Aboav's law for an assembly of discs of différent sizes

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Aboav’s law for an assembly of discs of différent sizes

N. Rivier

To cite this version:

N. Rivier. Aboav’s law for an assembly of discs of différent sizes. Journal de Physique I, EDP Sciences,

1994, 4 (1), pp.127-131. �10.1051/jp1:1994125�. �jpa-00246883�

Classification Physics Abstracts

05.40 05.90 64.60C 64.90

Aboav's law for

_an

assembly of discs of different sizes

N. Rivier

(*)

Blackett

Laboratory, Impenal College,

London SW7 2BZ, G-B-

(Received 29

September

1993, accepted 8 October

1993)

Résumé. Les _mousses foumies par des

empilements

de

disques

de différentes tailles ont une

relation d'Aboav (entre le nombre total de côtés des voisins d'une cellule à _n côtés et ni en femme de S, alors

qu'elle

est linéaire _{en n} dans les _mousses

engendrées

par des germes identiques ou dans les

mousses naturelles. On _montre que cette loi _en S _est le résultat de maximiser

l'entropie,

et une

mesure de la

dispersion

des tailles.

Abstract.-Polydisperse

froths obtained from assemblies of different size discs exhibit _an S- shaped Aboav's relation (between trie total number of sides of trie

neighbours

to an n-sides cell, and n), rather than the hnear relation round in monodisperse or natural froths. One show that

thi§

S-

shape

is _a result of maximum entropy mference, and _a signature of

polydispersity.

In _a recent paper, Annic,

Troadec, Gervois, Lemaitre,

Ammi and

Oger [Il

have studied

two-dimensional random mosaics _or froths obtained

by

radical tessellation

(a generalization

of Voronoi's to discs of different

sizes)

of

packings

of discs with two or many different sizes.

These

polydisperse packings

_are in statistical

equilibrium,

but their

equatiqns

of _state do net bave trie

simple,

linear

dependence

round in random frottis.

Equations

of state m frottis

[2]

are

Lewis's law, _a relation between average area of _an n-sided cell and _n, and

topological

correlations such _as Aboav's law

(or Aboav-Weaire's),

which relates the total number of sides of

neighbours

to an n-sided

cell, nm(n)

to n, and _{a more} detailed set of correlations between

neighbouring

_n- and k-sided cells

[3].

The average

topology

of _a cell is _a numerical constant

in)

=

6 in random

(3-valent)

froths

[2].

These

equations

of state _are characteristics of froths

as a statistical ensemble.

Annic et ai. obtain

non-linear, S-shaped

Aboav

([Il, Fig. 3)

and Lewis's

([Il, Fig. 5)

relations m

bi-disperse packings.

For

polydisperse packings,

Aboav's law

([ Il, Fig. 7)

shows _a

slight,

_{convex curvature or an}

inkling

of _an

S-shape

_; Lewis's

plot

is

always S-shaped ([ ii,

Fig. 9),

albeit less

markedly

than in

bidisperse packings

where it _is almost _a kink _at

n =

6. These deviations _{are not} due to poor

statistics,

and statistical

equilibrium

has been reached and maintained

by

constant motion of trie discs _{on an air} table

[4].

On the other

hand,

(*) Permanent address Laboratoire de

Physique

Théorique, Université Louis Pasteur, 3, rue de 1Université, F60874

Strasbourg,

France.

128 JOURNAL DE

PHYSIQUE

I N°

trie

(maximum entropy)

arguments

leading

to linear relations _are

solid,

and have been hitherto well

supported experimentally (exceptions confirming

the

rule).

So what is

going

_on ? This

note will

show, by applying

the traditional maximum entropy ^inference to

polydisperse

mosaics, that the

S-shape

discovered

by

Annic et ai.

Il

is _a result of

polydispersity,

and its signature.

Peshin et ai.

[3]

have shown

that,

for _«

totally

random _»

(1.e. monodisperse) mosaics,

Aboav's law is _a consequence of maximum entropy ^{inference. In}

monodisperse

mosaics, the

total number of sides of

neighbours

_{to an} n-sided cell,

nm(n),

increases

linearly

with

n

nm(n)= (6-a)n+6a+p2, (1)

p2 is the variance of the

shape

distribution

Il.J,

_eq.

(l ).

^In this note, Aboav's law for bi- and

polydisperse

mosaics is derived from the _same maximum entropy ^{inference. The}

S-shaped

relation between

nm~~P(n)

and _n, observed

experimentally

for

bi-disperse

_mosaics and discs with two sizes, is recovered.

