Polymers at the Θ-point on fractals : results of Flory approximations

HAL Id: jpa-00246327

https://hal.archives-ouvertes.fr/jpa-00246327

Submitted on 1 Jan 1991

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Polymers at the Θ-point on fractals : results of Flory approximations

S. L. A. de Queiroz, F. Seno, A. Stella

To cite this version:

S. L. A. de Queiroz, F. Seno, A. Stella. Polymers at the Θ-point on fractals : results of Flory ap- proximations. Journal de Physique I, EDP Sciences, 1991, 1 (3), pp.339-349. �10.1051/jp1:1991136�.

�jpa-00246327�

J.

Phys.

I1

(1991)

339-349 _MARS 1991, _PAGE 339

Classification

Physics

Abstracis

36.20E 64.40A 64.60K

PoIyn1ers at the >point

_on

fractals

_:

results of Flory approx1nlations

S. L. A. de

Queiroz (I,*),

F. Seno

f)

and A. L. Stella (3)

(1)

Department

of Theoretical

Physics, University

of Oxford, I Keble Road, Oxford OX1 3NP, G-B-

~) Dipartimento

di Fisica dell'Universiti di Padova, Via Marzolo 8, 1-35131 Padova, Italy (3)

Dipartimento

di Fisica dell'Universiti di

Bologna,

Via Irnerio 46, 1-40126

Bologna, Italy

(Received13

June 1990, revised 9 November 1990,

accepted

26 November

1990)

Abstract. We consider the

problem

of

establishing Flory-like

formulas for the ~exponent ^of

polymer

chains _on fractals. For the

B-regime

the

short-range screening

of

three-body

interactions,

together

with scale-invariance arguments, ^is invoked,

producing

a

Flory

formula with very

good

results. We

briefly

consider the extension of such arguments to branched

polymers

at the

&

point.

1. Inkoducfion.

The

Flory approximation [1-3]

for the

description

of the conformational

properties

of _a linear

polymer

chain in _a

good

solvent is known to

give

_very

good results,

both

qualitatively

and

quantitatively.

The upper critical

dimensionality

_d~

=

4 is

correctly predicted

while the exponent ^for the

scaling

of the end~to-end distance is estimated _{as v}

=

3/(d

₊ 2

)

for I _« d _« 4 _; this is exact in d

=

I, presumably

exact in d

=

2

[4],

not _more than 1-2 fb

higher

than the best estimate in d

= 3

[5],

and

again

exact

(apart

from

logarithmic corrections)

in d

=

4.

Numerous

attempts

^{have been} made to extend the

Flory approach

to a broad range of

physical

situations of interest. These include linear chains in _a poor solvent

[3]

^{and in} melts

[3,6];

branched

polymers

in

good

solvents and in melts

[6];

the

sol~gel ~percolation)

transition

[6];

directed linear

polymers [7,8];

directed branched

polymers

and directed

percolation [7, 9]

branched

polymers

in _a poor solvent

[10]

stiff

polymers (both

linear and

branched) [11] self-avoiding

walks

(SAWS) (which

model the exduded~volume effect of

polymer chains)

on fractals

[12-18].

It is

generally acknowledged

that the

approximations

involved in the calculation of internal energy and entropy ^{in the}

Flory approach

result in _a gross overestimation of both these

(*)

On sabbatical leave from

Departamento

de Fisica,

PUC/RJ,

Cx.P. 38071, Rio de Janeiro RJ, Brazil.

340 JOURNAL DE

PHYSIQUE

I N 3

errors then cancel each other _{more or} less

completely

when the free energy

(the physically

relevant

potential)

is evaluated.

Thus,

the criteria for numerical accuracy of _a

Flory

calculation cannot be too

stringent although

this does not _seem to be written

anywhere,

it is

more or less

expected

that _a

Flory

estimate will fall at most within 4~5 fb of the correct

value,

if it is to be deemed successful.

Many

of the above-mentioned

generalizations satisfy

this loose

rule; however,

in several _cases the absence of

experimental

data _or other theoretical estimates makes

comparison impossible.

For the

fKpoint

transition of

polymers

in _a bad

solvent,

_a

straightforward generalization

of the

Flory approach gives

results _more than _a few

percent

away from well-established values.

