Flory approximant for self-avoiding walks near the theta-point on fractal structures

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Flory approximant for self-avoiding walks near the theta-point on fractal structures

Iksso Chang, Amnon Aharony

To cite this version:

Iksso Chang, Amnon Aharony. Flory approximant for self-avoiding walks near the theta-point on frac- tal structures. Journal de Physique I, EDP Sciences, 1991, 1 (3), pp.313-316. �10.1051/jp1:1991133�.

�jpa-00246323�

Classification

PfiysibsAbsmwts

05.50 75.40

Shom Communication

Flory approximant for self-avoiding walks

_near

the theta-point

on

fractal structures

Iksso

Chang (I)

and AJnnon

Aharony

_(~>~)

(1)

School of

Physics

and

Astronomy, Raymond

and

Beverly

Sackler

Faculty

of Exact

Sciences,

lbl Aviv

University,

lbl Aviv

69978,

Israel

(~) Department

of

Physics, University

of

Oslo, Norway (Received 7Janua~1991, accepted

lo

Janua~1991)

Abstract. We present a

Flory approximant

for the size exponent and the _crossover exponent of a

self-avoiding

walk at the

theta-point

on ~actal structures. This

approximant

involves the three fractal dimensionalities for the backbone, ^the minimal

path,

and the resistance of the fractal structures.

There has been much interest in the statistics of _a

single

linear

polymer _{[1, 2]}

with

quenched

randomness

[3-16].

At

high temperature

^T or in

good solvents,

where the excluded volume inter- action between two non-bonded nearest

neighbor

monomers

prevails

over their attractive inter-

action, usually

this statistics is described

by

that of

self-avoiding

walks

(SAWS) [1,

2] ^{with the same} randomness. In the

presence

^{of dilution it has been shown that} for p ^> pc ~p is ^the

probability

of each bond

being occupied

and _pc is the

percolation threshold)

SAWS _{on a} dilute lattice

belong

to the _same

universality

class of SAWS _{on a} pure ^lattice

[3,

⁵

7, 10, 14].

^{For SAWS} on the

incipi-

ent infinite cluster at the

percolation

threshold

Aharony

and Harris

(AH) _[13] presented

a

Flory approximant

for the

exponent

v

describing

the

scaling

of the mean square ^end-to-end distance

(R~)

_~-

^N2v

^of SAWS of

N-steps

on fractal structures

[12,15].

For _a

long

time there had been _a debated

question

whether the randomness

changes

the

universality

class of SAWS _{at p}

= pc from

that _{on a} pure lattice.

Recently

Meir and Harris

[14]

^{answered this}

question

and showed

by

^the

renormalization

group theory

that it is described

by

a fixed

point

different from the _{one on a} pure lattice. Also their numerical

results,

obtained

using series,

_{are very} close _to the

prediction

from AH and exclude the pure ^{lattice values for} Euclidean dimensions

higher

than 2.

As T

decreases,

_or the solvent becomes poor, ^{there is} a

special temperature

^T = 8 at which the

(two body)

excluded volume and the attractive interaction

compensate

^each

^other,

^therefore

the three

body

interaction _{starts to}

play

a role

[1, 2, 17, _18].

As a

result,

the chain behaves closer to ideal. Also _a

single

linear

polymer

at its

e-point

is

usually

modeled

by self-attracting

SAWS on a lattice. It is

interesting

to see how the

quenched

randomness affects the

asymptotic

statistical

behavior of SAWS at the

e-point (8-SAWS).

In thin paper we

present

a new

Flory approximant

for the size

exponent

_we ^{and the} crossover

exponent #

of e-SAWS _on fractal structures.

314 JOURNAL DE PHYSIQUE I N°3

Our

starting point

^{is based} on the work of AH

ill, 13, _lti~

and follows the standard

procedure

for

Flory

formulae

[1, 2].

^In

general

one can write the

probability

of _a

single

linear

polymer

of N

monomers to have a linear dimension R as

exp(-F),

with

~d~,B

^°

N2

~g3

F ₌

~

⁺

jv (j)

+

jw

~~ +

II)

R B R B

The first term is the

entropic

contribution due to the

probability

of

finding

_a

regular

^{random walk}

whose end-to-end distance h R after

N-steps,

where

dw,B

=

DB

+

(R

is the fractal diJnension of random walks _on the backbone and

(R

is _a resistance

exponent.

