Simulation on Size Segregation: Geometrical Effects in the Absence of Convection

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Simulation on Size Segregation: Geometrical Effects in the Absence of Convection

S. Dippel, S. Luding

To cite this version:

S. Dippel, S. Luding. Simulation on Size Segregation: Geometrical Effects in the Absence of Con- vection. Journal de Physique I, EDP Sciences, 1995, 5 (12), pp.1527-1537. �10.1051/jp1:1995215�.

�jpa-00247156�

J.

Phys,

f France 5

(1995)

1527-1537 DECEMBER 1995, PAGE 1527

Classification Ph_Vsics Abstracts

46,10+z

64.75+g

81.35+k

Simulations

_on

Size Segregation: Geometrical Elllects in trie Absence of Convection

S.

Dippel(~,~)

and S.

Luding(~,~)

(~) Theoretische

Polymerphysik,

Rheinstr. 12, D-79104

Freiburg, Germany (~)(*)

HLRZ, KFA Jiihch. D-52425 Jiilich,

Germany

(~) Laboratoire

d'Acoustique

et

d'optique

de la Matière

Condensée(**),

Université Pierre et Marie

Curie,

4

place Jussieu,

75005 Pans, France

(Received

10

May

1995, revised 28

August

1995,

accepted

5

September 1995)

Abstract.

Using

_a modification of _an

algorithm

introduced

by

Rosato et ai., _we

study

the

segregation

behaviour of _a vibrated container filled ~vith monodisperse discs and _one

large

_m-

truder disc. We observe size

segregation

_on time scales

comparable

to

experimental

observations.

The

large

disc _oses with _a size

dependent

velocity. ^For size ratios smaller thon _a cntical value

we find metastable

positions during

the ascent, _in agreement with

geometrical

arguments by Duran et ai. This confirms the importance of geometry _{as one} of the

possible

_causes for _size segregation. We relate _Dur results to recent experiments ^{and discuss the mechanisms} active _m size segregation m the absence of convection.

l. Introduction

The

investigation

of the behaviour of noncohesive

granular

materials has received quite a lot of interest in the last years

iii.

Powder processing is of

great practical importance

and further-

more,

granular

media show

striking properties

of both theoretical and

experimental

interest.

One feature of the behaviour of

granular

matenals which is

usually

considered

annoying

in industrial processes is size

segregation [2-4].

The

demixmg

of

multidisperse

po~A>riens, ^where the

large particles

_use to the top, occurs m shear flow _as well _{as m} vibrated

powders.

The

latter is the effect _we

study

in the present ^work.

Recently,

there has been _a vivid discussion _on the

question

^{whether size}

segregation

_m vibrated

powders might only

be due to

convection,

another

phenomenon widely

observed in

granular

materials. This would

imply

that

segregation

should be

regarded

_as

being

_a

secondarjr

elfect rather than _a distinct

phenomenon [5.6].

^In

systems

where

strong

^{convective motion takes}

place,

convection

certainly

is the

dommating driving

^force for

segregation

_[5]. In _a vibrated

container with

rough

vertical

walls, large particles

_move to the

top together

with the small

(*)where correspondence

should be

addressed;

electronic mail:

s.dippel@kfa-juehch.de (**)URA

800

Q

Les Editions de

Physique

1995

1528 JOURNAL DE

PHYSIQtiE

I Nol?

ones, due to the

up,vard

con,>ective flux in trie middle of the container. The small

particles

mo,re do,vn _agam, but the

large particles _stay

at the

top

smce

they

_are not able to

penetrate

trie rather dense _array of small

particles

belo,v them. In

experimental setups,

convection is mitiated

already

for quite ^low ,;alues of the vibrational acceleration. For very small vibration

iiitensities,

convection is confined to a small

region

in the _Upper part of ^tire container

I?i.

^{As has} been sho,vn

experimentally [8, 9],

size

segregation

can be observed in this so-called

quasi-static regime,

too. i~Te concentrate on it in the

following.

It has been sho,vn that

geometric

elfects

play

_an

important

role in size

segregation

in the

quasi-static

_case

_[9].