In what

follows,

_we discuss the

physical

_reasons for this

discrepancy,

and show how _a modified

Flory

formula _can take corrections into account. In order to be able to

analyse

_as

many different _{cases as}

possible,

_we consider

underlying

fractal lattices. This makes it necessary first to discuss the extension of

Flory-like approximations

to fractal spaces. We do this

through

the

simpler

_case of

polymers

in _a

good solvent,

which _are described

by

SAWS.

We then return to the statistics of chains in _a bad solvent _{; we}

briefly

consider branched

polymers

_as well. In section 2 _we rederive the

approximation

of

Dekeyser,

Maritan and Stella

(DMS) [14]

for SAWS _on fractals and _compare its

predictions

for selected non-random fractals with those obtained from other

approaches.

In section 3 _we

generalize

for fractals _an

argument

recently developed

for the

screening

of

three-body

interactions at the

B-point

^of

two~dimensional linear

polymers [19]

; we then

apply

it to several fractals where exact results for the

fKpoint

of linear

polymers

_are available. In section 4 _we discuss branched

polymers,

both in

good

solvent and at the

&~point,

_on Euclidean lattices. In section 5 _our conclusions

are drawn.

2. SAWS _on fractals.

While for _an Euclidean lattice the

dimensionality

d is the

only

relevant parameter, ^fractals embedded in d-dimensional _space need at least three distinct

quantities

to be

specified

_: fractal

dimensionality /

dimension of _a random walk _on the fractal

d~,

and

spectral

dimension

I _(in

_{fact these}

are related

by d~

= 2

@iii [20, 21].

The

early _attempt by

Kremer

(K)

to write _a

Flory expression

for SAWS _on fractals _was the

simplest possible [12]

:

UK "

~

(i)

d ₊ 2

Later, Rammal,

Toulouse and Vannimenus

(RTV~ [13] proposed

_on

approach

which led to :

VRTV ~

II (2)

d d ₊ 2

A third

suggestion

_{was put} forward

by Ha,lin

^and Ben-Avraham

[15]

and elaborated

by Aharony

and Harris

(AH) [16]

_; ^Bouchaud and

Georges [17] provided

further

justification

for

it,

_as well _as

Roy

and Blumen

[18].

It reads _:

218+2f~-d~,~

VAH~

(~)

dm<n~R+~+dB~R

where

iB

is the fractal dimension of the backbone of the

fractal, f~

is the

exponent

^{for the}

scaling

of the resistance between two

points

and

d~;~ gives

the fractal

dimensionality

of the

chemical distance _{», or} minimum

path, along

the fractal.

N 3 POLYMERS AT THE fJ-POINT ON FRACTALS 341

A

comparative

discussion of these

proposals, together

with related

references,

_can be found in reference

[16]. Here,

_we recall _an altemative

formulation,

derived

by Dekeyser,

Maritan

and Stella

(DMS)

^{in the} context of diffusion _on fractals

[14].

It has the

advantage

of

being

very close in

spirit

to the

original Flory _argument,

while

taking

into account relevant features of fractals. Of _course, the ultimate verdict _on the

validity

of any such

approximation

must

rely

on the

quality

of the results

provided by

it. As _we shall _see

below,

the DMS

approach

stands

the test very well in many cases

further,

and _more to the

point

of the

present

paper, it _can be

generalized

to deal with other

problems

such _as the

B-point

of

polymers

_on

fractals,

_once

again

with

good

numerical accuracy.

In reference

[14]

^it was

argued that,

within the

Flory approximation,

the

repulsive

_energy

for _an SAW of N steps

stretching

over a distance R in _a space with dimension

I

is

given by

:

U~N~R~~ ₍₄₎

while for the entropy one assumes a Gaussian distribution for the distance R travelled

by

_a

random walk after N

R~" _steps

S~-R~N~~~" ₍₅₎

Of _course,

rigorously speaking

the distribution of the travelled distance _{on a} fractal is

expected

to be _a stretched

Gaussian,

rather than _a Gaussian. On the other

hand,

it is _not all clear _a

priori

whether such _more realistic features of random walk diffusion _{on a} fractal

should

necessarily

be

incorporated

into _a

Flory-like approximation scheme,

in which the fractal itself should be treated _as much as

possible

in _{a «} mean-field _{» way.} It is _{a common} feature to

Flory~like approaches

that such

questions

_can

only

be answered

aposteriori.

