A model of binary assemblies of dises and its application to segregation study

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A model of binary assemblies of dises and its application to segregation study

J. Troadec, A. Gervois, C. Annie, J. Lemaître

To cite this version:

J. Troadec, A. Gervois, C. Annie, J. Lemaître. A model of binary assemblies of dises and its ap- plication to segregation study. Journal de Physique I, EDP Sciences, 1994, 4 (8), pp.1121-1132.

�10.1051/jp1:1994242�. �jpa-00246972�

J. Phys. J France 4 (1994) 1121-1132 AUGUST 1994, PAGE l121

Classification Phystcs Abstracts

05.90 64.75 64.80G

A model of binary assemblies of discs and its application to

segregation study

J. P. Troadec

('),

A. Gervois

(2),

C. Annic

(')

and J. Lemaître

(1)

(')

_Groupe Matière Condensée et Matériaux (*), Université de Rennes l~ 35042 Rennes Cedex~

France

(2) Service de Physique Théorique, Direction des Sciences de la Matière CE

Saclay.

91191 Gif- sur-Yvette Cedex, France

(Re<.eii>ed 25 Februaiy /994, accepted in

final

foi-ni 3 May /994)

Abstract. We propose a theoretical model of random binary assemblies of discs which is the extension at any packing fraction of _a model

previously

described by ^{Dodds for} dense

packings.

In

dilute assemblies, the notion of _contact between the

grains

must be replaced by that of

neighbourhood

of the

grains.

We define the

neighbours

of _a disc through the radical tesselation.

The model is tested _on

packings

built grain after grain, obtained from numencal simulations. Its results are then

compared

with expenmental ^results obtained _on assemblies built _{on an} air table _: this shows that the

possibility

of rearrangement in those assemblies leads _to

segregation

at a local

scale.

l. Introduction.

It iS difficult to prepare mixtures of grains without

having

segregation.

Segregation,

which iS defined _{as a} non-random

spatial

distribution of the different Species of

grains

m the

medium,

has several engins and is influenced

by

differences _in

density, shape,

interaction and _size.

Segregation

_is

particularly

sensitive to size differences between grains and appears whenever the «

history

_» of the

packing

allows _a relative

displacement

of _one

species relatively

_ta the

others. Recent experiments on

binary

mixtures of

spheres

show that vibrations _cause the

larger grains

to go ^to the top ^{of the}

packing (Brazil

nuts

il ]).

In _a 2d

rotating

drum inside which there

is a

binary

_mixture of discs, the

larger

discs _go

rapidly

to the

periphery

while the smaller _ones concentrate at the center of the

packing [2].

In

3d,

for

large

_size ratios

(larger

than

6.5-7),

small grains may fait under

gravity

_at the bottom of the vessel

through

the pore space of the

large

grains,

leading

to a

separation

of the two

phases.

These _are

examples

of

«

global

_»

segregation, leading

to very

inhomogeneous packings.

In other circumstances, there may aise be _a segregation at a more local

scale,

in which the small

(*) URA CNRS 804.

particles position

themselves in _a non-random _way

(for example forming

chain-like _or compact

clusters), although

trie

packing

is

globally homogeneous. Segregation

effects _cari

bring

fortin serious

problems

in the _use of

granular

materais because

they strongly

affect the

properties

of the materials.

Segregation

_con be _more

easily

studied in

binary

mixtures,

by looking

at the way the

particles

_rearrange

according

to the

dynamics

of the

packing

_or the

building

rule. This has been

performed by

Bideau

[3]

for 2d compact ^{static mixtures of discs}

by examining

the local environment of the

grains

and

comparing

it with what would be obtained in _a random

packing

such _as

given by

the statistical model of Dodds

[4].

No attempt to

enlarge

the

question

to dilute assemblies has been

performed.

