Irreversible Growth Algorithm for Branched Polymers (Lattice Animals), and Their Relation to Colloidal Cluster-Cluster Aggregates

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Irreversible Growth Algorithm for Branched Polymers (Lattice Animals), and Their Relation to Colloidal

Cluster-Cluster Aggregates

R. Ball, J. Lee

To cite this version:

R. Ball, J. Lee. Irreversible Growth Algorithm for Branched Polymers (Lattice Animals), and Their

Relation to Colloidal Cluster-Cluster Aggregates. Journal de Physique I, EDP Sciences, 1996, 6 (3),

pp.357-371. �10.1051/jp1:1996161�. �jpa-00247189�

Irreversible Growth Algorithm for Branched Polymers (Lattice Animais), and Their Relation to Colloidal Cluster-Cluster

Aggregates

R-C- Bali

(*)

^{and J-R-} Lee

Theory

of Condensed Matter, Cavendish

Laboratory, Madingley

Road,

Cambridge,

CB3 OHE,

England

(Received

19 October 1995,

accepted

28

November1995)

PACS.61Al.+e

Polymers, elastomers,

and

plastics

PACS.61.43.Hv Fiactals;

macroscopic

aggregates

PACS.05.70.Ln

Nonequilibrium thermodynamics,

irreversible _processes

Abstract. We _prove that _{a new,} irreversible growth

algorithm,

Non-Deletion Reaction- Lim- ited Cluster-cluster

Aggregation (NDRLCA), produces equifibmum

Branched

Polymers, expected

to exhibit Lattice Animal statistics

iii.

We

implement NDRLCA, off-lattice,

_{as a} computer _sim- ulation for

embedding

dimension d

= 2 and 3,

obtaining

values for critical exponents, ^fractal dimension D and cluster _mass distribution exponent _T: d

= 2, D _m 1.53 + 0.05, _{T =} 1.09 + 0.06;

d = 3, D ₌ 1.96 + 0.04, _{T =} 1.50 + 0.04 in

good

agreement with theoretical LA values. The simulation results do not support recent

suggestions

_[2] that BPS _may be _m the

same

universahty

class _as

percolation.

We also obtain values for _a

model-dependent

critical

"fugacity",

_zc and in-

vestigate

the finite-size effects of _our simulation,

quantifying

notions of

"inbreeding"

that _occur in this

algonthm. Finally

_{we use an} extension of the NDRLCA

proof

to show that standard Reaction-Limited Cluster-cluster

Aggregation

is _very

unhkely

to be _in the _same

universality

class

as Branched

Polymers/Lattice

Animals unless the backbone dimension for the latter _is

considerably

less than the

published

value.

1. Introduction

In this paper we

investigate

the

relationship

between the ensemble of cluster

configurations

formed

by

irreuersible

pair-wise aggregation (such

_as observed in dilute colloidal

dispersions),

and the

eq~tiiibri~tm

ensemble of branched

polymers.

Dilute-limit branched

poiymers

in _a

good

solvent _may be

thought

of _as clusters _in

equilibrium

with

respect

to

exchange

of _monomers and

rearrangement

of structure,

subject

to exduded volume

constraints,

but _so well

separated

from each other that inter-cluster interactions may be

neglected. Off-lattice,

the chance _in this ensemble of

exactly closing

a

loop

is of _{measure zero.} Two

important

universal

exponents

for this

problem

_are the fractal dimension of the

dusters, D,

and the

exponent,

_{T, in} terms of which the ci~tster size distrib~ttion _is

expected

to be of form:

CN

+~

Y~N~~ (1)

(*)

Author for

correspondence (e-mail: rcbl©phy.cam.ac.uk)

Q

Les

Éditions

_de

_Physique

₁₉₉₆

358 JOURNAL DE

PHYSIQUE

I N°3

where N is the duster size and _y

depends

_on

system

details.

The ensemble of Lattice Animals

(LA'S)

is obtained

by disregarding

the

multiplicity

of

possible

connectivities with which on-iattice BP'S

(Lattice Trees)

_may be identified _{as a} tree.

It has

long

been

accepted

that dilute BP'S and LA'S _are in the _same

universality

Mass.

Lubensky

and Isaacson [1]

computed

values for LA critical exponents ⁱⁿ an e

expansion

about d~ ₌ 8.

Parisi and Sourlas _[3] demonstrated _a

relationship

between the

Lee-Yang edge singularity

of the

Ising

^{model in d 2} dimensions and dilute-limit branched

polymers

in d dimensions and

hence

derived,

from the _exact solution of the

former,

_T

= 1

(but

_no value for

D)

in d

=

2,

and

T =

3/2,

D

= 2 in d

= 3. A renormalisation group

approach by

Derrida and De Seze _[4]

gives

D _m

1.56,

in d

=

2;

there is evidence _[5] that lattice animals _are not

conformally

invariant.