Minimizing

the free _energy with respect to R

gives

R

N~~~§

where

v~~s =

2(1

₊

di1)/(2

₊

J) ₍₆₎

For Euclidean lattices where

1=d

_and

d~=2,

one recovers the

Flory

value

v =

3/(d+ 2).

The

resulting repulsive

_energy is

expected

to be relevant _as

long

_as

1~

2

d~ (7)

as can be _seen

by substituting

the result of

(6)

into

(4).

This is also to be

expected

from the _co- dimension

additivity

rule

[22], according

to which the intersection set of two fractals has _non-

zero dimension

only

if the dimension of the

underlying

fractal _space is less than the sum of the dimensions of the two fractals.

Again

for Euclidean spaces this

gives

the _upper critical

dimensionality

_d~

=

4 _; for

general

fractals is _seems that condition

(7)

is

always fulfilled,

_so

there is _no upper critical dimension

(note

that it is

equivalent

to

1< _4,

_whereas

usually

_one

has

1«

₂

; see Refs.

[20, 21]).

While

surprising

at first

sight,

this result _can be put ⁱⁿ context

by recalling

that _even random walks _on fractals do not behave

exactly

in _a trivial way : their

respective

fractal dimension

depends

_on what the

underlying

^fractal

is,

and sticks to different values at

high

dimensions for different

underlying

structures. Similar comments have been made in reference

[18].

At this

point,

it should be

clearly

stressed that all

existing

derivations of

Flory

formulas for SAW _on

fractals, including

the DMS one, never address _a fundamental issue.

Indeed,

the SAW

problem

is _a static and not a

dynamical

_one.

Thus,

_a

priori,

^it is not obvious whether in the

Flory

formulas

quantifies

like

d~

and

/

_which

_pertain

_{to random diffusion}

on the

JOURNAL DE PHYSIQUE T I, M3, MARS <99< j7

342 JOURNAL DE PHYSIQUE I N 3

structure, should enter. It has been shown

that,

in _some fractal lattices with nonuniform

coordination,

the fractal dimension d~ of a random chain

(static problem, equally weighted configurations)

is

completely

different from

d~ [23].

This

is,

_e-g-, the _case for the d

= 2

Sierpinski gasket

with _a generator ^of side

3,

in which sites of coordination 4 and 6 _are

simultaneously _present [23].

If

d~

#

d~,

one could

legitimately suspect

that _an

expression

of vs~w in terms of

/ _d~

or similar dimension could not be

appropriate.

In view of the above

difficulty,

the DMS

Flory formula,

like the others in

literature,

has to be considered _{more as}

an educated _guess, rather than _as the result of _a

deeply~founded

derivation _{; as}

such,

^{it should} be

judged

_on the basis of the

quality

of its

predictions.

In table I _we

display

the results of the several

approximations quoted (Eqs. (1), (2), (3)

and

(6))

for _some

representative

non-random fractals in which the exact exponent ^{is known. With} the

exception

of the

branching

Koch _curve

(a

7 fb

deviation),

the DMS

approach gives

results within 4 fb of the _{exact ones ;} the

extremely

accurate

prediction

for the 2D

Sierpinski gasket

is to be deemed somewhat accidental

(though similarly

accurate results _are found

throughout

Table I. End-to-end distance exponent v

for

SAWS _on non-random

fractals,

_as calculated

from

several

Flory approximations (see

text

).

For

dkfinitions of

the

fractal involved,

_see the

respective references quoted

in the column _« Exact _».

K

(Eq. (I)) (Eq. (Eq. (Eq.

Exact

Koch

curve 0.866 (~) ^0.785

(d)

0.855 (~) 0.822 (e) ^0.891

(d)

Sierpinski

gasket

0.837 (~) 0.768

(d)

0.825 (~) 0.7982 (e) ^0.7986

(d)

3D

Sierpinski

gasket 3/4

_(~) 0.654

(d)

^0.725 (~) 0.693 (e) ^0.674

(~

Modified

Rectangular

Lattice

3/4 _(b)

0.643

(d)

0.688

(b)

0,665 (~)

Modified 3D

Sierpinski gasket

0.732 _(C) 0.636 _(C) 0.675 _(C) 0.654

(h)

rms

deviation 9.6 fb 6.0 fb 5.3 fb 4.2 fb

(a) Reference [16].