Maximum entropy ^{inference is based} on the fact that _a random mosaic takes the

configuration

realizable

by

the

largest

number D of

microscopic,

local

arrangements

of cells, which is the

configuration

of maximum entropy S

=

Inn,

subject

to constraints.

(Here,

S and the

Lagrange multipliers

_are taken _as

dimensionless).

In

thermodynamics,

ehmination of

a constraint increases the entropy.

Here,

if _one constraint _can be made to

duplicate

others, it is

effectively

redundant. A mosaic with local arrangements

duplicating

_one constraint has

larger

entropy ^{and is much} more

probable

than any other mosaic without

duplication.

The condition of

redundancy

is Aboav's relation

(or

Lewis's law in the _case of

size-shape correlations) [3, 5].

For

topological correlations,

_one introduces the correlator

M~~(nfl) [3]. M~~(nfl )

is the

number of k-sided cells associated with _a disc of _size p,

neighbounng

_an n-sided cell

associated with _a disc of size

fl (nfl-cell

for

short).

It is _a conditional

probability,

_so

thàt Mka(nfl

_p~p is

proportional

_to the

probability

that _an interface separates a ka-cell from _an

np-cell [5].

_p~p is the

probability

of

finding

an

nfl-cell

in the mosaic.

M~~ (nfl

^satisfies two

identities,

3~~ M~~ (nfl

₌ n for ail

p

,

(2)

since an n-sided cell of any size has

n

neighbours,

and

Mka (~ô

_Pnfl

"

Mnfl

^(ktY pka

,

(3 by

obvious

symmetry

^{of the interface} between ka- and

np-cells. Apart

from these _two

identities,

the functional

dependence

of

M~~(np

_on

k,

_{n, a} and

p

is unknown. Aboav's relation is that

particular

functional

dependence

which makes _a constraint redundant. It is

therefore _an inference of maximum entropy.

The constraints _are the _{same as m} the random _{mosaic or}

monodisperse

disc

assembly,

except that the normalization has to be made

separately

for each disc size

p,

if _one

operates

at fixed

concentration cp ^{of each}

species. They

_are,

3~

_p~p

= cp

(normahzation) (4)

3np

npnp =

6

(topology) (5)

3np M,~ (np

_pnp

=

kp~~ (counting

the

edges

^of a ka -cell

from the outside and from

within) (6)

If the last set of constraints _are linear combinations of the first two,

Mk~ (nô )

_{= ak~,}

p +

nbk~ (7)

they

_are redundant and the random _mosaic is

overwhelmingly

_more

probable. Equation (7) gives

therefore

topological

correlations between

neighbouring

cells _m mufti-sized mosaics and discs assemblies which _are

random,

realizations of a statistical ensemble of maximum entropy.

Note that the _terni linear in _{n is}

independent

of

p.

This _comes from the fact that the

topological

constraint

in)

=

6

only

hoids for the mosaic _{as a} whoie. The individuai

in )

~ ⁼

3n np~p/cp (where p~p/cp

^{is the number of}

np-cens

divided

by

the total number of

p-ceiis)

_are

arbitrary

and _not constrained. The coefficients _are reiated

through kp~~

₌

3p

_{cp a~~}

p ⁺ 6

b~~ (8)

Symmetry (3) imposes

that

M~~(np)

^is

proportionai

to p~~, and biiinear in _n and k. In

fact, equations (2-8)

are summarized in the

generai

relation

M~~(np)=

_p~~

((n-6)ce(k-6)+p~(n-6)+pp(k-6)+ y~~p~) ⁽⁹⁾

where _{~r, p~} and the

symmetricai

_{y~~pj = yp~} are structurai

coefficients, independent

of the ceii

shapes

n and k.

(They

_are

Lagrange muitipiiers reiating

the

constraints).

The iast _{two are}

restricted

by

the

relations,

~Î~ C~ p~ = Î

(ÎÙ)

~a

_{~a l'(nfl)}

~

~'

(~

~)

[For

_one

singie species

^of

discs, they

are numbers _{: p}

= 1, y ₌ 6. If ci = c~, as in

figures

2-5

of reference

Ill,

there _are two

independent

_{parameters p} and _y, with pi = p, p~ =

2 _p,

Yii ~ Y22 " Y, Y12 " Y21 ~

l~

Y.l

One must

emphasize

that the form of

equation (9), specificaiiy

its

hnearity

m n and the

mdependence

on

p

of the first _{two ternis} of

(9)

and of the second _terni in

(7),

is a necessary consequence of maximum entropy ^{with the}

inescapabie,

mathematicai constraints

(4-6).

