Series studies of self-avoiding walks near the θ-points on 2D and 3D clusters at the percolation thresholds

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Series studies of self-avoiding walks near the θ-points on 2D and 3D clusters at the percolation thresholds

K. Barat, S. Karmakar, B. Chakrabarti

To cite this version:

K. Barat, S. Karmakar, B. Chakrabarti. Series studies of self-avoiding walks near the θ-points on 2D

and 3D clusters at the percolation thresholds. Journal de Physique I, EDP Sciences, 1993, 3 (10),

pp.2007-2016. �10.1051/jp1:1993228�. �jpa-00246848�

(2)

J. Pl~ys. I Hance 3

(1993)

2007-2016 OCTOBER1993, PAGE 2007

Classification Physics Abstracts

05.50 75.40

Series studies of self-avoiding walks

_near

the o-points

_on

2D and 3D clusters _at the percolation thresholds

K. Barat

(*),

S. N. Karmakar and B. K. Chakrabarti

Saha Institute of Nuclear Physics,

I/AF

Bidhannagar, Calcutta 700064, India

(Received

12 May1993, accepted in final form 22 June

1993)

Abstract. Enumerating all the N-stepped SAW configurations _on the infinite percola-

tion cluster of Monte Carlo generated bond diluted lattices

(in

dimension d

= 2 _as well _as

in d

= 3) at the respective percolation thresholds, the thermally weighted _average end-to-

end distance < RN > of self-interacting self-avoiding walks _are determined. The configura-

tionally averaged < RN >

(over

different percolation

clusters)

are then fitted to _a scaling form

<

R(

> _~J

N~"° f(N*r),

where _r

=

(T

o)

lo

denotes the temperature interval away from the

o-point, v° is the tricritical

(o-point)

size exponent, # is the _crossover exponent and f is the scaling function. From the best fit, the values of 0, v° and # _are obtained for the 2D and 3D

lattices considered. We find v°

ce 0.74 ~ 0.02 and 0.60 ~ 0.02 for the tricritical exponents _on the percolation clusters

(at

the percolation

thresholds)

in dimensions d

= 2 and d

= 3 respectively.

We also find theta-temperature (0) _ce 0.71~0. IS, 1.25 + 0.3 and 0.5~0,15 for bond dilute square,

triangular and simple cubic lattices respectively _on the critical percolation clusters. Our scaling

fit results for o-point and the v~ _values _for _various percolating lattices _are then compared with

some theoretical

(mean field-like)

estimates.

Introduction.

The

changes

in the conformational statistics of linear

polymers (in good solvent),

_as the temperature ^is ^lowered ^from ^the

high

temperature

limit,

have been studied

extensively

in the last two decades

Ill.

In the

high

temperature ^limit ^the

(lattice) self-avoiding

^walk

(SAW)

model is quite successful in

capturing

the universal behaviour of the conformational statistics. In such limit of

high

temperature, the effective interactions arise

mostly

from the excluded volume

considerations. With

lowering

of the temperature, a

point,

called the

o-point

temperature or

the tricritical

point,

is reached when the

two-body

excluded volume term

(or

the

#~

term in the effective LGW Hamiltonian of the n-vector model

ill

is

exactly

cancelled

by

the

growing

attractive interaction and the statistics is

governed by

the

higher (#6)

ordered excluded volume (*) Permanent address: Vidyasagar College for Women, 39 Sankw Ghosh Lane, Calcutta 700006, India.

(3)

2008 JOURNAL DE PHYSIQUE I N°10

terms. At this temperature a crossover occurs from the

high

temperature SAW statistics to _a tricritical

o-point

statistics. Below this transition temperature the attractive forces dominate

over the

(entropic) repulsive

term and force the chain to

collapse.

Recently

the statistics of linear

polymers

in porous media has been studied

extensively, using

the random lattice model [2-4]. ^In ^this ^model ^the ^SAW steps are allowed

only

_on the

occupied

bonds

(sites)

of the

randomly

bond

(site)

diluted

lattice,

where lattices still

percolate

_[5]. In the

high

temperature

limit,

the

percolation

effect _on the SAW

(excluded volume)

size exponent u~ is found _to be small both

theoretically

and in simulations [2, 3].

However,

the effect of

percolation

^fractal on the

8,point

size exponent

u°

_has _been _indicated _to _be

quite prominent.