Reaction-Limited Cluster-duster

Aggregation

is _a

growth

nlodel in which all

geometrically possible

bonded

configurations

between

pairs

^{of dusters} are

equally likely

and

sampled

in

an irreversible _sequence. It is

physically

motivated

by

colloidal

aggregation

under conditions where the

probability

of two clusters

joining

_upon contact is very

low,

_so there is _no

significant favouring

of

configurations

most

likely

to _occur first

during

_a duster-duster encounter. It has

long

been

speculated

that RLCA and LA'S share _sortie critical exponents. ^{This is}

partly

nlotivated

by

the convergence of

RLCA,

LA'S and BP'S for d < d~ ₌ 8 to give

Flory-Stocknlayer

clusters with D

= 4 and _T

=

5/2.

Ball et ai. [6] have looked at RLCA kinetics via the Smoluchowski rate

equation

for duster

mass

populations, CN(t):

~

CN(t)

=

~j _{K(P, Q)CpCQ CN} ~j _{K(N, P)Cp (2)}

~~ ~

P+Q=N P

where the rate kemel is

argued

to be of form:

K(P, Q)

m~

P~

_for

Q

_m P

(3)

K(P, Q)

+~

PQ~~~

for P >

Q (4)

and for > 1 the _mass distribution is

expected

to be of form:

CN

~

N~~

(5)

They argued

for _a value of =1 in d

=

3,

^for which the Smoluchowski

equation gives

T =

3/2 (in

the absence of subdominant terms in the kernel to which the solution is _now

thought

to be

sensitive

[7-9] ), nlatching

BP'S. In RLCA simulation for d

= 3 _on

lattice,

Brown and Ball

[10]

find = 1.06 + 0.02 and D

= 2.1+

0.03,

whilst Meakin and

Fanlily _[11]

report = 1.12 and

T m 1.69 for off-lattice RLCA simulation

Experiments

_on the RLCA of real _aqueous

gold

colloids [12] gave D

= 2.02 + o-1 and _T

= 1.5 +

0.2,

but in other

systems

_[13]

higher

values of

T and kinetics

equivalent

to

= 0 have been

reported.

In Section

2,

we present an

analytic argument,

^based on the Smoluchowski rate

equation [14],

which proves that _a

growth algorithm

which is

dosely

related to RLCA

produces equilibrium

BP'S. Dur

algorithnl

is one of irreversible

pair-wise, reaction-limited,

cluster-duster _aggrega- tion in

which, however,

the

parents

^of a new duster _are not deleted from the

sample;

NDRLCA

(Non-Deletion RLCA).

In Sections 3 and

4,

_a numerical simulation _is

presented

and

good

_agree-

ment with BP exponents is found. In Section

5,

we discuss the time-scale of the simulations

by studying

the

"fugacity"

_as it

approaches

critical. Finite-size

effects,

characterised _as "inbreed-

ing"

and

arising

from the

returning

of parent ^clusters to the

sample,

_are

analysed

in Section 6.

Finally,

in Section

7,

_we discuss whether the standard RLCA model and BP'S _are in the _same

universahty class,

and conclude that

they

almost

certainly

are not.

2. A New Proof that

Polydisperse

^RLCA without

Deletion, NDRLCA,

Gives Branched

Polymers

The model under consideration

here, NDLRCA,

_is

irreversible, polydisperse

reaction-hmited cluster-duster

aggregation

^{in which the}

parent

^{clusters which} go ^to

produce

_{a new} child cluster continue to exist thereafter _as

independent

clusters witliin the

sample.

Hence the amount of

matter present is not constant _as in standard

RLCA,

but increases with time. Here _we show

that such _a model follows

directly

from _a

simple

charactensation of branched

polymers.

Consider the number of

dusters, CM,

_{as a} function not

just

of their size N but rather _as

a function of their full

geometry tif,

for the

equilibrium problem

of branched

poiymers:

_up to some normalisation

conventions, CM

must be either 1 _or 0 _as the

configuration ^tif

is either allowed _or not. Consider _now

counting

all of the _ways in which _a duster of N atoms _can be broken into two

fragments:

since it will be held

together by precisely

N bonds _or nearest

neighbour connections,

this

gives:

(N 1)Cw

=

~j K(P, Q)CpCQ (6)

P+Q=tif

where the

right-hand

side of the

equation

counts the _same

quantity

in terms of the

possible

ways of

assembling

^duster

^tif

^from two

fragments.

The

kernel, K(P, Q),

_is the number of ways that dusters of

geometry

P and

Q

_can

join

to form _a duster of

geometry

P +

Q

=

tif.

Specified

here at the level of detailed duster

geometries,

^this kernel is identicai to that

governing

the

corresponding description

of RLCA

(see below, equation (9)).