In the present _paper, we present a theoretical model for segregation in

binary

mixtures of discs at any

packing

fraction. The model is tested bath _on

experimental

assemblies of discs

moving

on an air table

[5]

and _{on some}

packings

obtained from numerical simulations. A

study

similar _to that

performed by

Bideau

[3]

_{in compact} assemblies has been

undertaken,

the

analysis

^of contacts

being replaced by

that of trie

neighbours.

In section 2, we describe the

packings.

In section 3, _{we propose} a

generalization

of the statistical model of Dodds for the

case of dilute

binary

mixtures of discs. A local

study

m terms of percentages ^of

neighbounng

cells is

presented

in section 4.

2. Disc assemblies.

The

expenmental starting point

is _an

assembly

of discs _{set on a} horizontal _air table built _in

Rennes

university;

most technical and

experimental

points were

already explained

_in

reference

[5].

Because of the

heterogeneities

ⁱⁿ the _air

flux,

the discs rearrange

permanently

and the statistical

equilibrium

is

quickly

reached the

assembly

is

homogeneous

_at

large

scale.

The

experiments

_were

performed

with _{a size ratio} 2

(R~/Rj

= 2,

practically

2

Rj

= cm, the

index1

corresponds

to small discs and 2 _to

large ones)

^the

packing

fraction goes from

C

=

10 fb _to C

=

75 % and the ratio ni of the number of small discs _to the total number of discs _runs

approximately

from nj = 0.2 up ^to ni =

0.8. For

given packing

fraction C and numencal proportions n, of each

species (nj

_{+ n~}

=

1),

several

snapshots

_are taken. Position

of the _centers and the

length

of radii _are determined

through

an image treatment

analysis

and

any

quantity

of interest _can be obtained from this information

only.

For dilute

assemblies,

the

only

_way to appreciate without

ambiguity

whether _two discs _are

neighbours

is the radical tesselation

proposed by Gellatly

and

Finney [6]

which _is the

generalization

of Voronoi tesselation _{m monosize}

packings

_: each disc is surrounded

by

a

convex

polygonal

cell which is

generated by

radical _axes and two discs _are

neighbours

if their

cells have _one

edge

in _common. At _one vertex,

only

3 fines converge

(4

is

exceptional

and unstable)

(Fig.

l). Information _on the

repartition

of the discs _is

provided by analysmg

the

statistics of the cells, _common

edges

and vertices. The dual of the radical tesselation _{is a}

tnangular

network,

generalization

of the

Delaunay

network [7].

Some

complementary

studies _were

performed

with computer

generated packings

built grain

after

grain.

Dilute assemblies

(C ~509fi)

_are obtained

by

_a RSA

(Random Sequential

Adsorption) building

rule

[8]. Very

compact

packings

_are obtained

by

the Powell

algonthm [9]

in which each grain is

positioned

as low _as

possible,

in contact with _two

already

present ^discs

so that the

packing

fraction is

always

close _to the _maximum disordered _one, C

= 0.84 which is

the _maximum

possible

here. In the two

algonthms,

the species of the disc _to be

positioned

is

chosen

randomly.

We shall _see that numencal and

expenmental

assemblies do not behave

exactly

in the _same way.

Finally

_some RSA and Powell

packings

_were built with _an aspect ^ratio

R~/Rj

=

4 but _no different

qualitative

conclusions _were drawn.

N° 8 A MODEL OF BINARY ASSEMBLIES OF DISCS 1123

Fig. l. Radical tesselation _{in a}

binary

mixture (C 0.73~ _nj

=

0.5 ).

3. Statistical model of

binary

mixtures of discs.

We first define the

quantities

of interest with

possible generalization

to any number of

species then show how

they

can be derived from _a theoretical model which is an extension to any

packing

fraction of Dodds' model for dense

packings [4] by substituting

^the

neighbours

of _a given cell to the _contacts of the internai disc.

3.1 GENERAL QUANTITIES. For _a

binary packing,

and from _a

topological point

of view,

three kinds of

quantities

can be defined _at the local level _:

a)

the

(partial)

_average number _=j of

neighbours

of _a disc of species i (1 = 1~ 2 with the _sum rule obtained from Euler's relation

[10]

nj =j + n~ ?~ 6

(3. la)

and the

(correlated) probability,rj (resp.