Roy

et al. [6]

predicted

^u° G£ 0.68 _on the

percolation

cluster

compared

to

u(

G4 0.58 _on pure two-dimensional lattices

(using Flory

type

approximation;

_see

Appendix B). Chang

^and

Aharony

_[7]

predicted ^u°

Gt 0.72 _on the

percolation

cluster

compared

to

u(

G£ 0.66 with their

approximation

_on _a _pure two-dimensional lattice. Recent series studies of

thermally averaged

end-to-end distance of

(nearest neighbour) interacting

SAWS _on

percolation

clusters [8], ^when fitted to _a

scaling form,

indicated u° _G£ _0.74

(compared

to

u(

££ 0.57 _on pure

lattice)

and also

a finite

o-point

value

(8

% 0.71,

compared

to 80 * 1.54 _on pure square

lattice).

In

fact,

in this _case of SAWS _on

percolating

lattices at the

threshold,

the existence

(non-vanishing value)

of the

o-point

temperature is of utmost importance ^and

significance

and 8

being

_non-zero, it allows for the manifestation of the

collapsed phase

_on the

percolation

cluster.

In this paper, we will concentrate on the tricritical

(8-point)

behaviour of SAWS and extend

our

percolation

threshold. We

use series enumeration method and _our results for the SAW sizes, when fitted to the

scaling form, yield

^the estimates of

o-point

value and the size exponent u° _at _the

o-point-

These

results for

8,point

and u° for various two- and three-dimensional lattices _are then

compared

with those obtained from "mean-field like" estimates of

o-point (given

in

Appendix A)

and the

"Flory-like"

estimate for u°

(given

in

Appendix B).

Formalism.

The distribution function

GN(r)

which represents the number of

N-stepped

SAW

configura-

tions with end-to-end distance r, is not Gaussian for SAWS and its

scaling

behaviour is _now well-known

iii.

The total number of SAW

configurations

GN and the

thermally weighted

average end-to-end distance _< RN > grows with step size N _as

GN _"

£GN(r)

~J

p~N~~~

,

(1)

~

and

~

~~

_~~

IN li ~~~~~~~

~~~~

i~~

'~

~~~

' ~~~

where _e represents ^the nearest

neighbour (attractive)

interaction, M the number of non-bonded

nearest

neighbours

and T the temperature. Henceforth _we put

unity

for both the interaction

(e)

^and the Boltzmann factor

(k).

Here _p denotes the

connectivity

constant and its value

depends

_on the lattice type, _~t and _u _are the universal exponents ^which

depend only

_on the lattice dimension d

v=v~ forT>8

(2a)

= v° for T

= 8

(2b)

=

1Id

for T _< 8

(2c)

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N°i0 THETA STATISTICS AT PERCOLATION THRESHOLD 2009

where

(2a) corresponds

to the excluded volume

(random SAW) region, (2c) corresponds

to _a

collapsed phase

and at the tricritical _or

o-point,

the radius

(size)

_exponent is denoted

by

^v°.

There _are _many established

methods,

_e-g-, series enumeration and Monte Carlo _etc, methods [9], ^for ^the determination of

8-point

and tricritical exponent

v°

on different lattices.

We report here the results of series studies for SAWS _on Monte Carlo

generated percolation

clusters _on bond diluted _square,

triangular

and

simple

cubic lattices. The

thermally weighted

average end-to-end distances _< RN > are

fitted,

after

configurational averaging

_over

percola-

tion clusters

(denoted by

overhiad

bar),

_to the

scaling

form

<

R(

_>

=

N~~° f(N'T), ₍₃₎

where _T ₊

(T- 8) lo, f

is

scaling

function and

#

denotes the _crossover exponent. ^We enumerate all the

possible

SAW

configurations

_on the ercolation cluster for different steps and also for

different lattices. From the value of _<

R(

_> and

using

^the ^above

scaling

relation _we obtained

v°,

8 and

#

for different

percolating

lattices

(and

also _on _some

regular

lattices for

checking purpose).

Although

there _are numerical methods

(e.g.,

series enumeration, Monte Carlo etc. for esti-

mating

the

o-points

_on various

(regular)

lattices [9],

they

involve _a tedious numerical

analysis

and _are not

easily

extendable _to the random lattices.

Recently,

_a

simple (mean field-like)

_es-

timate for the

o-point

has been established

([10];

_see also

Appendix A),

where the

b-point

is shown _to be

linearly proportional

to the _average number Z~R ^of ^non-bonded

(nearest) neigh-

bours _on _a SAW _:

8 _« Z~R.

