A network simulation of anisotropic percolation

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A network simulation of anisotropic percolation

E. Guyon, J.P. Clerc, G. Giraud, J. Roussenq

To cite this version:

E. Guyon, J.P. Clerc, G. Giraud, J. Roussenq. A network simulation of anisotropic percolation. Jour- nal de Physique, 1981, 42 (11), pp.1553-1557. �10.1051/jphys:0198100420110155300�. �jpa-00209348�

A network simulation of anisotropic percolation (*)

E.

Guyon

Laboratoire de Physique ^des Solides, ^Bât 510, Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay, ^France J. P. Clerc (**), G. Giraud and J. Roussenq

Laboratoire de Physique ^des Systèmes Désordonnés, Université de Provence, 13397 Marseille Cedex 13, ^France

(Reçu ^{le 25} septembre 1980, ^{révisé le 22 mai} 1981, accepté ^le 3 juillet 1981)

Résumé. ²⁰¹⁴ Nous discutons le problème ^de percolation de liens de réseaux tridimensionnels anisotropes, ^carac-

térisés _par le paramètre d’anisotropie, ^{R =}

p~/p~.

^{Un exposant de} déplacement ^{de seuil} 2,3 ^± 0,1 compatible

avec l’exposant de la taille _moyenne des amas à 2D est obtenu _par simulation numérique. ^Une expression ^{de la}

conductivité électrique au-dessus du seuil est établie de part ^et d’autre de la probabilité ^{de «} crossover » 3 à 2D.

Cependant, près ^{du seuil de} percolation, ^{les effets de taille} anisotrope jouent ^un rôle crucial dans les expériences

sur systèmes ^finis.

Abstract. ²⁰¹⁴ We have studied the bond percolation problem ^{of 3D} anisotropic networks characterized by ^the anisotropy parameter ^{R =}

p~/p~.

A threshold shift exponent, ^{2.3 ±} 0.1, compatible ^{with the} exponent of the 2D clusters average size is obtained by ^a

numerical simulation. An electrical conductivity expression ^above threshold is established on each side of the 3 to 2D crossover probability. ^{However, near the} percolation threshold, anisotropic ^{size effects} play a very important

role in the finite network experiments.

Classification Physics ^Abstracts 05.50 ^- 72.60

1. Introduction. ^- A

major

aim of the

physics

^of phase transitions is the determination of critical exponents whose values

depend

^{on the}

dimensionality

of _{space D} [1]. ^{The term «} crossover » relates to

phenomena

^{where the} behaviour of the system ^is described

by

different values of D,

depending

^{on the}

distance to threshold. The

study

of films of finite thickness t

provides

^an

example

^{of a} first class of

crossover

phenomena

[2] : the behaviour is 2D-like

just

^{above a}

percolation

^threshold

pc(t) ;

3D-behaviour is recovered for

larger

values of _p such that

Pc(t) p’(t)

p where the crossover -,value

px(t)

is

roughly

characterized

by equating

^{the 3D corre-}

lation

length ç(p)

^oc

( p - pc3)- y3

^{and thickness t.}

Anisotropy

^crossover is another class of such

pheno-

mena. It is the purpose of the present ^{note to discuss 2}

to 3D crossover in this ^case. An

analysis

^{of the}

problem

was

proposed

recently

by

Redner and

Stanley [3]

using

^series

expansions

^on

anisotropic

^clusters.

The

anisotropy

^{crossover in a}

plane

^{was studied}

in two simulation

experiments :

(*) Supported ⁱⁿ part by ^{a contract of DGRST} Physique ^Elec- tronique ^{No 787277.}

(**) ^In partial fulfilment of a thèse d’état (1980).

- Smith and Lobb [4] measured the

conductivity

of two-dimensional conductor-insulator networks

generated photolithographically

^{from laser}

speckle

patterns. ^This

experiment

^would

correspond

^{to a}

network

problem

^with

equal probabilities

^{in ortho-}

gonal

directions but with different values of resis- tances. We do not consider here this type ^{of aniso-} tropy.

- Blanc et al. [5] measured the

conductivity

^{of a}

plane conducting grid

whose wires were cut at ran-

dom with different

probabilities

ⁱⁿ

orthogonal

^direc- tions,

together

^{with a} numerical simulation of the

problem.

