Transport through bootstrap percolation clusters

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Transport through bootstrap percolation clusters

Muhammad Sahimi, Tane Ray

To cite this version:

Muhammad Sahimi, Tane Ray. Transport through bootstrap percolation clusters. Journal de Physique

I, EDP Sciences, 1991, 1 (5), pp.685-692. �10.1051/jp1:1991162�. �jpa-00246363�

Classification

Physics

Abstracts

05.70J 62.20M M.60A

Transport through bootstrap percolation clusters

Muhammad Sahimi

(*)

and Tane S.

Ray (**)

Supercomputer

Center HLRZ,

c/o

KFA

Jfilich,

P.O. Box 1913, D-5170 Jfilich I,

Germany

(Received 2January

1991,

accepted

J7January

J99J)

Abs«act. In

bootstrap percolation (BP)

_on lattices sites _are

initially occupied

at random. Those

occupied

sites that do not have at least _m

occupied nearest-neighbors

_are then removed. For

sufficiently large

values of _m

(e.g.,

_{m m} 4 for the cubic

lattice)

first-order

phase

transitions _occur at the

percolation

threshold, _p~, while for small values of _m the

phase

transition is second-order.

We

study conductivity

of BP clusters as a function of _m, the

dimensionafity

of the system and its linear size L. This is relevant to

spin-wave

stiffness of disordered

magnetic

systems, e-g-, the dilute

Blume-Capel

model and, _{as we argue} here, it _may also be relevant to the behavior of disordered solids that

undergo

a brittle fracture _process, and to flow

through

_{a porous} medium.

On _a cubic lattice _we find that the

conductivity

critical exponent t for _m

=

3 is the _{same as} that of random

percolation (m

₌

0).

Since for _m

= 0-3 the correlation

length

_exponent also remains

unchanged,

but the critical exponent p of the

strength

of the infinite clusters is different for

m =

2 and 3, we argue that this indicates that for three-dimensional systems t cannot be related to

p. For _mm 4, the

conductivity

is discontinuous at p~, followed

by

_a

power-law jump,

_as the fraction of

conducting

^material is

increased,

with _a critical exponent ^{that is}

apparently

different from t.

1. Introduction.

For the

past

two decades

percolation

networks

ill

have been _an

important

tool for

investigation

of structural

phase

transitions and

transport

_processes in disordered

systems,

such _as porous

media, polymer gels, composite

solids and

magnetic alloys. Depending

_on the

specific application

of

percolation theory

to _a

problem

of

interest,

_many variants of the classical

(random) percolation

_process have been

developed [2]. Among

these _are continuous

percolation,

correlated

percolation,

directed

percolation, polychromatic percolation

and

bootstrap percolation (BP).

It is this last variant of

percolation

that is of

prime

interest in this paper.

(*)

Present and permanent address _:

Department

of Chemical

Engineering, University

of Southem Cafifomia, Los

Angeles,

Califomia

90089-1211,

U-S-A-

(**)

Present address _:

Department

of

Physics,

Clarkson

University, Potsdam,

New York 13676, U.S.A.

686 JOURNAL DE

PHYSIQUE

I N 5

In

BP,

sites _{on a} lattice _are

randomly occupied

with

probability

_p and then each site with fewer that _m

occupied neighbors

is rendered

unoccupied.

This

culling

_process is

repeated

until

a stable

configuration

of

occupied sites,

each with _{m or more}

occupied neighbors,

is obtained

or until _every site of the lattice is vacant, This

problem

_was first studied

by

Pollak and Riess

[3]

and

Chalupa,

Leath and Reich

[4].

^The

original

motivation for the invention of BP _was to

explain

the behavior of _some disordered

systems

^{in which}

magnetic impurities

_are

randomly

distributed in _a host of

non-magnetic

metals. In _some of such systems, e.g., the dilute Blume-

Capel model,

it is believed that _an

impurity

atom cannot sustain _a localized

magnetic

moment unless it is surrounded

by

_a minimum number of

magnetic neighbors [5],

But BP models have also been

applied

to the

study

of orientational order in

orthoparahydrogen

mixtures

[6]

and to the

dynamics

of

glass

transitions

[7j. They

_are also relevant to the

galactic spiral

structures

[8].

