27750, Advanced Characterization

1
Percolation, Cluster and Pair Correlation
Analysis31st March, 2005

• 27-750, Advanced Characterization
  Microstructural Analysis
• A.D. (Tony) Rollett

2
Objectives

• Introduce percolation analysis as a tool for
  understanding the properties of microstructures.
• Apply percolation to electrical conductivity as
  an example of the dependence of a key transport
  property on grain boundary properties and
  texture.
• Introduce cluster analysis via nearest neighbor
  distances
• Introduce pair correlation functions to analyze
  medium range correlations in positions of objects
  (e.g. precipitates).

3
References

• D. Stauffer, Introduction to Percolation Theory,
  2nd ed., Taylor and Francis, 1992.
• First mention of percolation theory is from S.R.
  Broadbent and J.M Hammersley Percolation
  processes I. crystals and mazes, in Proceedings
  of the Cambridge Philosophical Society, 53,
  629-641 (1957).
• Random Heterogeneous Materials Microstructure
  and Macroscopic Properties, S. Torquato, Springer
  Verlag (2001, ISBN 0-387-95167-9).

4
Definitions

• Percolation is the study of how systems of
  discrete objects are connected to each other.
• More specifically, percolation is the analysis of
  clusters - their statistics and their properties.
• The applications of percolation are numerous
  phase transitions (physics), forest fires,
  epidemics, fracture
• Applications in microstructure include
  conductivity, fracture, corrosion, clustering,
  correlation, particle analysis
• Transport Properties are particularly suited to
  percolation analysis because communication or
  transmission between successive neighboring
  elements is key.

5
Notation Percolation

• p probability of bond (connection) between
  sites
• ps probability of a site being occupied
• pc percolation threshold (critical probability
  for network to percolate)
• s size of a cluster
• S mean size of clusters, ltsgt
• ns number of clusters of size s.
• ws probability that a given site belongs to a
  cluster of size s.
• d dimensionality of the system.
• ? correlation length
• r radius, distance between sites.
• g(r) correlation function (connectivity
  function).

• c proportionality constant.
• ? critical exponent on cluster size, s
• ? critical exponent on proportionality
  constant, c
• P fraction of sites in the critical (infinite
  size) network, or power.
• ? critical exponent on size of critical
  network
• ? critical exponent on average cluster size, S
• L size of finite system.
• ? probability that a finite system will
  percolate
• pav average percolation threshold in a finite
  system
• ? critical exponent on average threshold
  probability, pav

6
Notation cluster analysis

• r radius, distance
• ltrkgt average distance to the kth nearest
  neighbor
• k, or, n order number of nearest neighbor (1
  first, 2 second etc.)
• pk Poisson process probability for occurrence
  of k objects in a given time or space interval
• ?t time (space) interval
• ? , or, ? expected value (e.g. a density)
• ?t time (space) interval
• X number of objects for evaluation of a
  cumulative probability

• d dimensionality
• ? Gamma function
• Wk cumulative probability of finding at least
  k objects in a given volume
• Vd volume of object in d dimensions
• Sd surface of object in d dimensions
• wk probability density of distance to the kth
  neighboring object

7
Notation Pair Correlation Function

• N total number of particles
• n chosen number of particles
• r radius, distance
• PN(rN) drN specific N-particle probability
  density function, PN(rN) drN
• ?n(rN) generic n-particle probability density
  function
• gn(rn) n-particle correlation function

• ? NV number density of particles
• g2(r12) 2-particle or pair correlation
  function

8
An example

• It is always easier to understand a concept with
  a picture, so lets see what clusters mean in 2
  dimensions on a square lattice. If we populate
  some of the cells (LHS) we can see that there are
  cases where the dots fall into neighboring cells.
  If we then draw in all the cases where these
  nearest neighbor links exist (vertical or
  horizontal bars, RHS) then we can connect cells
  together into clusters. One cluster is colored
  red, and the other one blue. Obviously they have
  different sizes. The isolated dots are left in
  black. This example is called site percolation.

9
Percolation Threshold

• A key concept is the percolation threshold, pc.
• For site percolation (to be defined next), there
  is a critical concentration (of occupied sites),
  above which a cluster exists that spans the
  domain, i.e. connects the left hand edge to the
  right hand edge.
• Example for a square 2D lattice and bond
  percolation, the percolation threshold 0.5.
  This same value is found for the triangular
  lattice and site percolation.
• In general, mathematically exact results are
  available for some lattice types in 2D but rarely
  in 3D.

10
Site vs. Bond Percolation

• For bond percolation, imagine that there is a
  bond or line drawn between each lattice site.
  Each bond has a certain probability, p, of being
  good or existing or closed, where the
  terminology depends on the field of enquiry.
• Conversely, the discussion so far has been on
  site percolation where there is a certain
  probability, p, of any site being occupied, but
  perfect connectivity (i.e. good bonds) between
  nearest neighbor occupied sites.
• As an example, when we discuss electrical
  conductivity in HTSC films, we will be dealing
  with bond percolation because it is the grain
  boundaries that determine the properties.

11
Lattice Types

• Although the use of different lattices is obvious
  to those who have written computer codes for
  numerical modeling of microstructures, the figure
  from Stauffer below illustrates 2 lattices
  triangular and honeycomb in 2D, simple cubic in
  3D.