_>-~, à-j +,r~ = 1) that _a disc of species

(resp. 2)

is present in a

triangle

^of

neighbours.

We have

6 _,r, z, =

(3. lb)

Of course,i, xii, ^and as the small species ^{is less} present in the

triangles,

we have

,ij ~ ni

(resp.

_{.r~ ~}

n~)

and _{=j w} 6 _w

z ~.

b)

The

probabihty

_t,~ that _a disc of

species

1and _a disc of

species j

are

neighbours.

We have

tri + t12 + t22 _~ l

(3.2a)

and

c)

The probabihty p (ijk )

for

havmg

p(1II

₊

p(l12)

₊

p(122)

₊

p(222)

= 1,

(3.3a)

with _sum rules

tj~ =

~

b~(l12)

₊

p(122)],

tjj =

p(1Il

₊

p(112) (3.3b)

t~~ =

p(222)

+

p(122).

If _{we now} consider the

Delaunay

network, the

only

^available

quantities

_are the average

distances

1,~

(C

between the centers of two discs _i and

j

at

packing

fraction C, _or

alternatively,

the average

angles

_9~~('(C with summit in _a

triangle (1, j, k).

We _are faced _to metric

quantities

depending

of _{course on} C and

(ni, n~),

^and in the _case of

binary

mixtures~ this

means

that,

_on the average,

Delaunay triangles

are isoceles

(triangles (l12)

or

(12?)j

_or

equilateral (triangles

(1

11)

_or

(222)

with

angle w/3),

which _was

precisely

the starting

point

m

Dodds'

analysis.

The link between the _two networks is achieved

by rewriting

z, =

2

w/6, (3.4)

where

6,

is the _mean

angle

^around a disc 1m _a

triangle.

We have

Angle

2 _{a is} the _mean

angle

^around the site 2 in

triangles (112)

and 2 fl ^the mean

angle

around the site1 in

triangles (122).

With the above

relations,

_{one cari} wnte

,r = nj

~

(p(l12)(1-

_a

-p(122)(1- _fl) (3.6)

« 6 6

3.2 THE MODEL. We _{are now} able _{to set Dur} model at any

packing

fraction. The argument ^is

divided into 2 steps.

3.2.1 Relation to

packing fi-action

C.

Quantities,<~

_tj~,

p(ijk)...

and the

angles

_a and fl

depend

on the

packing

fraction

C,

and it is _{Dur atm} to find this

dependence,

at least

approximately.

Then, _we will first try to evaluate the _mean distance

1,~ between the centers of _a disc of type i and _a disc of type

j.

The _mean distance

r(C

between any two

neighbour

sites

depends

on C

according

to the law

1(C

Il, C

(3.7)

When C

0,

the limits of the quantities ^{of interest} are known if _{we assume} the

assembly

to be random

(no

interaction between the disc~ _at C

=

0)

,1 = iii =j = z~ = 6, _{a =} fl ₌ w/6.

Moreover~ the _mean distances i,~ ^between two

neighbour

sites are ail

equivalent

ijj

~ij~~i~~~i(C).

At the other limit of

packing

fraction~ we assume that the

packing

con be described _{as a} dense ideal

packing~

in which ail the contacts are true

(Dodd's

model)~ i e. it is made

only

with

equilateral

(111 and 2??

types)

and isosceles

(112

and 122

types) triangles.

We denote

by

the

N° 8 A MODEL OF BINARY ASSEMBLIES OF DISCS l125

index _m the quantities ^related to this

packing

which has the maximal

packing

fraction

C~ (this quantity

_can be

easily

calculated when the radii of the discs and the

composition

of the

mixture _are

known).

We have _:

~llm"~~l, "12m"~l+~2, "22m"~~?.

