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CarnegieMellon

MRSEC

Percolation, Cluster and ���Pair Correlation Analysis (L22)

Texture, Microstructure & Anisotropy, Fall 2009

A.D. Rollett, P. Kalu

Last revised: 22nd Nov. ‘09

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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).

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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).

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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.

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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. ξ := correlation length r := radius, distance between sites. g(r) := correlation function

(connectivity function).

c := proportionality constant. τ := critical exponent on cluster size, s σ := critical exponent on proportionality

constant, c P := fraction of sites in the critical

(infinite size) network, or “power”. β := critical exponent on size of critical

network γ := critical exponent on average cluster

size, S L := size of finite system. Π := probability that a finite system will

percolate pav := average percolation threshold in a

finite system ν := critical exponent on average

threshold probability, pav

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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 α , or, λ := expected value (e.g. a

density) ∆t := time (space) interval X := number of objects for

evaluation of a cumulative probability

d:= dimensionality Γ := 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

ρn(rN) := generic n-particle probability density function

gn(rn) := n-particle correlation function (radial distribution function/RDF in 1D)

ρ =NV := number density of particles

g2(r12) := 2-particle or pair correlation function

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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.

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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 Site Bond Honeycomb (6) 0.6962 0.65271 Triangular (3) 0.5 0.34729 Square (4) 0.59275 0.5 Diamond (4) 0.428 0.388 Simple Cubic (6) 0.3117 0.2492 Bcc (8) 0.245 0.1785 Fcc (12) 0.198 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

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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 = Σsnss.

•  [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 / Σsnss.

•  Thus, the average cluster size, S, is given by: S = Σswss = Σs{nss2/ Σsnss}.

•  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.

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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, ξ •  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 ξ is the correlation length:

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Correlation Length, ξ, contd.

•  The correlation length below the transition is given by:

•  Interestingly, in 1D it is proportional to the cluster size, S ∝ ξ . In higher dimensions, the relationship is more complex.

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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 “τ” 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, σ:

•  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:

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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, β:

•  Finally, we find that for the average cluster size, S:

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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.

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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.

σ := critical exponent on proportionality constant, c

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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.

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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.

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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, Π, 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 Π(L<∞) has a finite slope over an appreciable range of p. The dashed line shows the probability density for the same quantity.

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Approach to Critical Point •  If one considers how to extract the critical probability, one

approach is to seek the inflection point in dΠ/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”, ν:

•  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.

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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).

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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.

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.

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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 ∆θ≥1° in (d). The next column shows a percolation map in (b) such that connected points are shown in a single color, with boundaries ∆θ≥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 ∆θ≥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.

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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.

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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 (ζ = 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.

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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).

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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” (α); 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, α, multiplied by a given time interval, ∆t, is the probability, α∆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 - α∆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.

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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, α, as follows.

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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, α. 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.

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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.

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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.

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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 λd 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 Γ is the gamma function:

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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:

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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.

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Nearest Neighbor Distances, contd.

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

(1D)

(2D)

(3D)

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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 (λ2 = 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.

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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.

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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.

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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.

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Pair Correlation: example

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)

Strict definition: conditional 2-point probability

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PCF: correlation lengths

ND / S TD / T

Tran #1 PCF Tran #1

Optical image of transverse plane (S-T) PCF of transverse plane

Example from Al 7075, aerospace aluminum alloy

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PCF: correlation length/ longitudinal

RD / L ND / S

Long #6 PCF

Long #6

Optical image of longitudinal plane (L-S) PCF of longitudinal plane

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PCF Analysis Mg2Si

BSE image PCF 139X139 µm

There is no correlation between the placement of Mg2Si particles

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PCFs on Orthogonal Planes

•  2.64 µm per pixel in the images; ratio of lengths in ND-TD section ⇒ correlation length = 35 µm // TD; similarly, correlation length = 130 µm // RD.

ND TD

RD

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PCF Analysis •  Given N particles in a given volume, one can

introduce a generic n-particle probability density function, ρn(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…=Πidri. Normalization means that:

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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.

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PCF Analysis, contd.

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

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

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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.

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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:

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•  Radial Distribution Function, g(r) •  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.

Radial Distribution Function (RDF)

1st=a

2nd=√a

3rd=(3).5a

lattice spacing

lattice spacing

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.

[Torquato]

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

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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)

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1283 grid

•  Microstructures generated by (kinetic) Monte Carlo grain growth simulation (isotropic boundaries), T=1.5, with periodic boundary conditions.

