Stability of two-dimensional packings of disks built with ballistic deposition

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Stability of two-dimensional packings of disks built with ballistic deposition

Rémi Jullien, Paul Meakin

To cite this version:

Rémi Jullien, Paul Meakin. Stability of two-dimensional packings of disks built with ballistic de- position. Journal de Physique I, EDP Sciences, 1991, 1 (9), pp.1263-1277. �10.1051/jp1:1991201�.

�jpa-00246410�

Classification

Physics

Abstracts

61.42 46.30 68.70

Stability of two-dimensional packings of disks built with ballistic deposition

Rkmi Jullien

(I)

and Paul Meakin

f)

(~) Bit. 510,

Physique

des Solides, Universitd Paris-Sud, Centre

d'orsay,

91405

Orsay,

France

f)

Central Research and

Development Department,

E-I- du Pont de Nemours and

Company,

Wilrnington,

DE 19880-0356, U-S-A-

(Received

6 March 1991,

accepted

in

final

_{form 23}

April1991)

Abstract. Two-dimensional

packings

of identical hard disks of unit diameter _are

generated using

_a ballistic model with

complete restructuring

and

starting

from _a first _row made with disks

horizontally regularly spaced by

a =

2 sin #, with #

ranging

from ^" to

I.

_The

_stability

_with

6 3

respect to random vertical

displacements

for the disks in the first _{row was} studied. A transition from _a

pseudo-regular

_array without defects _{to a} random structure with defects is observed when

the amount of disorder 8

(amplitude

of the vertical

displacements)

_{passes a} critical value

8~ ^and a

phase diagram

is measured in the

(#,

3

plane

which _can be understood in terms of

simple geometrical

arguments. The random _structure found for 3 _~ 3

~

does not

depend

too much

on #i and

strongly

ressembles to the _one obtained when

starting

from _a horizontal basal line the

density

of defects decreases _{as a} power law with

height

and the

large-height

defect-free structure is characterized

by

_a broad distribution of bond

angles.

1. Inboducfion.

Random

packings

of hard disks

(in

two

dimensions)

_or

spheres (in

three

dimensions)

have been the

subject

of considerable interest for _{many years.} The

major application

has been the

development

of _a better

understanding

of the _structure of

simple liquids [I]

and

glasses [2, 3].

The two-dimensional _case is

particularly interesting

since the structure of the random

packing

is

highly

unstable and

strongly depends

_on the

boundary

conditions

[4].

^In some cases, the structure of defects _may exhibit _some fractal _structure

[5].

Here _we

study

_{a very}

simple

two- dimensional

packing

which _was first introduced

by

Visscher and Bolsterli

[6]

and reinvented

by

_us in the context of random

deposition

_processes

_[7].

In this model hard disks _are released

one after

another, along randomly positioned

vertical

trajectories.

Each disk follows the

path

of steepest descent _on the

packing

until it reaches _a final

position,

stable

against gravity,

before the _next disk is released. lvhen the disks _were

deposited

onto _a horizontal basal

line,

the

density

of the

resulting

disordered

packing

_was found to be

equal

to 0.818±

0.001

[7].

In _{a more} recent work

[8],

we have

analysed

the

density

of defects _{p~, a} defect

being

defined _{as a} disk

having

_a number of _contacts different from four. This

density

_was found _to decrease with

height

h _{as a power}

law, p~(h)

~

h ~, with _an

exponent

a

numerically

close to

2. Moreover the bond

angle distribution,

in the defect-free random

packing

^obtained in the

asymptotic

limit h

- tx~, was found to reach _a characteristic stable broad

shape.

In this paper we extend the

preceeding study by considering deposits

_{on a} first _row of disk with _a controlled amount of disorder. This allow _us to

investigate

_a transition between _a

regular

_array of disks and _a disordered

phase.

2.

Description

of the model.