Duran et ai. [9]

proposed

_a model for size

segregation

based _on trie e,>aluation of trie

positions

that a

large

disc

(or sphere)

of radius R could take in _a

regular

array of

monodisperse

small discs

(or spheres)

of radius _r. In reference [9] a critical size ratio 4l~ =

RIT

_(4l~

= 12.9 in

2D,

4l~ ₌ 2.78 in

3D)

is deduced from these considerations. For size ratios belo~n. 4l~ trie stable

positions

for the

large

disc _are

separated by

_some

distance,

~Arhereas above this cntical ratio

they

_are continuons. From that _{one can} conclude that the intruder should rise

mtermittently

for small size ratios and

continuousljr

for size ratios 4l _>

4l~,

_a

finding

which is

supported by experimental

observations.

Trie existence of _a critical size ratio below which _no

segregation

is observed _was discussed

extensively [10,11]. Experimeiits by

Duran et ai. [8] show that _a size ratio exists below ,vhich

no ascent is observed within hours of experiment. ^The value of this critical _size ratio _~A.as

mentioned to

depend

_on the

shaking amplitude,

the size of the

container,

and _on the initial

position

^of the

particle.

Size

segregation

_was also related to the evidence of micro-fluctuations

m the

positions

of the

particles. Together

with

rougir walls,

that _may lead to micro-cracks _or

long

_range block

sliding

in the bulk. A

study

of these

phenomena

_m the 2D

monodisperse

_case

is

presented

in reference

[12].

Quite

_a fe~A> computer simulations of shaken

granular

media

exhibiting

size

segregation

bave been put

for,vard, including

à40nte Carlo like simulation methods

[13,14],

molecular

dynamics

simulations

[6,15]

or

sequential deposition

models

[16].

^{In this work} ~Are introduce _an extension of the

algorithm ^introdùced by

Rosato et ai.

[13]

^{and discuss the}

physical

relevance of the

parameters used. The

dependence

of the _ascent velocities of the

large

disc _on the _size ratio 4l

is

iuvestigated

^and the mechanisms causmg the ascent _ai-e discussed. ive find

good qualitative agreement

^,vith recent

experiments [8], sho,ving

that _size

segregation

_{occurs even} without

convection, only

due to

geometric

elfects.

2. Simulation Method

2.1. THE ORIGINAL ALGORITHM. We first give a short

description

of the

procedure

_as

introduced in reference

[13].

^The

algorithm

is

designed

to model the motion of _a 2~dimensional

array of IV hard discs in a

vibrating

container.

Initially

the discs _are

positioned

at random

mside the box and then

they

_are allowed _to relax

by falling

towards the bottom of the

container;

the details of which ,vill be

explained

belo,v. One

period

of the vibrational motion is modeled

by

^first

lifting

all discs

through

some

height

A

(called

the

shaking amplitude)

without

changing

their horizontal

positions. Secondly

the discs _are allowed to fall down. After relaxation has taken

place

the array is lifted

again by

-4.

Thus the

shaking

of the container resembles distinct

"taps" separated by long

time intervals

as used in reference [5].

Nevertheless,

it should be

possible

to compare the simulations to

experiments

in ,vhich the vibration

frequencies

_are

higher,

_as

long

_as the

assembly

of beads

can relax

sufficiently

~vithin _one

penod.

In the simulation, the do,vnward

part

of the vibration

cycle

_is realized _as follows. A disc _is selected

randomly

and _a trial

position

^for this disc _is

generated.

If this _ne~Ar

position

leads to

N°12 SIMULATIONS ON SIZE SEGREGATION 1529

an

overlap,vith

another

disc,

the motion is

rejected

and the old

position

is

kept:

otherwise the disc is moved to the _ne~A>

position.

^After this the next disc _is chosen at random until _a

specified

number of trials _{nmax ai-e}

rejected (nmax

= IV in Ref. [13]

).

^The

assembly

is then considered

as relaxed and _{a new}

shaking cycle

starts.

The trial

position

for _a disc is chosen

by generating

two

independent

and

equally

distributed random numbers

(~

E

[-1, Ii

and

(y

E

[-1, 0].

^The ne~A>

position (z', y')

of _a disc is then

given by

Z'

# X +

(xô (1)

v'

"

v+(Yô (2)

where _z and _y denote the old horizontal and vertical

position respecti,>ely.