In BP the

percolation

threshold _{p~ is} defined to be the critical value of p below which _no

sample-spanning

cluster is found in the infinite

system

in the limit of

infinite

time. For _any value of _m, the value _{of p~ is}

larger

than _or

equal

to the threshold of random

percolation

in the

same lattice. With _m

=

0 _one has the usual random

percolation,

whereas with _m

=

I,

isolated sites and for _m

=

2 the

dangling

sites of

percolation

clusters _are removed. For _m

=

I and 2

one expects ^{the value of} p~ ^to remain _as in random

percolation

_on all lattices. For

m ₌ I critical exponents

characterizing

various

percolation properties

_{near p~ are} the _{same as} those for _m

= 0. Adler and co-workers

[9-11]

have reviewed various

properties

of BP and have classified them

nicely.

In

particular, Adler,

Stauffer and

Aharony [10]

showed that for _a

d-dimensional lattice of linear dimension L and _mm 4 _one has

lnL

=

~

+

~lnp~+A, _(I)

Pi

^~

where _a

=

(d- 1)~

_~, A

=

a~(d-

I In 2 da

),

A

= In 16

~'~a~ _~(2 gr)~~+ _~],

and _a and b 2

are constants of order

unity,

_so that _p~

- I _as L

- co.

Moreover,

for

large

values of _m

(e.g.,

m = 3 in two dimensions and _{m m} 4 in three

dimensions)

_one has _a first-order

phase

transition

at p~. Adler and Stauffer

ill] recently

studied BP _{on a}

simple-cubic

lattice with

m =

3,

and estimated _p~

= 0.5717 _± 0.0005 and

p

=

0.6 _± 0.

I,

where

p

is the critical

exponent

^{of the}

strength

of the

percolation

cluster

S~p)

_{near p~.} These should be

compared

with p~ w 0.3116 and

p

w 0,4 for random

percolation

_on the _same lattice.

Complete

details

are

given

in _a recent review

by

Adler

[12].

In the

present

_paper we

study

transport processes

through

BP clusters in _two- and three- dimensional lattices.

Specifically,

_we consider conduction processes in BP clusters and

study

the behavior of the

macroscopic conductivity

_« of the system as m and the

dimensionality

of the lattice _are varied. There _are several _reasons

why

such _a

study

_can be useful and

important.

(I) Transport

_processes in disordered systems are sensitive to the

connectivity properties

of the

system. Varying

_m allows _one to

change

the

connectivity properties

of the

system

^and

study

their effect _on the conduction _process,

(2)

An unsolved

problqm

in

percolation

_processes

[13]

^{is the}

possible

relation between the

conductivity

critical exponent t defined

by

«~p)

_{~p p}

c)' (2)

and the

geometrical exponent

of

percolation

such _{as v,} the correlation

length exponent, p

and

so on. We shall show that _our

study provides insight

into whether _t

might (or might not)

be related to v,

p

and any other

exponent.

(3)

An

important property

^{of disordered}

magnetic

systems ^is their

macroscopic spin-wave

stiffness

D~p). According

to Last

[14]

and

Kirkpatrick [15] D~p)

and

«~p)

_are related

through

the

following relationship

@=s~p)@. ₍₃₎

Since BP _was

originally

invented _to

explain

the unusual

properties

of _some

magnetic _systems,

our

study provides

the first calculations of the

spin-wave

stiffness of such systems.

(4)

As mentioned

above,

for

large

values of _m the

geometrical phase

transition at

p~ is first-order. On the other

hand,

_one also observes first-order

geometrical phase

transitions in _some of the recent models of fracture of disordered solids

[16, 17].

As _we shall _argue

below,

the behavior of

conductivity

of BP clusters for

large

values of _m is very similar to that of

transport properties

of fractured systems, so that _our

study

_may

provide

_some

insight

into the behavior of fractured

systems.

(5) Finally,

_a variant of BP _was used

by

Lenormand and Zarcone

[18]

to model flow in porous media.

Therefore,

_our

study provides

estimates of

conductivity

_or fluid

permeability

of such model

systems,

^In

fact,

in BP the

shape

of the

sample-spanning

clusters of

occupied

sites and that of the holes that _are created

by

the

empty sites, through

which _no transport takes

place,

_are

similar, respectively,

to those of the open pore space and the solid matrix of the medium in the model

system

^{that Rothrnan}

ii 9]

^{used in his cellular} automata simulation of flow in porous media

(see below).

With these

motivations,

we now

proceed

to describe _our

simulations and the results.

2.

Description

of shnnlations.

The BP clusters _were

generated according

to their standard definition

[10]. Initially,

sites _are

occupied

with _a

probability

_p.

