Random sequential adsorption of line segments : universal properties of mixtures in 1, 2 and 3D lattices

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Random sequential adsorption of line segments : universal properties of mixtures in 1, 2 and 3D lattices

B. Bonnier, Y. Leroyer, E. Pommiers

To cite this version:

B. Bonnier, Y. Leroyer, E. Pommiers. Random sequential adsorption of line segments : universal

properties of mixtures in 1, 2 and 3D lattices. Journal de Physique I, EDP Sciences, 1992, 2 (4),

pp.379-384. �10.1051/jp1:1992150�. �jpa-00246491�

Classification Physics Abstracts

05.70L 68.10J

Short Communication

Random sequential adsorption ^{of line} segments

_:

^universal

properties of mixtures in 1, 2 and 3D lattices

B.

Bonnier,

Y.

Leroyer

and E. Pommiers

Laboratoire de Physique Th60rique*, Universit6 de Bordeaux I, Rue du Solarium, F-33175

Gradignan Cedex, France

(Received

5 December 1991, accepted 24 January

1992)

Resum4. Nous _exposons les r6sultats d'une simulation Monte Carlo du _processus d'adsor-

ption al6atoire s6quentielle _sur des r6seaux _en deux et trois dimensions, de segments de difl6rentes

longueurs choisis _avec diverses probabilit6s. La limite asymptotique du taux de recouvrement du r6seau, consid6r6e _comme

une fonction de la longueur des segments et de leur probabilit6,

a le m6me comportement quelque soit la dimension. De plus, la cin6tique du _processus suit

exactement les lois obtenues dans le _cas du modle r6solu unidimensionel, oh le recouvrement du

r6seau croit exponentiellement _{vers sa} ^limite asymtotique _{avec un} taux 6gal 1la probabilit6 de tirage du plus petit segment.

Abstract. Monte Carlo results _are reported for _{a process} of random sequential adsorption

on two and three dimensional lattices, of line segments ^of two different lengths chosen with various probabilities. The jamming limit, when considered _as

a function of the segment lengths

and of the probability, is found to have the _same behaviour in both two and three dimensions

as was observed in _one dimension. In addition, ^{the kinetics} of the _process exactly follows the laws found in the solvable _one dimensional _case, where the rate of late stage deposit is given by the shortest segment probability.

Recently

there has been _a renewal of interest in the random

sequential adsorption (RSA) model, especially

in the _case of well-defined mixtures of

deposited objects [I].

In

particular,

a model for the

deposition

of line

segments

^of two different

lengths,

chosen with

equal prob- abilities,

has been

numerically

simulated _{on a square} lattice [2]. ^{In the} mean

time,

_{a one}

dimensional

analog

of this

model, generalized

to _any

mixture,

has been solved [1, 3], ^thus pro-

viding

_a useful

guide

for the

study

of this _process in

higher

dimensions.

Many

features of the

ID exact solution [3] are indeed

already

present ^{in the results}

reported

for the 2D Monte Carlo

experiment

of reference [2].

*Unit6 Asssocide _au CNRS, U.A. 764.

380 JOURNAL DE PHYSIQUE I N°4

The _purpose of the present work is to

generalize

the numerical simulation of reference [2],

to the _case of

arbitrary

mixtures in two and three

dimensions,

in order to check whether the main

properties

of the ID model hold in

larger

dimensions.

The model is defined _as follows segments of

given length

_are

sequentially

and

randomly placed

_{on a} D-dimensional lattice

according

_to the

following

rules

:

I) randomly

^{select the}

length

of the segment among ^two

predefined

values

li

and _{12 >}

li (measured

in number of lattice

sites),

with

probability

_a for the value

lit

it) randomly

select _a site _on the lattice and _a direction

along

_one of the _axes;

iii)

if the selected site is empty, try to put the segment on the lattice

along

the selected direction with its

origin

_on the selected

site;

iv

)

if the segment can be

placed

such that it does _not

overlap previously deposited

segments, it is left _at this

position

and the attempt is

successful;

v)

if the selected site is

occupied,

_or ^if there is _no

place

for the

deposited

segment, ^the segment ^{is withdrawn and the} attempt ^is

unsuccessful;

vi

)

^whether or not the attempt is

successful,

the time counter is increased

by

St and _{a new} try is realised.