•  Snapshots at 105 and 106 MCS have about 7,000 and 450 grains, respectively, and sphere equivalent diameters of 8 and 20.5 pixels, respectively.

•  RDF calculated from the grain centers (of gravity). •  The first peak occurs just below the average diameter, with a

shallow minimum at <D>sphere. Recall that the grain size distribution peaks to the left of the average (median < mean).

•  RDF similar to that of liquid.

← Truskett et al. Phys Rev. E, 58

Microstructure from simulation, ~450 grains, “ph16”

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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.

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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)

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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)

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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)

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10i05 Initial Ellipse Statistics

p1=B X sectionp1=A Y sectionp1=A Z sectionp2=c X sectionp2=C Y sectionp2=B Z section

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uenc

y

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10i05 Final Ellipse Statistics

p1=B X sectionp1=A Y sectionp1=A Z sectionp2=C X sectionp2=C Y sectionp2=B Z section

Freq

uenc

y

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Cross-plot

Initial vs. Final section distributions

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.

0.05

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ABC

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70 Comparison of PCFs for Original and Reconstructed Particle Distribution

Rolling plane (Z) - Transverse (X) - Longitudinal (Y)

From CA

Reconstructed

71

Reconstructed 3D particle distribution

72

•  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.

Summary

73

Other references 1. D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed., Taylor and Francis, 1992; MR 87k:82093. 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 90j:60109. 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 96h:05001. 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 90g:62001a. 8. R. T. Smythe and J. C. Wierman, First-Passage Percolation on the Square Lattice, Lecture Notes in Math. 671, Springer-Verlag, 1978; MR 80a:60135. 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. 20. M. F. Sykes, D. S. Gaunt and M. Glen, Perimeter polynomials for bond percolation processes, J. Phys. A 14 (1981) 287-293; MR 81m:82045. 21. A. Margolina, H. Nakanishi, D. Stauffer and H. E. Stanley, Monte Carlo and series study of corrections to scaling in two-dimensional percolation, J. Phys. A 17

(1984) 1683-1701. 22. D. C. Rapaport, Monte Carlo experiments on percolation: The influence of boundary conditions, J. Phys. A 18 (1985) L175-L179. 23. D. C. Rapaport, Cluster size distribution at criticality, J. Stat. Phys. 66 (1992) 679-682.

74

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 92f:82027. 25. J. Güémez and S. Velasco, A probabilistic approach to the site-percolation problem, II: Standard lattices, Physica A 171 (1991) 504-516; MR 92f:82028. 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 86f:60118. 29. S. A. Zuev, Bounds for the percolation threshold for a square lattice, Theor. Probab. Appl. 32 (1987) 551-553; MR 89h:60171. 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 91h:60116. 31. T. Luczak and J. C. Wierman, Critical probability bounds for two-dimensional site percolation models, J. Phys. A 21 (1988) 3131-3138; MR 89i:82053. 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 86j:

60222. 33. B. Bollobás and Y. Kohayakawa, Percolation in high dimensions, Europ. J. Combinatorics 15 (1994) 113-125; MR 95c:60092. 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 89b:82061. 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 89b:82062. 38. E. R. Scheinerman and J. C. Wierman, Infinite AB percolation clusters exist, J. Phys. A 20 (1987) 1305-1307; MR 88e:82018. 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 89m:60259. 41. J. C. Wierman, AB percolation on close-packed graphs, J. Phys. A 21 (1988) 1939-1944; MR 89m:60258. 42. M. J. B. Appel, AB percolation on plane triangulations is unimodal, J. Appl. Probab. 31 (1994) 193-204; MR 95e:60101. 43. M. J. B. Appel and J. C. Wierman, AB percolation on bond-decorated graphs, J. Appl. Probab. 30 (1993) 153-166; MR 94a:60133. 44. S. Janson, An upper bound for the velocity of first-passage percolation, J. Appl. Probab. 18 (1981) 256-262; MR 82c:60178. 45. D. J. A. Welsh, An upper bound for a percolation constant, Z. Angew Math. Phys. 16 (1965) 520-522. 46. J. van den Berg and H. Kesten, Inequalities for the time constant in first-passage percolation, Ann. Appl. Probab. 3 (1993) 56-80; MR 94a:60134. 47. J. A. Fill and R. Pemantle, Percolation, first-passage percolation and covering times for Richardson's model on the n-cube, Ann. Appl. Probab. 3 (1993) 593-629; MR 94h:

60145. 48. D. H. Redelmeier, Counting polyominoes: Yet another attack, Discrete Math. 36 (1981) 191-203; MR 84g:05049. 49. S. G. Whittington and C. E. Soteros, Lattice animals: Rigorous results and wild guesses, in Disorder in Physical Systems: A Volume in Honour of J. M. Hammersley, ed.