Explorating

small scale simulations in which hard disks _were

deposited

onto a row of

equally spaced

disks with the _same

height

indicated that ordered

deposits

_were

generated

for center to

center disk

spacing

a in the range

do

_~ a _~

/ _do

_where

_do

_{is the disk diameter. However}

as

the

spacing

a

approaches do

or

/ _do (corresponding

to a

triangular lattice)

the hard disk

packing

becomes very sensitive to

perturbations

and in the limits _a=

do

and _a=

/ _do

even the

perturbations resulting

from the round off _errors in the double

precision

arithmetics used in _our models _was sufficient to induce _a transition from _a structure with

long

range translational order to a disordered structure that in the

asymptotic (h

- tx~

)

has

long

range orientational order but

only

short range translational order. To

explore

this transition from _« ordered _» to disordered structures in _{a more}

systematic

_{manner a} model in which the substrate disorder could be

continuously

tuned _was

developed.

In this model the first _row is made of L disks of unit diameter whose centers are

positioned

at horizontal coordinates

x~ =

(I

^{a with I} =

1, 2,

,

L 2

and vertical coordinates _:

Zj =

3fi

where

f~

are

independent

random variables

uniformly

distributed between 0 and I. In the

following

_we will often make _use of _an

angle

parameter #i related to the horizontal

spacing

_a

through

_:

a =

2 sin _$~

(1)

$~ is the

angle

from the vertical of the bond directions of the

regular

_array obtained for 3

=

0,

whose

density

is _:

g~ fir

(2)

~ 4 sin

(2 $r) fi~

Here bonds _are defined _as lines

joining

the centers of

contacting

disks in the

deposit.

We will

vary $r between the two limits 30° and

60(

which

correspond,

for 3

= 0 to the

regular

« horizontal _»

(a

=

I

)

and vertical _»

(a

=

/) _{triangular lattices,} respectively.

The

density

is maximum and

equal

to p =

"

m 0.907 in these two limits _while it is minimum and

equal

2

/

to

(

m 0.785 for the square lattice obtained for #i

=

45°

(a

=

/).

Then the

packing

is built within _a

strip

of width La

(with periodic boundary

conditions at the

edges

of the

strip) by using

a

procedure

in which disks of unit diameter are released _{one at}

a time. To

deposit

a disk _we first choose _a random number between 0 and La to define _a random vertical

trajectory.

Then _we release the disk from above the

top

^{of the}

deposit along

this

trajectory.

When the

moving

disk contacts _a

previously deposited

disk it _{starts to} roll down _on it until it either reaches _an

equatorial position

where the two disks have the _same vertical

coordinate,

_or it finds _a second _contact with another disk of the

deposit.

If it reaches

an

equatorial position,

it looses its

previous

_contact, falls

vertically

until it

again

contacts a

particle

in the

deposit,

etc... As _{soon as} the

moving

disk has found _two contacts, we

analyse

the

stability

of its

position

under

gravity by checking

if the horizontal coordinate of its center lies between those of the

contacting

disks. If the

position

is

stable,

the

moving

disk stops ^and

another disk is

deposited.

If not, ^the

moving

disk looses its _contact with the upper

contacting

disk to roll _on the lower _one and the motion is continued _as

long

_as the

moving

disk does not reach _a stable

position.

We

always

wait until the n-th disk has reached _a stable

position

before

releasing

the

(n

₊ I

)-th

disk.

Moreover,

when the stable

position

is

reached,

the disk stays

permanently

in this

position (we neglect many-particles

effects such _as

avalanches).

In

practice,

in _our

largest simulations,

_we have taken the

quantity

La _{as an}

integer,

La

=

16

384,

and _we have varied the

integer

L in order to most

closely approximate

the selected

$~ values. For each _$r

value,

_we have

deposited

108 disks and _we have calculated various

quantities

such _as the coordinates of the

disks,

their number of contacts, ^the

orientations of the

contacting

bonds

compared

to the vertical and the

density

of the

packing

in the

large height regime.

3.

Qualitative

results.

In

figure la, b,

_c and

d,

_we show small

regions

the first _rows for

typical packings

obtained with

$~ =

35(

and

increasing

values of 3

(3

=

0, 0.1,

0.2 and 0.5

respectively).

The bonds between

contacting particles

_are shown _on these

figures

and

only

the defect

particles,

I-e- those

having

a number of contacts different from

4,

_are indicated

by

circles. For 3

=

0,

the

expected

stable

periodic

_array is found, For 3

=

0.

I,

this array is

only slightly

distorted and all the disks inside

the bulk _conserve four contacts i the structure is _«

regular

» and does not exhibit any defect.