If _one observes deviations from

iinearity

m n for

M~(n

which _{are not}

statisticaiiy insignificant

_events

due to

disappear

with better statistics _or with time

(since M~(n )

must be

positive,

_no tires with

n such that trie

extrapolation

of

M~(n )

in

(9)

is

negative

shouid appear m statisticai

equihbrium,

and _one

singie

tire

aiready sports

the

iinearity),

this is either because (1) the froth is not in statisticai

equiiibrium,

or

(ii)

trie effect of _an additionai

constraint,

or it indicates

(iii)

the presence of correiations

beyond

nearest

neighbours.

Aitemative

(iii)

is

aimqst certainiy what happens

m the statisticai modei of Le Caër and

Deiannay [6, 7],

which shows

M~(n ) deviating considerabiy

from

Îinearity.

_Their modei iifts the

degeneracy

^{of the} vertices of _a

reguiar

iattice at

random,

and these vertices

gather

ceiis which _{are not} ail _nearest

neighbours

as

they

do not share _an interface. Correiations

beyond

nearest

neighbours (iii)

_may aiso be the resuit of steric exclusion _m the

triangular

cerfs of the

concentrated, monodisperse

discs assembiies of Gervois _et ai.

[8], aithough they

aiso suggest

an additionai constraint in 1In

(ii)

_as the _cause of the

non-iinearity

of Lewis's iaw which

they

observe.

We _are interested in

nm(n, p )

=

3~~ kM~~ (np ),

which is _n times the average

shape

of the

lieighbours (of arbitrary species

_a to an

np-ceii,

_or the total number of sides of ail

neighbours

to an

np-ceii,

nm(n, ~ )

=

(n

⁶

) (p2

_{~ +}

~a

_C~

(k)

~

p~ +

(p2

_{Pff +}

~a

_Ca

(~)

a

Y(afl)1 ⁽¹²⁾ )

This _is Aboav's relation for

species p.

It _is iinear in _n. The

siope (6 -a)

₌

[p~~r+3~c~ik) p~]

is

independent

of the

species,

but the

iitercept _Xp=

130 JOURNAL DE PHYSIQUE I N°

[p~

_p

p +

3~

_c~

ik)

~

y~~

p~] depends

_on

fl,

_as found

experimentaiiy Ii ].

_p~ is the variance of the

shape

distribution for the whoie

mosaic,

_p~

=

3~~ p~~(n -6)~.

_{There is}

an

identity 3~

_c~

X~

= p~ +

36, independentiy

of the structurai parameters.

From _{now on, we} shah consider _two

species

^of

discs, p

= 1,

2,

with

denoting

the

iarger

size.

Cieariy, in )

~

in )

~.

Less

cieariy, Xi

_~

X~,

since _a

iarger

6-sided ceii wiii have _a

iarger

penmeter to accommodate 6 discs with _a

iarger

number of sides than those around a 6-ceii

generated by

a smaiier disc.

(Xi

~

X~

wouid indicate _a strong

tendency

for

langer

discs _to be surrounded aimost

exciusiveiy by

smaiier discs with

in )

~,

whereas _a smaii disc with the _same

n has _more

iarger

_{ones as}

neighbours.

It is indeed the _case if ci ^« c~, and if the _two

species

of discs have very different

radii.) Aiso,

_q~i is the

oniy

_parameter of the

partition

between disc sizes, since q~~ = q~j. Generaiization to a

poiydisperse assembiy

of discs iike that studied

m the iast section of reference

Ill,

is

straightforward.

The

S-shaped

_curve of

figure

3 of reference

Ii ], nm~~P(n ),

is the

plot

^of n times the average

shape

of

neighbours

to any n-ceii,

regardiess

of the _size of the disc generating it. For every value of _n, the iinear nm(n, p is

weighted by

q~p = p~p/~~Y~j ⁺

pn2],

with

3p

_q~p

=

(13)

q~j increases

monotomcaiiy

with _{n, as} the

proportion

of ceiis

generated by iarger

^discs

increases with

increasing

_n. If the

shape

distribution p~ =

3~

_p~~ is bimodai, _q~j increases

sharpiy

with _n between _two

plateaux,

from 0 _at

n w 4 to for _{n m} 7, _{say. If p~ has} one

singie maximum,

the increase is much smoother, the iower

plateau

at n ₌ 3 may not be reached and the

higher

_one

oniy asymptoticaiiy.