G. R. Grimmett and D. J. A. Welsh, Oxford, 1990; pp. 323-335; MR 91m:82061. 50. V. Privman and N. M. Svrakic, Difference equations in statistical mechanics, I: Cluster statistics models, J. Stat. Phys. 51 (1988) 1091--1110; MR 89i:82079. 51. S. Mertens, Lattice animals: A fast enumeration algorithm and new perimeter polynomials, J. Stat. Phys. 58 (1990) 1095-1108; MR 91d:82021.

75

Other references 52. S. Mertens and M. E. Lautenbacher, Counting lattice animals: A parallel attack, J. Stat. Phys. 66 (1992) 669; MR 92i:82056. 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 82j:05049 and

MR 83c:05041. 57. J. L. Cardy, Critical percolation in finite geometries, J. Phys. A 25 (1992) L201-L206; MR 92m:82048. 58. G. M. T. Watts, A crossing probability for critical percolation in two dimensions, J. Phys. A 29 (1996) L363-L368; MR 98a:82059. 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 81h:05050. 68. H. N. V. Temperley, Graph Theory and Applications, Ellis Horwood Ltd., 1981; MR 83d:05001. 69. R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, 1982, pp. 322-341; MR 90b:82001. 70. N. Biggs, Algebraic Graph Theory, Cambridge Univ. Press, 2nd ed., 1993; MR 95h:05105. 71. F. Y. Wu, The Potts model, Rev. Mod. Phys., 54 (1982) 235-268; MR 84d:82033. 72. H. Kunz and F. Y. Wu, Site percolation as a Potts model, J. Phys. C 11 (1978) L1-L4. 73. H. N. V. Temperley and S. E. Ashley, Site percolation problems and multi-site Potts models, J. Phys. A 15 (1982) 215-222; MR 82k:82030. 74. M. L. Glasser, D. B. Abraham and E. H. Lieb, Analytic properties of the free energy for the "ice" models, J. Math. Phys. 13 (1972) 887-900; MR 55 #7227. 75. S. W. Golomb, Polyominoes, 2nd ed., Princeton Univ. Press, 1994; MR 95k:00006. 76. D. Eppstein, Polyominoes and Other Animals (Geometry Junkyard). 77. M. P. Delest, Polyominoes and animals: some recent results, J. Math. Chemistry 8 (1991) 3-18; MR 93e:05024.

76

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 97g:

60136. 79. B. D. Hughes, Random Walks and Random Environments, v. 2, Oxford, 1996; MR 98d:60139. 80. I. Vardi, Prime percolation, Experim. Math. 7 (1998); preprint; MR 2000h:11081. 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 99b:60165. 82. H. Kesten, On the non-convexity of the time constant in first-passage percolation, Elec. Commun. Probab. 1 (1996) 1-6; MR 98c:60142. 83. S. E. Alm, A note on a problem by Welsh in first-passage percolation, Combin. Probab. Comput. 7 (1998) 11-15; MR 99a:60112. 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

2000j:60118. 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 89b:82107. 87. R. K. Guy, The strong law of small numbers, Amer. Math. Monthly 95 (1988) 697-712; MR 90c:11002. 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 97b:05012. 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 91k:82035. 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 91m:82123. 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 82i:60183. 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. 97. E. A. Bender, Convex n-ominoes, Discrete Math. 8 (1974) 219-226; MR 49 #67. 98. D. A. Klarner and R. L. Rivest, Asymptotic bounds for the number of convex n-ominoes, Discrete Math. 8 (1974) 31-40; MR 49 #91. 99. D. A. Klarner, Polyminoes, Handbook of Discrete and Computational Geometry, ed. J. E. Goodman and J. O'Rourke, CRC Press, 1997, pp. 225-240.

77

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 97g:

05007. 103. D. Hickerson, Counting horizontally convex polyominoes, J. Integer Seq. 2 (1999) 99.1.8; MR 2000k:05023. 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 82c:60179. 111. J. C. Wierman, Bond percolation on honeycomb and triangular lattices, Adv. in Appl. Probab. 13 (1981) 298-313; MR 82k:60216. 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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