The _mean distance between any two

neighbour

sites in this

packing

is then

'm ~ Îll

m

~~l

₊

Î12m(l~l

+

~2)

_+12?m ~

~2

~ '

(~m),

and _i

(C

can be written as

jC~

i

(C

= r~

(3.8)

C

Ai mtermediate

packing fractions,

let _us consider the _mean distance ijj between _two

neighbours

of type ^The ratio ^'

~~

~ 0 when C

~ 0 because its denommator goes to

i

(C )

co. An approximate ^{form for} rjj is the

following

jC~ ^{Ri (R)} j

~" ~~

C ^~ ~~

(R) C~' ^~~'~~~

In the _same way,

jC~ ^Ri

^{+ R~ 2}

^(R) ~f

~'~ ~~

C ~~~

2)R) _C~' ^~~'~~~

~~ _~

~ÎÎ

_~

~

~~ RÎ~~

~/Î~ ^~~'~~~

(R)

is the _mean radius of

neighbour

discs at the

packing

fraction C _:

Rj

₊ R~

(R)

= tj

Rj

_{+ tj~ + t~~} R~

(3.10)

In these

expressions,

the coefficients of

fihave

_{been chosen}

in such _a way that

i (C ) is

effectively

the _mean distance between any two

neighbour

sites

J

(~)

~ Îll1Il + il? _'12 +1?2 '2?,

the _exact result is obtained for _a monosize

packing,

i jj ⁼ 2

Rj,

_i

j~ =

Ri

₊

R~

^and _{i~~ =} ²

R~

^{for C} ₌

Cj~.

Of _course, the above relations _are

only approximations

thon are

expected

to work ail the

berner _since the size ratio _is closer to 1.

They

cannot describe

correctly

_a dense

packing

with _a

large

size ratio

(forger

thon

6.5)

and _a weak

proportion

of small discs _in this _case, the small disc~ will be free inside the _cavities of the

packing

^of the

large

_ones~ and the representation ^of

contacting grains

by triangles

will be inaccurate. One _{can note} thon relations (3.9)

imply

ii + r~~ = 2 rj~, for ail the

packings

_we have studied

(size

ratios 2 and 4, ai ail

packing

fractions and

composition~)

this

equality

is verified within _or 2 fi.

As

already

noticed ai the

beginning

of

3.2,

the

analysis

^for

Delaunay triangles implies

thon, like for C

=

C~,

the

packing

_can be modeled _{as an}

assembly

of

equilateral

_or isosceles

triangles

at intermediate

packing

^fractions. Within this

approximation,

the

angles

a and pare given

by

where the _1,~ are

given by equations (3.9).

Notice that, in these

expressions,

_parameters

t,~ ^occur

implicity

in

(R)

_so that the

coupled equations (3.6)

and

(3.9)-(3.11)

cannot be solved

without any further information _on the conditional

probabilities.

3.2.2 Random mode/.

Equations

(3.6) and

(3.9)-(3.11)

can now be solved if _{we assume} that the

packing

_is random, 1-e- if the relative

positions

of the two species of discs _are not

correlated. Then ail

probabilities

factorize

tjj

=.i~

_tj~

=

2,1(1

-.r), t~~ ₌ (1

-,1)~ _(3.12)

p(111) =.r~, _p(l12)

=

3,r~(1--1), _p(122)

=

3,r(1-,r)~ _p(222)

=

(1-,r)~. _(3.13)

Equations (3.6)

and

(3.9)-(3.13)

have been solved

numencally. They

are the basis of _our random model. Let _us note that ail _sum rules _are

obeyed,

in

particular

_{nj ÷j + n~ =~}

= 6.

SO

far,

some comments on the relation _to Dodds' model _are necessary. In Dodds'

model,

ail

neighbours

_are first assumed _to be in contact, then _some contacts are cut

randomly

_so that the

actual coordination number ₌ satisfies the

(approximate)

relation _{iij =j} +n~=~ ==~ with

experimental

₌ 3.5 4.