For 3

= 0.2 and 3

=

0.5 _a

strongly

disordered _structure is observed with _many defects _near the bottom. The

density

of defects decreases

rapidly

with

increasing height.

A similar transition between _a

pseudo-regular

_array without defects for low-a values and _a disordered

packing

with defects for

large-

3 values is observed for all values of _$~. In the

large-

3 disordered

phase

the

particular

structure of the defect and the

height dependence

of their

density

is _more

clearly

seen in

larger

scale

pictures

such _as that shown in

figure 2a,

b and _c for 3

= 0.5 and

#

₌

35(

45° and

55( respectively.

The defect

pattern

consists of streaks whose

mean orientation with the vertical is _$r. The decrease in the defect

density

with

increasing height

is

quite _apparent

in these

figures.

4.

Quantitative study

of tile Wansition.

A convenient _way to define the location of the transition is to

plot

the total number of defects

as a function of 3 for _a

given #

^{value and} to determine the value

3~

at which the first defect appears. In

practice

_we measured the number of defects

N~

^{found within} a distance of 8 disk diameters from the substrate. Because the defect

density

decreases

quite rapidly

with

increasing height

this

quantity

is

approximately proportional

to the total number of defects

and _can be measured much _more

quickly.

The numerical estimates of

3~,

for

#

₌

32.5( 35(

37.

5( 45(

55° _are shown

by

dots in the _«

phase diagram

_» of

figure

3. Moreover _we have tried _to fit the behavior of

N~

near the transition

by

_{a power} law of the

type

:

N~

= C

(3 at

_)Y

(3)

The estimate of the exponent y appears ^to be

roughly equal

to

3,

for

# ranging

from 30° to

35°, 0.0 35°,0,1

« ~ < >

30 DIAMETERS 30 DIAMETERS

a)

b)

35°1 0.2 35°, 0.5

. 30 DIAMETERS ~ « _3o DIAMETERS >

c)

dj

Fig.

I.

-Typical examples

of

packings

obtained with $r =

35[ Cases _a, b, c and d

correspond

to 3

= 0.0, 0.1, 0.2, 0.5,

respectively.

v 35°

'.

v 45°

0.5 ~l _0.5

'~

« - « ,

l 024 DIAMETERS 1024 DIAMETERS

a) b)

v 55°

= o.5

'~", ../

fl

* I024DtAMETERS ^~

C)

Fig. 2.- Large scale view of the

packings

in which

only

defects _are shown

by

black dots. These

packings

are built with 8

= 0.5 and parts _a, b and _c

correspond

to #i

= 35°, 45° and 55(

respectively.

DA

03

/

o-z

o-i

O-O

30 35 40 45 50 55 60

lt _(deg.I

Fig.

3.-Phase

diagram

in the #i-3~ plane. The dots

correspond

to numerical results and the

continuous _curve to

analytical

results _as

explained

in the text. The _error bars _on 8~

0ess

than

0.001)

are

always smaller than the dot thickness.

about 43° and

roughly equal

to 2 for

# ranging

from 43° to 601

Examples

of fits _are

given

in

figures

4a and b where

log N~

has been

plot

_{as a} function of

log (3 3~)

for # ₌ ^{35° and}

# ₌ 55°

respectively.

All these numerical results _can be understood

using simple geometrical

_arguments. The idea is that _a defect is created _{as soon as} the relative

heights

between

neighboring

disks in the first _{row are}

sufficiently high

to create _an

equilateral triangle

of bonds when other disks _are

deposited

_on the

top

^{of them. The critical}

configurations

_are of

quite

different _nature when

#

is close to 30° _or when it is close _to 60[ In the first _case, the

equilateral triangle

is formed _as

soon as two

neighboring

disks of the second _{row come} into _contact. Since the random vertical

displacements

are

uniformly

distributed between 0 and

3,

the smallest value of 3 which _can

creates such event

correspond

to the

symmetrical

situation

depicted

in

figure

5a which

involves three

neighboring

disks

(I,

I ₊

I,

I ₊

2)

of the first _row with _z~

= z~~~ = 0 and

z~~i = 3.

