Scaling behaviour in size segregation ("Brazil Nuts")

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Scaling behaviour in size segregation (”Brazil Nuts”)

Pierre Devillard

To cite this version:

369

LE JOURNAL DE

PHYSIQUE

Short Communication

Scaling

behaviour in size

_{segregation ("Brazil}

_Nuts")

Pierre Devillard

HLRZ c/o

Forschungszentrum

Jülich,

Postfach

_1913,

D-5170 Jülich

_1,

F.R.G.

(Reçu

le 11 décembre

_1989,

_accepté

le 19 décembre

₁₉₈₉₎

Résumé. 2014 En utilisant une méthode de Monte Carlo due à

_{Rosato, Prinz,}

_Strandburg

et Swend-sen, nous avons effectué des simulations

_numériques

en deux dimensions de la

_{ségrégation}

_par

taille,

phenomène qui

se manifeste

_lorsqu’un

matériau

_granulaire

constitué de

_particules

de tailles

diffé-rentes est soumis à des mouvements vibratoires. Nous _proposons une loi d’échelle reliant la vitesse de

_{ségrégation v à}

_{l’amplitude A}

de la vibration et au

_rapport

R du diamètre des

_grands

_disques

sur

celui des

_{petits disques.}

_{Lorsque A}

est

_{plus grande qu’une amplitude critique Ac,}

une loi d’échelle est

valable: 03C5 ~

(R2-1)03B1 f

_{((A -Ac)}

(R2-1)z)

où 03B1 et z sont des

_exposants.

Abstract. 2014

Using

a

_previously

introduced Monte Carlo

method,

due to

Rosato, Prinz,

Strandburg

and

_Swendsen,

we _carry out in two dimensions numerical studies of size

_segregation,

a

_phenomenon

which occurs when a

_granular

medium made of

_particles

of different sizes is shaken. We _propose a new

_scaling

law

_relating

the

_segregation

_velocity

03C5 to the

_amplitude

of

_{shaking A}

and to the ratio R of the diameters of the small disks to the

_large

disks. ForA

_larger

than a critical

_{amplitude Ac,}

a

_scaling

law holds: 03C5 ~

(R2 - 1)03B1 f

((A -Ac)

(R2-1)z)

where 03B1 and z are

_exponents.

1

_Phys.

France 51

₍₁₉₉₀₎

369-373 ler MARS

_1990,

PAGE 369

Classification

PhysicsAbstracts

46.10 - 02.70 - 81.20E

Size

_segregation

is a

_phenomenon

which occurs when a box

_containing

a

_powder

made of

particles

of différent sizes is

_subject

to some vibration.

_{Upon shaking,}

the

_{larger particles}

tend

to rise to the

_top.

This

_phenomenon

seems to

_{depend mainly}

on

_geometrical

factors such as the size of the

_particles

and the _way the box is

_shaken,

and

_depends

little on other factors such as the différence in the

_density

of the

_particles

for

_example.

This has led Rosato et al.

_{[l, 2]}

to

study

a

_simple

idealized two-dimensional case where a

_large

disk is

_placed

on the bottom of a box

containing

many small hard disks. Rather than

_solving

the exact mechanical

_equations

of motion like in a molecular

_dynamics

_simulation,

a Monte Carlo method is used

_[3].

We first recall their method and indicate a few modifications we made.

370

In a two-dimensional

_box,

hard disks are

_placed

at random

_{nonoverlapping positions}

one after the other. When the desired number of disks has been

_placed,

we

_begin

to simulate their fall

to the bottom of the

container,

this is called the

_pouring

_part.

In order to simulate the random

collisions between

disks,

all disks are moved

_{successively by}

random

_{displacements.}

If the move leads to

_overlap

of

_particles,

the move is

_rejected

_anyway.

If no

_overlap

_{occurs, the} difference DE in

_{gravitational}

_{energy after} and before the move is calculated. The move is

_{accepted only}

with

_probability

_exp

_(-

_{A IkT),}

where T is some

_temperature.

When all the disks have been tried to be

_moved,

one _says that a

_pouring

_pass

has been

_completed.

One starts to

_try

to move the first disk

_again

and resume the former

_procedure.

After _many

_pouring

_passes,

a random

_packing

of hard

_spheres

is obtained. This ends the

_pouring

_part.

One can

_modify

the above

_procedure

slightly

to obtain a random

_packing

with

_only

one disk

_larger

than the others

_lying

on the bottom of the container. After the

_pouring

_part

cornes the

_shaking

_part.

7b simulate the

_shaking

of the

container,

one lifts ail the

_{spheres by}

a certain amountA called the

_shaking

_amplitude

and then let the

_system

relax as in the

_pouring

_part.