12
Percolation Thresholds
Note that 2D grain structures can be regarded as
being very close to hexagonal tilings, like the
honeycomb lattice, or that the boundary networks
contain mainly tri-junctions, like the triangular
lattice. 3D grain structures can be regarded as
being similar to the fcc lattice. Thus properties
that are sensitive to percolation thresholds
(e.g. fracture at weak boundaries) often exhibit
thresholds that are similar to the values
displayed in the table
13
Cluster Size - 1D

• In order to discuss clusters, we need some
  definitions.
• This is most easily done in ONE dimension because
  exact solutions area available for 1D (but not,
  of course, for higher dimensions!) every site
  must be occupied for percolation, so pc 1!
• ns is the number of clusters per lattice site
  of size s. Note how the definition is given in
  terms of each individual site. This quantity is
  also called the normalized cluster number.
• The probability that a given lattice site is a
  member of a cluster of size s is given by the
  product of the cluster size and ns.
• These probabilities are related to the occupation
  probability in a simple way ps ?snss.
• Stauffer p. 21

14
Cluster Size - 1D, contd.

• The probability, ns, that a given site belongs to
  a cluster of size s, is given by dividing the
  probability that that site belongs to that size
  class, ws, and dividing it by the occupation
  probability, ps ws  nss / ps nss / ?snss.
• Thus, the average cluster size, S, is given byS
  ?swss ?snss2/ ?snss.
• This definition of cluster size is also valid for
  higher dimensions (dgt1) although the infinite
  cluster must be excluded from the sums.
• Lattice animals are very similar but derived from
  graph theory.

15
Cluster Size in 1D

• To obtain the mean cluster size in terms of the
  transition probability, p, significant work must
  be done, even in 1D.
• The result, however, is very simple and elegant
  (p lt pc)
• It tells us that the cluster size diverges (goes
  to infinity) for probabilities as they approach
  the critical value, pc, which of course equals 1
  in 1D.

16
Correlation Length, ?

• It is useful to define a correlation function,
  g(r), that is the probability that a site, that
  is at a distance r away from an occupied site,
  belongs to the same cluster. In 1D, this means
  that every site in between must be occupied, so
  the probability is equal to the p raised to the
  rth power, pr g(r) pr.
• Thus the correlation function (or connectivity
  function) goes exponentially to zero with
  distance, where ? is the correlation length

17
Correlation Length, ?, contd.

• The correlation length below the transition is
  given by
• Interestingly, in 1D it is proportional to the
  cluster size, S ? ? . In higher dimensions, the
  relationship is more complex.

18
Higher Dimensions

• To discuss higher dimensionalities, we need to
  explain that we are interested in behavior near
  the critical point, i.e. what happens when a
  system is about to make a transition from
  non-percolating to percolating. More precisely,
  we say that p-pcltlt1. Leaving out much of the
  (important, but time consuming) detail, the
  probability that a given point belongs to a
  cluster of size s, turns out to be given by an
  expression like this

19
Higher Dimensions, contd.

• Note the appearance of an exponent t that turns
  out to be one of a set of critical exponents.
  The proportionality constant, c, is also
  described by an equation with another critical
  exponent, s
• Then we can write a similar expression for the
  fraction of sites in the critical (infinite)
  network, P, which will be of particular interest
  for conductivity

20
Higher Dimensions, contd.

• By further derivations, one can find that there
  is a simple relation between P and the deviation
  from the critical transition probability, with a
  new critical exponent, b
• Finally, we find that for the average cluster
  size, S

21
Critical Exponents

• The table shows values of the critical exponents
  for a variety of situations.
• Note that the values of the exponents do not
  depend on the type of lattice but only on the
  dimensionality of the problem.

http//www.ibiblio.org/e-notes/Perc/perc.htm
22
Critical Exponents

• Not only is it remarkable that these exponents
  depend only on the dimensionality of the problem,
  but there are definite, theoretically derivable
  relationships between them. We give two of the
  basic relationships here.

23
Finite Size Systems

• The percolation threshold becomes a probabilistic
  quantity for systems that are not infinite in
  size. In plain language, there is a certain
  probability that a spanning cluster exists in a
  given realization of a lattice with a specified
  filling probability.
• The next step in this analysis is to analyze
  probabilities of the occurrence of spanning
  clusters.

24
Percolation Cluster Examples

• A spanning cluster is one that crosses completely
  from one side to the other (or top to bottom).
• Non-spanning cluster shown in the picture.
• See http//www.physics.buffalo.edu/gonsalves/ComPh
  ys_1998/Java/Percolation.html for a java applet
  that allows you to experiment with percolation
  thresholds in 2D.

25
Finite Size Systems

• For systems of finite size, L, the transition
  from non-percolating to percolating is fuzzier.
  More precisely, there is a finite probability,
  P, that a large enough cluster (of the right
  shape) will occur in a given realization of a
  system that spans the domain.
• As the system increases in size, so the
  probability of this happening decreases for a
  given deviation, pc  p, of the transition
  probability below the critical value.
• The graph reproduced from Stauffer shows the
  behavior schematically the solid line for P(Llt8)
  has a finite slope over an appreciable range of
  p. The dashed line shows the probability density
  for the same quantity.

26
Approach to Critical Point

• If one considers how to extract the critical
  probability, one approach is to seek the
  inflection point in dP/dp. More properly, one
  must integrate the slope of the spanning
  probability.
• Then one finds that the approach of the measured
  probability approaches the true critical value as
  a function of the system size that includes one
  the the critical exponents, n
• Although actual values of the exponent are known,
  in practice one has to find the best value by
  inspection of, say, simulation results. It is
  possible (e.g. in certain 2D systems) for there
  to be no variation with system size for cases in
  which the transition is symmetrical.