Taking

the

origin

_as indicated _on the

figure,

the coordinates of

point

A _are 2 sin

#

and 3 and the first coordinate of

B,

_{x~ can} be then calculated to be _:

x~ = sin

#

3

i~ _~

(4)

~~~sin ~

Then the critical condition becomes _:

xB #

(5)

In the other _case, the first

equilateral triangle

_occurs when _a

deposited particles

touches

y ~35°

60= 0.106

#~

£ /

/ O-102

SLOPE

m 3.0

~

~) ^~

l/&-&oj

^{~ ~}

V=~~

6o= 0,136

f

~

l35 /

/ Slope=2.0

~

~ h

9 8 7 6 5 4

b) In(b -6c)

Fig.

4.- Plot of

log N~

as a function of log (8

8~).

Parts _a and b

correspond

to #i ₌ 35° and

# ₌ 55[

respectively.

simultaneously

a disk of the first row and _a disk of the second _{row, as} shown in

figure

5b. Now the critical

configuration

involves

only

two

neighboring

disks

(I,

I ₊

I). Using

the notations of the

figure,

the critical condition is _:

x~ =

2 sin _$r

~

(6)

2

The two conditions lead to _a second

degree equation

in

al,

_which

can be solved _to

give

_:

3j

=

2(cos~ _#

_A^~

~/(cos~ _{# A~)~}

4 A^~

sin~ _# ) (7a)

with:

A

= sin

#

in the first _case

(7b)

2 and _:

A

=

~

sin

#

in the second _case

(7c)

2

This

equation

determines the

phase diagram

shown

by

the continuous _curve in

figure

5. In the intermediate

region,

the lower of the two

possible

values for

3~

obtained from

equation (7)

ti

A

6 6

o

a)

c

ti

A 6

o 2 sin y

b)

Fig.

5.- The critical

configuration corresponding

to the appearence of _a defect. Parts _a and b

correspond

to #i values close _to 30° and

60( respectively,

and _are characteristic of the two branches of the

phase diagram.

must be selected. Thus there is _an

angular point

which is located _near 43t The

expansions

of the

analytical

formulae at first order _near 30° and 60°

give

3~

_{= $r}

)

=

0.0175($r

30°

) (8a)

3~

=

I #

=

0.0302

(60 #

^°

(8b)

The fact that

3~

^{tends to} zero when

#

^tends to either 30° _or 60° _means that the

triangular

_array,

which is the most compact

arrangement

of

disks,

is unstable _{as soon as an} infinitesimal amount of disorder is included. This _was

already

noticed in _our

previous

work [7] ^where we

observed

that,

when

starting

from _a

regular

line of tangent

disks,

the round off _errors of the

computer

were sufficient to induce the random structure.

The value of the

exponent

y can be also understood from the above

geometrical arguments.

In the first _case, when 3 is

slightly larger

than

3~,

^{the condition} to obtain defects writes _:

zi+zi~~-2z~~i >23~. (9)

Since z~, z~~ i and z~~~ are distributed

independently

with _a flat

probability

distribution between 0 and

3,

the

probability

to have _a

configuration satisfying

the above condition is

proportional

to

(3 3~)~.

^{Since the number of defects is}

proportional

to this

probability,

y =

3. In the second _case, the condition for

creating

_a defect involves

only

two

adjacent disks,

so that defects _are formed if _:

z~ -z~~i

>3~

leading

to a

probability proportional

to

(3 3~)~

and _y

=

2.

S.

Density

of defects.

Some results for the variation with

height

of the

density

of

defects, p~(h),

in the disordered

phase,

are

given

in

figures 6a,

b and _c, for different values of

#: #

₌

35(

45° and 55° and different values of

3,

3

=

0.5,

1-o and

I-o, respectively.

The results _are consistent with a

power law behavior of the

type

:

p~(h)

~h~"

with _a

=1.68,

1.65 and 1.80 for

#

=

35(

45° and

55( respectively.

The _same kind of behavior is found for all

wand

3 values within the disordered

phase

and the estimated _a

exponent

^is

always lying

in the _range 1.6

~ a ~ 2.0. The

exponent

a

might

_very well

depend

on the model parameters

but,

due to the numerical _{errors, we} have not

strong

^numerical evidence to support ^this conclusion. It is worth

noticing

that the value

a = 2

corresponds

to the _one found when

starting

from _a basal line

[8].

6. Distribution of bond orientations.