The condition P +

Q

"

tif requires, amongst

^other

things,

^that the _{masses are} matched _as P ₊

Q

= N. As _a

result,

the

equation

^above is invariant under the transformation

CM

_-

(y'/y)~Cw, confirming

that the

duster _mass distribution of branched

polymers

is

only

universal up ^to a factor of

g~

Consider _{now a} different

generalisation

of the cluster numbers defined _as:

Ww(z)

₌

^z~-~ Cv (7)

where _{z is} in effect _an extra

fugacity

factor associated with each of the

(N ₁₎

bonds _in the duster.

Differentiating

and

substituting CM

from

(equation (6)) gives:

)Ww(z)

=

£ KIF, Q)Wp(z)WQ(z) ₁₈₎

This _is akin to the Smoluchowski _rate

equation

^{less the} deletion term, if _we think of _{z as}

analogous

to _a time. The

analogous

and exact

equation

of evolution for RLCA is:

(Cwlt)

=

~j Kli', Q)Cp (t)Colt) 2Cwlt) ~j K(tif, i')Cp lt) (9)

where the

differing

factor

of1/2

is _an

insignificant

matter of convention here.

The

important point

is that

equation (8)

defines _a stochastic _process with "time" _z which

we can

implement, by analogy

with

RLCA,

_as an irreversible duster

aggregation

model. The absence of the deletion terms

(which

in RLCA

give

_mass

conservation)

_means that the model must be _as RLCA

e~cept

that when two

(parent)

dusters succeed in

aggregating,

their child cluster _is added to the

sample

without the

parents

themselves

being

removed. The value of _z in the simulation _can be

simply tracked,

_as discussed in Section 5. Note that

given

the

expected

360 JOURNAL DE

PHYSIQUE

I X°3

form of the cluster size distribution for

equilibrium

branched

polymers (Eq. (1)),

the

analogous z-dependent quantity

behaves at

large

N _as:

WN(z)

_m N~~ ^~

...(10) (10)

zc~~

where the critical value of _z is given

by

_{zc =}

~.

Strictly speaking,

the discussion above starts from the

(infinite)

ensemble of branched

poly-

mers and maps this onto a

growth

model for _a

time(z)-dependent

ensemble. In

practice

_we

can

implement

the

growth

model

numerically only

_{on a} finite

sample,

the members of which must not become too correlated with each other if the results _are to be _a fair

sampling

of the ensemble. This is _an

increasingly

_severe constraint with

increasing

time due to

dusters,

_as parents,

spawning dosely

related

children,

which _we

quantify

below under

"inbreeding".

3.

Computer

Simulation of Irreversible

Polydisperse

NDRLCA

The

starting point

is _a

sample

of M atoms each of which is _a cluster of size 1. These

single-atom

clusters _are

aggregated

to form further clusters of atoms of fixed

geometry.

The

original single-

atom clusters will be referred to as monomers. To

sample

with

equal probability

all

possible

ways of

joining

two

clusters,

_we choose at random two atoms and

place

them in contact in a random direction.

0nly

if this results in _no

overlap

between any of the atoms of the two associated dusters

(parents)

is the

attempted aggregation accepted.

In this _case the _new

(child)

cluster is created with _new atoms which _are

duplicates

of those of the two

parent

^{dusters. The}

two

parent

^{dusters and their} atoms remain in the

sample,

which is the

only

^difference between this model and RLCA.

In _a finite

sample

it is necessary ^to decide to what extent a duster _can

aggregate

^with itself

(or

indeed clusters related to it discussed under

"inbreeding",

Sec.

6);

we disallowed

only aggregation

of _a duster with its identical self. Because _our simulations _were limited

by

available store rather than

time,

_we stored dusters

implicitly

that

is,

in terms of which two

parent

^dusters a

given

chiid cluster had been born from and how the _two

parents

^{had been}

joined.

This reduces the

storage requirement

to of order the number of

dusters,

rather than of order the total number of atoms. To _test

possible aggregation

events _we therefore had _to

build the atom lists of the parent ^dusters

by going through

this data

recursively.

This costs

only

of order _{ni + n2} where _ni and _{n2 are} the _masses of the

parent dusters,

but with

large

coefficient. To test for

overlap,

_we searched for

pairs

of atoms, one from each

duster,

doser than

unity separation:

^this costs of order _nin2, but _was somewhat

mitigated by checking

first for

overlaps

between _atoms close _in the duster _atom lists to the _pair of atoms

being joined.

Values for

calculating

radius of

gyration

statistics and _mass distribution _were stored. In addition values for the

following quantities

_were recorded at

regular

intervals: total number of

attempts

made to

join clusters,

total number of clusters

formed,

and the

f~tgac~tg~

z

(see

Sec.

5).

A further

quantity

that _was recorded _was

inbreeding, Xcju~,

_{a measure} of the _extent

of _common

parentage

in clusters

being

created.

Aggregation

_was

stopped

when the a,~erage

inbreeding

of the last hundred dusters exceeded _a given value. We report results from three data sets, shown _in Table 1.