27
Electrical Conductivity

• One obvious application of percolation analysis
  is to electrical conduction in materials with
  weak links, e.g. high temperature
  superconductors. Although the application of
  percolation may seem straightforward, the actual
  dependence of conductivity on the transition
  probability is not as simple as equating the
  conductivity to the strength, P, of the infinite
  (spanning) network. Think of the strength as the
  fraction of sites/cells that are part of the
  spanning network. Stauffer and Aharony quote a
  result from Last Thouless (1971) in which the
  conductivity (solid line) increases considerably
  more slowly from the critical level than does the
  cluster strength (dashed line).

28
High Tc Superconductors

• As a result of the development of technologies
  that deposit (ceramic oxide) superconductors onto
  long lengths (gt1 km!) of metal substrate tapes,
  analysis of the percolative nature of
  microstructures has been actively investigated.
• The orientations of the grains in the nickel
  substrate are carried through to the grains in
  the superconductor layer (via epitaxial growth).

29
Boundaries in Hi-Tc Superconductors

• The critical property of interest in the ceramic
  oxide superconductors is the strong inverse
  correlation between misorientation and ability to
  transmit current across a grain boundary. This
  plot from Heinig et al. shows how strongly
  boundaries above a certain angle effectively
  block current.- Appl. Phys. Lett., 69, (1996)
  577.

30
Magneto-optical Imaging

• Feldmann et al. Appl. Phys. Lett., 77 (2000)
  2906 have used magneto-optical (MO) imaging to
  great effect to reveal the effects of
  microstructure on electrical behavior.
• The micrographs show EBSD maps of surface
  orientation for the Ni substrate in (a) and
  boundaries with ??1 in (d). The next column
  shows a percolation map in (b) such that
  connected points are shown in a single color,
  with boundaries ??4 in (e). The MO image of
  current density in the overlaying YBCO film
  (1µm) is shown in (c) - light color indicates
  low current density. (f) shows boundaries with
  ??8.

31
Crystallographic Effects on Percolation

• Schuh et al. Mat. Res. Soc. Symp. Proc., 819,
  (2004) N7.7.1 have shown that, although standard
  percolation theory is applicable to analysis of
  materials properties, the existence of texture
  results in strong correlations between each link
  of the network, where properties depend on grain
  boundary characteristics.
• Standard percolation theory assumes that the
  strength (or probability of a connection) for
  each link is independent of all others in the
  system.
• Grain boundaries meet at triple junctions
  (topology of boundary networks) and so one of the
  3 boundaries must be a combination (product, in a
  sense) of the other two.
• The impact is significant. To paraphrase the
  paper, for conductivity in simulated 2D networks
  of grains and associated boundaries, the
  percolative threshold from non-conducting to
  conducting is between 0.31 and 0.336 for
  different standard texture types, whereas the
  theoretical threshold for a random network
  (triangular mesh) is 0.347.

32
Other Analyses Neighbor Distances, Pair
Correlation

• Many examples of microstructures involve
  characterization of two-phase systems.
• If the material contains a dispersion of
  particles in a matrix, there are many types of
  analysis that can be applied.
• If we are interested in the clustering (or
  separation) of the particles, we can examine
  inter-particle distances.
• If we are interested in the spatial distribution
  of the particles, we can characterize pair
  correlation functions.

33
kth-Neighbor Analysis

• If particles are clustered together, the
  distances between them will be small compared to
  the average distance.
• Therefore, it is useful to measure the average
  distance, ltrkgt, between each particle and its kth
  neighbor, as a function of k.
• If particles are placed randomly, this quantity
  can be described analytically (equations to be
  described).
• In the simplest, 1D case, this quantity is
  proportional to the neighbor number.
• In 2D, the function is more complicated but ltrkgt
  varies approximately as vk.

34
Kth Neighbor Example

• In this example from Tong, this analysis was
  performed on nucleus spacing during
  recrystallization to examine the dimensionality
  of nucleation, i.e. whether new grains appearing
  on lines, were effectively random, or whether the
  restriction to lines was significant. The result
  shown by circles (? 10) is for closely spaced
  nuclei on boundaries for which the latter was the
  case.

W.S. Tong et al. (1999). "Quantitative analysis
of spatial distribution of nucleation sites
microstructural implications." Acta materialia
47(2) 435-445.
35
Kth-Neighbor Analysis

• There are a series of equations that are needed
  in order to understand the theoretical basis for
  ltrkgt
• The first is taken from standard probability
  theory for the Poisson Process. This theory
  gives us a quantitative basis for predicting the
  probability that a given event will occur in a
  given interval of time or space.
• Useful examples for application of the concept
  include radioactive decay (how likely is it that
  a decay event will be observed in a specified
  time interval, based on an average count rate) or
  counting trees in a forest (how likely is it a
  specified number of trees will be found in a
  given area, based on an average tree density).

36
Poisson Process Probability

• We begin by summarizing the basic theory of the
  Poisson process for predicting the probability
  that a given event will occur within a certain
  interval of time or space. This is easiest to
  understand with the help of practical, physical
  examples. As an example of a time-based Poisson
  process, consider radioactive decay. We know
  that if we measure a sample of a radioactive
  substance with a Geiger Counter such as a uranium
  bearing mineral, we will obtain a certain number
  of counts per minute. In statistical terms, the
  count rate is the expected value, or rate of
  process, for the event of interest, i.e. a single
  radioactive decay event. We will call this
  expected value alpha (a) in some texts this is
  written as ltngt. The critical assumptions that
  permit us to apply the basic theory for the
  Poisson process are as follows.
• 1. The expected value, a, multiplied by a given
  time interval, ?t, is the probability, a?t, that
  a single event will occur in that time interval.
• 2. The probability of no events occurring in
  that same time interval is 1 - a?t,
• which requires that the probability of more than
  one event occurring in that same interval is of
  order ?t (o(?t)).
• 3. The number of events in the given time
  interval is independent of the events that occur
  before the given time interval. Another way to
  say this is that the events are uncorrelated in
  time.