Straley

^{in reference} [6]

distinguishes

^{5 different}

kinds of

anisotropic problems.

^{The work of refe-}

rence

[5]

^{and the} present ^{work are} type (d) ^{in the} classification whereas Lobb and Smith’s one is type

(b).

We have done simulations in the same

spirit

^as

Blanc et al. in reference

[5]

but in the three-dimen- sional _case, in order.- to

emphasize

^{some new}

quali-

tative features of the

percolation problem.

^The para- meter

controlling

^{the crossover is the ratio R =}

p"lp.l

where

symbols //

^{and 1 refer to the} directions

along

a

given

axis Oz and at

right angles

^to it. The aniso-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198100420110155300

1554

tropy

problem

is relevant to the transport

properties

in _many real systems ; ^we consider ^here

essentially

the limit R ^{1 of}

conducting

^and

weakly

^connected

layers.

The limit of

large

^R [7] ^would

correspond

^to

weakly

connected chains (1 ^{to 3D} crossover) ^{and has}

pathological

aspects ^{due to} those of the 1D limit where _Pc ⁼ 1.

The

theory

^{can be}

approached by analogy

^with

the

magnetic anisotropy problem

which has received considerable attention both

theoretically [8]

^and

experimentally [9] :

^{let us consider a 3D}

Ising

magnet with an

anisotropic exchange

^{energy. R =}

J II /Jl : J

^Il

(resp. J1)

^is the interaction _energy between nearest

neighbours along

^z

(resp.

^{is the xy}

plane).

^{The limit}

R ⁼ 0 is the classical 2D

Ising

model. The finite R

case has been studied

[8] using

^a

homogeneous

expres- sion for the 2D Gibbs function of the system ^{in terms} of the variables R, r =

[T - Tc(R

⁼

0)]

and expres-

sing scaling

relations in the limit where both ^T and R go ^to 0. A similar

approach

is taken here in the

analy-

sis

of percolation

threshold and

conductivity.

2. Percolation thresholds. ^- 2. 1 MONTE CARLO

CALCULATION. ^- We have used a computer program for 3D cubic

percolation

with different

probabilities along

the 3 directions x, y, ^z. Here we

take pz

⁼ _Pli,

pjc = py ^{= pl. First a bond square} network of size n

containing

^N elements,

(N

^{= n2 +} (n -

1)2

^{as in}

reference

[10])

is created with a

pseudo

random number generator (RANDUD

type)

^of

large period (231).

Neighbouring

active bonds are assembled in clusters line

by

line and labelled as

they

^{are created}

along

^x first, ^{and then}

along

^{y. Bonds}

parallel

^{to z in a}

given

line are drawn after a line has been created and

graft-

ed on the

plane

^clusters.

They

^are labelled with the label of the

plane

cluster. Once the

plane

^{has been}

analysed,

construction of the

following plane

^is

initiated. The

parallel

bonds of the

previous plane

are connected to the

corresponding

sites of the new

plane

^{and the content of their} tags

kept.

^{We draw M}

such

layers.

At the end of the

operation

^{we have a}

knowledge

of the 3D clusters.

Conductivity

is tested between two

opposite plane probes parallel

^{to xOz which} group in

parallel

^the

set of xOy planes.

A direct measurement uses the onset of

cônductivity

in the network. However this introduces a

systematic

error, due to the finite size of the system.

2. 1. 1 T wo-dimensional limit (R ^{= 0} _{or p} 1 ⁼ 0). ^-

Let us consider first the limit R ⁼ 0 such that

Pcl-(R

⁼ 0) ⁼

1/2.

^{When M}

planes

^{are in}

parallel,

the effective threshold measured in the 1 direction is smaller than 0.5 if the size of the

planes

^{is finite.}

Such a

deficiency

^can be corrected. Let

F’n( p)

^be

the

probability

of conduction of a

plane

^of size n ; the

probability

^{that M}

planes

ⁱⁿ

parallel

^{do not}

conduct is :

We define an apparent ^threshold

by :

thus

In the numerical simulations we have used the cube size 9 x 9 x 9

(which corresponds

^{to the exact}

series calculations of reference

[8])

^{and 25 x 25 x 25.}

Using

the data obtained in a

previous

paper

(Ref. [10])

^we get for the shift of the threshold

These results are in _very

good

agreement ^{with those} calculated

by

^{Stauffer for the same}

problem using

^the

approximation given by

Lobb and Karasek

[11]

and

by

Esbach, Stauffer and Hermann

[12].