4. Simulation Results

Results for the radius of

gyration

_vs. duster size show dear _power law

scaling, RN

+~

A~~/~,

where D is _a fractal dimension.

Figure

1 shows

In[RNÎ

_vs.

ln[N] plots

for d ₌ 2 and 3. A

Table I. Data sets

for

NDRLCA sim~tiations.

set

embedding

number initial number of average

inbreeding

dimension,

d _{monomers, m} additional dusters of last 100 dusters

2 4.660 _x

3 5 _x 8.100 x

3 x

+

o

+

O 2 3 4 S 6 7 8

lniNi

Fig.

1.

Average

radius of gyration _vs. cluster _size for NDRLCA clusters:

(o)

two dimensions

(set 2d.1); (+)

three dimensions

(set 3d.1) (see

Tab. I for set

details).

The fines drawn have slopes:

1ID

= 0.65 for d

= 2 and

1ID

= 0.51 for d

= 3.

gradient equivalent

to D

= 1.96 + 0.04 is

dearly

_seen between

In[N]

_m 2 and

In[N]

_m

7,

consistent with the

expected

value of D

= 2 obtained from field theoretic considerations

[3].

The

large

number of dusters available for each data

point helps

to

keep

random fluctuations down to

insignificant

levels. In two

dimensions,

from renormalisation

theory _[4],

_{we expect} the

fractal dimension to be D _m 1.56. The

gradient

measured between

In[N]

_m 2 and

In[~T]

m 6.5 is

equivalent

to D

= 1.53 +

0.05,

in

good agreement

^{with this.}

The cluster _size distribution _{is more} dillicult _to

analyse,

not least because _we expect the power law to be

heavily weighted by

_an

exponential

tail for

large

N _or short time _as

CN

"

N~~(z/zc)~. _Figure

2 is the

log-log plot

^of duster _size distribution for d

= 3 using ^set 3d.1.

It exhibits the

expected scahng

of _T

= 1.50 + 0.04

(theoretical

_T

=

I.à, [3])

^{for the "dose}

to

equilibrium"

state of the system, _up to

In[N]

_m 5. Note that _a cut-off size of N

= 1000

was introduced for this set to _{save on} computer ^time

ii-e-

all dusters

exceeding

N

= 1000

were

discarded).

This cut-off should not affect statistics of N _<

1000,

_as

clearly

_no smaller cluster _may be made from _a

larger

_one, but it does _mean, in

principle,

that _{z can} evolve

past

zc.

Figure

2 shows the time evolution of the distribution _{as z} increases ~vith time.

Figure

3 addresses the two dimensional _case

(set 2d.1)

where _we obtain

scaling

close _to that

predicted,

362 JOURNAL DE

PHYSIQUE

I N°3

+

O

Fig.

2. Cluster size distribution _vs. cluster size for NDRLCA in three dimensions

(set 3d.1),

evolving

_m "time" towards critical

fugacity,

_zc. Initial _{monomers, m}

= 5 _x 10~

were used. Successive

plots

alternate

symbols

for

clarity.

From left to

right

each

graph

has _m 10~

more clusters

represented

than the last. Left most 10~ _extra clusters;

right

most 8.1 _x 10~ extra clusters. Sohd line has

gradient

-T = -1.5.

o

+ +

+ +

+

+ +

+

+

+

+

f

+

)

~

2 3 4 5 6 7 8

lnfNJ

Fig.

3. Cluster _size distribution

vs. cluster _size for NDRLCA in two dimensions

(set 2d.1), evolving

m "time" towards critical

fugacity,

_zc. Initial monomers, m = 10~

were used. Successive

plots

alternate

symbols

for

clanty.

From left to

right

each

graph

has _m 10~

more clusters

represented

than the last.

Left most 10~ _extra clusters;

nght

most 4.66 _x 10~ _extra clusters. Solid fine has

gradient

_{-T =} -1.09.

T = 1.09 + 0.06

(theoretical

_{T =}

1,

_[3] ^for the hnear section of the

graph

closest to

equilibrium.

The _power law

approximation

_appears to be valid _out to

In[N]

m 5

again,

which is well before the

point

where cluster

inbreeding, X~jus (N)

exceeds

15%.

5.

Measuring Fugacity-Theory

and Results

In this section _we consider the

relationship

between the model

system

"time" scale _as measured

by fugacity,

_{z, as} it appears in

equation (8)

and the "real" simulation time scale _as measured

by

the number of

attempts

^made to

join

two dusters

together

to form _{a new one.} An

understanding

of the role of

fugacity

in the

dynamical system

and _some

knowledge

of

scaling

of

(z z~)

close to z~ is useful in

interpreting

the results.

First let _us consider the evolution of the total cluster

numbers, WTOT(z), (from Eq. (8)):

dlwToTlz))

₌ dz

~j _Kli', Q)Wplz)Wolz) (11)

Now

assuming

that the cluster numbers _are normalised _so that

Wi

=

1,

then

d(WTOT (z)

=

ds,

where ds is the number of successful

aggregations

_pet

original

_monomer

during

the interval.