The observable

quantity

of interest is the _coverage

I(I;li,12,a),

defined _as the fraction of sites covered

by

the

deposited

_segments after

a time I. This function has _an infinite time

limit,

the

jamming

limit boo

(li,12, a), which,

from the exact solution of reference [3], has the

following properties

in _one dimension

. for

[

and _a fixed it is _an

increasing

function

of12

. for ₁₂ and _a fixed it is _a

decreasing

function of li

. for li and ₁₂ fixed it is _a

decreasing

function of _a

. this last property leads to the discontinuous behaviour

boo(12) "

boo(li

_{>12> a =}

0)

< lim

boo(li,12> a)

Furthermore,

the

jamming

^limit is approached

exponentially

with _a rate

simply

related to the

probability

_a and

independent

of the

lengths

li and ₁₂₁

b(t; li,12 a)

₌

boo(li>12, a) C(li,12, a)

^e~" for at

large (1)

The first _two

properties

^{have been} observed in the 2D simulation of the model with the

segment length li

and ₁₂ drawn with

equal probability (a

₌

1/2)

[2]. In the _same reference the difference in

long-time

behaviour between the _case of

equal probability mixing

and the

"pure case", li "12

_{or a}

=

I,

has been noticed.

In order _to check the

validity

of the

previous properties,

_we have

generalized

this Monte Carlo simulation to various

probabilities

_a in two and three dimensions.

We take lattices of size L ₌ 256 and S12 in D ₌ 2, and L ₌ 32 and 64 in D ₌ 3. The

lengths

of the

deposited

segments

(in

number of

sites)

_range from 1= 2 to ₌ 12 in 2D and 8 in

3D,

_so

that,

the ratio

I/L being small,

_we expect the finite size effects to be

negligible.

The time scale is fixed

by setting

St ₌

I/DL~

_{in order to allow}

a

comparison

with the exact ID model of reference [3], ^{where the time unit}

corresponds

to the

(averaged)

time needed for

reaching

_a

given

site with _a

given

orientation.

Actually

to _save computer

time,

the selected site is

(randomly)

chosen in the list of

jfee

sites and the

corresponding

time step is St

/pf(I)

where

pf(I)

₌

(I b(I;

li,12>

a))

is the

probability

for _a site to be

unoccupied

at time I.

~ 10

I

9 O a«0.25

$

. a=0.50

8 8 a a=0.75

©

. a=1.0

£ ₇ exact

6

5

4

3 D=1

2

1

~0

2 3 4 5 6 7

at

Fig.I.

The quantity In(Sac

8(t))

_{as a} function of at for the

one dimensional _case with li

= 2,

12 " 4 and for four values of _a. The data points result from _our simulation and the continuous lines

from the exact solution of reference [3].

li

_or l~

At each successful attempt the coverage is increased

by

the fraction

~

On _a finite lattice the

jamming

limit is

exactly

reached in _a finite time which

depends~n

_{the lattice size}

and which _can be determined

easily.

For all the _{cases we} have

investigated,

_we found that this time of saturation is of order is ₌

71a

in _our time units.

For each set of parameters

(a, li,12),

we

perform

from 50 to 200

independent experiments,

and average

b(t;li>12, a)

over this

sample.

The fluctuations _are

small, allowing precise

_mea-

surements.

We first fix _our

procedure by

_a

comparison

of _a ID simulation with the exact formula of reference [3], for the

particular

choice

li

_"

2,12

_" 4 and _{a =} 0.25,

0.50,0.75,1.

We set the lattice size to L

= 2~~,

large enough

for the finite size and finite saturation time effects to be

negligible. Actually,

the

jamming

limit coverage,

boo(li,12> a)

that _{we measure,} coincides

with the exact value _up to four

digits.

Furthermore _we have checked the time

dependence by

comparing

the measured and exact values of _:

r(t)

₌

lI1lb«(li>

_12>tY)

b(t;

_11>'2>tY)1

(2)

In

figure

I, we have

plotted

this

quantity

_{as a} function of _{r =} at, for several values of _a. From the

long-time

behaviour of

boo(li,12>a) expressed

in

equation (I),

_we expect ^these curves to be

parallel straight

lines. The agreement between exact and measured values is

total,

confirm-

ing

that the finite size effects _are

negligible.

However, if _we want to extract from these data the

exponential long-time

behaviour of

equation (I),

_we find that for 2 < _r < 5 the

slope

of the

straight

line is

slightly

overestimated with respect to the exact

asymptotic value,

due to corrective terms in e~~°~. _{But for} the simulation to

give

accurate

results,

the time _r must be less than the saturation

time,

which

depends

on the size of the lattice.

Therefore,

finite size effects _occurs

through

the

long

time behaviour and

precise

determinations of the rate of the

exponential

decrease and of the

prefactor,

need

large

scale simulations.

382 JOURNAL DE PHYSIQUE I N°4

~ 0.9

°~ mixed _case 1,-4, ~-6

pure .