Note that trie choice for the _new

position corresponds

to do~Arn~vard motion with fluctuations in the horizontal

direction. The

parameter à,

which

corresponds

to the maximum step ~,>idth for _a

disc,

_was set to à

= r in reference

[13].

2.2. MODIFICATIONS OF THE ALGORITHM. ~Te introduce two modifications to the

original

algorithm.

The first and less

important

of the two consists of

allowing

small

upward

motions of the discs _m the

falling part

^{of the}

shaking cycle.

Trie _reason for this modification _is the

following.

The Rosato

algorithm

models the fluctuations in the

system by

random

displace-

ment. Because the fluctuations stem from

interparticle collisions,

the

algorithm

should also

allow

upward

motion _{m some cases.} For this _{reason, ~Are} take

(y

E

1-1, ôo],

^~A>here ôo ^is some

small number

larger

than _zero. The value of ôo should not be too

large, either,

since _m that

case motion

might

_{never cease m} the

system.

^{From another}

point

of

vie,v,

ôo

might

aise be considered _{as some measure} of the

elasticity

of the discs.

Laige

ôo values

correspond

to

large

fluctuations and thus to ~Areak

dissipation. However,

it is unclear how ôo ^is related to

physical quantities

hke the coefficient of restitution. We take ôo ₌ 0.05 in our

simulations,

,vhich _cor-

responds

to

only

_very

slight upward

motion and

mamly

makes it easier for the system to relax mto a final state.

The second modification of the

algorithm

consists of

choosing

a much smaller value for the

maximum

displacement

à thaii in the

original

work. This

changes

trie behairiour of the

system

significantly.

To illustrate

this,

_we

performed

simulations with values of à

rangiiig

from 0.05r to 0.5r. In all simulations referred _{to in} this paper, we use N

= 500 discs of radius _?. with

one

large

disc of radius

R,

1. _e. the intruder. The

assembly

_was considered _as relaxed when

no move could be

accomplished

for _nmax

= 5N trials.

Clearly,

this does not

provide equally good

relaxation for different _values of

à,

since the

larger à,

the _{more moves} will be

rejected.

We chose the abovementioned value for _nma~

bit shaking

an

assembly

of discs and

checking

for which value of _nm~x the

density

of the

assembly

could trot be increased

significantly by

mcreasing

_nmax. These tests also showed that

including

the

possibility

for the discs to move

upwards substantially

enhanced the relaxation of the

assembly.

The ,ralue _{nmax we}

finally adopted

showed reasonable relaxation for the whole range of à used

here, although

it _was best for the smallest ,naines. We should mention here that

although

trie

algorithm

does not

provide exphcitly

for

rolling

of discs _on each

other,

discs _on the surface

always

"roll" into _a local minimum

by

_a succession of Monte Carlo steps for _our choice of parameters.

The initial

configuration

in these simulations _was

provided

for _as described above for the

original algorithm,

_i. e. the

large

disc _~vas

placed

at _some

height

above trie bottom of trie

container,

then random

positions

were chosen for trie

background

discs

(see _[13])

This

config-

uration _was then allowed to relax _m the _{same way as m a} normal

shaking cycle,

after which trie

shaking procedure

started.

1530 JOURNAL DE

PHYSIQUE

I Nol 2

50.o

40.0

~

~,

~3 30.0 Éi )

~n

z'é

io.o

~

of

cycles

Fig.

1. Influence of the choice of à _on the

speed

of ascent of _a

large

disc of 4l

= 2 in simulations with A

= 2r. The

corresponding

^{values of à} _are

(from

_top to

bottom)

o-à, 0.3, 0.2, o-1, 0.05.

Figure

1 shows the

strong dependence

of the

speed

of ascent for _an intruder of size ratio 4l

= 2 _on à.

"Speed"

_is to be understood in _terms of

height

_per

shaking penod.

The

period

is the

only

time scale defined in this kind of simulation. For small values of à it _even looks

questionable

if trie

large

^disc uses at all.

The parameter à determines ho~v much the whole

assembly

_can

expand during

the

falling part

^{of the}

shaking cycle.

The

larger à,

the _more it

expands

and thus

larger

relative motions of the discs _are

possible.