2.1.2 Anisotropic ^case

(P ll

ll :0 0). ^{- The numerical}

calculation of threshold was done

by selecting

^a

given

^{value of} pi and

using

5 values of pl, ^{around a} value such that

conductivity

^is established. One hundred cubes are drawn in the 9 ^x 9 ^x 9 case and 10 in the 25 ^x 25 ^x 25 one. For each value of _{P 1-’}

the

probability F(p)

^is calculated for each of the 5 values

ofpjj.

^The threshold is defined

by interpola-

tion on values of pl,. ^{In the limit} p ^{ll =} 1, ^{the exact} value is _pei ⁼ 0.

A shift of threshold can also be

expected

^{in this}

case. Let pal be the apparent threshold. In this _case, all the bonds

parallel

^to Oz exist for _any line of M active bonds. The

probability

^{that two such lines are}

not connected

by

^bonds

parallel

^{to the}

plane

xOy

is : (1 ^-

Pa1-)M.

We have a square bond

problem

^{with bond pro-}

bability

^{1 -} (1 -

Pa1-)M

and threshold

1/2.

Thus :

The

resulting displacement

^{of pc± is :}

The

resulting phase diagram

^is

presented

ⁱⁿ

figure

¹

for both cubes. The corrected threshold values _pc,

(p ll)

are estimated

by correcting

the values for networks with n ⁼ 25

by

^{the value}

0394Pal

^obtained

by

^{a linear}

interpolation

(insert ^of

figure

1) between the values for R ⁼ 0 and oo. A more accurate correction could be obtained in the

spirit

of the work of reference

[10]

using

^a

scaling hypothesis

^on networks of different sizes n for the different values of R.

Fig.1. - Phase diagram : critical line in (Pü Pli) plane, symbols

e ^are related to the numerical calculation ; 0 ^are fitted values as

indicated in text ; * values for ^{R =} 0 and R ⁼ oo ; (p.lJR ⁼ 0) = 1/2 ^and P .lJR ⁼ 1) 0.25). ^For each value _Pc.l, the correction

- APa.l is obtained by ^linear interpolation ^between apal (R ⁼ 0)

and APa.l (R ⁼ oo) ^{as indicated on} the insert.

2.2 DISCUSSION. - In the

percolation problem,

one can define [3] ^{a crossover} exponent

by

where A is a constant

independent

^{of R.}

In the

corresponding Ising problem [8],

^{lfJ2, had}

been found to be

equal

^{to y2 the 2D}

susceptibility

exponent. ^{It seems} reasonable to use the same identi- fication in the present

problem :

the extension of clusters in a

given plane

is measured

by

^the average cluster size which

diverges

^when

Pc2 is approached

from below as :

If no p ^{ll bond is} present, the cluster must extend to

infinity

^{in order to establish}

connectivity

^{across an}

infinite

sample

^and

Pc.1(R

^{= 0) = P,2-}

However, ^if _pl, ^is non-zero,

connectivity

^{will be}

established between two

neighbouring planes

^{if two}

clusters

overlap. Assuming

that all clusters have

a number

S( pl)

^{of bonds, the} average number of links between

parallel planes

^{varies as} +pjj

- S(p, ).

Equating

^{this to one is}

equivalent

^to

stating

^that

conductivity

^{will be} established across the whole

sample using

three-dimensional bonds. This relation leads to an

expression

of the formula (1) ^and may ^serve

as a

plausibility

argument, ^{which can be referred}

to the exact result of Redner and

Coniglio

^[13].

Our threshold results are

plotted

ⁱⁿ

log log

^units

on

figure

^{2. A}

single

line is obtained for both size networks (this coincidence

being possible

^{due to the}

corrections on

pc).