37
Poisson, contd.

• Based on these assumptions we can write the
  probability, pk, of k counts being recorded in
  the time interval ?t, based on an expected value,
  a, as follows.

38
Poisson, contd.

• For space-based analysis, consider mapping out a
  forest and counting trees. The expected number
  of trees in a given area can expressed as so many
  trees per hectare, a. Then the probability of
  finding 10 trees in two hectares, for example, is
  simply the same expression but with the area, A,
  substituting for the time interval.
• So, if we count 131 trees per hectare then the
  probability of finding only 10 trees in 2
  hectares (clearly very unlikely) is
• Note that the formula contains unwieldy
  quantities from a numerical perspective so it may
  be necessary to re-scale the number of interest,
  the area or time interval and the expected value
  in order to make it possible to calculate a
  probability.

39
Poisson, contd.

• Now, often a more useful probability is the one
  that describes how likely it is that at least k
  events will be observed in the specified
  interval. Since this probability, p(k?X), is
  effectively a measure based on a cumulative
  distribution, a summation is required in order to
  arrive at the desired answer.

40
Cumulative Probabilities

• Another way to understand this approach is to
  consider precipitates in a material. If the
  particles are located in the material in a
  completely random fashion, then we can model
  their positions on the basis of a Poisson
  process. Thus we can write the probability of
  observing n particles in a given area by the
  following. In this version of the equation, ltngt
  is the expected value, i.e. the expected/average
  value for that number of particles in the
  specified region or time interval.

41
Nearest Neighbor Distances

• Now we can extend this approach to relate it to
  nearest neighbor distances between particles.
  Lets define a density of particles as ld so that
  the standard notation in 3D would be NV. For a
  given volume, Vd, where d denotes the dimension
  of the space (normally 2 or 3), then the expected
  value that we are interested in is given by ltngt
  NV Vd. From statistical mechanics Pathria, R.
  K. (1972). Statistical Mechanics. New York,
  Pergamon Press, we know that the volume, Vd, and
  surface area, Sd, of a region of size (radius) r
  is given by the following, where G is the gamma
  function

42
Nearest Neighbor Distances, contd.

• From these one can find the cumulative
  probability function, Wk, for the probability of
  finding at least k particles in the volume of
  interest (see above for the basic equation).
• From this, we can find the probability density of
  the distance to the kth nearest neighbor by
  differentiation (the clever trick in all this!)
  of this quantity.

??????????
43
Nearest Neighbor Distances, contd.

• This probability density function is not
  immediately useful to us so we have to make an
  average by taking the first moment, i.e.
  integrating the density, w, by the radius from 0
  to infinity.

44
Nearest Neighbor Distances, contd.

• Evaluating this expression for 1, 2 and 3
  dimensions, we obtain

(1D)
(2D)
(3D)
45
2D Examples

• Here are the results from Tong et al. of
  calculating the 2D nearest neighbor average
  distance using both the theory given above and
  results from distributing points at random over a
  plane and measuring interparticle distances
  directly. Note that in order to accommodate
  different particle densities (l2  NA) the
  vertical axis is normalized by the density. The
  theoretical result and the numerical ones lie
  essentially on top of one another, i.e. the
  agreement is near perfect.

46
Application

• Tong et al. exploited this analysis to diagnose
  the degree to which nuclei for a phase
  transformation examined in cross section
  (therefore 2D) were behaving as clusters on grain
  boundaries (and thus obeying the 1D nearest
  neighbor characteristic) versus being effectively
  dispersed at random throughout the material (thus
  2D behavior).
• The plot shows the normalized average distance to
  the kth neighbor for two sets of nuclei
  distributed randomly along grain boundaries in a
  2D microstructure. The circles correspond to a
  case with a high density of points such that
  points within klt10 cluster as if on lines. In
  the second case, triangles, the low density of
  points means that the nuclei all behave as if
  they were randomly scattered throughout the area.

47
Grain Morphology

• The difference between the low (a - triangles)
  and high (b - circles) nucleus densities
  illustrated is shown by these images of the fully
  transformed structures.

48
Pair Correlations

• A simple concept that turns out to be very useful
  in particle analysis is that of pair correlation
  functions.
• This is also an important concept in particle
  physics.
• Conditional (2-point) probability function given
  a vector whose tail is located within a particle,
  what is the probability that the other end (head)
  of the vector falls inside another particle (of
  the same phase)?
• The average probability (over all vectors) is
  just the volume fraction of particles.
• If particles are highly correlated in position in
  a given direction, then this probability will be
  higher than the volume fraction for vectors
  parallel to the correlated direction.
• We can illustrate the idea with an example in
  aerospace aluminum.

49
Pair Correlation example
Strict definition conditional 2-point probability
Input (500X500) Center of 1 dot to end of 5th dot
is 53 pixels
Output (401X401) Center of image to end of red
dot is 53 pixels
Color in a PCF is scaled from black (low
probability) to white (high)
50
PCF correlation lengths
Optical image of transverse plane (S-T)
PCF of transverse plane
Tran 1 PCF
Tran 1
ND / S
TD / T
2 particles per stringer
51
PCF correlation length/ longitudinal
Optical image of longitudinal plane (L-S)
PCF of longitudinal plane
Long 6
RD / L
ND / S
Long 6 PCF
10 particles per stringer
52
PCF Analysis Mg2Si
BSE image
PCF 139X139 mm
There is no correlation between the placement of
Mg2Si particles
53
PCFs on Orthogonal Planes
ND
TD
RD

• 2.64 µm per pixel in the images ratio of lengths
  in ND-TD section ? correlation length 35 µm //
  TDsimilarly, correlation length 130 µm // RD.