Actually,

here, we

keep

the

Delaunay triangles

so that the average number of

neighbours

_is

exactly

6

(identity above),

which is _more

satisfactory.

The

remaining

is a

generalization

at any

density~

^with main

assumption

_in

equations (3,9)

and

(3.

il ).

4. The smaJl scaJe

segregation.

The random model

presented

in the

previous paragraph

is first tested

by comparison

with the results obtained _on the

packings

obtained from numencal simulations these

packings

are built grain after grain and _no

reorganization

is

possible,

_{so we expect a} small segregation. On the contrary, companson of the model with the

packings

built _on the _air table show _a clear

segregation ^{in the latter.}

4,1 COMPUTER _GENERATED PACKINGS.

Figure

2 show the variations of r~j, ij~ and

ii~ as a function of C for RSA and Powell

packings

with _{a size ratio} 2 and

equal

numbers of small and

large

discs. The agreement ^between

experimental

ion computer

generated packings)

and theoretical values is fair. For ail the

packings

_we have studied~ _{size ratio} 2 _or 4 and whatever the

composition,

the

discrepancy

_is less thon 5 fl.

In

figure

3, the

partial

coordination numbers _=j and _{=~ are}

represented

as a function of ni for dense Powell

packings

with _size ratio 2. Here again, the agreement ^{with the random}

model _is very

good.

In

figure

4 the same

quantities

are shown _{as a} function of C for

packings

with a size _ratio 4 and

equal

numbers of small and

large

discs.

Agoni

there _is a

good

agreement

with the theoretical results. The variations with C _are linear

(it

_{is so} for ail the computer

generated packings)

and suggest an

approximation by

a first order

development

in

C/C~~

of equations

(3.9)

and

(3.10), leading

_to the basic expression

~'

'Î

~~~~

~'~~ÎRÎ~ Îm ^~~'~~

with

(R)

~ ^{= n}

Rj

+ n~

R~,

^{and from} which ail other quantifies can be obtained. This _is indeed

an excellent

approximation.

N° 8 A MODEL OF BINARY ASSEMBLIES OF DISCS l127

izo

ru, ru, rzz

+

40 rzz

ru

~

C

0 0~ 0A 0.6 o-g 1

Fig. ^{2. Variation of the} _1,~ ^{with the}

packing

fraction for numencal

packings

with _{a size ratio} 2 and nj 0.5. Fuit fine theoretical results, (+)

expenmental

points.

z1, zz

~o

Zz

#e

6°

~ee ^~

:

O*°

à ^o

0 0~

ig. 3. - Plot of he

ncreasmg of _small discs. (OI Theoretical oints,

(+)

xpenmental points.

8

c

0

ig. 4. - Plot of

the

packings with size _ratio 4 and ii~ =

0.5.

_(O) Theoretical (+) points.

However the

packings

obtained from numerical simulations _{are net}

totally

random because of stencal exclusions. Both in RSA and Powell

algorithms,

it appears to be _more difficult to add _a

large

disc thon _a small _one. The small scale

segregation

_con be studied

by

examming the

mean environment of the grains of the two

species,

m

particular

the t,~. ^These

quantities

_are

compared

with the theoretical _ones for _a size ratio 4 and nj = n~ = 0.5

(Fig. 5).

As C increases, trie small disc cells become smaller and smaller

relatively

to the

large

disc cells

,ij >.~

and tjj decrease, while and t~~ increase. The agreement ^between

experimental

and

Jll J12

theoretical t,~ appears

globally

_correct. However, a

segregation

is present, as mdicated

by

the

expenmental

values of tj~,

slightly larger

than the _maximum value 0.5 than _can be obtained in

a random

packing

from the relation tj~ =

2

à-i1

x). As

expected,

the

discrepancy

between theoretical and

expenmental

^values is

larger

for the dense

(Powell) packing.

In

figure 6,

_we have

plotted

the t,~ ^{as a} function of _n for Powell

packings

with _a size _ratio 2. The

probabilities

tll, t12,tzz

o t t

+

~

tu tzz

o à

w ~

v

Ù-1 ~u

C o

o o~ oA 0.6 off 1

Fig.

5.- Plot of the

t,~ ^{as a} function of C for numencal

packings

with _{a size} ratio 4 and

iii ~ 0.5. (O) Theoretical points, (+)

experimental points.

~

tu, tzz

off

tu

0.6

0.4

o~

0

0 o~ 0.4 0.6 off 1

Fig.

6. Variations of the t,~ with _ii~ for Powell

packings

with _a size ratio 2. Fuit fine _: theoretical results. (+)

expenmental

points.

N° 8 A MODEL OF BINARY ASSEMBLIES OF DISCS l129

p(ijk)

of the different types ^of

triangles

are aise well

given by

the model

(Fig.

7 for _a size ratio

2),

with the _same remarks _{as to a} small

segregation. Indeed,

the factorizations

(3.12)

and

(3.13)

_are well

obeyed

in the

packings

obtained from numerical simulations. From ail these observations, we conclude that _Dur model

gives

a

good representation

of _a ramdam two-size mixture of discs.