^The

slope gives

^a

value 72 -

^2.3

closer to the exponent y2 ⁼ 2.40 ± ^0.03

[14],

^than

Fig. 2. - Variation of R versus (Pc.l(R) - 1/2) ^{in a Log Log} plot for : $ cube size (25)3 ; * cube size (9)3. ^The slope gives ^an exponent ll(P2 ^with (P2 ⁼ 2.3 ± 0.1.

the value 1.7 obtained in reference

[3].

This value 2.3 agrees with the results obtained in reference [13].

2.3 CONDUCTIVITY. ^- The

qualitative analysis developed

^above suggests the existence of a crossover

effect above the

percolation

^value

Pc.l (R) :

^close

enough

^to threshold, ^{clusters are} three-dimensional

as

conductivity implies

conduction across the

layers.

However far above

PcJ.JR),

the behaviour should be

governed by

2D clusters. Crossover

properties

^are

similar to those deduced iri the thickness crossover

problem,

the inverse thickness t-1

being replaced

here

by R

^{with one}

major exception :

in the size effect,

crossover takes

place

towards 3D farther from thre- shold.

In an attempt ^{to obtain a}

systematic

form for the conductance

Ql(R,

e) ^we have written it as a homo- geneous function of the two parameters,

analogous

to the

equation

(7) of reference

[8] :

The formula (2) ^{can be used to obtain a relation}

for the exponents

by considering

^the

asymptotic

forms valid for the 2 and 3D

regimes :

i)

^{In the limit R =} 0, it should

extrapolate

^{to the 2D}

conductivity expression

^oc

(et2

^{if one} excludes the immediate

neighbourhood

^of

Pc.l(R).

^Thus

for

large

^{z. A}

scaling

^{relation :}

insures the overall

independence

^{of R.}

1556

ii)

On the other hand, very close to the shifted threshold

p,,,(R)

^{with e’ =} P.L -

Pc.L(R)

^{- 0 at fixed R,}

we expect ^the

conductivity

^to vary ^as [/3 as the infinite threshold should be 3D in this limit.

The

replacement

^{of e}

by

^{e’ in the}

scaling expression (2)

^{should not}

strongly

affect its form : the

scaling

factor z ⁼

elRm

^is

compatible

with the result for the shift of threshold if we take

m ⁼ 1/-v-, (4)

as this will

give

^{a value of z}

independent

^{of R for the}

shifted threshold

Pc.l(R)

^but

Replacing

^e

by

^{8’ in}

(2) gives

which does not affect the

scaling

^form (2).

Expressing

^{that 03C3 oc E’t3 as 8’ -+ 0 leads to :}

with

The forms (5) ^and (6) ^{are similar to} those obtained for the

expression

^{of the}

magnetic susceptibility

ⁱⁿ

the

anisotropic Ising problem

^{not too far from}

Tc(R)

(R ⁼

J Il IJ,). ^Using

^a

scaling expression

^to

characterize the

approach

of both R and 1 - 0

(corresponding

^{to our e --+ 0),}

Hankey

^and

Stanley

[8]

obtained the value for a critical exponent :

where ₇₂ and ₇₃ are the 2 and 3D

susceptibility

^cri-

tical exponents. ^{We note the}

change

^of

sign

^between

the two

expressions :

^the

magnetic problem

^{is asso-}

ciated with a

diverging quantity

^{- the}

susceptibility -

whereas in the

percolation problem

^the

conductivity

goes ^{to zero at} threshold. The value of 0’ in the _per- colation

problem

^is

negative :

expressing

that the increase of

conductivity

^above

the shifted threshold should become

sharper

^{as the}

anisotropy

^{increases (and} the threshold decreases).

The

scaling

^form (2) with the values of exponents 0 ⁼

t2/y2,

^{m =}

1/y2

could be tested from

experi-

ments

giving

the variation of

Ql( pl)

^{for a}

large

range of values of R

by loocking

^{for an universal form} [2]

in coordinate axis

Ji/R°

^versus 81 R m. ^In

particular, right

^at

PC1-(O),

^we expect Ql to vary ^as R t2lY2. Another

possible

^{test uses the form} (5) ^and measurements of

o’_L ^versus R at the same distance E’ for the shifted threshold.