Next,

sites _are

successively

removed if

they

have fewer that _m

occupied neighbors

until the system reaches _a fixed

configuration. Depending

_upon the value of _m and the type or size of the

lattice,

the

resulting configuration

_can be either _a

spanning

cluster,

a group of small clusters which do _{not span or an}

empty

^lattice.

If _a

spanning

cluster is

generated,

the

conductivity

of the cluster is measured

according

to the

following procedure.

The

occupied

sites in the cluster _are assumed _as

being

nodes

through

which _{current may} flow. The current can pass between

neighboring occupied

sites

through

_a

resistor of conductance _g. In this

work,

_{g was} set

equal

to I for all such resistors. The current

Q;y through

_a

given

resistor is

linearly dependent

_upon the

potential

difference between the two sites I and

j.

^{Since the} total current

reaching

.a node is

conserved,

one has

z _Q;j

₌ ^°

,

(4)

where

(I)

denotes the set of

nearest-neighbor

sites of I. This leads to a set of N

coupled,

linear

equations

in the nodal

potentials

where N is the number of sites in the BP cluster. As the

boundary

conditions _we

applied

_a nixed

potential gradient

_across the whole cluster in _a

given

direction. This _was

accomplished by connecting

all sites _on either side of the cluster to

planes

of sites at _a fixed

potential.

A standard

conjugate gradient

method _was used to solve the

equations. Finally,

the total _current

through

the cluster _was measured and the

macroscopic conductivity

_{« was} calculated. The

conductivity

_was

always

normalized

by

its value for _a

completely

connected cluster

~p

₌ ^I

).

For _a

large enough lattice,

the result is

independent

of the size of the cluster.

688 JOURNAL DE PHYSIQUE I N 5

The simulations _{were run on} the

Cray-YMP

supercomputer ^located at KFA Jfilich. Helical

boundary

conditions _were used both to

generate

the BP clusters and in the calculation of _«.

We used square and

simple-cubic

lattices in _our simulations. For the _m

= 5 _{case on} the cubic lattice and the _m

=

3 _{case on} the square

lattice,

it _{was necessary} to

generate

at least

200dusters for each value of p in order to obtain reasonable statistics. For the

m =

4 _{case on} the cubic

lattice,

100 clusters _were

generated.

For the _m

=

2 _{case on} the cubic

lattice,

10 dusters _were

generated.

All of the

conductivity

_{averages were} done _over those realizations of clusters which

spanned

the

system

; if the cluster did not span the

system

^the

run was discarded.

3. Results and discussion.

A

typical

BP cluster _on the square lattice is shown in

figure

I. It _was obtained with p = 0.939 and _m

= 3. As _can be _seen,

rectangular

islands of

empty

sites _are formed in _an

ocean of

occupied

nodes. If _we

imagine

that the islands _are part ^{of the solid matrix of} a porous

medium

(through

which _no fluid flow takes

place),

and the

occupied

sites make up the _open pore space of the

medium, through

which _a fluid

flows,

the system would be _very similar to the model of porous media used

by

RothJnan

[18]

in his simulation of fluid flow in porous

media.

Thus,

the system can be _a useful model of disordered _porous

media,

at least those that have

relatively high porosities.

j ^~

j , ^.

'~ ,

~.,"

.

'

, ,

,

o ~

,',

m . , ,o;

Fig.

I. A BP cluster _on the square lattice vith _p

= 0.939 and _m

= 3. Solid

rectangles

_are the empty space.

In

figure

2 _we show how the

conductivity

of the BP cluster in the square lattice with

m ₌ 3 varies with the

occupation probability

_p, for _two different lattice

sizes,

L

= 45 and 75.

Two

points

_are worth

noting. First,

_as L

increases,

the

conductivity

_curve terminates at a

higher

value of p, consistent with the fact that for m ₌

3,

one must have p~- I _as L- _co. In

fact,

for

L=75,

the

conductivity

curve terminates _{at p}

m0.94, (I.e., p~(L=75) w0.94),

which is consistent with

equation (I). Secondly,

for

L=75,

the

conductivity

of the BP cluster

just

^above the effective _{p~ is} still

relatively large (about 0.7),

but it

discontinuously

vanishes at p = p~, consistent with the fact that the BP transition _on the

square lattice with _m

= 3 is first-order. Had _we calculated the

conductivity

of the BP cluster for still

larger

values of

L,

_we would have found that the

conductivity

vanishes at still

higher

values of p, and that _{a very} small fraction of

unoccupied

sites is

enough

to make the

conduction _{process cease} to exist. This

dependence

of _won L and the fact that for

large

values of L

conductivity

of the

system

^is

large just

above _p~ is

completely

similar to the behavior of _a

disordered solid which is

undergoing

_a brittle fracture process. ^In such _{a process}

[16, 17],

^the transport and mechanical

properties

of the system

change

_very

little,

_{as a} crack is initiated and grown in the

system,

^{and do} not indicate that the

macroscopic

failure of the system ^is

imminent.

