Finite fractal diffusion-limited aggregates

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Finite fractal diffusion-limited aggregates

M. Arturo López-Quintela, M. Carmen Buján-Núñez

To cite this version:

M. Arturo López-Quintela, M. Carmen Buján-Núñez. Finite fractal diffusion-limited aggregates. Jour-

nal de Physique I, EDP Sciences, 1991, 1 (9), pp.1251-1261. �10.1051/jp1:1991204�. �jpa-00246409�

Classification

Physics

Abstracts 05.40 68.70

Finite fractal diffusion-limited aggregates

M. Arturo

L6pez-Quintela

and M. Carmen

Bujhn-Nfiiiez

Biodynamical Physics Group, Department

of

Physical Chemistry, University

of

Santiago

de

Compostela,

E-15706

Santiago

de

Compostela, Spain (Received

9

July

1990,

accepted

in

final form

6

May 1991)

Abstract. Particle-cluster

aggregation

has been studied

by

computer simulation. It has been observed that the fractal dimension of _a diffusion-limited aggregate ^is

only

constant when the number N of its

elementary

units is

sufficiently

large. For small N, the fractal dimension is _a function of N. The concept ^of a differential fractal dimension

D(N)

=

-d[logn(e/ro,N)]/

d[log (e/ro)]

allows the deviation from ideal behaviour to be

quantified

and

interpreted.

Introduction of

D(N)

in the

scattering

equations allows characterization of the deviation from

linearity

of

plots

of

log

I

against log

_q, where I is the

scattering intensity

and _q the

amplitude

vector.

1. Inboducfion.

The formation of

large

structures from small units is _{a common} process of

importance

in many scientific and

technological

fields. The

discovery

that most of such structures _can be described in terms of fractal

geometry [I]

has in recent years stimulated

growing

interest in these kinds

of systems. ^One

aspect

^{of this interest is the}

development

of _new

techniques

for research _on the

growth

of these structures. In

particular,

_computer simulation methods for

studying

both structural and kinetic issues have received much attention. The _use of

extremely

idealized models for simulation _purposes

brings

out features of fractal processes that may be obscured in real

physical examples,

and _so

help develop

intuitive

understanding

of _more

complex

systems

and,

in many cases,

provide

_a framework for

relating apparently

unconnected processes.

The first

computer

model of _an

aggregation

_process was

developed by

Witten and Sander

[2,

3] ⁱⁿ an attempt to

explain

Forrest and Witten's

[4] experimental finding

that small metal

particles suspended

in _a dense _gas clustered into fractal structures. In the Witten-Sander

algorithm, particles

_are added to the cluster _one

by

_one

along

random

trajectories

to

produce

fractal structures.

Though

_many

non-equilibrium

_{processes are} not

explained by

this model

(including

the Forrest-Witten

phenomenon),

there _are many for which it does

provide

the

key

to

understanding, including

fluid-fluid

displacement

in Hele-Shaw cells

[5-7]

and porous media

[8, 9],

^the breakdown of dielectrics

[10], electrodeposition [11-14],

random dendritic

growths [15-18],

the dissolution of porous media

[19, 20]

and the

growth

of

biological

structures

[21]. Subsequent computer

^{models have either}

involved,

like the Witten-Sander

algorithm,

the addition of

particles

_one

by

_one to clusters

[2, 3, 22-24],

_or have allowed the

aggregation

of

multi-particle

clusters to each other

[25, 26]

_as in the

experimental

observations of Forrest and Witten.

In this article _we discuss the

particle-cluster aggregation

_process

by

_means of _a simulation program based _on Ermak and Mccammon's Brownian

Dynamics algorithm [27].

In section 2

we describe the simulation

procedure,

in section 3 _{we use} the

concept

^of differential fractal dimension

(DFD)

to characterize the

dependence

of the fractal dimension of _our

aggregates

on scale when the number of units is

small,

and in section 4 _we examine the influence of the DFD _on the structure factors of

aggregates

^{with small numbers of units the} introduction of the DFD in the

scattering equations explains

a curvature in

log

I _vs.

log

_q

plots

that is unrelated to the

non-linearity corresponding

to upper and lower cut-off

points.