(4)

Z~R in turn can be calculated _as Z~R ci

p[(Z i) pal (see Appendix A),

where _p is the lattice bond

occupation

concentration and Z is the

(pure)

lattice coordination mumber. _po is the

connectivity

constant of the pure lattice.

Similarly,

the

rigorous (renormalization

_group

etc. theories for the tricritical behaviour

(8-point

_exponent

v°) iii

_are quite difficult _to extend for random

lattices,

in

particular

to

percolation point;

and _are _not yet ^available. ^There are

thus several attempts to

improve

_upon the

Flory-type

theories for the tricritical behaviour _on fractal lattices [6, 7, 2].

Indeed,

_one such

Flory-type approximent

for the size exponent v is sketched in

Appendix

B. This

approximent

[6, _2] ^reduces to well-known and established

Flory

type ^results ⁱⁿ ^the appropriate ^limits. ^Our

scaling

fit results for

o-point

values and v° for

various

percolating

lattices _are

compared

with these _mean

field-type

estimates.

Simulations and results.

Using

Monte Carlo method first _we generate ^the ^bond ^diluted ^lattices at the

respective

_perco- lation thresholds (p~ = O-S, ^0.3473 and 0.2488 for _square,

triangular

and

simple

cubic lattices

respectively [5]).

^Then we search for the infinite cluster which is the

spanning

cluster and touches all the boundaries. If _we do not find the infinite

cluster,

_we omit that

configuration

and generate a new

configuration

until _we get a

spanning

cluster.

Isolating

the infinite cluster with the bond diluted

lattice,

_we take _a suitable

origin

and generate ^the ^SAWS on that cluster and find

G~j~~(r),

the number of

N-stepped

SAWS with M nonbonded

neighbours

with _a fixed

end-to-end distance _r and for _a

particular spanning

_or

percolating

cluster

configuration (I).

From the sets of

Gjj~~(r)

_we compute

Gjj~~

=

£~ Gjj~~(r).

FYom

G~j~~(r)

we calculate the

average

end-to-end distance for _a

particular configuration (I)

which is

given by,

<

(R()(~)

_>=

£M~ r~G~j~M(r)e~'/T

£M~ G~~ (r)eM/T ⁽⁵⁾

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2010 JOURNAL DE PHYSIQUE I N°io

2.0

lS0

= 2.47

= 0.80

~ ~

loo ⁼ o.33

<R~>

~ so

1.0 20 40

f(X)

~"''~

^~ ~

o.s

~°~

-l 0 1 2 3 4 5 6

x

Fig.

i. Plot of

f(~) (e

_<

R( >/N~"°)

_against

_~(+ N*r)

for the series data of SAWS _on _a bond diluted _square lattice _at the percolation threshold. The inset shows the plots of _<

R(

> against N at different temperatures for the _same lattice.

The values of _<

(R§)~

> are evaluated _over _a wide _range of temperature T

(within

_say

0.25 < T _< 5.00 at the interval of

0.05)

for all the lattices. We then

generate

^about ¹⁷⁵ different

(percolation

_or

spanning cluster) configurations

^and ^the step size N of SAWS is varied

(9

< N _< 31 for _square, 7 _< N _< 30 for

triangular

and

simple

cubic

lattices).

We then calculate _<

R§

_>, the

configurational

_average of _<

(R§)~

> at different temperature.

These data for _<

R§

_> for various N and temperature ^for _any

particular

lattice is then fitted _to the

scaling

relation

(4).

We

plot

^the scaled variables

fix) [e< R§

_>

/N~~°] against x(n N'T)

for various choices of

v°, #

and 8. The choices for these exponent

(v°

and

#)

and 8-

point

values _are determined

by

the best

collapse

of the

"experimental" points

_on _a

single

_curve

representing

the

scaling

function

fix),

_as shown in

figures

1, 2 and 3 for _square,

triangular

and

simple

cubic lattices

respectively (the

insets show the <

R§

_> _versus N _at _some

typical temperatures).

It may be noted that each such _curve _comes from _a

collapse

of about 900 to 1800

points (coming

from about 20 different step ^sizes ^N at each temperature ^and

nearly

45 different temperatures ^considered ^for _square ^lattice ^and 90 for

others).

The best fit values _are

given

in table

I,

where _we have also included the _square lattice results from _our

previous study

18].