After the

_system

has relaxed

_{sufficiently,}

a

_{shaking cycle}

has been

_completed.

One then lifts all the

_{particles again}

and

_begin

another

_{shaking cycle.}

The

_temperatures

of interest are

_practically

zero

_temperature,

which means that

_upward

moves are forbidden

_(except

when one lifts all the

_disks).

Time t is discrete and will be defined as the number of

_{shaking cycles,}

one

_{shaking cycle involving lifting}

all the disks and

_letting

them fall. Former studies

_[2]

have shown that the

_large

disks rise

_linearly

with time t, the

_speed

at which the

_large

disk rises is called the

_{segregation velocity}

v. v

_{depends mainly}

on two

_parameters,

the

shaking

amplitude

A and the ratio R of the diameter of the

_large

disk to the diameter of the small

disks. The non-zero value of v arises because small disks can

_easily

fill a hole below a

_large

disk. On the other

hand,

the

_large

disk can move downward

_only

if a much

_larger

hole is

_present,

which is a more

_unlikely

event. It is the

_purpose

of this short communication to examine

_{quantitatively}

how the

_{segregation velocity depends}

on the

_parameters

R and A.

We followed the method of reference

_[1]

with a few différences. The

_pouring

_part

has been done at zero

_temperature

and a

_packed

structure has been obtained. After

_lifting

all the disks

simultaneously by

the

_quantityA,

each disk is

_sequentially

moved

_by

some

_randomly

chosen vector.

The

_length

of this

vector

is chosen

_randomly

in an interval

_{[ 6 min,}

_6maux].

The direction of this

vector makes an

_angle

with the horizontal axis which is random in

_[-

7r,

0],

(the

move is

_always

downward).

One tries to move each disk and

_reject

the move if it

_would

lead to

_overlap

or if a disk

would

_go

out of the container. Even if the move is

_rejected,

we

_go

then to the next disk. After one has tried to move all the

_disks,

one

_says

that one

_pass

has been

_performed.

In reference

_[1],

_passes

are

_stopped

when the

_energy

difrerence between two

_passes

is less than 0.1

_percent

of the total

energy.

We think that it

may

happen

that,

when the disks are

_densely

_packed,

all

_attempts

made in one

_pass

to move the disks fail because it would lead to

_overlap.

This would involve no

_change

in the

_potential

_energy.

_According

to the

_procedure

of reference

_[1],

one should

_stop

and

_begin

a new

_{shaking cycle}

_(i.e.

lift all the disks

_again).

We think that one should continue the

_passes

and

try

to move the disks downwards before

_lifting

them

_again.

It _may

_happen

that the directions of the vectors we chose to move the disks were not the

_"right"

ones and that another choice would lead to

_permitted

moves and to a decrease of

_potential

_energy.

We

_prefer

to fix the number of

passes

to a

_{"sufficiently" high}

number

_(one

hundred seems

_adequate)

and check that the results

do not

_depend

_{significantly}

_upon

this number.

The

_algorithm

we used is a cell method. When one wants to check for no

_overlap,

one knows

in which cell the disk we want to check is. One needs

_only

to check if the disk

_overlaps

with the disks which are in this cell and in the

_neighbouring

cells. The disks are classified

_according

to

their ordinate and one tries to move the lowest first. We used a box of size 24 x 48

_(in

units of the

Figure

1 shows the

_{segregation velocity v}

versus the

_amplitude

of

_shaking

_A,

for différent values of R.

_Averages

have been taken

_typically

over 40

_samples.

For

_amplitudes

smaller than a value

Ac,

around

_0.3,

_segregation

is _very slow. This seems to hold for all values of R. If one does not

_average

over

_samples,

one sees that the

_height

of the

_large

disk increases with time in a

stepwise

fashion. The

_large

disk remains at the same

_height

for some amount of time and

_then,

quickly

moves

_up

_by

an amount close to

’l,3.

As noted in reference

_[1],

this

_quantity

’13

is the 2

distance between

_planes

in a

_hexagonally

closed

_packed

structure.

However,

upon

averaging

over

several

_samples,

the

_stepwise

movement seems to

_give

_way

to a

_straight

line. When A becomes

very small

(typically 0.1),

for each

_sample,

the time between the

_steps

becomes

_larger

and

_larger.

As a

_consequence,

it becomes harder and harder to obtain

_good

statistics.

Fig.

1. -

Segregation velocity v

as a function of

_{shaking amplitude}

A for various values of R. Stars : R

=

_{1.1 ;}

downward

_{pointing triangles : R}

=

_{1.2 ;}

_{pluses :}

R =

1.5 ;

circles : R =

2.0 ;

_{upward pointing}

triangles : R

= 4.0. The vertical lines denote the _{error bars.} The box has a width L = 24 and a

_height

h =

48,

it contains 500

disks,

_including

the

_large

one. 1000 _passes were used to do the

_pouring

_part.