On the other

hand,

_{we can}

_compute

the ratio of successful to

attempted

_{moves as:}

~ K(l', Q)Wp(z)Wolz)

)

=

~'~~

~

(12)

where the numerator counts ail the _ways in which two dusters _can be

successfully aggregated

and the denominator counts the number of _ways in which the simulation _can

attempt

an

aggregation;

here _{natm, is} the total number of atoms

(per original monomer).

Combining

these two results then

gives

^the evolution of the

fugacity

_as:

)

"

~&L (13)

where both the nunlber of

attempted

_{moves a} and trie number of

atoms,

natm, are

expressed

_per

original

_{monomer in} the

sample.

In _terms of unnormalised number of

attempted aggregations,

number of

atoms, Natm,

and the number of

original

_monomers,

M,

this _grues:

Zsim " M

~ N£ÎJ l14)

attempted aggregations

where _we have assumed the initial condition of _{pure monomer}

corresponding

to _{z =} 0.

We _now ask how

z

approaches

_{z~ as} the number of

attempted mojej

^becomes

^large. tope

to _z

= z~ the number of _{atoms varies as: natm m}

~

N~~~

^~ _{dN m 1-} ^~

,

Î ^zc

zc from

which, using equation (13),

_{we can}

integrate

the number of

attempted

moves:

z 2T-4

a m

/ _dz'

^~

₍₁₅₎

0

C~

The

resulting

behaviour

depends

_upon how

T compares with

3/2.

For

T <

3/2,

_{z - z~ as}

a - oc and _we obtain: _{z~ z m}

(a

+

const.)~~/(3~~~)

for _{T <}

3/2

and _{z~ z m}

exp(-const. a)

for _T

=

3/2.

For _T >

3/2

the model would reach _z~ at finite _a

= a*:

z~ z m

(a* a)~/~~~~~~ (16)

364 JOURNAL DE

PHYSIQUE

I N°3

whereupon

_our

expressions

break down and what

happens

in _a finite simulation _{is an open}

question.

It would be

interesting

to

explore this,

for

example by considering

the infinite

dimensional _case,

equivalent

to

neglect

of

overlap,

where _T

= 5

/2

is known for

Flory-Stockmayer

dusters.

There _are various ways ^to arrive at _an estimate for _z~, if _{we are} able to measure a value of

T. From

equation (10),

note that the

gradient

^of the

plot

of

In[Civ(z) N~]

vs. N is given

by:

~~~~~~Î~

^~~~~ ~

~~

ÎÎÎ

~~~~

For _T <

1.5,

_{z~ may} also be obtained _as the

intercept

of _a

plot

of _{z vs.} a~P where _-p is:

~

(3

2T) ~~~°~ ~~'

_{~~~~~ ~~~~}

From

this,

and _a

knowledge

of the value of

z reached in the simulation _so

far,

_{we may} obtain

a value for _zc.

z~ is

expected

to be invariant with respect to the number of initial _monomers, but

depend

_on the

embedding dimension,

and this _was born out

by

measurement. We obtained:

z~ m 0.371 for d

= 3

and

z~ m 0.497 for d

= 2

using

the methods

represented by equations (17)

and

(18),

_as

appropriate.

6. Finite

Sample

Efiects

Inbreeding

and Fluctuation Measurement

Inbreeding

arises

naturally

_{in our}

irrite

non-deletion mortel of RLCA. Whilst _{a more} standard finite deletion mortel _may suffer from fluctuations _in the _mass distribution due to effects such

as run-away

clusters,

our mortel _can also suffer from at least two,

possibly serious,

forms of correlation that may affect the statistics.

They

_are both based around

counting

^the number of times

particular

_{monomers are} used. Recall that the _{monomers are} the

original single-atom

dusters that seed the

simulation,

and

they effectively duplicate

themselves as

they _aggregate

with

others,

_so that _one cluster will

typically

contain many ^atoms derived from the _same

monomer.

The first form of correlation considered is duster

inbreeding, Xcius.

This is the ratio of the number of

distinct, original

_monomers

represented

in _a

cluster,

_mcjus, to the total number of atoms, _natcius, in that cluster. Hence

inbreeding, X~jus,

is defined:

Xcius

₌ 1- ^'~~crus

~~~~~~~

(i~~

which

gives X~ius

" 0 for _a duster _{size natcius} made up of

atoms,

^{all of which}

originate

from

distinct initial _monomers, and

Xcju~

_{- as} fewer and fewer distinct _{monomers are}

represented

within the cluster. From this definition _we could monitor the average

inbreeding

of the last 100 dusters

produced,

_<

X~jus

_>ioo, and

stop

^the run when this became

greater

than _some cut-off

(1%

for d

= 3 and

15%

for d

= 2 simulations used to obtain critical

exponents). Figure

4 shows

X~ju~

vs. duster _size, N for t,vo three dimensional sets and _a two dimensional set.