Analog

simulations were carried [15]

using

^{a 3D}

grid

of Cu wires in a cubic lattice form cut at random with

probabilities

_p I I

and P1-

( >

Pli)

^{in the}

^spirit

^of

an

experiment by

Watson and Leath

[16].

^{In this}

case, the x0y

grids

^{are soldered to two}

opposite

^bus

wires of direction Ox. These wires are short circuited to realize the same contact

configuration

^{as in M.C.}

simulation. However the

experiment

^{failed to} repro- duce the results of the above

analysis

because of a

smeared variation around threshold. This can be attributed to the effect of finite size in the finite con-

ducting grid

^used (n = 25, M ⁼ 18). As this is a

crucial factor in all simulations let us comment on it.

The

smearing

effect has the same

origin

^{as the}

continuous variation

of Fn(P)

ⁱⁿ § ^{2.1. It occurs when}

the correlation

length

^becomes

comparable

^{with the}

sample

^size.

We express the

anisotropy

^{in terms of an aniso-}

tropic

correlation

length

^{such that}

(see

^for

example

^ref.

[17], equation (7))

where

v3 is the 3D correlation

length

exponent.

The formula (8) suggests the existence of finite clusters extended in the direction of the

layers

^below

the 3D threshold. Above it, the

superlattice picture

[18]

would

correspond

^{to a}

rectangular

^mesh

elongated

in the 1 direction. In a finite

sample,

^when pi increases towards

Pc1-(R)

finite clusters _may start

touching

^the

bus wires in the 1 direction of the

sample.

^{This takes}

place

^before any size effect in the // direction is

occurring

^if

Mln

^is

larger

^{than R 1/2. In}

particular,

at the bulk

Pc1-(R)

^the

conductivity

^obtained

by

averag-

ing

^{over several}

samples

^{with the same} values of _p and R is

appreciable.

^{This was the case in the}

experi-

ments of

[15]

^where

Mln

^{= 0.72 and R 1/2 varied}

from 0.2 to 0.4. The

analysis clearly

shows that the size limitation in the

experiment

^was

mostly

^{due to}

the size of the

planes

^{and not to the number of}

layers.

Preparing larger samples

should be made while

fulfilling

^{a criterion such as} R 1/2 - Mln ^to

optimize

the number of elements. Direct numerical calcula- tions of the _average

conductivity ( 03C3n(p)>

^{are in}

progress both for

isotropic

^and

anisotropic

^networks.

By using

^a

scaling approach

^{for the} variation both

as function of n and _p one can expect ^to get ^accurate determinations both on the threshold and on the

conductivity

exponent.

3. Conclusion. ^- Simulations ^{on 2 to} 3D anisotro-

pic percolation problem qualitatively

indicate the strong influence of the

out-of-plane connectivity

^on

the

percolation

behaviour. A

satisfactory quantita-

tive

description

is found from a direct extension of the results on the

anisotropic Ising

model. The

expression

for the variation of

conductivity (formula ⁽⁵⁾⁾

^can

be

compared

^{with the} result found in a size effect

crossover

problem

[2] ^where

However in the latter

problem

^{the crossover is bet-}

ween 3 and 2D as one approaches the threshold

pc(t)

function of the film thickness t. In the size effect _case,

we were able to check the

scaling

^form (9). ^{In the}

simulation of reference [15], ^{due to the}

importance

^of

size effects in the

plane

^{of the} film, ^{such a}

comparison

would not be

possible.

^The

anisotropy

^{of the} geo- metry

together

with that of

probability

^{must be}

considered in finite size

samples.

Acknowledgments. ^{- We were} introduced to the

anisotropy problem by

^P.

Pfeuty.

^We thank S. Ale- xander, ^D. Stauffer, ^H. Ottavi, ^{and E.}

Stanley

^for

several discussions

along

this work. In

particular

S. Alexander has

provided

^{us with an}

(unpublished) analysis

^{based on an}

anisotropic

correlation

length

whose results on the crossover exponent ^{and on the}

conductivity

agree the present

approach.

^{Last but}

not least we

acknowledge

several crucial criticisms of a referee report.

References [1] STANLEY, ^H. E., ^{see for} example ^Phase transitions and critical

phenomena; (Oxford University Press) ^1971.

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