1
Percolation, Cluster and
Pair Correlation Analysis (L22)
Texture, Microstructure & Anisotropy,
Fall 2009
A.D. Rollett, P. Kalu
Carnegie
Mellon
MRSEC
Last revised: 22nd Nov. ‘09
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 0-38795167-9).
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, <s>
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
<rk> := 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 (radial distribution
function/RDF in 1D)
 =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 let’s 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.
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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.
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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.
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Percolation Thresholds
Lattice
Honeycomb (6)
Site
0.6962
Bond
0.65271
Triangular (3)
Square (4)
Diamond (4)
Simple Cubic (6)
0.5
0.59275
0.428
0.3117
0.34729
0.5
0.388
0.2492
Bcc (8)
Fcc (12)
0.245
0.198
0.1785
0.119
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 are 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]
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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 by:
S = swss = s{nss2/ snss}.
• This definition of cluster size is also valid for higher
dimensions (d>1) 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, S, in terms of the
transition probability, p, significant work must be
done, even in 1D.
• The result, however, is very simple and elegant (p <
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.
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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 probability,
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-pc|<<1.
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:
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Higher Dimensions, contd.
• Note the appearance of an exponent “” 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, :
• 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 (fraction of sites in critical
network) and the deviation from the critical transition
probability, with a new critical exponent, :
• 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.
• The Bethe Lattice is a
special type of lattice (not
very realistic!).
http://www.ibiblio.org/e-notes/Perc/perc.htm
Note: a Bethe lattice or Cayley tree, introduced by Hans Bethe in 1935, is a connected cycle-free graph where each node is
connected to z neighbours, where z is called the coordination number. It can be seen as a tree-like structure emanating from a central
node, with all the nodes arranged in shells around the central one. The central node may be called the root or origin of the lattice.
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.
 := critical exponent on proportionality constant, c
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/ComPhys_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, , that a large enough cluster (of the right shape)
will occur in a given realization of a finite 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 (L<∞) 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 d/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”, :
• 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 (>1 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), see e.g.
http://www.ornl.gov/HTSC/htsc.html and
http://www.lanl.gov/orgs/mpa/stc/index.shtml .
• See http://www.amsc.com/ for engineering
applications.
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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°.
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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, <rk>, 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 <rk>
varies approximately as √k.
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 <rk>
• 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” (); in some texts this is written as <n>. The
critical assumptions that permit us to apply the basic theory for the Poisson
process are as follows.
1. The expected value, , multiplied by a given time interval, ∆t, is the probability,
∆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 - ∆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, , 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, . 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, <n> 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. Let’s define a density of
particles as d 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 <n> = NV Vd. From statistical
mechanics [e.g. 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  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.
Probability
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 (2 = 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 to Nucleation
•
•
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
k<10 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 Correlation Functions
• 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
Example from Al 7075, aerospace aluminum alloy
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
~10 particles per stringer
Long #6 PCF
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 NDTD section  correlation length = 35 µm // TD;
similarly, 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 position, ri, 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, ppt2, at a separation
r, given a particle, ppt1, already located at the
tail of r. Given a spherical shell of area s(r),
this probability, p, is:
59
Radial Distribution
• Radial Distribution
Function, g(r) Function (RDF)
• Basic idea: calculate the probability of finding another grain center
at a given (1D, radial) distance from a given grain center.
• Useful in atomic physics because it is a reasonably sensitive
measure of both solid and liquid structure and is easily measured.
Also useful for developing interatomic potentials.
• The example below shows that the RDF is particularly simple
when particles are arranged on a regular lattice (simple cubic),
with a=20 voxels.
2nd=√a
lattice
spacing
1st=a
lattice
spacing
3rd=(3).5a
60
PCF Analysis, example
• 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).
• The example below is of a radial correlation function for a system
of disordered interacting particles, in a liquid-like state (where the
interaction occurs 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]
61
2-point Probability Function
•
•
•
•
The pair correlation function and
conditional probability function discussed
previously are closely related to the 2point 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)
63
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.
64
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.
65
Sections through 3D Image
66
Generated Particle Structure:
Sections
Ellipsoids were inserted into the
domain with a constant aspect
ratio of a:b:c = 3:2:1. The target
correlation length was 0.07x400 =
28, with 10 particles per colony
Rolling plane (Z) - Transverse (X) - Longitudinal (Y)
67
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)
68
2D section size distributions
10i05
• A comparison of the shapes of
ellipses shows reasonable
agreement between the fitted set
of ellipsoids and initial crosssection statistics (size
distributions)
0.7
B/X/Istat
A/Y/Istat
A/Z/Istat
C/X/Istat
C/Y/Istat
B/Z/Istat
0.6
Initial Statistics
0.5
0.4
0.3
0.2
0.1
10i05 Final Ellipse Statistics
10i05 Initial Ellipse Statistics
0.7
0.7
p1=B X section
p1=A Y section
p1=A Z section
p2=c X section
p2=C Y section
p2=B Z section
Frequency
0.5
0.4
0.3
0.6
0.5
Frequency
0.6
0
0
p1=B X section
p1=A Y section
p1=A Z section
p2=C X section
p2=C Y section
p2=B Z section
0.4
0.2
0.1
0.1
0
2
4
6
Size
8
10
0.3
0.4
0.5
0.6
Cross-plot
0
0
0.2
Final Ellipse Statistics
0.3
0.2
0.1
0
2
4
6
8
10
Size
Initial vs. Final section distributions
0.7
69
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.
10i05:Ellipsoids:Fitted
10i051:Ellipsoids:Injected
0.4
0.4
A
B
C
0.35
0.3
Frequency
Frequency
0.3
0.25
0.2
0.25
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
2
4
6
Size
8
10
A
B
C
0.35
12
0
2
4
6
Size
8
10
12
70
Comparison of PCFs for Original and
Reconstructed Particle Distribution
From CA
Reconstructed
Rolling plane (Z) - Transverse (X) - Longitudinal (Y)
71
Reconstructed 3D particle distribution
72
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.
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