We thus find finite

nonvanishing o-points

for

percolating

lattices: 8 _ci 0.71, 1.25 and 0.5

(compared

to iA ~ 0.1, ^2.8 ^~ ^0.8 ^and ^{2.0 ~ 0.5} on pure lattices [9]) ^for square,

triangular

and

simple

cubic lattices

respectively.

These estimates for the

o-points

_are

compared

in

figure

4 with the theoretical

(mean field-type)

relation

(4),

which is derived in

Appendix

A. Our best fit values for v° _on _the

percolation

cluster suggest v° ci 0.73 ~ 0.02 and 0.60 ~ 0.02

(compared

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N°i0 THETA STATISTICS AT PERCOLATION THRESHOLD 2011

2.0

T = s.oo

T ₌ i.oo

1.5 2 T ₌ 0.25

<R~>

~

~~~,~,, ii iir,>,zx; ^.:

~mmwwg~s~a~,+;;>m~_oil»., N ^{4 o}

.3m'%~

,-<%~

fl~~

o.s

0.0

0 2 4 6 8

X

Fig.2.

Plot of

f(~)

_(% _<

R( >/N~"°

against

~(% N*r)

_for _the series data of SAWS _on _a bond diluted triangular lattice at the percolation threshold. The inset shows the plots of <

R(

> against N at different temperatures for the _same lattice.

2.5

= s.oo

= o.so

T = 0.25

2.0

<R~>

1 5

2 0

~

4 0

~

,d'~~°

,~jl("'.'

1 0 _=.

o.s

~°~

0 5 lo IS 20 25

x

Fig.3.

Plot of

f(~)

(% <

R( >/N~"°)

_against

~(% N*r)

_for _the _series _data _of _SAWS

on a bond

diluted simple cubic lattice at the percolation threshold. The inset shows the plots of <

R(

_> against

N at different temperatures ^for ^the same lattice.

(7)

Table 1 Parameter values for the best fit to the

scaling

relation

(3 )

for different

percolating

lattices.

Percolating

lattice type ⁸ uo

~

Square

0.71~0.15 0.74~0.02 0.20~0.10

~iangular

1.25 ~ 0.30 0.73 ~ 0.01 0.20 ~ 0.10

Simple

cubic 0.50 ~ O-is 0.60 ~ 0.02 0.20 ~ 0.05

4

1

~~

3.5

/~

~ ~

~/

~ 0.5 /

, Q _/

/

3 b fir

Z~f,

_i

~'~i~

b

c Q /

2.5 ~

e 0.5 1

~eff

/

C /

8 ^~

/

~ ~ _l &~Q

b /

1

~~

/~

~l

Ql

0.5 /

/

~

0.2 0.4 0.6 0.8 1

Zeff

Fig.4.

Plot of o-point values for different regular _as well _as percolating lattices

(bond

diluted) against ZeA

(calculated

from Eq.

(A.2)).

The inset gives the plot of the ZeA values against the _same

obtained from enumeration method

(denoted

by

Z]~).

Here _a, b and _c denote the results for _pure square, triangular and simple cubic lattices respectively and a', b' and c' respectively the _same for pc clusters.

(8)

N°10 THETA STATISTICS AT PERCOLATION THRESHOLD 2013

0.30

triangular

0.20

Zeff

o.io

Simple

^Cubic

0.00

o~o2 0.04 0.06 °.°8 ~°~

1/ N

Fig-S-

Plot of

Z$

vs.

i/N

for different bond diluted percolating lattices. The extrapolated values

(for

N _~

cc)

_are taken for the enumeration estimate Z]~ in the inset of figure 4.

to 0.57 and 0.50 in pure

lattice)

in d

= 2 and d

= 3

respectively.

Our theoretical estimates from

Flory-type approximant (indicated

in

Appendix B) gives

^v° ci 0.68 and 0.61 in d

= 2 and

d

= 3

percolation

cluster

respectively.

In

figure

4, we have

plotted

the various

o-point

values obtained here for different

percolating

lattices

(along

with those for _some

regular

lattices; taken from Ref.

[9]), against

^the ^theoreti-

cally

estimated Z~R

(ci p[Z

i

pal

^for various

percolating (and regular)

lattices. This part is done to check the

proposed

linear

relationship (4)

between 8 and Z~R. In

fact,

the dotted

straight

^line

passing through

the

origin

indicates the fit to linear

relationship

and

gives

the

slope

of the

straight

^line to be about 4.1

(for

the value of the

proportionality

constant in

(4)).

However, Z~R _can also be

directly

estimated from enumeration results. Z~R is calculated

from,

zN_£M~~NM

eR~

£

~ ^'

M NM

where GNM

=

(i/n) £]=i

_G~j~~. This average number

Z$

of

bridges

or number of _non- bonded nearest

neighbour pairs

is

plotted (in Fig-S) against

i

IN

and _we make _a least _square fit of these points. The

extrapolated

value Z~R of

Z$

for

large

step ^size

(N

_-

oc)

is obtained from the

intercept

^of the best fit line. The inset in

figure

4 shows the

plot

of Z~R ^values

obtained from the above enumeration

(denoted by Z]~)

versus its theoretical estimate from _p.