However,

after _a small fraction of the

microscopic

elements of the

system (whose

value

depends

_on the

sample size)

fails under _an extemal stress or

strain,

a

single macroscopic

fracture is

formed,

the

system

fails and

transport

^{and mechanical}

properties suddenly vanish, I.e.,

the

system experiences

_a first-order

phase

transition. This is

completely

similar to the behavior of the

conductivity

of BP

clusters,

for

large

values of _m, found here. This is also

supported by

_a recent result

[20]

that shows that there _are certain relations between

percolation

processes and brittle fracture in disordered solids.

The variations of the

conductivity

of the cubic lattice with the

occupation probability

_p, for L

= 25 and _m

=

2 and

3,

_are shown in

figure

3. It is clear that for both values of _m the

conductivity

transition is continuous and second-order.

Moreover,

_as this

figure indicates, p~(m

=

2

)

>

0.31 and

p~(m

=

3

)

w

0.57,

consistent with the available data

[I1, 12].

To check the effect of

sample

size _on the

results,

_we also calculated the

conductivity

of the cubic

1. i

i

i~

i

~

jz

~

^l

jll

~

g

^~~ _~~

i

~

n

fl

~

Z

o '

8

~x~ ~

x &

x

, x

x

& x

~

a

~~.9

0.92 0.94 0.96 0.98 1.

~

0 0.2 0.4 0.6 0.8 1.

P P

Fig.

2.

Fig.

3.

Fig.

2.-

Conductivity

of BP clusters _versus

occupation probability

_p in the square lattice with

m =

3, where

(x)

_are the results for L

= 45, while

(/L )

_are those for L

= 75.

Fig.

3.-

Conductivity

of BP clusters _versus

occupation probability

_p in the cubic network of size L

=

25, where

(/L

) are the results for _m

= 2, while

(x)

are those for _m

=

3.

690 JOURNAL DE PHYSIQUE I M 5

network for _m

= 3 and L

=

?5,

the results of which _are shown in

figure

4. A

comparison

between

figures (3)

and

(4)

indicates _no

major sample

size ef§ect. This is

typical

of systems that

undergo

_a second-order

phase

transition.

Figure

5 presents the

conductivity

of the lattice for L

= 25 and _m

= 4 and 5.

Now,

the

behavior of the

system

^is

totally

dif§erent from that shown in

figure

4 for _m

=

3. The

conductivity undergoes

a first-order

phase

transition

and,

_as

figure

5

indicates,

it

abruptly

vanishes at p~ ₌ 0.87 for _m

=

4 and p~ = 0.98 for _m

=

5. To

investigate

the effect of

sample size,

_we

repeated

the

computation

for _m

= 5 and L

= 45. In

figure

6 _we compare the results for L

= 25 and 45. As this

figure indicates, p~(L

= 45

=

0.988,

and

although just

above this p~ the

conductivity

of the

system

is _very

large (about

96 fib of its value at p =

I),

it

abruptly

vanishes at p~. Had _we not known about the nature of

phase

transition in this system ^{and the} value of its p~, we could not have

predicted

that the

system

^would

undergo

such _a sudden

conductor-insulator transition and that its failure is imminent. As discussed

above,

this

qualitative description

of

conductivity phase

transition in BP is

completely

similar to the

catastrophic

failure of disordered solids in _a brittle fracture _process. In

fact,

if instead of _awe calculate the elastic moduli of BP clusters in this system, ^{the results} will be

essentially

identical with those of _a solid in brittle fracture.

i i.

A A A A

O-9

~

~

~

A

~

~

j

~ ~

O ,

Q Q ~~

~

~

~ ~

fl

^Q ~

Z ,

8

'

8

O.7

~ O

O ~

. ~

O

~O

0.2 DA O.6 0.8 1.

~~.86

O.888 O.916 0.944 0.972

P P

Fig. 4.

Fig.

5.

Fig.

4.-

Conductivity

of BP clusters _versus

occupation probability

_p in the cubic network for

m ₌ 3 and L

= 75.

Fig.