We have calculated the distribution

Ni (0

of the

angles

0

giving

the orientation of the bonds from the vertical

direction, Ni(0)

do is the number of bonds whose orientation

angle

lies

between 0 and 0 + do. In

figures 7a, b,

_c and d _we report ^these distributions for

#

=

35° and 3

=

0.1, 0.25,

0.5 and

I, respectively.

In

figures 8a, b,

_c and d _we report ^the

corresponding

_curves obtained for

#

=

55t In each

figure,

four different

distributions,

obtained with _a number _n of

deposited particules lying

in the

respective

_ranges n ₌ 0.5 _x 10? 1.0 _x

lo?,

n = 1.5 _x

lo?

_2.0

x

lo?,

n = 3.5 _x

lo?

_4.0

x

lo?,

n =

7.5 _x

10?

_8.0

x

10?

_have been

superimposed.

^While for the 3 value below the

threshold,

_we obtain _a

relatively

narrow

peak

centered at 0

=

#,

^with a

symmetric shape

and _a finite

width,

for 3 values above the

threshold,

we

get

a broad

distribution,

which is _more and _more extended from 30° to 60°

when 3 increases. The small tails below 30° and above

60(

_are

only

observed for

relatively

small

heights

and _are due to the presence of defects _at the bottom of the

packing. They disappear

for

larger heights

where the distribution reaches _a stable

asymptotic

limit

characteristic of the disordered _structure. It appears

that,

in the

large

3

(strong disorder) limit,

the

shape

of the distribution is almost

independent

on

wand

ressembles to the _one found in the _case where _we start from _a horizontal basal line

[8].

^The

peak

at 30° is however _a little bit less

pronounced

here.

Also,

for _a

given sample,

the fine structure seen in the _curve does _{not vary} with

height

and is thus

typical

of the nature of the disorder

put

in the first _row

(it changes

when _one

changes

the seed of the random

generator).

This memory effect _comes

from the fact

that,

when there is no

defects,

successive bonds stay

strictly parallel

from _rows to

rows. This fine structure is

averaged by increasing

L _so that _we expect to get a continuous

curve in the infinite-L limit. Thus it appears that the disordered structure in the

large-h

limit is

of

quite special

nature since it exhibits strong

long

_range orientational order and _no

long

_range

,-Slope=-1.68

C

y=35$

6=O.5

2 3 4 5 6 7 8 9

In(h) j)

,~slope=-1.65

C

y=45$6=1.o

2 3 4 5 6 7 8 9

In(h)

b)

2

~

^slope « -1.80

C

y =

55$ 6 =1.o

2 3 4 5 6 8

In(h) c)

Fig.

6.-

Density

of defects

pd(h)

_{as a} function of the

height

h

(log-log plot).

Parts _a, b and _c

correspond

to

(#i

=

35[ 8

=

0.5), (45(

1.0) and

(55(

1.0)

respectively.

1800000

y 35$ b 0.1

1400000

6~ 1000000

£ 800000

200000

~

30 35 40 45 50 55 60

al 6

y=35$ 6= 0.25

30 35 40 45 50 55 60

b)

o

y 35$ 6 0.5

30 35 40 45 50 55 60

C) ⁰

y=35i 6=1.0

30 35 40 45 50 55 60

d)

₀

Fig.

7. Distributions

Ni (8 (not normalized)

for the bond orientation

angles

8 from the vertical for

#

=

35[ Parts _a, b, _c and d

correspond

to 8

= 0.1, 0.25, 0.5 and 1.0,

respectively.

JOURNAL DE PHYSIQUE I T I,M 9, SEPTEMBRE 1991 ^5o

v"55t _6=0.1

30 35 40 45 50 55 60

a) o

V "55$ b 0.25

~f

~400000

200000

30 35 40 45 50 55 60

b)

⁸

v =55°, 6 =o.50

30 35 40 45 50 55 60

~) ^o

v*55i &=1.0

30 35 40 45 50 55 60

d) o

Fig.

8. Distributions

Ni (9 ) (not normalized)

for the bond orientation

angles

9 from the vertical for

#

=

55°. Parts _a, b, c and d

correspond

to 8

= 0.1, 0.25, 0.5 and 1.0,

respectively.

positional

order.

Moreover,

since the

density

of defects decreases

algebraically,

the

long

range orientational order takes

place quite quckly

when h increases and thus _we have

quite

good

confidence that _we have here

enough

informations _on the

asymptotic limit,

in

contradiction with the conclusions of

Delyon

and

Levy _[9]

_on the _same kind of model.

7.

Packing density.

We have also determined the distribution

N~(# )

for the

angle #

between the two bonds that _a

given

disk makes with the two