Exact

equality

of the two requires a

straight

line

passing through

the

origin

with

slope unity

and _as _can be _seen from the inset, this is indeed

closely

the _case.

(9)

Conclusion.

We have enumerated all the

N,stepped

^SAWS on the infinite

percolation

cluster of the Monte Carlo

generated

bond diluted _square,

triangular

and

simple

cubic

lattices,

at their

respective percolation

thresholds. The

thermally weighted

_average end,to-end distance _<

RN

>

(as

determined

by (2)

^and

(5))

after

configurational

_average _over different

percolation clusters,

_were fitted _to the

scaling

form

(3).

The best fit choices for the enumeration data _to the

scaling

relation

give

finite

nonvanishing 8,point

values _on the

percolation

clusters: 8 _ci

0.71,

1.25 and 0.50

(compared

to iA ~ 0.1, ^2.8 ~ 0.8 and 2.0 ~ 0.5 _on pure lattices [9]) ^for square,

triangular

and

simple

cubic lattices

respectively.

It _may be noted here that the

comparatively

accurate results _on the

percolation

cluster _are due to the

possibility

^of

enumerating

all the

configura-

tions upto

larger

step sizes N (~J

O(30))

_on

percolation clusters, compared

to that _on _pure lattices

(where

N _<

O(20)).

In

fact,

this

advantage

even

helps

to make up for the _errors due to

configurational

fluctuations in the

percolation

clusters. The

scaling

fits also

give

^v° _ci 0.73 and 0.60

(compared

to 0.57 and 0.50 in pure

lattices)

in d

= 2 and d

= 3

respectively

which _are in

good

agreement ^viith ^the ^values ⁱⁿ ^reference [9]. ^It should be mentioned here

that, although

the

fitting

to the

scaling

relation

(3)

is not very sensitive to minor

changes

in the value of

#,

it is

quite

sensitive to the

changes

in the values of v° _and _8. _The

nonvanishing o-point

values obtained for

(highly raniified) percolation

fractals _are then

compared (in Fig.4)

with _a

(mean field-like)

theoretical _estimate

(Eq. (4);

_see

Appendix A).

The

qualitative

agreement is

encouraging.

Our best fit values for the tricritical exponents

v°

on the

percolation

cluster also compare well with the

generalised Flory approximant (see Appendix B)

for the

SAW,

obtained from the

polymer

^radius ^of

gyration

distribution.

Appendix

A.

Chakrabarti and

Bhattacherjee

_[10]

suggested that, coming

from the

high

temperature

(T

>

8)

SAW

phase,

the

o-point

should _occur, _on _an _average

(in

the _mean field

sense),

when the average number of non-bonded

neighbours _(Z~R)

_on _a SAW

(giving

the attractive

interaction)

is balanced

by

the thermal noise

(kT,

at T ₌

8)

_:

8 _« Z~R

(Ai)

where Z~R is the _average number of non-bonded _nearest

neighbour pairs

^of _any lattice

point

on a SAW.

Now,

at any intermediate

point

_on _a SAW there _are _p

(connectivity

constant as

defined in

(i))

number of options _on the average ^to make _a further forward step. If Z be the coordination number for the pure

lattice,

the maximum number of such

options

at each

point

will be Z i;

taking

_care of the fact of

avoiding

the visit to the site just

previously

visited.

The

self-avoiding

restriction for all the

options

to p. Hence the average number of the non-bonded nearest

neighbour

pairs is

given by

Z~R=Z-i-p (A2)

In

percolating

lattices with increase of lattice dilution I.e. with decrease of lattice bond _con- centration p, p will be affected. The

change

^of _p will be prominent at the

percolation

threshold pc. If the SAW is not confined to the

percolation

cluster alone then

p(p)

= ppo

exactly,

where pa =

p(p

₌

i).

In

fact,

the linear

relationship

is very

closely

maintained [3] upto the _perco, lation threshold _even for the

(infinite) percolation

cluster average. Then for the dilute lattice the estimate of Z~R ^is

zeR

£~

P(z i) ll(P)

"

P(z

i PO)-

(A3)

(10)

N°10 THETA STATISTICS AT PERCOLATION THRESHOLD 2015

This relation

obviously

reduces to

(A.2)

for _p

= i.

Extensive numerical

checking

^of these two relations

(A.i)

and

(A.3)