Final conclusions _are drawn in section 5.

2. Simulation

proceJlure.

The

aggregation

model used is _a version of Witten and Sander's

particle-cluster

mechanism

[2, 3],

the

general plan

of which is _as follows. The space in which the process occurs consists of

an inner _zone bounded

by

_a

sphere

B of radius _r~ and _an outer _zone bounded

extemally by

_a concentric

sphere Q

of radius

r~.

^{At the}

beginning

of the simulation there is _a

single particle

located _at the centre of the

spheres,

and simulation

proceeds by releasing particles

_one

by

_one

from random

positions

on

B,

whence

they

describe random

paths

until

eventually they

either

al

reach _some

point adjacent

to _one of the

particles

of the

aggregate,

to which

they

_are then

regarded

_as

adhering,

_or

b)

wander outside

Q,

in which _case the

particle

is abandoned. In most DLA

simulations, including

the

original

Witten-Sander

algorithm, particles

_{move on a}

lattice, usually

a square one,

by

steps ^of one lattice unit in _a random

direction,

which restricts the

angles

between successive

path steps

and between

adjacent aggregate particles (in

the _case of _a square

lattice,

to

0( 90(

180° and

270).

A _more realistic

approach

is _to simulate

particle

motion

by

_means of Errnak and Mccammon's Brownian

Dynamics algorithm [27]

because

this

algorithm

does not introduce restrictions of

angles allowing,

_on the other

hand,

_{a very} easy

incorporation

of

particle

interactions like electrostatic _or

DLVO,

and then _can be used to simulate the

aggregation

of

spherical

colloidal

aggregates [28-34].

This

algorithm

constructs

stochastic

trajectories

whose steps are solutions of Smoluchowski's

equation

for _a short time

interval,

_as follows.

From the

Langevin equations,

Errnak and Mccammon obtained the

following expression

for the

displacement

of _a

diffusing particle

:

~

3Dg D( F)

r~ = r~ +

£

At ₊

£

At ₊ R~

(At) (I)

J

~9

J

~B ^~

where

r)

and _{r~ are} the

positions

of the

particle

at the

beginning

and end of _a time interval of duration

At,

_D~~ is the diffusion tensor,

l§

is the total external non-Brownian force _on the

panicle

and R~ is a random

displacement

due to Brownian _causes

(superscript

0's indicate values at the

beginning

of At and «I,

j»

label

coordinates).

This

equation

is valid for At

large compared

with

mD(/k~

T

(where

_m is the _mass of the

particle,

taken _as

unity

in _our

work)

and small

enough

to ensure

negligible

variation of the extemal forces and the diffusion tensor

gradient.

The random Brownian

displacements Ri

are calculated _as

R~

(At )

=

£ _{«q xj} (2)

where the

%

^{are obtained from} a random number generator ^with

(,)

₌ ⁰ and

(xi xj)

=

2

3q

^At

(3~j being

^{the Kronecker}

delta)

and the

weights

_«~j

satisfy

Ii

^{-1 1/2}

"11 "

~~ £ "~k (3~)

k =1

lj

^-1

«~~ ⁼

D( £

_{«i~. «y~}

/«jj

I _>

j (3b)

k=1

Equations (1)-(3) imply

that the first and second moments of the distribution of Ar~ = r~

r)

_are

given by

(Ar~(At))

=

£ ^~~°

+

~° ~)

At

(4a)

j

~~j

^{~B T}

(Ari (At ) _Ar~ (At))

= 2. D

~~

At

(4b)

In this work _we assumed that _no extemal forces _were

present,

so that F

=

0 and the diffusion tensor is

given by

k~

T

Dij

=

(5)

car1~ro

where ro is the radius of the

diffusing particles,

_1~is the

viscosity

coefficient of the solvent and _c is _{a constant} whose value is 4 _or 6 when stick _or

slip

conditions _are considered

respectively.