So,

Aboav's iaw for ail

species

is

given by nm~~~(n)

=

zp

_{qnfl ~'~~}

(~, ô

"

(6 ~) (~ 6)

_{+ ~nl}

Xl

+

(1

_~nl

) X2 (14)

It is non-hnear _since q~j is non-iinear in _n. It has the

S-shape

obtained

experimentaiiy

in

figure

3 of reference

Ill,

where the _common

siope (6

_a

=

[p2

_~r +

3~

_c~

ik)

~

p~ is

cieariy noticeabie,

_as is the

rapid

transition between smaiier and

iarger

^disc

populations.

Even for the

poiydisperse assembiy,

the

shght,

convex deviation from

iinearity ([1], Fig. 7)

is the

resuit of

this,

_now smooth transition,

parametrized by 3~p~~~~jj~~~~q~p,

^which

replaces

q~i in

poiydisperse

assembiies. The iower value is _never reached

(there

_are 3-sided ceiis

generated by

other than the smaiiest

discs),

and _one

begins

to see that the upper

plateau

^{wiii be}

reached

asymptoticaiiy.

One

anticipates

simiiar non-iinear behaviour for the average size of _a

n-ceii,

_A~

(Lewis's iaw),

^with the

non-iinearity

controiied

by

the

partition

_q~j, since Lewis's iaw _is aise the consequence of

dupiicating

constraints

(in

this _case, the total _area of the mosaic, with

(4)

and

(5)).

The

partition

_q~j

(3~p

~ ~~~jj~~~~q~p ^m

poiydisperse assembiies)

can

easiiy

be extracted from

the data _:

subtracting

the contribution of smaiiest discs _nm

(n,

2

)

from the

expenmentai

Aboav

plot nm~~P(n ), yieids

:

qui "

inm~~P(n)

nm

(n,

2

il iXj X21 (15)

It is

monotonicaiiy mcreasing

with _n, between 0 and i if the

shape

distribution _is bimodai.

The

non-iinearity

of Aboav's

plot

indicates the

species dependence

of the structurai parameters p~ and y~~py ^{The same}

procedure gives 3~p~~~~jj~~jjq~p[Xp

X~~~jj~~~] m

polydisperse

assemblies.

For any

binary assembly,

the

easily

observable features of Aboav's

plot,

its

slope

(6-

ai and

[Xi -X2]

are

expressed

m terms of ci, of the structural parameters ce, p, y, and the

experimental

_measure of the

dispersion

A

=

in) in)

~. If ci = c2 ₌ 1/2,

(6-a)=

_p20e

+6+A(p -1)/2

Xi -X2

=

2

p2(p -1)

+

(y 6)

A.

(16)

The dense

binary assembly

^studied in reference

il (Figs.

2,

3),

_ci

= c~, with

packing

fraction C

=

0.73,

has

in)

= 7.09,

(n )~

₌

4.91,

and p~ =

0.77

(J.

P. Troadec,

private

communi-

cation).

One would require an additional

observation,

such _as the width of the

kink,

to

determine the three structural parameters.

By

contrast, a

monodisperse

_mosaic has _one

surgie

structural parameter ce,

fully

determined

by

the

slope

of the linear Aboav

plot.

Acknowledgments.

The author is

grateful

to the Rennes group for

sending

him their paper

Il

_pnor to

publication.

He is

especially

indebted _to A. Gervois for

msisting

_as

early

_as m 1990 that Aboav's law _was not

always

linear.

References

[Ii Annic C., Troadec J. P., GervoisA., LemaitreJ., Ammim. and OgerL., J. Phys. I France 4

(preceding article).

[2] Weaire D, and Rivier N.,

Contemp. Phys.

25 (1984) 59.

[3] Peshkin M. A.,

Strandburg

K. J, and Rimer N., Phys. Rev. Lett. 67 (1991) 1803.

[4] Lemaitre J., Gervois A., Peerhossami H., Bideau D. and Troadec J. P., J. Phys. D

:

Appl. Phys.

23 (1990) 1396.

[5] Rimer N., Disorder and Granular Media, D. Bideau and A. Hansen Eds. (Elsevier, 1993) ch. III.

[6] Le Caër G. and

Delannay

R., J.

Phys.

A 26 (1993) 3931.

[7] Delannay R., Le Caër G, and Khatum M., J. Phys. A 25 (1992) 6193.

[8] Gervois A., Troadec J. P. and Lemaitre J., J. Phys. A 25 (1992) 6169.