0.S

~

a

-

probability a of the

pure

Table I. - boo

histories

on lattices of size L =

on

a

2 ₄ ^.92880.9208 0.9138 0.9067

2 6 0.9328 ⁹²³⁴ _0.9153^.9070

2 8 .9241 .9157

0.9068

2 ₁₀ 0.9349 .92420.9160 .9068

2 12 _0.9351 0.9242

4

2D 4 8 0.8520 0.8352 0.8225

4 10 0.8567 0.8376 0.8237 0.8106

4 12 0.8600 0.8390 0.8240 0.8103

6 8 0.8027 0.7890 0.7782 0.7702

6 10 0.8127 0.7935 0.7806 0.7697

6 12 0.8196 0.7982 0.7835 0.7700

8 10 ⁷⁷⁷⁶ 0.7656 0.7559 0.7478

8 12 0.7899 0.7722 0.7575 0.7483

4

D 4 6 0.8053 0.7945 0.7868

4 8 0.8088 0.7968 0.7874 0.7809

s s

7 7

~ 6 ~ 6

#

-

- -

@ 5 ~ t ~f

~ 4 ~ 4

§ 3

~~

~

~

₂ =3

1 1

~ o

(a)

"~ ^b) "~

Fig.3.

-

The ^uantity

and

dimensions (Fig.

for the

_{case li} " 4, 12 " 6 _and the _fourvalues

of

from the top curve

(a

= 0.25) _{to the}

lowestone (a ₌

1).

Turning

_now to 2D and 3D, we have measured boo

(li,12> a)

for _a set of values of the _param- eters

li,

₁₂ and _a. The results _are collected in table1.

We may

verify, using

this

data,

all the

properties

^of

boo(li>12,a)

described above in ID.

Furthermore,

_we find that as12 > li> ^it

quickly

reaches _a

limiting

value

depending

on li ^and

O.

The discontinuous behaviour of

boo(li,12, a)

_{as a}

- 0 _appears

clearly

in

figure

2, for li

" 4

and _{12 "} 6 in the 2D _case. When _a decreases from I, ^which

corresponds

to the pure case1 ₌

4,

to

0,

boo increases towards _a

limiting

value much

larger

than the _o

= 0 _one which

corresponds

to the _pure case1= 6.

Another

striking

result _concerns the time

dependence

of

b(t; li,12, a).

On the basis of the ID results _we expect it to be

exponential

in the variable _{r =} at. In order to check this

assumption

we have

plotted

the

quantity r(t)

defined in

equation (2)

_{as a} function of _r for all the values of a, li and ₁₂ of the table I.

Figure

3

displays

the _case li _" 4, _{12 "} 6 for the four values of _a in 2D

(Fig. 3a)

^{and 3D}

(Fig. 3b).

It

clearly

_appears that the _{curves are}

parallel,

and linear for _r

large enough.

The

slopes

of all these _{curves range} from 0.95 to

I.I,

whereas the

prefactor slowly depends

_{on a.} As _we have _seen in the ID case, an unbiased determination of the

slope requires

a

high ^precision

measurement. We have realised this

experiment

for the _case

(

₌ 4, _{12 "} 6

on lattices of linear size L

= 512 in 2D and L

= 64 in 3D with 200

independent

histories. We

perform

_a linear fit of

r(t) (Eq.(2))

in _an interval oft which satisfies the

following

conditions

(I)

t

sufficiently large

that corrective terms to the

leading

behaviour of

equation (I)

_are

small,

and

(it)

t be less than the

averaged

saturation time in order to avoid finite size effects. The

irrors

on the

slopes

are obtained

by displacing

this time window _across the

region

0.sts < t < 0.7ts where is ^{is the saturation time. The results} _are 0.99 ~ 0.02 in 2D and 1.00 ~ 0.02 in 3D.

In conclusion, _we have confirmed

by

_a simulation that the salient features of the

exactly

solved ID model of random

sequentially deposited

_segments of different

lengths

_are present in 2 and 3 dimensions. The trend for the

asymptotic

_coverage to grow with the

length

^ofthe

largest

segment, ^{but to decrease with the}

length

of the shortest _one, is recovered.

Furthermore,

_we check that the

asymptotic

_coverage

corresponding

to

only

_one segment

length, discontinuously jumps

to _a

larger

value _{as soon as an} infinitesimal mixture with shorter

segments

is allowed. If

IOURNAL DE PHYSIQUE t T 2, N'4, APR~ 1992 16

384 JOURNAL DE PHYSIQUE I N°4

the mixture is realised with

larger segments,

^the _coverage

continuously

_{grows as} the

proportion

of

large

segments

increases,

_a behaviour not _so close to

physical

intuition _as the

previous

_ones.

Finally,

the

exponential

rate which dominates the

long-time

behaviour ofthe _coverage, is shown to be

exactly equal

to the

probability

of

drawing

the shortest segment.

References

[1] Bartelt M.C. and Privman V., Phys. Rev. A44

(1991)

2227 and references therein.

[2] §vrakib N. M., Henkel M., J. Phys. I France1

(1991)

791.

[3] Bonnier B., preprint LPTB 91-8

(Bordeaux-I University).