contacting

disks

placed

below. This

angle corresponds

to the top

angle

of the rhombus made with four

neighboring

disks when there is _no defect. The

knowledge

of this distribution allows _a

precise

calculation of the

density

of the

packing through

the formula _:

~

N2(# d#

~ ~

N~(#)

sin

# d#

~~ 31.25$32,5$35°,37.5°,40$42,5$45°

o.86

c~

0.2 0.4 0.6 0,8 1-o 1.2 1.4

a)

6

0.88 4~a 47_50, sol 52.5$57.5°,58.75°

o.86

~

o.84

0.82

o.8o

~ i 2 1.4

0.7§

_{~ ~ ~ o.4 °.6 ~'~}

~ ~

Fig.

9. Bulk

density

_p calculated in the

large-h asymptotic regime

plotted _{as a} function of 8. In a) _are shown the _curves

corresponding

to #

= 31.25(

32.5( 35(

37.5( 40( 42.5° and 45[ each _{curve can} be

recognized by

the fact that _p

(8

=

0)

decreases

monotically

when # increases from 30° to 45°. In

b)

are

shown the _curves

corresponding

to #

=

45( 47.5( 50( 52.5( 55(

57.5° and

58.75(

each _{curve can} be

recognized by

the fact that _p

(8

=

0)

increases

monotically

when # increases from 45° _to Ml

Alternatively

the

packing density

_{p can} be obtained

directly

from the

particle

coordinates and this

approach

_was used _to obtain the data

presented

here. In the

large

disorder

(large-3)

limit

we obtain _p

= 0.806 _±

0.001, independent

of _{$r, a} value

definitively

smaller than the value p =

0.818 _± 0.001 found for the disordered

packing

obtained when

starting

from _a basal line.

This is _a

small,

but

significative,

difference which

might

reflect the fact

that, although apparently

universal

(I.e. independent

_{on $r and} 3 for

large 3)

^the

present

disordered model

might

not be in the _same

universality

class than the model which starts from _a basal line. The transition _can also be _{seen on} the _p

(3

_curves which _are

reported

in

figure

9a and b. These

curves exhibit _a

sigmoidal shape

_near the transition but the

singularity

at

30

is

certainly

weaker than in the _case of

N~.

It is worth

noticing

that while for $r

lying

ⁱⁿ the range 30~37.5°

or in the range 55~60° the order-to-disorder transition

corresponds

to a

sharp

decrease of the

density,

for 37. 5° _~

#

_~

55(

it

corresponds

to an increase of the

density,

this _means

that,

_near

45(

the disordered

phase

is denser than the ordered

phase

_~p45

=

~

=

0.78539...).

4 8. A model to describe the structure of defects.

In order _to understand the characteristic

stripped

structure and

algebraic

decrease of the

density

of defects in the disordered

phase

we have built _a

simplified

model which contains most of the features of the defect formation. The model _starts from _a first _row made of L sites located _at the

positions

_x~

(I

=

1, 2,

...,

L).

Two semi-infinite

straight lines,

oriented to the left and _to the

right,

with the _same orientation compare ^to the vertical _are

propagated

from each site. The intersections of these lines define _a

regular

_array. Periodic

boundary

conditions

are considered. Bond oRentations of

fit (I

and

0~(I )

_are associated with the fines

emanating

to the left and

right

directions

respectively

from the I-th site. For the first _{row, we} set

0t(I)

and

0~(I)

to be

randomly

and

uniformy

distributed between 0° and 901 Then _we

investigate

all the line intersections

sequentially,

row

by

_row,

starting

from the second row.

At _a

given intersection,

which

corresponds

to a

crossing

between _a left line I and _a

right

line

j,

we calculate the

quantity #

=

fit (I )

+

0~Q

which represents the

angle

between bonds at this

site in the model. If

#

lies between 60° and

120(

there is _no

«interaction»,

I-e-

0t(I )

and

0~(I ) keep

their

original values,

the intersection is not _a defect

(this corresponds

in the

original

disk

packing

model to the fact

that,

when there is _no

defect,

the top

angle

of the rhombus made with four

neighboring particles

lies between 60° and

120].

If

#

is smaller than 60° _or

larger

than

120(