~~~ju~ is

important

as it _measures the extent to which _our

system

is a fair

sample

of the ensemble of branched

polymers. X~ju~

_is

typically high

for _a cluster that is

produced

from _two

o

o ^o

*

+ +

o °

~ + ^~

o +

~ o ₊ ~

, o

o ~

~ o ù ^~

o "

+

~

*

~

w o ~

-~ *

O +

$ f *

~

_o

~z *

$ ^~

o

o o *

o

+ _* ^~

+ +

o +

~

*

*

~ o + ^*

+ *

o

~

Î

O 2 3 4 S 6 7 8

infNJ

Fig.

4. Cluster

inbreeding

_vs. cluster _size. Cluster

mbreeding,

Xcius

= 1 ^~°~~~~ is _{a measure} of

,

natcius

the fairness of

sampling

from the ensemble of BP _s

by

_our simulation.

(o)

two dimensions

(set 2d.1),

initial _{monomers m}

=10~,

_extra clusters grown = 4.66 _x 10~;

(*)

three dimensions

(set 3d.1),

initial

monomers m = 5 _x 10~, extra clusters grown = 8.10 _x 10~j

(+)

three dimensions

(set 3d.2),

initial

monomers m

=10~,

extra clusters gro,vn = 1.759 x10~.

clusters that _are

closely

related and of similar size. The chances of

picking

t~vo

similar,

related clusters is much

higher

in _our

system,

than if the choice _{was over} the whole ensemble. For this

reason we must be careful that

X~ju~

is not too

high

or our

sample

will be

untrustworthy.

We have also

investigated

_{a measure} of

inbreeding

bet,veen similar sized clusters. For _pur- poses of

analysis

the dusters _were counted in

bins, loganthmically spaced by

_mass. Then to

quantify

the

inbreeding

between all dusters within _a size bin _we defined

Xbin

in _a similar

manner to

X~ju~, except quantities

_{are now} counted _over all the dusters in the bin:

~in t i m~in

~

~~~~~n

j~~~

where _{mbin is} the number of distinct _monomers used in

constructing

all the dusters that _are found in the

bin,

and _natbin is the total number of _{atoms m} the bin.

Xbin

_measures correlation

across the set of dusters

produced. High Xbin implies

that the dusters within the bin _are

similar. The

typical high Xbin creating

event is when _a very small duster attaches itself to a

larger

one. Since there is _no

deletion,

the bin will _now contain _one of the parents ^{and the}

offspring.

Similar events _can continue to _occur to these

dusters, heavy

contributions

being

made _to

Xb~n,

until _a duster

eventually

leaves its

parent's

bin

by being

too

large.

The

problem

is

compounded by large

duster-small duster

pairings being

the most favoured.

Clearly Xbin

and

X~ju~

are related because _many

closely-related,

similar-sized

dusters,

_as measured

by Xbin,

_are

ideal conditions for

creating high

X~ju~ dusters. In

Figure

5 we see that

Xb~n quickly

becomes

large,

for the _reasons described above.

By

around

In[N]

m

3.5, Xbin

m

90%

for the _m

= 1 _x

105

sample.

Xbin

does not, in

itself,

call in to

question

the

fairness

of the

sample

from the ensemble of branched

polymers

that _our simulation

represents.

Rather it tells _us how many

independent

366 JOURNAL DE

PHYSIQUE

I N°3

o

O ° ° $ _*

+ ~ ~

~ +

O °

+ + ( _o

o O

° +

, o

O + _o " *

. +

o + o

C

+ o

~

t ~

"

O

~ _~

Î

~ Î

~ S

à

~ ^O

~M +

,~ ^O

ÎÉ

$ .

~

o

~

u~~

2 3 4 5 6 7 8

lnfNJ

Fig.

5. Cluster bin

inbreeding

_vs. cluster size. Cluster bin

inbreeding,

Xb~n

= 1- ^~~~~ _{is a}

natb~n

measure of

inter-dependence

of clusters _{m a} bin.

(o)

two dimensions

(set 2d.1),

_m

= 10~, extra clusters _grown

= 4.66 _X 10~.

(*)

three dimensions

(set 3d.1),

_m

= 5

x10~,

_extra clusters _grown

=

8.10 x

10~, (+)

three dimensions

(set 3d.2),

m

=10~,

_extra clusters _grown

= 1.759 _x 10~.

O

Fig.

6. Cluster size distribution CN _vs. cluster _{size in} three dimensions for set:

(+)

set 3d.2:

m =

10~,

extra clusters _grown

=

1.759x10~;

error bars calculated from Xb~n

analysis

of

"independence"

of clusters _{m a} bin and modified

by

_an

empincal

factor of 4

(see

text for

details).

Sohd fine has

gradient

-T = -1.5.

Oo~

*..

~°~ag

.

Sao

~

*~~O

~ OO

-~

#

~O~

-

$

O

$

~~ _O~

*+

°OOOOOO°°°°~~~

~

*+

~Î

Î

~4

~.~~

++

~~~ ~++++~~~

O

* ~

~

Î

~ o

o

" *

o * o o * * ~ * *

~

~O 2 4 6 8 la

in

fkJ

Fig.