(b) With data from references

[25-27].

(C) With data from reference [28].

(d) Reference

[13].

(C) With data from reference [13].

(§

Reference

[29].

(~) Reference [26].

(~) Reference

[28].

M 3 POLYMERS AT THE B-POINT ON FRACTALS 343

the whole

family

of 2D

gaskets,

_see Tab. I of Ref.

[14]). Compared

with the other

estimates,

the DMS

approach performs quite well,

with the least average square deviation. It would be

interesting

to have the AH results for the last two entries of table I. The

problem

with the

branching

Koch _{curve seems} to be related to the existence of one-dimensional _« links _{; on}

writing

the entropy,

equation (5),

_we cannot take into account that the contribution from the links is

actually

_zero

(once

the SAW has crossed _a

link,

there is _no

retum). Owing

to the _same

reason, for _a

topologically

one-dimensional fractal the DMS formula does not

automatically give

_v

=

I

II

_as is the _case for the RTV and AH

expressions.

This should not be _a matter of

concern in the present paper, as we shall have _more

complex (that is,

not

topologically

_one-

dimensional)

structures in mind. It must be

pointed

out

that,

whenever the links _{are a}

vanishing

fraction of the total number of

bonds,

such _as for

percolation

clusters at

(he threshold,

the DMS formula

gives

_very

good

results

[24].

In what

follows,

_we shall _use the DMS

approach

and

adaptations thereof;

in _our

opinion,

the above results _ensure that the

physical picture underlying

this scheme

captures

^{the essential} features of the behaviour of

polymers

in _a

variety

of conditions.

3.

Polymers

at the

&point

_:

screening

of

three-body

interactions.

For

polymers

at

high

temperatures in a bad

solvent,

the excluded volume condition dominates and SAW behaviour is observed. At low

temperatures

the net monomer-monomer attraction becomes _more

important

and the chain

collapses.

At the so-called

B-point,

the

boundary

between

high

and low temperature

regimes,

_a third

(intermediate)

behaviour sets in.

It is

usually accepted that,

_at the

fKpoint

the

two-body

attraction cancels the excluded volume effect and _a

three-body

net

repulsion

then becomes the

leading

term in the energy.

The standard

Flory approximation

then

gives (see,

_e-g- Ref.

[3])

^{for the} energy

U~N~R~~~ (8)

for _a chain of N

steps

^with an average linear dimension R in space dimension

(Euclidean)

d.

The

entropy

^{is assumed} to be derived from _a Gaussian distribution

S

R~/R(w (9)

with

R(w

N. Minimization of the free _energy

gives

R~N~~, v~=2/(d+I),

d«3.

(10)

Indeed, experiments

and numerical work

[30]

_agree with d= 3 _as the upper critical

dimension,

with

ve(3 d)

=

1/2.

On the other

hand,

for d

=

2 the

Flory

result _v

~(2 d)

=

2/3

is overestimated

by

far the

accepted

values _are around 0.55-0.58

[31, 32],

with

mounting

evidence

poinfing

towards the exact value

4/7

=

0.5714...

[32].

For the _case of

polymers

at the

fJ-point

in two

dimensions,

_a

coarse-grained approach originally developed

for SAWS in reference

[33]

_was

used, together

with scale~invariance

arguments, ^{to obtain} a modified

Flory

formula _ve

=

7/12

=

0.583...

[19].

The additional

physical

feature included _{was a}

screening

^of the

three-body interaction,

related to the

persistence

of the SAW condition at short

distances,

thus

lowering

the overall

repulsion.

Here,

we

generalize

the

approach

of reference

[19]

to the _case of

polymers

at the

fKpoint

on fractal

lattices,

and

apply

the

resulting expression

to several fractals where results _are available.

We consider _a chain with N _{monomers, on a} fractal whose

respective

fractal and random walk dimensionalities _are

land d~.