from the available data and also numerical estimates of

8,point

and Z~R _on various

regular

and

percolating fractals,

have been made

by

Barat [10], ^who estimates the

proportionality

constant for

(A.i)

in 8 to be about _{3.6 ~ 0.3} for all the

(simple)

^lattices considered.

Here,

the

figure

4 shows the variation of

8-point

values

against

the Z~R estimates

(both using

the relation

(A.3)

and also numerical estimates _see the

inset)

for various pure [9] and

percolating

lattices

(as

obtained in this

paper).

The variation _over this wide _range _can indeed be

approximated

_as alinear _one, _as

suggested by equation (A.i),

and the

proportionality

constant turns out to be about 4,i.

Equation (A.3)

is confirmed

by

the inset in

figure

_[4].

Appendix

B.

This

Appendix gives

_a

Flory

type

approximation

for the

polymer

size

(radius

of

gyration)

exponent.

Following

Lhuillier

iii]

and

Roy

et al. [6, 2], ^the ^form ^of ^the distribution function

P(R)

of the

polymer

radius of

gyration

is

given by P(R)

~-

exPi-Nl(N~/R)"

+

(R/N~)~ II, (Bl)

where _a, _> 0. The

proportionality

^factors _are omitted for

simplicity.

This form of

P(R)

ensures that the distribution is very small if R is outside the bounds N~ < R < N~. The

above form of

P(R)

shows that it

decays exponentially

to _zero _as R _crosses the above bounds.

The distribution function is maximum at the most

probable

size RN when

dP(R)

dR

R=RN

~

If _we _express the _most

probable

size _as RN

°J N~ then _we get v =

(a

₊

~b)/(1+ ~),

where

~ =

&la.

Let _us first consider the

regular

lattices

I)

For SAWS in _a pure lattice in the

high

temperature ^limit

(random

SAW

limit) ^N~/~

<

R < N. In this _case the

two-body

interaction term in the

Flory

free energy

(F(R)

~J

In

P(R))

is ensured

by

_a

= d.

Similarly

the elastic term in

F(R)

comes from

= 2. In the SAW

limit, therefore,

_we get v =

vi

₌

3/(d

₊

2),

the normal

Flory

estimate for SAW size exponent.

ii)

At the

o-point,

the

appropriate

bound _was

suggested

to be [6, 2]

N~/~

_< _R _<

NW

_Here

the

three-body

term in

F(R)

is ensured

by

a = 2d and the elastic term in

F(R) again

_comes

from = 2. One thus gets v =

v(

=

(d

+

5)/[(d

+

2)(d

₊

i)].

It

gives v(

=

7/12

G£ 0.58

compared

to the exact value

4/7

Gt 0.57 in d

= 2. This d

= 2 result for VI

(and

also the

F(R))

is the _sortie _as that obtained from various

screening

considerations [12]. ^Also VI =

1/2

when

three-body

term vanishes

(becomes independent

of

N)

at d > dc G4 2A in this

approximation.

Now _we consider the

percolation

clusters:

I) ^Here ⁱⁿ ^the

high

temperature

limit,

the bound _on R is N~/~'B _< _R < N~/~m"», ^where dB is the

percolation

backbone dimension and dm;n is the shortest chemical

path

dimension [5]. ~

(+

&la)

should incorporate here the

spectral (random walk)

dimension of the

percolation

cluster for the elastic energy ^term and is

given by

_[6] _~ ₌ dwdm;n

/dB

(dw -dm;n

),

where dw is the random

walk dimension _on the

percolation

cluster. One then gets _v ₌ v~

=

(dm;n

+ kdB

)/dBdm;n(1+ ~).

The _same

expression

was obtained

by Aharony

and Harris [13] ⁱⁿ a different _way. This

gives

the values of

v~(> vi

G£ 0.77,

0.66,1/2

in d

= 2, d

= 3 and d > 6

respectively.

ii)

For SAWS at the

o-point

_on the percolation

cluster,

the appropriate bound is assumed to be N~/~'B < R <

N'

_The _value _of

~ here

corresponding

to the

three-body

interaction

JOURNAL DE PHYS>DUE -T 3 N' '0 OCTOBER >993 74

(11)

gives

_[6] u = u°

=

(1+1cdBu~)/dB(1+1c),

where1c

=

dwdm;r~/2dB(dw

dm;r~). ^This

gives v°(> u()

££ 0.68, 0.61 and

1/2

in d

= 2, d

= 3 and d > 6

respectively.

It is _to be noted that here also the _upper critical

dimensionality

^for the

S-point

_on the

percolation

cluster shifts _to dc _# 6.

In _an alternative

approach Chang

and

Aharony iii

extended the

Flory approximant

for the theta

point

_on

regular

structures to the _cases of theta

point

_on fractal structures.

They

obtained

"~ dB~~~w,B