7.

Probability

distribution function,

nm(k),

for _an initial _monomer to be used k times

by

virtue

of the atoms derived from

ii,

1-e-

nm(k)

_{monomers appear}

precisely

k times _as atoms in the final set of

clusters. Sets used:

(o)

"2d.1" _m

=

10~,

extra clusters _grown

= 4.66 _x 10~.

(*) "3d.1",

_{m =} 5 _x 10~,

extra clusters grown = 8.10 _x

10~, (+) ~'3d.2",

_m

=

10~,

extra clusters _grown

= 1.759 x 10~.

clusters _{are m} the bin. In other

words,

the _amount of

independent sampling

of the ensemble

is not _as

large

_as the number of clusters

produced

would

suggest.

^An estimate of the number of

"independent"

clusters in _a bin is: n]j~~~~~ ^m

il Xbin)

_n~iusbin, where _n~ju~bin is the

total number of dusters _m the bin. This

suggests

an estimated _error for the duster _mass distribution:

~iÎ

~

~N il

~

(Rllusbin)

^~_~Î

'(~l) (~l)

In

practice,

we found that the _errors

predicted by equation (21)

_were of the

right form,

but

we needed to introduce _an

(arbitrary) "empirical"

factor of 4 to achieve sullicient

magnitude

of _error for the

sample

3d.2 in

which,

due to

large Xbin

and sparse

dusters,

the _{errors were}

significant,

_see

Figure

6.

Finally,

_we looked at the

probability

distribution

function, nm(k)

for _{a monomer} to be used k times

by

virtue of the atoms that _are derived from

it,

1.e. _nm

(k)

_{monomers appear} k times _as

atoms in the final set of dusters. 0ne

might expect

an

exponential

distribution for

nm(k),

_as

the much

simpler

distribution of the

number,

_n, of times _an

object

is

picked

and

replaced

from

a set of _s items scales like

m~

el~"~~(~~l. However,

the creation of the distribution

nm(k)

is very much _more

complicated

than this and the

semi-log plot

of

In[nm(k)]

_vs. k did not

suggest

an

exponential.

The

log-log plot gives

_some

suggestion

of _{a power}

law,

_or several regions of _power law behaviour. When

nm(k)

is normalised

by

the number of _monomers, the

log-log graphs

of different sized 3d sets

overlap dosely,

whilst the 2d set has _a different

gradient, Figure

7.

Conjecturing

that _{a power} law _is present;

nm(k)

m~

k~~

we find:

d=3, ~fim1.5

d =

2,

_{~fi m} I.1

(22)

between

In(k)

_m 2 and

In(k)

_m 6. We have not made progress in

interpreting

these results.

368 JOURNAL DE PHYSIQUE ^I N°3

~

(

^w

Fig.

8.

Diagram

of _a cluster _{as a} "backbone'~ with side-branches

hanging

off ii. The backbone has curvilinear

length,

_s, and _an end-to-end

length (.

7. Is Standard RLCA in the Same

Universality

Class _as Branched

Polymers?

with

Reference to the Backbone

Dimension, DB

In this

section, by

consideration of the duster backbone and the reaction

kernel,

_we derive _a

scaling exponent equation

that links several

important exponents

^{and hence show that standard} RLCA is

unlikely

to be _in the _same

universality

Mass _as branched

polymers

and NDRLCA.

We may think of _a duster _{as a} backbone of curvilinear

length,

_s, and end-to-end

length, (,

with the rest of the duster

hanging

off this backbone _{as a} collection of side-branches

(see Fig. 8).

Each link in the backbone is _an atom of the duster. The smallest side-branches _are of

size

1,

1e.

just

the atom on the

backbone,

and the

largest

is

expected

to be

of

order the size of the whole duster. The curvilinear

length

^{of the backbone} is

expected

to be related to the

end to end

length, (, by

a power law:

s m~

(~b (23)

where

Db

is the backbone

dimension,

also called the

resistivity

exponent

[15].

We expect the total _mass,

N,

of the duster to scale with the end-to-end

distance, (,

_as:

(~

m~

N

(24)

where D is the fractal dimension. Hence the

scaling

of the average side-branch _mass,

~,

is

given by:

s

~

m~

(~~~~

m~

N(~~~b/~l.. (25) (25)

s

The _average side-branch _{mass is} also the average interval bet,veen

possible fragment

_masses into which the duster _can be divided

along

the backbone. To _see

this,

consider

starting

with _one

possible

division

point along

the

backbone;

the nearest division to this will be one

step

^further

along

the

backbone,

and of order _one side-branch _mass will have been transferred between the

two

fragments by

this step. ^It follows that the

probability

that the duster _can be cut into two

(exactly) equal fragments

of _mass ^~ is given

by:

2

P~ut

=

))

=

)

+~

N(~~/~~~) (26)

Strictly speaking

the above

only applies

to

cutting along

_a

particular backbone,

but it is

easily

shown that all cuts more

equal

than 2:1 must be

along

_{one unique} backbone.