In

experiments

it _~vas observed that in the

quasi-stat.ic

situation the

discs _move

nearly

_{as a} solid block with

only

small fluctuations around their _mean

position

_m the

assembly _[8]. Unfortunately,

_no data _on these fluctuations _are available.

Thus,

_so far it is

impossible

to relate à

directly

to

physical quantities. However,

it should be set to _a small value still

sufficiently larger

than _zero to

provide

observable fluctuations around the _mean

positions

of the discs in

falling.

We

investigated

the

segregation

behaviour of the

system

^{for à} =

o.lr,

which gives

quite good agreement

^of our simulations with

experimental

results.

Because small values of à make the simulations _more

time-consummg

there _{are some} limita- tions to the _size of the

system

^and to the _maximum number of

shaking cycles.

The maximum

size ratio _we used _was 4l

= 12, because for

larger

size ratios the number of small discs would have to be

increased, leading

to

longer computation

times.

In the

following

simulations the

starting configuration

was obtained

differently

from the

procedure

descnbed

above,

_m order to be _m close

agreement

with the experiments ^of [8].

Instead of _{usmg a} random initial

configuration

of small

discs,

the box _~Aras filled with small discs _{m a}

triangular pattern

as

regularly

_as

possible,

after first

having placed

the intruder at some well-defined

height.

For the

larger

intruders _{(1. e.} 4l >

8)

this _was

usually

chosen _as

(4l

+

1)r.

We will discuss the

implications

of this choice _m the

following section,

_as

they

_are

related to the mechanism of _size

segregation.

3. Results

3.1. THE MECHANISM _OF SIzE SEGREGATION IN THE

QUASI-STATIC

CASE. In this section

we describe _m detail the mechanisms

by

which size

segregation

takes

place

_{m our} simulations.

N°12 SINIULATIONS ON SIZE SEGREGATION 1531

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a) b)

Fig.

2.

Snapshots

of the ascent of _an intruder (4l ₌ 10.5, ^A ₌

0.5r). a)

The circles denote the position of the discs after 445

shaking cycles,

the _arrows the final

positions

after 470

shaking cycles, always

taken after

complete

relaxation of the

assembly. b)

Situation after 900

shaking cycles.

Besides,

_we discuss

quahtatively

the

dependence

of the

rising

behaviour of the intruder _on the size ratio. A

quantitative

discussion will be

presented

in the

following

section.

Figure

2 shows two

snapshots

of the ascent of _an intruder of 4l

= 10.5. Trie mechanism

by

which this ascent is

accomplished

can be _seen

clearly

from

Figure

2a. Due to the fluctuations in the

falhng

_part of the motion of the whole

assembly,

a small disc _is able to squeeze m beneath the

large

disc and rolls down mto the hole below it, followed

by

_an avalanche of other small discs. The intruder

is then stabilized in this _new

(higher) position.

In the motion of the small discs _no convection

was observed.

Usually,

_an avalanche _as the _one in

Figure

2a is followed

by

another _{one on} the other side of the intruder _as to make _up for the

asymmetry

^induced

by

the first _one.

The small discs whose

arrangement

^above the intruder _was disturbed before _arrange them- selves _{m an}

orderly

fashion below the

large

disc _as it rises.

Besides,

_a

triangular

hole forms belo~n. the intruder _{as can} be _seen from

Figure

2b. This hole is very stable and

persists through-

out the further rise of the intruder. We will discuss the relevance of this effect in the

followmg

subsection.

In

Figure 3,

_we

plot

the vertical

position

of _a rather small intruder

(4l

₌

2)

_versus the number of shakes and observe the intermittent ascent _as also

reported expenmentally

_[9].

Figure

3

shows that ascent takes

place,

but very

slowly.

The

plateaus

_on which the intruder rests for very

long

times

correspond

to the stable

positions

^{calculated in reference} [9]. ^The

speed

^of

rise is

approximately

4

layers

of small discs

during

10 000 shakes. In

expenments,

see

Figure

4 of reference

[9],

^trie

speed

is 7

layers dunng

54 000 shakes

(1h

with _a

frequency

of15

Hz).

This

is a very crude

evaluation,

but it shows that the time scales observed in _our simulation _are of the _same order of

magnitude

_as those observed

expenmentally.