For the

shaking

part,

we used 100 _passes. To measure v, results were

_averaged

over

_typically

40

_samples,

each one

having undergone typically

100

_{shaking cycles. (More shaking cycles}

were carried for small values of

_v).

For A

_larger

than

_Ac,

the

_{segregation velocity}

first increases

_linearly

with A -

As,

and for

larger

A,

seems to saturate. The

_shape

of the curves

_suggest

that one could

_expect

some kind of

scaling.

When R is

_1,

no

_segregation

should occur. Since the

_segregation

occurs from the fact that a

_large

_disk,

in order to move downward needs a

_larger

void than a small

disk,

a natural

372

f

is a

_scaling

function and a and z are

_{exponents. Since,}

from

_figure

_{1, v -}

_{(A - Ac)}

for small A -

Ac,

one must have

For fixed

A,

v must increase with R

_thus,

At fixed

_R,

whenA becomes

_{large, v}

tends to some constant value V3at. V3at increases

with R, thus,

limx -+ 00

f (x)

= Constant.

and

Figure

2 shows a

_plot

of v

(R2 -

_{1)- a}

versus

_{(A -}

Ac)(R2 -

_1)’

with a =

0.6,

z = 0.1 and

Ac

= 0.3.

These values of a and z were the ones for which the best data

_collapse

occured. From

_figure

_1,

we

_{obtained Ac}

_{approximatively}

0.3. For R = 1.1

(stars), R

= 1.2

(downward pointing triangles),

R = 1.5

_(pluses)

and R = 2

(circles),

the data seem to

_fall,

within error

_bars,

onto a

_single

curve.

However,

for R =

4,

this does _not seem to be the case. This can be due to a finite size

_scaling

effect,

since our box has

_only

a width of L = 24. To see whether this is a finite size

_effect,

we

_repeated

the simulations for R = 4 on a box twice as small

₍₁₂

x

₂₄₎

and

_containing

four times less disks

(125 disks). Though

there are some finite size

effects,

they

seem to be too small to account for the

_{non-collapsing}

of the data for R = 4 observed in

figure

2. The reason for this

_{non-collapsing}

is

_probably

that the

_scaling

law

₍₁₎

is not valid for

_large

values of R.

Fig.

Z - A data

_{collapse plot}

for the scaled

_{segregation velocity}

_{v(R2 -}

_1)-a

versus

_{(A -}

Ac)(RZ -

1)Z

with Ac =

_0.3,

a = 0.6 and z = 0.1. The

symbols

have the same

_meaning

as in

_figure

1.

In

_experiments

on

_{phase segregation,}

the

_{segregation velocity}

has been observed to tend to

[4].

Since the model

_parameterA

should increase with the

_applied

acceleration,

our

_result,

that v tends to a constant for

_large

_A,

is in

_qualitative

_agreement

with

_experiments.

For A close

_toAc,

comparison

with the

_experimental

results of reference

_[4]

becomes more difficult.

We have carried out Monte Carlo simulations of the

_segregation

_process

in two dimensions. We studied how the

_{segregation velocity v depends}

on the

_amplitude

of

_shaking

A and on

_R,

the ratio of the diameter of the

_large

disk to the diameter of the small disks. As was hinted in reference

_[1],

we find a

_phase

transition at some critical

_valuea,,

below which v is

_very

small.

_A,

is

independent

of R. In the

_{regime A}

_larger

_{than Ac,}

we

_propose

a

_scaling

law. Our numerical data show that this law seems to hold for R _up to

_2,

with effective

_{exponents a}

= 0.6 and z =

0.1,

and

fails for

_larger

values of R.

Acknowledgements.

i

We thank J.S.

_Ho,

H.E.

_Stanley

and D. Stauffer for

_stimulating

discussions and D. Stauffer for a critical

_reading

of the

_manuscript.

References

[1]

ROSATO

_A.,

STRANDBURG

_K.J.,

PRINZ F. and SWENDSEN

R.H.,

Phys.

Rev. Lett. 58

₍₁₉₈₇₎

1038.

[2]

ROSATO

_A.,

STRANDBURG

_K.J.,

PRINZ F. and SWENDSEN

R.H.,

Powder Technol. 49

₍₁₉₈₆₎

59.

[3]

BINDER

_K.,

Monte Carlo Methods in Statistical

_Physics,

Ed. K. Binder

_{(Springer Verlag,}

Berlin,

Hei-delberg,

New

_York)

1979.