Now _we consider the

scaling

of the

diagonal part

^of the reaction

kernel, K(N, N)

and its

significance

for the

universality

Mass

question.

We start from

equation (8) (Section 2)

written with

respect

to ci~tster sizes _as

opposed

to duster

geometries:

)LdN(z)

⁼

~ K(P, Q)LdP(z)Ldolz) (2?)

P+Q=N

where

~ON(z)

^{is the}

fugacity-dependent

duster size distribution and

K(P, Q)

is the kernel in terms of cluster size rather than

geometrg, given by:

~j _{K(P, Q)OpCQ}

K(P, Q)

=

~~'~~~~~~~~

~j _CpCQ (28)

lPl=P>loi=Q

from which the

z-dependence completely

cancels. The

probability

that _a cluster may be

split

in half is

given by P~ut

^~

(Eq. (26) ),

from which it follows that:

2

CNP~Ut

=

l~)

= Ii

~~, ~) CN/2C~l12 (29)

2 2 2

and K

~,

^~

,

scales _as:

2 2

N N

N-TN(D~/D-1)

K

~-, -)

m~

N~

m~ m~

~i(~b

~~~+~)

(30)

2 2 N~~N~~

This _means that branched

polymers

and NDRLCA

satisfy

the

exponent equation:

À=T-1+~~

(31)

If RLCA is _in the _same

universality

Mass it too must

satisfy equation (31).

In Section we

looked at the kinetics of RLCA _as discussed

by

Ball et ai. [6].

They

^observed that for _< 1 the Smoluchowski

equation

does not

give

_{a power} law for the cluster _mass distribution and for

>

1,

T > 2 is

obtained,

_m

disagreement

with the branched

polymer

value of _T

= I.à. This

leaves

= 1 _as the

only possibility

for

agreement, requiring:

Db(req.)

"

D(À

₊₁

_T)

=

(32)

This is well below the value

Db

₌ 1.36

[15] reported

for branched

polymers

_m d

=

3,

al-

though

it agrees well with numerical results obtained for

polydisperse

RLCA

simulation, Db

"

0.960 + 0.033

[16].

^{Both of these values relate} to duster _sizes of order

10~.

For RLCA and branched

polymers

to be in the _same

universality Mass, Db

" 1 would have to be satisfied

by

both: _m the

light

of the values cited _{we must} condude that this is most

unlikely.

8. Conclusions

We have introduced _{a new}

proof

that

polydisperse

RLCA without deletion of parents,

NDRLCA,

gives branched

polymers

which exhibit lattice animal statistics.

Computer

simulation has been

3m JOURNAL DE

PHYSIQUE

I N°3

shown to give values for critical

exponents highly

^consistent with those

predicted by

field the- ory and renormalisation group method

approaches il, 3,4]

to trie

study

of lattice animals and

dilute-limit branched

polymers, namely:

d = 2 _: D

= 1.53 + 0.05 _T

= 1.09 + 0.06

d = 3 _: D

= 1.96 + 0.04 _T

= 1.50 + 0.04

where the theoretical values _are:

d=2: Dm1.56 T=1

[4,3]

d=3: D=2

T=3/2

_[3]

As _an

algorithm

for

exploring

the statistics of Branched

Polymers,

the

key

asset of _our method is that it

generates

a whole cluster size distribution.

This, however,

is also its

greatest

drawback where

only

the

spatial

statistics of individual clusters _are of interest. We have also considered the concepts ^of

inbreeding

that arise _in finite NDRLCA and used them _to

explam

fluctuations in the duster size distribution. Further work would be useful in

understanding

_some of the

functions introduced to this end.

Using

the

approach developed

for

proving

that NDRLCA

gives

dilute-hmit branched

poly-

mers, we were able to show that standard RLCA is very

unhkely

to be _in the _same

universahty

class _as branched

polymers

_as it

probably

cannot

satisfy

the

exponent equation

derived

above, namely:

= T -1+

j ₍₃₃₎

which

implies

that the backbone

dimension, Db

for RLCA clusters must be 1 where _as the value for branched

polymers

has been

reported

_as

Db

" 1.36

[15].

Acknowledgments

The authors

acknowledge pilot

studies of the NDRLCA

algorithm by

B-L- Parkes and M.T.

Wilson,

^and research

funding by EPSRC, grant GR/K511948.

JRL

acknowledges

earher _sup-

port

^under an EPSRC

studentship.

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