The

positions

from which the

intruder falls down

again

_once in _a while

correspond

to the less stable of the two

positions

_an

intruder of 4l

= 2 _can take _on the

triangular

lattice formed

by

the small discs

(see

Ref.

[9])

1â32 JOURNAL DE

PHYSIQUE

I s?12

15.o

Î

~

g io.o

j

à a

£

Z 5.o ~

o-o

~~~~ loooo

Number of shaking cycles

Fig.

3. Ascent

diagram

for

an intruder of size ratio 4l

= 2 at ~

= r and à

= 0.1r.

Intermittent ascent of _an intruder with 4l

= 3

already

_was

reported

in reference

[13]

^{for small} vibration

amplitudes A,

but _on much shorter time scales. For

larger

intruders. intermittent ascent _cari still be observed in _ouf

simulations,

_as well _as continuous ascent for 4l

= 12

(see

Fig. 4).

~Ve

investigated

the influence of 4l _on the

segregation speed

for 4l < 12 in _some detail ,vhich led to

surprismg

results that will be discussed in the

follo,ving

subsection.

3.2. INFLUENCE OF THE DIANIETER RATIO ON SEGREGATION SPEED. The influence of

the diameter ratio _on the

segregation speed

of _a

large

intruder in the non-convective regime has been

investigated experimentally

_[8]. A hnear

dependence

of the

speed

of ascent on the

diameter ratio has been

found,

_as well _{as a} critical diameter ratio 4l~ m 3.5 for the

gi,>en expenmental

situation.

An

analysis

of the

dependence

of the

segregation speed

_on the diameter ratio

by

_means ^of

the

original algorithm

of Rosato et ai. has been conducted before

[14],

^but

mainly

to derive

an

amplitude dependence

for

amplitudes

much

larger

than the

typical

values _{we use.} The time scales _on which

segregation

occurred in their ~A.ork _were of the order of time scales in the

original work,

bath of i,>hich _are much shorter than _m trie

experimental

work of reference _[8]

and in our simulations. For further

comparison

of _our simulation to

experimeuts.

_we

performed

simulations for mtruders of size ratios 8 < 16 < 12 _{to nleasure} trie

segregation speed.

In

Figure 4,

_we

plot

the

position

of trie intruder _{as a} function of the number of shakes for different _size ratios for A

= 0.5r and à

= 0.lr. As _can be _seen from

Figure 4,

it is almost

impossible

to deduce _some exact ascent

speed

fi.om these

diagrams.

iiTe calculate the

velocity by averaging

it over the whole simulation _i-un

(except

in _cases where trie intruder reaches the surface

earlier).

Due to the

long computation

^{time each simulation took} we

performed only

a few _runs for each 4l value.

Figure

5 shows the

dependence

of

segregation speed

_on 4l _as extracted from simulations of the type of those shown in

Figure

4.

Averages

_are taken from 2 to 3 simulation _runs.

However,

the values for different realization for fixed

parameters

are

consistent.

ive checked that the initial

position

of the intruder dia trot affect the simulation results.

When

placed

_as described _at the end of Section 2, the intruder ,vould in all _{cases rise}

quickly

on a first

layer

of

particles. During

this

time,

the whole

assembly

of small discs relaxed into trie

typical configuration

that _can be _{seen m}

Figure

2. ~A>ith the intruder still

resting

on trie first

N°12 SIMULATIONS ON SIZE SEGREGATION 1533

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20.0 çp m

~

OEm100 10.0

i

~ ~

o soc i ooo

Number of shaking cycles

Fig.

4. Ascent

diagrams

of _an intruder for _vanous values of 4l at A

= o-ST and à

= 0,lr.

0.20

e

0.15

j

ÎÎ

fi

Î

_0.10

>

~

~ ^~

~ Ô

«

0.05 j

o o

o o

o ~ o

~~~

8 9 10 11 12

a

Fig.

5.

Dependence

of ascent velocity _on ^4l. S denotes the number of

shaking cycles.

layer

of small discs.

Thus,

ail intruders started from stable and well-defined initial

positions,

which _can be

recogmzed

_as small

plateaus

in the

begmmng

of the simulation at

height (4l

+

2)r.

The intruder _was considered to start its ascent when it left this

plateau. Starting

the intruder not in the middle of the

box,

_{or on more} than _one

complete layer

dia not

change

the simulation

results.