~~~~

where

o=dm;r~/(dw,B

dm;r~) and dw,B " dB +

(r, (r being

the resistance exponent. ^This

expression

reduces to the

Flory

result u°

=

v(

₌

2/(d

₊

I)

in the pure lattice limit

(giving v(

Gt

0.66, compared

to exact result

4/7

Gt 0.57 in d

=

2).

On the infinite

percolation

cluster at p = p~, this

gives

^v° Gt

0.72,

0.62,

1/2

in d

= 2, 3 and d > 6

respectively.

It may be noted that the _upper critical

dimensionality

shifts to d ₌ 6 in this

approximation

also.

References

ill

See _e-g-, de Gennes P. G., Scaling _concepts in polymer physics

(Cornell

University Press, Ithaca- NY,

1979).

[2] ^See e-g-, Chakrabarti B. K., in Polymer physics: 25 years of the Edward's Model, S. M. Bhat-

tacherjee Ed.

(World

Scientific, Singapore, 1992) p.167.

[3] Roy A. K. and Chakrabarti B. K., J. Phys. A 20

(1987)

215;

Lee S. B. and Nakanishi H., Phys. Rev. Lett. 61

(1988)

2022;

Meir Y. and Harris A. B., Phys. Rev. Lett. 63

(1989)

2819;

Lam P. M., J. Phys. A 23

(1990)

L831;

Nakanishi H. and Lee S. B., J. Phys. A 24

(1991)

1355

Muthukumar M. and Baumgartner A., Macromolecules 22

(1989)

1937;

Baumgartner A., in [2] p.127.

[4] ^Obukhov ^S. P., Phys. Rev. A 42

(1990)

2015;

Vanderzade C. and Kamoda A., Europhys. Lett. 14

(1991)

677; Phys. Rev. A 45

(1992)

_5335;

Smailer I., Machta J. and Redner S., Phys. Rev. E 47

(1993)

262.

[5] See _e-g-, Staulfer D. and Aharony A., Introduction to percolation theory, 2nd edition

(Tayler

lc

Francis, London, 1992).

[6] Roy A. K., Chakrabarti B. K. and Blumen A., J. Stat. Phys. 61

(1990)

903.

[7] Chang I. and Aharony A., J. Phys. I France1

(1991)

313.

[8] Barat K., Karmakar S. N. and Chakrabarti B. K., J. Phys. A 25

(1992)

2745.

[9] ^Privman V., J. Phys. A 19

(1986)

3287;

Seno F. and Stella A. L., J. Phys. France 49

(1988)

739;

Meirovitch H. and Lim H. A., J. Chem. Phys. 91

(1989)

2544.

[10] ^Barat K., Die Makromolekulare Chemie: Theory lc Simulations

(1993)

_to be published;

Chakrabarti B. K. and Bhattacherjee S. M., J. Stat. Phys. 58

(1990)

383.

[iii

Lhuiller D. J., J. Phys. £Yonce 49

(1988)

705.

[12] ^de Queiroz S. L. A., Seno F. and Stella A. L., J. Phys. I £Yance 1

(1991)

339;

Lhuiller D. J., J. Phys. II £Yance 2 (1992) 1411.

[13] Aharony A. and Harris A. B., J. Stat. Phys. 54

(1989)

1091;

Roy A. K. and Blumen A., J. Stat. Phys. 59

(1990)

1581.