alexander, N Wagner - Physical Review B, 1984.

A good place to start on fractals is Benoit B. Mandelbrot’s book “A fractal geometry of Nature”. W. H. Freeman (1983). Other good books include D. L. Turcotte, “Fractals and Chaos in Geology and Geophysics”, Cambridge University Press; 2 edition, 1997.