Figure

5 illustrates the

strange

behaviour _we found and should be taken _as a

qualitative

rather than _a

quantitative

result. The dashed line is

merely

meant to

guide

the _eye. The

speed

of _ascent exhibits _an

irregular

behaviour where certain _size ratios _seem to rise

preferentially

which _can be

explained by purely geometrical _arguments.

To _our

knowledge,

_no

experiments

m 2D have covered this range of size ratios _in such detail. We observe no size

segregation

^for 4l _< 9 _on these time

scales,

_i e. for 1000

shaking cycles. Naturally,

it is

possible

and _even

probable

that smaller intruders

might

_{use as} well in

longer

simulation runs. In

experimental

situations _as described in

[8],

^{for 4l} = 5.3

waiting

times of several minutes

occurred,

after ~Arhich

1534 JOURNAL DE

PHYSIQUE

I N°12

T

irÎ3

~

~

_ir ~

Fig.

6. Schematic

drawing

of the situation encountered by the

large

disc after

having

_nsen

through

some distance

m the simulations.

the intruder resumed its ascent. This

corresponds

_{to a} few thousand

shaking cycles,

which is

already longer

than the simulation _runs

presented

in

Figures

4 and 5.

TO

explain

^the

irregular

behaviour _we observed in _our simulations in the ascent

speed

of the

intruder,

_one has to look into the mechanism

by

which the _ascent takes

place.

As mentioned

before,

_a ^small disc has to squeeze in between the mtruder and the

supporting

discs in order to stabilize it in _a

higher position.

After _some

time,

a

triangular

hole bas formed below the intruder.

Figure

6 shows _a schematic

drawing

of the situation. The walls of this hole _are

geometrically

_very

stable,

_so that the

only

discs

likely

to promote ^the ascent of the intruder _are the discs labeled _{a m}

Figure

6. As we now

proceed

to

show,

trie size of the small hole bet~n.een the intruder and the disc labeled is _a non-monotonous function of 4l. Since the disc _a has to fit into this

hole,

the

typical

_size of the hole

strongly

affects the

speed

of _ascent.

We _now calculate the

position

of the disc _{m a}

geometry

as shown _m

Figure

6. When this

position

is

known,

the distance between the disc1 and the mtruder _can be calculated. One _can

assume that the center of the intruder is at

height 2(R

+

r)

with respect to the center of the

disc labeled 1 in

Figure

6. The center of the disc1 _is at

height

j =

irvà, ₍₃₎

because it is the 1-th disc _m the wall. The value of1 _can be

expressed

_as

=

là _(R

+

ni

₁₄₁

The square brackets denote the

integer part,

e. the

largest

natural number smaller _or

equal

to the

argument.

^This

corresponds

to the disc

being

the

highest

of the discs _m the walls of the

triangular

hole whose center is still below the _center of the intruder. The

corresponding

value of the x-coordinate of the

position

of the disc is _x~

= ir

(always

with

respect

to the

center of the disc

1).

For the distance d between the disc1 and the intruder _we

get

d =

(2(R

⁺

^r) irvi)~

₊

_{(ir)~) là)}

N°12 SIMULATIONS ON SIZE SEGREGATION 1535

3.0

~

j

2.0

§

l 0

~~

~~

8 9 10 11 12 13

a

Fig.

7. Distance between the disc and the

large

disc.

(R

+

r)

_was subtracted from d to draw

attention to the size of the hale between the two discs.

In

Figure 7,

we

plot

the distance between the surfaces of the disc1 and the intruder in units

of _r for certain values of 4l

according

to

equation (5).

The

jumps correspond

to a

change

of

the

supporting

^disc

(and

thus of the value

of1)

due to _an increase _m the _size of the intruder.

It _occurs whenever

2(4l +1) Il

is _an

integer (see Eq. (4)).

The

irregular

behaviour of the

ascent

speed

of the intruder _{as a} function of 4l _{con now} be

explained

_as follows. For _size ratios

slightly

below such _a

jump value,

the _size of the hole is _too small for the disc _a to squeeze m.

Above the

jump value,

the hole is much

larger,

thus

promoting

_a fast ascent of the intruder.

As the size of the intruder is increased

further,

the size of the hole decreases

again.