the site is considered _{as a} defect and _new values of

01(I)

and

0~@

are

assigned

to the

intersecting

lines. In the

original

model these _new values

depend

not

only

_on the

previous

values of

01(I)

and

0~Q)

but also _on the bond orientations _at the

neighboring

columns which

certainly depends

_on the

original

3 and

#

value. We _assume that the main feature is that the _new distribution of the bond

angle Starting

from _a defect is _no

longer uniformly

distributed between 0° and 90° but is concentrated into a narrower range so

that in the

asymptotic (large height)

limit all

angles

will lie between 30° and 60t In the

simplified

model the bond

angles of (I)

and

0~Q) following

_an interactive intersection at a

defect _are selected _at random from _a uniform distribution between _q and 90° _q

(where

_q is

a parameter ^{of the}

model).

In the actual disk

packing

_process lines _are created _or

destroyed

at a defect and this is _not taken into account in the

simplified

model.

With this

simplified

model _we have found

that,

_{as soon as q} is

larger

than _a critical value

q~ of order

7(

the

density

of defect decreases

algebraically

with the

height

with _a

characteristic exponent a that varies

continuously

with _q.

a tends to zero when _q tends _to q~. The

algebraic

law is tested with

considerably

_more

precision

than in the

original

model.

An

example

of the _structure of defects obtained with the

simplified

model is shown in

figure

10 for L

=

000 and _q

= 12° for which the _a

exponent

^{is close} to 2.

Although

there _are less correlations between

defects,

the

stripped

structure of

figure

lo look like those shown in

y

~.

t ,

_/.~,

i~'

:

?

? ;.^'

1,

.

, ^$'' ₍ _ ^'

~ ^{' .}

~ l

I

-

'/

_ ^{/ ,' ' P}

,

( $ ^, '

'

' ~ ^'

.

' '

o '_

' ~

L' . f

' ' -

Fig.

10. Structure of defects of the

simplified

model for L

= 1000 and

~D ⁼ 12[

figure

2. The results of this model indicate that the exponent a

might

_vary

continuously

with

parameters 3 and

#

in the whole disordered

phase.

However the transition found at

q~ is of

completely

different nature than the _one found at

3~

^{in the}

original

model since the

geometrical ingredients

which

explain

this transition _are not taken into account in the

simplified

model.

9. Conclusion.

We have obtained evidence for _an order-disordered

phase

transition in _a two-dimensional

packing

of disks when

including

disorder at a

boundary.

We have been able to _recover the

phase diagram by simple geometrical _arguments.

Since _we

always

obtain _a defect-free

structure in the

large-height regime,

we are convinced that the disorder

coming

from the

random choice for the successive vertical

trajectories

is irrelevant in the mechanism of this transition and thus _our results

might

be very

general.

We intend _to

apply

the _same

procedure

to _a variant of the Bennet model

[3]

in the

strip

geometry. ^{In this model disks} are added

sequentially always

at the lowest

position insuring

two contacts with

previously deposited particles

: the

procedure

is _now

completely

deterministic _so that _{we can} be confident that

disorder _comes

only

from the first _row.

References

[Ii

BERNAL J. D., Proc. Roy. ^{Sac. London A280} (1964) 299 _; FINNEY J. L., Proc. Roy. Sac. London A319

(1970)

479.

[2] CARGILL S., J.

Appt. Phys.

41

(1970)

2248.

[3] ^BENNETT C. H., J.

Appt. Phys.

43

(1972)

2727.

[4] ^BIDEAU D., GERVOIS A., OGER L. and TROADEC J. P., J. Phys. France 47 (1986) 1697.

[5] ONODA G. Y. and TONER J.,

Phys.

Rev. Len. 57

(1986)

1340.

[6] VISSCHER W. H. and BOLSTERLI H., Nature 239

(1972)

504.

[7] MEAKIN P. and JULLIEN R., J.

Phys.

France 48

(1987)

1651.

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Europhys.

Lett. 14

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667.

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Phys.

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4471.