54
PCF Analysis

• Given N particles in a given volume, one can
  introduce a generic n-particle probability
  density function, ?n(rN).

• This is based on the specific N-particle
  probability density function, PN(rN) drN, which
  is the probability of finding the center of
  particle 1 in volume element dr1, the center of
  particle 2 in volume element dr2 , etc., with
  drN dr1 dr2 dr3?idri. Normalization means that

55
PCF Analysis, contd.

• The n-particle probability density is not
  strictly a probability density because its
  normalization is as follows.

• Now we assume that we have a statistically
  homogeneous material so that we can pick an
  arbitrary and measure all vectors relative to
  that position rij ri - rj.

56
PCF Analysis, contd.

• Thus, the one-particle density function is just
  equal to the number density of particles, ? NV

• Next we define an n-particle correlation
  function, gn(rn)

57
PCF Analysis, contd.

• As the distance between particles goes to
  infinity, and provided the material is
  homogeneous, any of the n-particle correlation
  functions tends to unity. Therefore deviations
  from unity reveal the extent of spatial
  correlation between particles (unity means no
  correlation).
• The most important higher-order quantity is the
  2-particle or pair correlation function.

58
PCF Analysis, contd.

• Given a statistically isotropic material, the
  direction of the vector r connecting the two
  particles is irrelevant and the function depends
  only on the separation distance.
• We can define a conditional probability of
  finding another particle at a separation r, given
  a particle already located at the tail of r.
  Given a spherical shell of area s(r), this
  probability is

59
PCF Analysis, contd.

• In the examples that preceded this derivation,
  the quantity being plotted was the (2 particle)
  conditional probability derived from the pair
  correlation function (rather than the PCF
  itself).
• An example is given below of a radial correlation
  function for a system of disordered interacting
  particles (via a potential energy function).
  Note that many equilibrium properties of dynamic
  systems of particles can be computed/derived
  based on a knowledge of the pair correlation
  function.

Loosely speaking, one can think of the pair
correlation function as giving information on the
likelihood of finding a particle relative to the
average density
Torquato
60
2-point Probability Function

• The pair correlation function and conditional
  probability function discussed previously are
  closely related to the 2-point probability
  function, S2(r).
• The 2-point probability function is found by
  dropping a test line (vector) onto the
  microstructure and calculating the fraction of
  drops for which the ends of the line fall in the
  same phase.
• Note that there is a signal (the volume fraction)
  for a single particle (which is excluded in the
  conditional probability).
• Obviously there are n-point probability functions
  for as high an order of correlation as is of
  interest.

Examples of 2-point correlation function (radial)
from Torquato (fig. 2.7)
61
Artificial Digital Particle Placement

• To test the system of particle analysis and
  generation of a 3D digital microstructure of
  particles, an artificial 3D microstructure was
  generated using a Cellular Automaton on a
  400x200x100 regular grid (equi-axed voxels or
  pixels). Particles were injected along lines to
  mimic the stringered distributions observed in
  7075. The ellipsoid axes were constrained to be
  aligned with the domain axes (no rotations).
• This microstructure was then sectioned, as if it
  were a real material, the sections were analyzed,
  and a 3D particle set reconstructed.
• The main analytical tool employed in this
  technique is the (anisotropic) pair correlation
  function pcf (again, strictly speaking, we use
  a 2-point conditional probability function, in
  2D).
• The length units for this calculation are pixels
  or voxels.

62
Simulation Domain with Particles

• Particles distributed randomly along lines to
  reproduce the effect of stringers.
• Series of slices through the domain used to
  calculate pcfs, just as for the experimental
  data.
• Averaged pcfs used with simulated annealing to
  match the measured pair correlation functions.

63
Sections through 3D Image
64
Generated Particle Structure Sections

• Ellipsoids were inserted into the domain with a
  constant aspect ratio of abc 321. The
  target correlation length was 0.07x400 28, with
  10 particles per colony

Rolling plane (Z) - Transverse (X) -
Longitudinal (Y)
65
Generated Particle Structure PCFs

• Pair Correlation Functions were calculated on a
  50x50 grid. The x-direction correlation length
  was 29 pixels (half-length of the streak), in
  good agreement with the input.

Rolling plane (Z) - Transverse (X) -
Longitudinal (Y)
66
2D section size distributions

• A comparison of the shapes of ellipses shows
  reasonable agreement between the fitted set of
  ellipsoids and initial cross-section statistics
  (size distributions)

Cross-plot
Initial vs. Final section distributions
67
Comparison of 3D Particle Shape, Size

• Comparison of the semi-axis size distributions
  between the set of 5765 ellipsoids in the
  generated structure and the 1,000 ellipsoids
  generated from the 2D section statistics shows
  reasonable agreement, with some leakage to
  larger sizes.

68
Comparison of PCFs for Original and Reconstructed
Particle Distribution
From CA
Reconstructed
Rolling plane (Z) - Transverse (X) -
Longitudinal (Y)
69
Reconstructed 3D particle distribution
70
Summary

• Percolation analysis is very useful for transport
  properties and for fracture propagation in
  solids.
• Cluster analysis can be performed using nearest
  neighbor distance analysis.
• Pair correlation functions are useful for
  analyzing alignment of, say, particle positions
  over small multiples of the average spacing.
• When fitting particle distributions with
  particles of arbitrary shape and size, it is
  generally necessary to use numerical methods,
  e.g. a simulated annealing algorithm to fit a 3D
  distribution to 2D cross section information.