The above n-dimensional Brownian

generalized algorithm _[35]

_was used to simulate the formation in two dimensions of DLA

aggregates

^made up of various numbers of

particles

N for each

N,

simulations _were

performed using

various values of the _mean free

path

d defined

1/2 as d

=

(£ _Ar))

^{This value} can be varied

using

_a constant value of

D~ ^(taken

as

unity

in

,

our

work)

and

changing

the value of At.

In DLA simulations it is

usual,

for each _new

particle

released from

B,

to

update

B and

Q

_so that

r~/2

_ro

= J1~~~~/2 ro + 5 and

r~

=

3

R~~~,

^where

l~~~

^{is the}

length

of the

longest

radius of the

aggregate

at this stage of the simulation. In the work described here _we chose to

update

B and

Q

with reference to the radius of

gyration

of the

aggregate, 1§,

^which as a random

variable

parametrized by

the number of

particles

in the aggregate ^{fluctuates less than} l~~~~, ^{I-e- is} a more stable index of the

relationship

between the size of the aggregate ^and the

number of

particles

in it.

Specifically,

the initial values of _r~ and r~ ^were

51ro

and

71ro respectively,

and _as the aggregate grew, r~ and

r~

were

updated

to

keep (r~ R~) ^Id, (r~ r~)/d

and

d/R~

^{constants at the} same time l~~~~ was checked to make _sure it did _not exceed r~

(since

it _never

did,

_we refrain from

describing

the

adjustment

to be made in this

event).

When the

previous

relations _are constants the _mean number of steps for

trajectories

obtained with _a

given

value of At is constant. This means

[36-38]

that _we obtained

trajectories

with _a fractal dimension

d~

constant

during

the

growth

of _an aggregate.

In

particular,

in this paper we will focus _our

study

_on two

aggregates

^obtained with the conditions

(r~ R~) Id

₌ ^{25 with}

d/R~

m I

(d~

m 1.9

)

and

(r~ R~)/d

=

5 with

d/ll~

_m 5

(d~ m1.6).

It has been checked that the value of the constant d/11~ ^{does not have} any influence _on the results

obtained,

at least in the _range 0.5 _«

d/ll~

< 5 _we have

performed

the

simulations. A value _{of ro}

=

0.51

was used.

3. Fractal

analysis

of swan

aggregates.

The fractal dimension of each aggregate constructed _was

computed by

the box

counting

method

[39].

This method consists of

covering

the aggregate with a lattice of side _e

(in

units of

ro), counting

the number

n(e, N)

of cells

occupied by

the aggregate, ^and

repeating

this

procedure

for

successively

smaller values of _e. The fractal dimension is defined _as

Figure

I

shows,

for _a two-dimensional aggregate, ^the linear

scaling

behaviour of

log

_n

(I.e.

the existence of _a

scale-independent Di)

for each of several values of N. For _an ideal fractal

aggregate,

^{the fractal dimension}

given by equation (6)

should be

independent

of N. Our simulation results

show, however,

that this situation should

only

be attained for

sufficiently large

N. For small

N,

the fractal dimension is

N-dependent

the lines of

figure

I have different

slopes. Analogous

behaviour has also been observed in other fractal processes,

including

Brownian

trajectories [36, 37].

To characterize such

behaviour,

_{we use a} differential fractal dimension

(DFD) [36, 37],

I-e- _a fractal dimension defined

locally

with respect to the scale

parameter.

^{In the} present ^context we can define the DFD _as

~ p~

d

log

_n

(

_e

_fro,

N

)

dlog (e/ro) ^(7a)

6 N=isoo

/~

~S

~

~

=200

4

4

» ,

'

O~

log

_e

Fig. I.- Log n(e,N) (number of occupied boxes) versus Loge (e ^{is the} side of the box) for bidimensional aggregates

composed

of different numbers N of

elementary particles.