~~~~~~' ~~~~~~'

P(122), p(222)

. t ^t

Î

P(122)

~ ~

6 o

oJ à p(l12)

0~ _o

~ _~ ^.

~

0,1 $

g

g p(Ill)

~

C

o o~ 0A 0.6 o-g 1

Fig.

7. -Plot of the p(ijk) as a function of C for numerical packings ^with a size ratio 2 and

n~ =

0.5. (O) Theoretical points~ (+) experimental

point~.

4.2 AIR TABLE PACKINGS. For the air table

packings~ comparison

with the random model is

difficult _{: ai} very short distances, discs

repell

each other _so that the _air flux _con be

ejected

and

there is _{no true contact} between the

particles consequently~

in _{a monosize}

packing,

the

regular

arrangement ^{of the}

grains

is obtained for a

packing

fraction less

by

a factor 1.16 thon

the maximum value 0.91 for a dense

packing

of

equal

discs il ii. To take this effect into

account, it is sufficient _to divide

C~

_m ail relations of

paragraph

³

by

^1.16 and

multiply

i~

by ,/1.16, assuming

however that the factor _is the _{same m}

binary

mixtures. Ail results below _are

given

first _at

given proportions (nj, n~)

^and

increasing packing

fraction

C,

then ai fixed C and

increasmg

_ni.

The

partial

coordination numbers zj and z~ are

plotted

as a function of

packing

fraction C for

an

equal

mixture of small and

large

discs (n ₌ ^{0.5 in}

figure

8. The agreement ^{with the model}

is not as

good

as in the

packings

built

by

^numerical simulations. The _same

study

was

performed

for other proportions ^{of the mixture}

(nj

₌ ^{0.75 and} _nj

=

0.25).

In ail _cases, the

variations _are

roughly

linear, with fluctuations that _con be attributed to fluctuations in the

composition

of the

packings

for different values of the

packing

fraction. The

large

discs have _a

coordination number

larger

than _in the model. This suggests a

larger

number of small

neighbours

for _a

large

disc, m accordance with the structure shown _in

figure1,

where the chain-like clusters of small discs favours mixed bonds. This _is confirmed

by

the

study

of the t,~. ^{The latter are}

reported

in

figure

⁹ for _an

equal

_mixture

(nj

=

0.5 _at increasing

packing

fraction. The percentage ^{of mixed}

neighbours

tj~, ^much

larger

than its random limit, 0.5, shows _a

large

segregation. The

study

^of the _p

(ijk

leads to the same conclusions. As shown _in

figure

10 for ni = 0.5, which _con be

compared

with

figure

7 for the

packing

built

by

numencal

simulations,

the

triangles (112)

and

(122)

_are much more numerous than _in the random _case.

The factorizations

(3.11)

and

(3.12)

do not hold in these

packings.

z~,zz

~+j

+ ,

t~~~~

^~~

, ^o

~ ~~~

iii

g~

v

0 0.1 0~ 0~ 0A 0~ 0.6 0.7 0fi

Fig.