5.-

Conductivity

of BP clusters _versus

occupation probability

_p in the cubic network for

m =

3 and L

= 75, where

(/L )

_are the results for _m

=

4, while (x) are those for _m

=

5.

We should

point

out that

Kogut [21]

^{has studied}

microscopic conductivity

_«~ of BP clusters

on a Bethe lattice of coordination number _z, which is the mean-field limit of

percolation.

Kogut

shows that for 3

~ m « z

-1,

_«~

undergoes

_a first-order

phase

transition and _near PC

«~ = c + e

~p p~)~ (5)

where _p

=

1/2

and _c and _{e are}

z-dependent

constants.

Straley (as quoted

in

Kogut [21])

has shown that for

macroscopic

_«

conductivity (which

is the

quantity

we calculate

here)

of the

Bethe lattice the _same type of transition

(unlp discontinuity

followed

by square-root

power- law increase _{as p} is

increased)

exists. It would be

interesting

to determine the value of the

exponent

_p of

equation (5)

for

macroscopic conductivity

of three-dimensional lattices for those values of _m for which there is _a first-order

phase

transition.

Finally,

_we estimated the

conductivity

critical exponent t

(Eq. (2))

for the cubic lattice and

m ₌ 3. Since _a BP cluster has to go

through

a

culling

process before it reaches _a stable

configuration,

it would be

extremely

difficult to _use finite-size

scaling [22]

to determine t.

Thus,

in

figure

7 _we

plot

_{« versus}

8p

_{= p} p

~,

for L

=

75,

where _we take _p~

= 0.57175 from Adler and Stauf§er

[I I].

A fit of the

data,

_«

8p

^,

yields

t n~

2,

and since the size of the lattice used is

large,

we believe this estimate oft is reliable. Our estimate of t is consistent with the value of t for random

percolation [23, 24],

t

=

2-2.05,

which indicates that for _mm 3 the

universality

class of _« does _not

change.

I

li'

~a ,

~a xa

a o

a~

x o

~a

, o

x x,

, xa o

, x

Q

Q i3

i~ ~

C4 ^~

~

,

$$ O

O

x Q ^'

Q

x

o

p DELTAP

Fig. 6.

Fig.

6.- onductivity of

m

=

5,

_where (xl are _the

results for

_L = 25, while (/L )

Fig. 7.- of

BP

_{clusters versus} 8p = p-

p~

in the

cubic for m = 3

and

L

=

75,

where p~ = _0.5717.

Our oft for _m

= 3

between t and the _of perco14tion such

as v

and _fl.

According to

dler

and [11] on

a

_cubic

lattice

and

for _{m =} 0,

1, 2,

and 3,

nchanged, whereas

the

_{fl (m} = 0) = p _(m

=

I ) =

p _(m = 2) # _p (m

three-dimensional systems t

is

ndeed

related

to

the xponents of it

cannot

be

to p, since for m

= ₂

and

3,

v

least _partly because _{of the} fact that _in

a system_the of he entire

sample-spanning

cluster

is the_same as that of _its

backbone, since

^the

dangling

_ends do not

contribute

to _the flow of

current.

Thus, at _least

for

three-dimensional systems,

any

relation

that

relates

t to the geometrical _exponents of percolation should be invariant under the

ransformation,

cluster-backbone- the _exponent

v _does

_obey

such

a

one

has p

(cluster

)

_# p

(backbone ). _If

this

692 JOURNAL DE

PHYSIQUE

I M 5

dimensional systems, for which it _was

recently proposed

that

[25],

t ₌ v

p /4

=

187/144

=

1.2986...,

in

complete agreement

^{with the} most accurate numerical estimate

[26],

t

=

1.2993 _± 0.0015.

4.

Summary.

We have studied

conductivity

of BP clusters in both two and three dimensions. We have found that for three-dimensional

systems

and _m

=

3,

the

conductivity

critical

exponent

t is the

same as that of random

percolation (m

=

0

),

and that for _m

~

3,

the behavior of the system ^is similar to that of disordered solids that

undergo

_a brittle fracture _process. From _our

data,

_we

have also

argued

that for three-dimensional

systems,

t is

probably

_not related to the

percolation _exponent p.

Acknowledgenlents.

We would like _to thank Joan Adler and Dietrich Stauffer for many useful

discussions,

and for

making

available to us their results

[11] prior

to

publication.

We would also like _to thank Dietrich Stauf§er and the

Supercomputer

Center HLRZ at KFA Jiilich for _warm

hospitality.

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