71
Other references

• 1. D. Stauffer and A. Aharony, Introduction to
  Percolation Theory, 2nd ed., Taylor and Francis,
  1992 MR 87k82093.
• 2. S. Havlin and A. Bunde, Percolation,
  Contemporary Problems in Statistical Physics, ed.
  G. H. Weiss, SIAM, 1994, pp. 103-146.
• 3. M. Sahimi, Applications of Percolation
  Theory, Taylor and Francis, 1994.
• 4. G. Grimmett, Percolation, Springer-Verlag,
  1989 MR 90j60109.
• 5. C. Godsil, M. Grötschel and D. J. A. Welsh,
  Combinatorics in Statistical Physics, Handbook of
  Combinatorics, v. II, ed. R. Graham, M. Grötschel
  and L. Lovász, MIT Press, 1995, pp. 1925-1954 MR
  96h05001.
• 6. J. W. Essam, Percolation and cluster size,
  Phase Transitions and Critical Phenomena, v. II,
  ed. C. Domb and M. S. Green, Academic Press 1972,
  pp. 197-270 MR 50 6392.
• 7. J. C. Wierman, Percolation theory,
  Encyclopedia of Statistical Sciences, v. VI, ed.
  S. Kotz and N. J. Johnson, Wiley 1985, pp.
  674-679 MR 90g62001a.
• 8. R. T. Smythe and J. C. Wierman,
  First-Passage Percolation on the Square Lattice,
  Lecture Notes in Math. 671, Springer-Verlag,
  1978 MR 80a60135.
• 9. D. Stauffer, Scaling theory of percolation
  clusters, Phys. Reports 54 (1979) 1-74.
• 10. A. R. Conway and A. J. Guttmann, On
  two-dimensional percolation, J. Phys. A 28
  (1995) 891-904.
• 11. H. N. V. Temperley and E. H. Lieb,
  Relations between the 'percolation' and
  'colouring' problem and other graph-theoretical
  problems associated with regular planar lattices
  some exact results for the 'percolation' problem,
  Proc. Royal Soc. London A 322 (1971) 251-280 MR
  58 16425.
• 12. M. F. Sykes and J. W. Essam, Exact
  critical percolation probabilities for site and
  bond problems in two dimensions, J. Math. Phys. 5
  (1964) 1117-1121 MR 29 1977.
• 13. J. W. Essam and M. F. Sykes, Percolation
  processes, I Low-density expansion for the mean
  number of clusters in a random mixture, J. Math.
  Phys. 7 (1966) 1573 MR 34 3952.
• 14. M. F. Sykes and M. Glen, Percolation
  processes in two dimensions, I Low-density
  series expansions, J. Phys. A 9 (1976) 87-95.
• 15. M. F. Sykes, D. S. Gaunt and M. Glen,
  Percolation processes in two dimensions, II
  Critical concentrations and mean size index, J.
  Phys. A 9 (1976) 97-103.
• 16. C. Domb and C. J. Pearce, Mean number of
  clusters for percolation processes in two
  dimensions, J. Phys. A 9 (1976) L137-L140.
• 17. P. L. Leath, Cluster shape and critical
  exponents near percolation threshold, Phys. Rev.
  Letters 36 (1976) 921.
• 18. P. L. Leath, Cluster size and boundary
  distribution near percolation threshold, Phys.
  Rev. B 14 (1976) 5047.
• 19. A. Coniglio, H. E. Stanley and D.
  Stauffer, Fluctuations in the number of
  percolation clusters, J. Phys. A 12 (1979)
  L323-L327.

72
Other references

• 24. J. Güémez and S. Velasco, A probabilistic
  approach to the site-percolation problem, I
  Bethe lattices, Physica A 171 (1991) 486-503 MR
  92f82027.
• 25. J. Güémez and S. Velasco, A probabilistic
  approach to the site-percolation problem, II
  Standard lattices, Physica A 171 (1991) 504-516
  MR 92f82028.
• 26. J. Adler, Series expansions, Comput. in
  Phys. 8 (1994) 287-295.
• 27. J. Adler, Y. Meir, A. Aharony and A. B.
  Harris, Series study of percolation moments in
  general dimension, Phys. Rev. B 41 (1990)
  9183-9206.
• 28. B. Tóth, A lower bound for the critical
  probability of the square lattice site
  percolation, Z. Wahr. verw. Gebiete 69 (1985)
  19-22 MR 86f60118.
• 29. S. A. Zuev, Bounds for the percolation
  threshold for a square lattice, Theor. Probab.
  Appl. 32 (1987) 551-553 MR 89h60171.
• 30. M. V. Menshikov and K. D. Pelikh,
  Percolation with several defect types. An
  estimate of critical probability for a square
  lattice, Math. Notes Acad. Sci. USSR 46 (1989)
  778-785 MR 91h60116.
• 31. T. Luczak and J. C. Wierman, Critical
  probability bounds for two-dimensional site
  percolation models, J. Phys. A 21 (1988)
  3131-3138 MR 89i82053.
• 32. M. Campanino and L. Russo, An upper bound
  on the critical percolation probability for the
  three-dimensional cubic lattice, Ann. Probab. 13
  (1985) 478-491 MR 86j60222.
• 33. B. Bollobás and Y. Kohayakawa, Percolation
  in high dimensions, Europ. J. Combinatorics 15
  (1994) 113-125 MR 95c60092.
• 34. R. M. Ziff and B. Sapoval, The efficient
  determination of the percolation threshold by a
  frontier-generating walk in a gradient, J. Phys.
  A 19 (1986) L1169-L1172.
• 35. R. M. Ziff, Spanning probability in 2D
  percolation, Phys. Rev. Letters 69 (1992)
  2670-2673.
• 36. M. J. Appel and J. C. Wierman, On the
  absence of infinite AB percolation clusters in
  bipartite graphs, J. Phys. A 20 (1987) 2527-2531
  MR 89b82061.
• 37. J. C. Wierman and M. J. Appel, Infinite AB
  percolation clusters exist on the triangular
  lattice, J. Phys. A 20 (1987) 2533-2537 MR
  89b82062.
• 38. E. R. Scheinerman and J. C. Wierman,
  Infinite AB percolation clusters exist, J. Phys.
  A 20 (1987) 1305-1307 MR 88e82018.
• 39. H. Nakanishi, Critical behavior of AB
  percolation in two dimensions, J. Phys. A 20
  (1987) 6075-6083.
• 40. J. C. Wierman, On AB percolation on
  bipartite graphs, J. Phys. A 21 (1988) 1945-1949
  MR 89m60259.
• 41. J. C. Wierman, AB percolation on
  close-packed graphs, J. Phys. A 21 (1988)
  1939-1944 MR 89m60258.
• 42. M. J. B. Appel, AB percolation on plane
  triangulations is unimodal, J. Appl. Probab. 31
  (1994) 193-204 MR 95e60101.