8.- Variations of the =, with the packing fraction for air table packing~ with n~ =0.5.

lO) Theoretical points, (+) experimental points.

0.6 tl2,t22

~ ~ ~

++ ^{+ ~ +}

+ +

+

o~ o o o o o o oo o o o ~~ tu

oA

0~

* $ g g g

~t~

0~

f

O

f Î ( ^f

^~

+ o o ~

+ °

~

° o o ~ ~

~ ~ +~

+ + ~

~~

o-i

C 0

0 0~ 0A 0.6 0fi

Fig.

9. Plot of the t,, as a function of C for _air table

packings

with n~ 0.5. (O) Theoretical points, (+)

experimental

_points.

°~ p(122),

p(222)

~ +

+ +

~

+ +

~

+ +

~

p(122)

0.4 ^~

+ ~

+ + + + +

~~ p(l12)

o~

~ ~ + ~

p(222)

o-1

~

~

~ +

+ +

+ ~~

p(Ill)

~

~ ~ ~ ~

C

0 o-1 0~ 03 0.4 0~ 0.6 0.7 0fi

Fig.

10. Plot of the p(i jk) as a function of C for _air table

packings

with ni = 0.5. (O) Theoretical points, (+) expenmental points.

N° 8 A MODEL OF BINARY ASSEMBLIES OF DISCS l131

The same is _{true at constant}

packing

fraction C and

increasing

_ni. As _an

example,

^the

t,~'s

are

plotted

in

figure

^{for C} ₌ 63 fb. This

figure

_is to be

compared

^{with the} same one for Bideau _et ai.

[3]

in compact ^assemblies

(C

= 0.84), built in _a collective _manner, but where

contact percentages were taken _{mto account.}

Segregation

is clear for both _air table and

Bideau's

experiments.

On the contrary, ^{Powell assemblies at the} same

packing

fraction

C

= 0.84

yield

data doser to that of the random model

(Fig. 6).

This comforts the idea that segregation may ^exist in systems ^where rearrangements are

possible

and is

negligible

in numencal models where

grains

_are

positioned definitely.

~ tzi

tll

+ +

+

+ +

+ 0.4

~ +

o-z

~ +

0

0 0~ 0.4 0.6 0.8 1

Fig. il. Plot of the t,~ as a function of n~ for air table packingh with

packing

fraction

C 0.63. Fuit fine _: theoretical

points,

(+)

experimental

points.

5. Conclusion.

We have

presented

a theoretical model for the

description

of the

possible geometrical

correlations _m

binary

2d disc assemblies. It is derived from the Dodds model for compact

packings,

the

neighbour replacing

the contact, and holds at any

packing

fraction.

Experimental

results show that assemblies built

grain

after

grain by

numerical

simulations,

without

reorganization,

with _size ratio

R~/Rj

=

2 behave like random

packings

at low

(RSA)

or

high (Powell) packing

fractions. Even for

larger

size _ratio

(R~/Rj =4)

_a first order

approximation (Eq. (4.1))

_grues

good

^{results. On the} contrary, ^{assemblies built} on the air table show _a clear

segregation

due _to the

building procedure

that allows local

reorganization.

in the above

analysis,

we were interested in very local arrangements ^of

grains,

_i-e- the _mean

environment of each type ^of grains. At _an intermediate scale, _{it seems} that small discs

rearrange ^into chain-like clusters

along large

discs

though

the

packing

as a whole _remains

globally homogeneous.

This other aspect ^of

segregation

will be studied in _a

forthcoming

_paper.

Acknowledgments.

We would like _to thank M. Ammi and L.

Oger

for

providing

_us with the programs for

building

the numerical

packings.

This work _was

partly supported by

the GDR CNRS

«

Physique

des

Milieux

Hétérogènes Complexes

_»

References

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L. and Troadec J. P., J. Phys. France 47 (1986j 1697.

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