The aggregates

were obtained

using trajectories

with the initial simulation condition

(rs R~)/d

₌ 25. The simulation results (A) have been fitted to a

straight

line

(lines).

in which _we have

explicitly

stated that _e is

expressed

in units _{of ro.}

Alternatively,

_an

analogous

definition of the DFD _can be introduced _{as :}

d

log M(r/ro,

N

)

~ ~~'~°~ ~

d

log r/ro ) ~~~i

where

M(r/ro,

N

)

is the number of

particles

within _a radius _r of the centre of the

aggregate (ro

_~ r _~

R~~~).

^We have found that these DFD functions defined for clusters formed

by

diffusion-limited

aggregation

_can be fitted

by

similar

equations

to those

proposed by Takayasu [36],

Tsurumi

[38]

and the _own authors

[37]

to random motion

D(N)

=

Di

~

(8a)

+

k~.

N ^~

D

(r fro i

= D

f ~

(8bi

+

k~. (r/rot

^'

where

Di

is the

limiting

value for

large aggregates,

^and k~,

k~, fl

~

and

fl

~ are

parameters

^that

depend

_on the conditions of the

aggregation

process

(I.e. aggregation model,

fractal dimension of

trajectories

and interactions between reactive

particles

and

cluster). Likewise, Dr depends

_on the conditions of the

aggregation

_{process :} for

example,

for the two limit _cases,

when the

trajectory

is _a two-dimensional ballistic motion

(d~

=

I

Di

=

2

[40],

while for two- dimensional ideal Brownian motion

(d

-

0, d~

=

2) Di

_- 1.6

[2, 3].

^With our

algorithm large

clusters with _any fractal dimension between these two limits _can be obtained.

So,

for

example,

in

figure

2 the fractal dimension

Di

for different

large

clusters

(N

_m

10~)

is

plotted

_versus the value of

d/(r~ R~)

^{used in the}

simulation,

I-e- _; for different values of the fractal dimension of the

trajectories, d~.

It _can be _seen that for

d/(r~ R~)

^{S 0.01 the usual DLA fractal is}

obtained.

For _a

specific aggregation

model

(for example particle-cluster, cluster-cluster, etc.)

it is assumed that the parameters

fl~, k~ (or fl~, k~)

must have the _same values when the fractal

dimension of the reactive

trajectories

and the eudidean _space,

d~

in which the aggregate ^is

embedded,

_are the _same, I-e-

they

_are universal parameters ^{like the fractal dimension}

Di,

which has been studied

by

other authors

[41] showing

that it

depends

_on

both,

2,I

Di

2,o

,""

l 9 ~,"

l, 8

,~

;i

1.7

"

1.6

o-i

d/(rB.Rg)

Fig.

2.-Fractal dimension

Dr

for

large

aggregates versus the relation

d/(r~-R~)

used in the simulation. The

symbols

_are the simulation results, The dashed lines _are to

guide

the _eye,

d~

and

d~. So,

for

example,

Honda et al.

[41]

have studied this

dependence

for _a

particle-

cluster model

showing

that

Di

=

(d(

₊

_d~

_I

_)/(d~

₊

_d~

_I

_).

_In

_figure

₃

we have

plotted,

for

aggregation

_processes with two different values of the fractal dimension of the reactive

trajectories,

the

N-dependence

^of the fractal dimension of the

aggregate

;

symbols

indicate fractal dimensions obtained

using equation (6),

and the continuous _curve is the result of

fitting equation (8a).

The results of

k~, fl~

and

Di

_are summarized in table I for the two

aggregates.

This table shows that

flN

decreases when

Di

^decreases and that

k~

has the inverse behaviour.

However,

_more studies _are

required

to obtain _a definite relation between the behaviour of these parameters

(k~, fl~

or k~,

fl~)

and the fractal dimension

Di.