73
Other references

• 52. S. Mertens and M. E. Lautenbacher,
  Counting lattice animals A parallel attack, J.
  Stat. Phys. 66 (1992) 669 MR 92i82056.
• 53. A. G. Guttmann, On the number of lattice
  animals embeddable in the square lattice, J.
  Phys. A 15 (1982) 1987-1990.
• 54. D. A. Klarner, Cell growth problems,
  Canad. J. Math. 19 (1967) 851-863 MR 35 5339.
• 55. D. A. Klarner and R. L. Rivest, A
  procedure for improving the upper bound for the
  number of n-ominoes, Canad. J. Math. 25 (1973)
  585-602 preprint MR 48 1943.
• 56. B. M. I. Rands and D. J. A. Welsh,
  Animals, trees and renewal sequences, IMA J.
  Appl. Math. 27 (1981) 1-17 corrigendum 28 (1982)
  107 MR 82j05049 and MR 83c05041.
• 57. J. L. Cardy, Critical percolation in
  finite geometries, J. Phys. A 25 (1992)
  L201-L206 MR 92m82048.
• 58. G. M. T. Watts, A crossing probability for
  critical percolation in two dimensions, J. Phys.
  A 29 (1996) L363-L368 MR 98a82059.
• 59. D. W. Erbach, Advanced Problem 6229, Amer.
  Math. Monthly 85 (1978) 686.
• 60. V. Adamchik, S. Finch and R. Ziff, The
  Potts model on the square lattice (1996).
• 61. V. Adamchik, S. Finch and R. Ziff, The
  Potts model on the triangular lattice (1996).
• 62. R. Ziff, S. Finch and V. Adamchik,
  Universality of finite-size corrections to the
  number of critical percolation clusters, Phys.
  Rev. Lett. 79 (1997) 3447-3450 preprint.
• 63. R. J. Baxter, Potts model at the critical
  temperature, J. Phys. C 6 (1973) L445-L448.
• 64. R. J. Baxter, S. B. Kelland and F. Y. Wu,
  Equivalence of the Potts model or Whitney
  polynomial with an ice-type model, J. Phys. A 9
  (1976) 397-406.
• 65. H. N. V. Temperley, The shapes of domains
  occurring in the droplet model of phase
  transitions, J. Phys. A 9 (1976) L113-L117.
• 66. R. J. Baxter, H. N. V. Temperley and S. E.
  Ashley, Triangular Potts model at its transition
  temperature, and related models, Proc. Royal Soc.
  London A 358 (1978) 535-559 MR 58 20108.
• 67. H. N. V. Temperley, Lattice models in
  discrete statistical mechanics, Applications of
  Graph Theory, ed. R. J. Wilson and L. W. Beineke,
  Academic Press 1979, pp. 149-175 MR 81h05050.
• 68. H. N. V. Temperley, Graph Theory and
  Applications, Ellis Horwood Ltd., 1981 MR
  83d05001.
• 69. R. J. Baxter, Exactly Solved Models in
  Statistical Mechanics, Academic Press, 1982, pp.
  322-341 MR 90b82001.
• 70. N. Biggs, Algebraic Graph Theory,
  Cambridge Univ. Press, 2nd ed., 1993 MR
  95h05105.