^{The value of} a, for

typical

random

walks,

has been

argued by

^{Harris and}

Aharony [Ill

to be

" "

~

,~~~

~

(~)

m~

where

dma

is the fractal dimension of the minimal

(or chemical) path.

The second term in equa- tion

(I)

is the two

body

interaction

contrAution,

which may ^be

repulsive

_or attractive

depending

on the

sign

of the interaction

parameter

_v, ^and the third term is the three

body

interaction _con-

tribution,

with the interaction

parameter

w. The last _two terms use the mean-field

approxima- tion,

in the sense that _one

replaces

a concentration _c of _monomers

by

the average concentration

? ₌

N/ROB,

where

DB

is the fractal dimension of the backbone

(the polymer

^is restricted to the

backbone,

since its self-avoidance

prevents

^it from

coming

^back out of

dangling bonds).

^Both _u

and _w contain factors of I

/kBT,

and thus become

unimportant

at

high

T. For

regular SAWS,

with

v > 0 and _w

=

0,

AH

[13]

^{minimized F in}

equation (I)

and found that

~~ "

Dmw

^"

D~

+

adw,~

^~~~

The _same

expression

was also found

by

others

[12, @, using

different

arguments.

At the 8

-point,

v vanbhes since the

(two body) repulsive

and attractive interactions

compensate

^each other. Min- imization of

equation (I)

with

respect

to R

immediately yields

our new

Rory approximant,

"~

De 2DB+ adw,B

^~~~

Note that for the non-dilute _case ~p =

I), DB

₌

d, _dw,B

= 2 and _dn~~n

=

I,

and

equations (2) (4)

recover the usual

Floty

results _vmw

=

3/(d

+

2)

and _{Me =}

2/(d

+

I).

In table I _we lit the

predictions

kom

equation(4)

for e-SAWS _on the infinite

percolation

clus- ter at p = pc

together

with those for SAWS kom AH

~Eq.(3))

ford ₌

2, 3, 4,

and > 6

respectively.

First,

we notice that _vmw > we for every

d,

which makes _sense

physically. Second,

_we is

always larger

than the

pure

^{lattice values}

(we

₌

)

_and

)

for d ₌ 2 and >

3),

^{which tells} us that the dilu-

tion of sites swells the

polymer.

^Thin h to be

expected:

the dilution of sites increases

effectively

the excluded volume interaction

(or

introduces _a

long

_range excluded volume

interaction)

between

monomers, since it eliminates some

spatial region

where a chain _can be embedded. Therefore it makes _a chain more swollen than _a chain on a pure ^{lattice for a}

given

^{Euclidean dimension} d. We also observe that the

inequalities,

_d~n;n <

DsAw

<

De

<

DB

hold for every d. These

inequalities

are exact for _our

approximants,

and follow kom the fact that

dm~

<

DB.

on

regular lattices,

the

Flory

formulae

quoted

above work

only

for d <

du,

with the upper critical dimensions

du

₌ 4 for SAWS and

du

₌ 3 for e-SAWS. For d _>

du,

one recovers the

lbble 1. Esdhlates

for

the

Floly approdmant for

the she

e~ponent

vmw and _we, and the _crossover e~ponent

#

on the

incipient

in

finite pelcolation

cluster _{at p}

= p~ The values

of dn~a,

_DB~ and

dw,B

are cited

Jtom r~fierence [13].

d

dm;n DB dw,B DSAW

_"

I/VSAW De

_"

I/Ue #

2 1,13 1.62 2.61 1.31ci

1/0.76

1.39 _~

l/0.72

0.17

3 1.36 1.83 3.14 1.53 _ci

1/0.65

1.62 _ci

1/0.62

0,14 4 1.62 1.94 3.53 1.73 _ci

1/0.58

1.79 _ci

1/0.56

0.09

/

6 2 2 4 2 ₌

1/0.5

2

=

1/0.5

0

non-interacting

behavior N _«

R~.