1.8

m

D(~/)

m

m

m

1,6

a

, a

1.5 a

1.4

O

N

Fig. ^{3. Box}

counting

fractal dimension D

(N)

of two bidimensional aggregates obtained by _computer simulation,

plotted against

N.

Aggregates

_were obtained

using

reactive

trajectories

with

(rs R~)/d

=

25

(A)

and (rB

R~)/d

₌ ⁵

(.).

Solid lines _are the results of

fitting equation (8a).

Table I. Values

of p~, k~

and

Di

obtained

by fitting of

^the simulation results _to

equation (8a).

Aggregate flN ~N ~f

0.350 ± 0.010 0.285 _± 0.005 1.79 _± 0.01

II 0.462 _± 0.005 0.121 _± 0.002 2.00 _± 0.01

The small-N

N-dependence

_or small-r

r-dependence

of the fractal dimension of diffusion- limited

aggregation

clusters also arises with other methods of measurement, such as the radius of

gyration

method

(log R~

₌ const. ₊

D(N). log N)

or the correlation function method

(log

_g

(r/ro)

= const. +

(D (r/ro) d~). log r).

As _an

example,

in

figure 4,

in which

log

N is

2~7

2.3

»

. »

* »

. 4

* 4

* »

»

«

1.9

O.7 1-O

log _Rg

Fig. 4.- Log ^N (number of

particles

of the

aggregate)

_vs.

Log

_R~

(radius

of

gyration)

for the bidimensional aggregates of

figure

3. Symbols as figure 3. Solid fines represent ideal fractal behaviour (behaviour at large

scales).

plotted against log _R~

for the _same

aggregates

as in

figure 3,

the

N-dependence

of the dimension measured

using

the

gyration

^radius is shown

by

the deviation from the

straight

lines

corresponding

to the ideal behaviour of

large

aggregates. A similar deviation from

linearity

is observed in

plots

of

log g(r) against log

_r.

4.

Scattering

from swan diffusion limited

aggregates.

For _an

aggregate

of fractal dimension

Di,

the correlation function

g(r)

is

given by [42]

g

(r)

_{cc r}^{~~ ~~}

(9)

which leads

[43]

to the

following expression

for the

scattering

structure factor

S(q)

_:

i(q)

_cc

s(q)

_cc

q~f _(lo)

where _q is the

scattering amplitude

vector, ^and

I(q)

is the

scattering intensity.

In real systems

equation (10) only

holds _{over a} finite range ro ~ r _~

f (f

is the size of the

aggregate).

Several modifications have been

proposed

to model the effect of _an upper cut-off _at the

aggregate size,

the most

widely

used

being [43-45]

g(r)

= A _r ~~~~~

exp

(- r/f) (I I)

which leads

[43]

to

S(q)

=

~~ ~~~~

~

sin

i(Df

I arctan

(q, f

)1

(12)

(1

₊

1/(q, f )~)~~~~

^~~'~

(q ro)

^~

where r is the _gamma function. The lower cut-off at ro ^can be

largely

taken into account

by introducing

_an additional form factor

f~(q)

=

exp(-q~r(/6)

in the

expression

for the

scattering intensity I(q)

_:

1(q)

= P

VI _(Ps

_Po)~

f~(q _{S(q) (13)}

where _p is the

panicle

number

density

of the

aggregate, (p~ po)

is the contrast in

scattering length density

and

Vo

^{is the volume of the}

elementary particles.

When

Di,

in

equation (11),

is

replaced by

the

scale-dependent

fractal dimension

D(r/ro)

_{so as} to model the behaviour of non-ideal

fractals,

it becomes difficult to obtain _a

correspondingly

sensitive

expression

for

S(q)

unless it is assumed that q cc

I/r.

With this

assumption, equation (8b)

_can be written

D(q)

=

Di

~

(14)

+

k~.