74
Other references

• 78. J. C. Wierman, Substitution method
  critical probability bounds for the square
  lattice site percolation model, Combin. Probab.
  Comput. 4 (1995) 181-188 MR 97g60136.
• 79. B. D. Hughes, Random Walks and Random
  Environments, v. 2, Oxford, 1996 MR 98d60139.
• 80. I. Vardi, Prime percolation, Experim.
  Math. 7 (1998) preprint MR 2000h11081.
• 81. J. van den Berg and A. Ermakov, A new
  lower bound for the critical probability of site
  percolation on the square lattice, Random
  Structures and Algorithms 8 (1996) 199-212
  preprint MR 99b60165.
• 82. H. Kesten, On the non-convexity of the
  time constant in first-passage percolation, Elec.
  Commun. Probab. 1 (1996) 1-6 MR 98c60142.
• 83. S. E. Alm, A note on a problem by Welsh in
  first-passage percolation, Combin. Probab.
  Comput. 7 (1998) 11-15 MR 99a60112.
• 84. S. E. Alm and J. C. Wierman, Inequalities
  for means of restricted first-passage times in
  percolation theory, Combin. Probab. Comput. 8
  (1999) 307-315 MR 2000j60118.
• 85. R. Ahlberg and S. Janson, Upper bounds for
  the connectivity constant, unpublished (1981).
• 86. M. L. Glasser, V. Privman and N. M.
  Svrakic, Temperley's triangular lattice compact
  cluster model exact solution in terms of the q
  series, J. Phys. A 20 (1987) L1275-L1280 MR
  89b82107.
• 87. R. K. Guy, The strong law of small
  numbers, Amer. Math. Monthly 95 (1988) 697-712
  MR 90c11002.
• 88. A. M. Odlyzko and H. S. Wilf, The editor's
  corner n coins in a fountain, Amer. Math.
  Monthly 95 (1988) 840-843.
• 89. A. M. Odlyzko, Asymptotic enumeration
  methods, Handbook of Combinatorics, v. I, ed. R.
  Graham, M. Grötschel and L. Lovász, MIT Press,
  1995, pp. 1063-1229 MR 97b05012.
• 90. V. Adamchik, A class of logarithmic
  integrals, Proc. 1997 ISSAC Maui conf. preprint.
• 91. N. Madras, A pattern theorem for lattice
  clusters, Annals of Combin. 3 (1999) 357-384.
• 92. S. A. Zuev, A lower bound for a
  percolation threshold for a square lattice,
  Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1988
  59-61 MR 91k82035.
• 93. C. Moore and M. E. J. Newman, Epidemics
  and percolation in small-world networks, Phys.
  Rev. E 61 (2000) 5678-5682 preprint.
• 94. N. Madras, C. E. Soteros, S. G.
  Whittington, J. L. Martin, M. F. Sykes, S. Flesia
  and D. S. Gaunt, The free energy of a collapsing
  branched polymer, J. Phys. A 23 (1990) 5327-5350
  MR 91m82123.
• 95. R. T. Smythe, Percolation models in two
  and three dimensions, Biological Growth and
  Spread, Proc. 1979 Heidelberg conf., ed. W.
  Jäger, H. Rost and P. Tautu, Lecture Notes in
  Biomath. 38, Springer-Verlag, 1980, pp. 504-511
  MR 82i60183.
• 96. W. F. Lunnon, Counting hexagonal and
  triangular polyminoes, Graph Theory and
  Computing, ed. R. C. Read, Academic Press, 1972,
  pp. 87-100 MR 49 2439.

75
Other references

• 100. M. Bousquet-Mélou, Counting convex
  polyominoes according to their area (1991-1992
  INRIA lecture summary).
• 101. D. Gouyou-Beauchamps, Enumeration of
  remarkable families of polyominoes (1998 INRIA
  lecture).
• 102. M. Bousquet-Mélou, A method for the
  enumeration of various classes of column-convex
  polygons, Discrete Math. 154 (1996) 1-25 MR
  97g05007.
• 103. D. Hickerson, Counting horizontally
  convex polyominoes, J. Integer Seq. 2 (1999)
  99.1.8 MR 2000k05023.
• 104. D. Zeilberger, Symbol-crunching with the
  transfer-matrix method in order to count skinny
  physical creatures, preprint (2000).
• 105. A. J. Guttmann, I. Jensen, L. H. Wong and
  I. G. Enting, Punctured polygons and polyminoes
  on the square lattice, J. Phys. A 33 (2000)
  1735-1764.
• 106. I. Jensen and A. J. Guttmann, Statistics
  of lattice animals (polyominoes) and polygons, J.
  Phys. A 33 (2000) L257-263.
• 107. S. R. Broadbent and J. M. Hammersley,
  Percolation processes. I. Crystals and mazes,
  Proc. Cambridge Philos. Soc. 53 (1957) 629-641
  MR 19,989e.
• 108. J. M. Hammersley, Percolation processes.
  II. The connective constant, Proc. Cambridge
  Philos. Soc. 53 (1957) 642-645 MR 19,989f.
• 109. J. M. Hammersley, Percolation processes
  Lower bounds for the critical probability, Ann.
  Math. Statist. 28 (1957) 790-795 MR 21 374.
• 110. H. Kesten, The critical probability of
  bond percolation on the square lattice equals
  1/2, Comm. Math. Phys. 74 (1980) 41-59 MR
  82c60179.
• 111. J. C. Wierman, Bond percolation on
  honeycomb and triangular lattices, Adv. in Appl.
  Probab. 13 (1981) 298-313 MR 82k60216.
• 112. T. Mai and J. W. Halley, AB percolation
  on a triangular lattice, Ordering in Two
  Dimensions, ed. S. K. Sinha, North Holland, 1980,
  pp. 369-371.
• 113. J. M. Hammersley and D. J. A. Welsh,
  First-passage percolation, subadditive processes,
  stochastic networks, and generalized renewal
  theory, Proc. 1965 Berkeley Seminar,
  Springer-Verlag, pp. 61-110 MR 33 6731.
• 114. I. M. Gessel, On the number of convex
  polyominoes, Annales Sci. Math. Québec 24 (2000)
  63-66 preprint.
• 115. I. Jensen, Enumerations of lattice
  animals and trees, J. Stat. Phys., to appear.

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