That way the

probability

for _a hole that is

large enough

to accommodate the disc _a to open up decreases and therefore the

speed

of _ascent decreases. Simulations

performed

for diameter ratios

slightly

below and above these

jump

values show the

expected

behaviour

(see Fig. 8), implying

that

50.o

OE=11 5

40.0

$ _OE=9.5

~ 30.0

~i

à

~ ^OEmll.0

~ ^20.0 _~

10.0 ^~*~.~

~'~

o soc iooo

Number of shaking cycles

Fig.

8. Ascent

diagrams

for A

= o-ST and à

= 0.lr. The

corresponding

jump values for the _size d of the hole _are at 4l

= 9.39 and 4l

= 11,12,

respectively.

1536 JOURNAL DE

PHYSIQUE

NO12

this _is the _reason for the

irregularities

in the

speed

of ascent. In

experiments,

_no such behaviour

coula be observed because the

development

of _a

triangular

hole below the intruder does not take

place

in trie

experimental

situation.

Therefore,

_{we assume} that fluctuations will wash out this

purely geometric

^effect in most

experiments.

^Small holes do

develop,

but

forger

_{ones are}

too instable trot to be filled

eventually, mainly by

horizontal block motions of trie small beads.

Trie ~n.alls of such _a hole _are toc feeble to resist trie forces exerted _on them

by

trie other discs from trie

sides,

forces which _are absent from _our simulation.

One should be able to reduce this

"locking"

behairiour of the intruder

by introducing

_{a size} distribution of

relatively

_narrow width _on the small

particles

to disturb the

regular arrangement

observed in

monodisperse packings.

~~nother

possibility might

be _{to use a} different

probabil-

ity distribution for the choice of the

displarement

of the

single

discs.

Experiments

evidence

fluctuations of the discs around their _mean

position

_[8]. If _more about these fluctuations _were

known, they might

be

incorporated

in the simulations to render them _more realistic.

4.

Summary

We studied _a

periodically tapped system

^{of discs with} one

large

intruder disc _{m a}

background

of

monodisperse

smaller discs

by

_means of _a Monte Carlo like simulation method which

emphasizes geometric

effects.

By using

a much smaller value for the maximum

displacement

in _one Monte Carlo step than in previous work

[13],

^the

agreement

with

experiments

coula be enhanced

substantially.

lN~e observed size

segregation

without

accompanying

convective motion of the

background

discs. Trie

typical

ascent velocities _are similar _to the _ones observed in

expenments

conducted in this

quasi-static regime [8],

^due to a suitable choice of the maximum step width à in the simulations.

The ascent of the mtruder takes

place

_{m an} intermittent _or continuous

fashion, depending

on the _size ratio 4l

=

RIT.

In the _case of intermittent ascent we find well defined stable

positions

of the intruder. In simulations with _{a very} small

intruder,

the

system

may stay in such _a stable

configuration

^for _{a very}

long

time _as observed in

experiments _[9].

For very

large

values of

4l,

close to the theoretical threshold for continuous ascent of 4l

=

12.9,

the 4l-

dependence

of the

stability

of these

positions

affects the ascent

irelocity significantly.

We grue

a

purely geometrical explanation

for

this,

based _on the mechanism

by

which the ascent of the mtruder is

accomphshed

in _our simulations. In

experiments,

other mechanisms besides this _are active. As recent

experiments sho,v, large

scale horizontal block motions _are

hkely

to untrap the intruder

Ii?i

These do not _occur m our

simulations,

~A>hich

emphasize

the local influence of

geometry

instead of

long-range

interactions. Their absence _may be _an

explanation

for the

irregulanty

of the ascent

velocity

_even for

large

intruders.

Thus,

the simulation _can contribute to the

understanding

^of the mechanisms active _{m size}

segregation by clearly showmg

which

effects _are due _to the

geometry

^of the

problem

and which _are not.

Acknowledgments

We thank A.

Blumen,

E.

Clément,

J.

Duran,

^J.

Schàfer,

H. Schiessel and D. Wolf for

helpful

discussions. The support ^{of the Deutsche}

Forschungsgememschaft (SFB 60)

and of trie Fonds der Chemischen Industrie _is

gratefully acknowledged.

N°12 SIMULATIONS ON SIZE SEGREGATION 1537

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