In our case we never recover the

analogous

behavior N _«

R~»>B. However, equations (3)

and

(4) yield DsAw

₌

De

₌

DB

for d > 6. This is _a direct

consequence

^{from the fact that}

loops

are irrelevant for d _>

6,

and the

polymer simply

follows the

singly

connected

backbone,

as in _one dimension. Thin

probably

results in the absence of a

e-point

for d _>

6,

and identifies d

= 6 as an upper critical dimension.

We _{now turn} to the

vicinity

of the

e-point,

with a small _{v «}

(T 8).

One _can describe the

crossover from the 8-SAWS behavior to that of SAWS

by minimizing equation (I)

with

respect

^to

N, keeping

both two

body

and three

body

interaction terms

(v # 0,

_w

# 0).

This

yields

~ ~

~DB W

R " N"~

(j

+

p~j ₍₅₎

Thh

expression

can be written _as the

scaling

form

R

~w

N"e

f ((T e) N~)

,

(6)

and the crossover

exponent #

b identified _as

#=DBwe-1. (7)

The

resulting

values of

#

for the the

incipient

infinite

percolation

cluster _{at p = pc are} also listed in table I. The

scaling

function

f(z)

behaves as a constant for _z

-

0,

and as

z~/(DB+°dw,B)

_for

z - +cx>. The _crossover will _occur when

vN4

_{h of} order _w, I.e. when N _«

(T 8)~~/~.

For T _<

8,

I.e. v <

0,

we

expect

a

collapsed phase

with N _«

R~m",

where _dn~ax h the fractal dimension of the

longest

SAWS _on the cluster.

However,

this

collapse

h not contained in

equation (I).

The fact that

#

> 0 for all d _<

6,

and

#

vanhhes at d ₌

6,

also identifies d

= 6 as a border line dimension at which _we may

expect logarithmic

corrections.

Recently Roy, Chakrabarti,

and Blumen

[19] attempted

to construct _a

Flory approximant

for the size

exponent

^{of SAWS and 8 -SAWS} at p = pc.

However, they

started witli _a

particular

choice of the

probability

dhtribution function based _on ad hoc

assumptions

^which have _no

convincing physical justification.

In

particular,

their

argument yields

_a

discontinuity

at d

= 3 for SAWS and

df

= 3 for

8-SAWS,

where

df

_{is the}

_spectral

_{dimension on the}

_incipient

_infinite

percolation

cluster

316 JOURNALDEPHYSIQUEI N°3

at p = pc.

Rho,

unlike our

expression,

^their results do not

reproduce

the correct

entropic

behavior

(tile

first term in

Eq.(I))

at

high

T.

Finally

we consider e-SAWS _on the

Sierpinski gasket (SG)

in _two and three dimensions. Real-

space

renormalization group calculations _were used to show that _no

e-point

exists on the two dimensional SG

[20, 21].

On the three dimensional

SG,

tliese calculations

yielded

_{we t}

0.529,

and 0.507 for _a

length rescaling parameter

^b =

2,

^{and 3}

respectively [20, 22].

^Our

equation (4),

for tile

SG, yields

_we ci

0.762,

and 0.749 in two

dimensions,

and _{we Ci}

0.645,

and 0.622 in three

dimensions for

=

2,

and 3

respectively.

At

present,

we are not aware of any numerical evaluations of _{we on} the

incipient

infinite

per-

colation cluster _{at p}

= pc, with which we can compare our results. We _are

currently carrying

out

exact series enumeration

calculations,

which may

yield

some such

comparhon

in the future.

In

conclusion,

we have

presented

a new

Rory approximant

for the

vicinity

of the

e-point

_on

fractals,

based on an

approach

which unifies the treatment of SAWS and 8.SAWS.

Acknowledgements.

The work at the lbl Aviv

University

was

supported by

tile Israel

Academy

of Sciences and

by

the U-S--Israel Binational Science Foundation. The work at the

University

of Oslo has been sup-

ported by VISTA,

a research

cooperation

between the

Norwegian Academy

^{ofscience and} Letters and Den norske stats

oljeselskap

_a-s-

(STATOIL),

and

by

the

Norweghn

Council of Science and

Humanities

(NA~F~.

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