_q 9

Ignoring

the upper cut-off and

assuming

further that

D(q)

varies

slowly

with q,

S(q)

_cc

(ro.

_q

)~~l~l,

and the

scattering intensity

will be

given by

the

equation

I(q) CCf~(q)(ro. _q)~~~~ (15)

for I

If

~ q ~ l

fro.

For observations at a

large

scale

(q

« I

fro),

D

(q

-

D~

and

equation (15) predicts

the

usually

assumed linear

dependence

of

log

I _on

log

_{q ;} but for small _r this is not _so,

Figure

5 shows calculated

scattering

data for the

computer-simulated

fractal aggregates of

figure

3.

4

~~

i~

3

'._ '>._

° _""

-1,5 -1.I -O.7 -O.3

log _q

Fig.

5.-

Log

I

(scattering intensity)

_vs.

Log

_q

(scattering

vector

amplitude)

for the aggregates of

figure

3.

Symbols

_as

figure

3. The dotted fines _are the results of

fitting equation (15) ~f~(q)

=

constant)

and the

straight

lines

correspond

to ideal fractal behaviour.

It should be realized that the non-linear behaviour reflected in

figure

5 has

nothing

to do with the non-linearities associated with the _upper and lower cut-offs. The upper cut-off effect

occurs for _r

>

f(q

~ l

If ) (even

in the linear

region

of

log

I vs.

log

_q

plots,

_so

long

_as

f

^is

finite),

and the lower cut-off effect _{occurs r}

~

ro(q

_> I

fro)

_; the effect shown in

figure

5 _occurs in the range ro ~ r ~

f.

This is evident in

figure 6,

which

shows,

for _an ideal aggregate ^of

constant fractal dimension

Di

and for _an aggregate ^with a

q-dependent

fractal dimension

given by equation (15),

the

(log

_q

)-dependence

of

log I(q)

over a range of _q

including

both cut-off

points (the

values of

D~, k~

^and

fl~

^used are those obtained for the

aggregate

with the smaller fractal dimension in

figure 5,

and

f

has been chosen

large enough

for ideal fractal

behaviour to have been reached before

cut-ofo.

Note that the ideal and the non-ideal fractals behave

identically

in both cut-off

regions.

4.9

~_ q_

v- ~-

3.O

f )

~ u

~

i ^I

Q+

~$

^' ~

~

bo

I

'

° ~ ~

v fi

I-I tr ., t~

-O.8

-3.2 -22 -1.2 -O.2 O.8

1/j

i~~ _~

^i/r

Fig.

6.

Log

I _vs.

Log

_q for _an ideal fractal aggregate of constant fractal dimension

Dr (solid line)

and _a non-ideal fractal aggregate

(dotted line),

with the upper and lower cut-offs

given by

the

equations (12)

and

(13). (It

is used the values of p, k and

Dr

obtained

by

fitted of the aggregate ^with smaller fractal

dimension,

fis assumed

large enough

that for _some value of _q

D(N)

₌

Dr.)

The

region

into the square

corresponds to figure 5.

S. Conclusions.

In this communication _we have shown that aggregates

produced by

diffusion-limited

aggregation only

_{possess a} definite fractal dimensiod

D~

^if

they

_are

sufficiently large.

When

N,

the number of

particles making

_up the

aggregate,

^is

small,

the fractal dimension of the aggregate increases with N

approaching D~

in the limit. This behaviour has been observed

experimentally

in the

growth

of ammonium chloride

crystals [46].

Deviation from linear

scattering

behaviour in the

region

between the upper and lower cut-off

regions

is also

predicted

for small N and should be taken into account to

interpret experimentally

observed deviations

[47].

Acknowledgments.

This work has been

supported by

the

Spanish

Direcc16n General de

Investigac16n Cientifica

_y Tkcnica under

Project

No. PB86-0651-C03-03. M-C-B-N- thanks the Ministerio de Educac16n y Ciencia for the

fellowships

awarded.

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