To discuss the

fKpoint,

_{we assume} that the _energy

344 JOURNAL _DE

PHYSIQUE

I M 3

depends

on

three~body

interactions. We first reobtain the unscreened energy in the _coarse-

paining interpretation

of reference

[33],

then

proceed

to include

screening

within the _same context.

We divide the chain into segments with

f

_{monomers each} I _«

I

_{« N}

).

In the

spirit

of the

Flory approach,

^each

segment

^{is considered} as a

noninteracting

random

walk,

with linear

extent

~i~~~".

_{In space dimension}

f

_{the number of}

tiree~segment

_encounters

_(self-

intersections of the chain _{as a}

whole)

will be

# of

3-segment

encounters

~v

(N /f)~ (R/f~~" )~

^~~

(unscreened) (I I) again

in the mean-field context. In order _to obtain the number of three~monomer interactions

(which gives

the

energy),

_we have to look _at what

happens

at each

three~segment

encounter at this

level,

_we have intersections of three

independent replicas

of segments

(mutual

intersections).

This is

given by

the average linear extent of _a segment

(~ _i~~~")

raised to the fractal dimension of the set of mutual intersections. This latter is

given,

for

general k-multiple

intersections of fractals with dimension

d~

on an

underlying

_space of fractal dimension

f by _{[22, 33]}

d

(k~multiple intersections)

=

kd~ (k 1) ^1. (12)

Thus for k

=

3,

# number ofmutual intersections

i

^{~ ~~~"}

(unscreened) (13)

the unscreened three

body

interaction _energy is then

given by

the

product

of

(11)

and

(13)

_:

3-body

interaction _energy N R~ ^~~

(unscreened) (14)

which is

independent

of the

segment length I,

_as it should _be.

The

entropy

^{in this}

picture

is

S

(Rli~~~")~/(Nli)~'~"

=

R~N~~~~" ₍₁₅₎

again ?-independent

_{and the same}

as in the DMS

approach

for SAWS

[14].

The

generalization

of the DMS

approach

for the

B~point

would then be

(by minimizing

the free energy obtained from

(14)

and

(15))

_:

R

~

N

~~,

v 8 ⁼

~ _~

~~~~

(unscreened) (16)

2(1

₊

d)

As _we show

below,

this

expression

is

unsatisfactory, usually giving

_errors of

~v

15 fb _{or more} for fractals in which the exact exponent is known. To correct

this,

we make _use of the

screening concepts developed

in reference

[19]

^for the two~dimensional _case.

The additional

physical

feature to include is the

fact,

which has been noticed in reference

[30],

that

(at

least in d

=

3) short-range

chain stiffness

persists

_{even at} the

fKpoint.

As the authors of reference

[30] point

out, at the

fKpoint two-body

attractions and

long~range

SAW condition

cancel,

but the

short-range

SAW condition remains. In the

following,

_we

assume that this is _so in other

(fractal)

dimensions _as

well,

and calculate the effects

arising

from such

short-range

condition _on the

three~body (long-range)

term.

The idea is to

incorporate

the

short-range

SAW condition _{as a} sort of

perturbation

to the standard _energy

expression, given by

the

product

of

(I I)

^and

(13).

The

advantage

of _{a coarse-}

M 3 POLYMERS AT THE B-POINT ON FRACTALS 345

grained description

lies

exactly

in that _{we can} take into account the short range SAW character of the walks

by properly modifying equation (13).

In the

spirit

of _a first-order

perturbation approach,

the SAW nature of the walks at short distance _can be taken into account

by considering

the

triple

mutual intersections in

equation (13),

not

just

_as

pertaining

to these

independent

random walks but _as

appropriate,

_e-g- to two random walks and one

SAW. In other

words,

_we consider _one of the fractals _{as an}

SAW,

with _an

unperturbed

background

of two

random-walks,

and search for the set of

triple

intersections.

The mutual double intersections of the two random walks have _a fractal dimension

2d~ /

and their number is

~vf~~~~", recalling

that each random walk has _a radius i~~~". _{We have} then to consider the intersections of these double

points

with the third

(SAW) segment.

^{For the} intersections the fractal dimension would be

(2 d~

+

$~w)

2

/

where

(~w

is the fractal dimension of the SAW itself

( vj2w). However,

in order _to find their number _we have to make _an additional

approximation

since the average radius of the SAW is

diff~rent _(larger)

_{than that of the random walks}

(~v

i~~~~~),

_we decide _to write _:

#

triple points

_oz (i~~~")~~^{~" ~}

(f~~'~~)~~~

^~

~v

f~

^~~~" "

(17)

where

a

mll~ ⁽ (18)

ds~w

W

That

is,

_we take into account, in _an average sense, the fact that the SAW has _a

larger

extent than the random walk. Of _course, there is _some

degree

of arbitrariness in this choice.

Alternative ways ^to write

approximate expressions

for the number of

triple points

lead to

quantitatively comparable

results. Note that _a similar

lowering

of the number of mutual

intersections,

due to _an

asymptotic

SAW

condition,

_was also found in reference

[33],

in _a

discussion of 5-tolerant walks

(in

which _{up to} 5 visits to a same site are

allowed).

Now,

for the energy term to remain

f~independent (which

is _a necessary

requisite

in this

interpretation

of the

Flory approach [33]),

the self-intersection term

(Eq.(ll))

must be

multiplied by ^f".

This cannot be done

by

itself: _at this level _{we are}

looking

at the _coarse-

grained properties

of the

chain,

_so the dimensionless variables _are

Nil

and

R/f~~~"

Since it is the number of intersections

(and

not e-g- a

distance)

which is

being renormalized,

the proper

combination is

(Nli)~" So,

the

short-range (I) _properties

_{reflect themselves}

on

large (N~

scales. One has _:

# of

3-segment

encounters

(N If

_)~ ~

(

R/f~~~"

)~

^~~

(screened) (19) Thus,

the

(f~independent)

screened _energy is

given by

the

product

of

(17)

and

(19)

_:

3-body

interaction energy N ^~ " R~ ^~~

(screened) (20)

The usual

minimization, together

with the entropy from

(15),

leads to

3 _{a +}

2/d~

V8 "

(screened) (2j)

2(1

₊

d)

with _a

given by equation (19).

Of _course, with

equation (21)

we lose the _exact upper critical dimension d~ = 3

instead,

if

we use

~Aw

"

(d+2)/3

and

d~

=2 _we obtain

d~=2.6.

^It

might

be

argued

that at

346 JOURNAL DE PHYSIQUE I M 3

d

=

3, screening

does not take

place

because the

long~range

_energy is

already

irrelevant anyway

[19].

In table II _we show the results of the

application

of both

equations (16)

and

(21)

to fractals where the exact exponent _ve ^{is known. The} inclusion of

screening systematically brings

the

error down from 17-20 §b to less than 2 fb.

Thus,

it _seems that the correct

ingredients

have

been included in _our formulation.

Regarding

the value

quoted

as the exact

exponent

^{for the modified}

rectangular

lattice

(MRL),

this has been obtained in reference

[27]

at a tricritical

point

which arises

only

in the

isotropic

limit of

(approximate)

infinitesimal recursion relations.

Although

the MRL itself is

inherently asymmetric,

Dhar and Vannimenus _argue that in

taking

this limit it is

possible

to get rid of the _extreme

anisotropy

induced

by

this asymmetry

(which gives rise,

_among other

things,

to a rod-like

phase),

while still

analyzing

the behaviour of the fractal. In

doing

_so, the

exponents

^related e-g- ^to the SAW and _{to a} different tricritical

phase (which

_are

exactly

calculated for the MRL as

respectively

0.66503 and

0.80503)

_are

slightly

underestimated

(0.6616

and

0.8024).

If the _same

happens

at the

isotropic

tricritical

point (which

is _not

accessible

directly through

the _exact

analysis

of the

MRL),

the actual value of ve should be 0.538~0.539.

This, however,

does not

change

the

qualitative

_way in which _our results compare ^to theirs.

Before

closing

this

section,

it is

interesting

to

point

out that the blind

application

of

equation (21)

to the 2D

Sierpinski gasket (with / d~

and vs~w as

given

in Ref.

[13]) gives

v =

0.634.

However,

it is known

that, owing

to the

specific

geometry ^of this

lattice,

the fJ-

point

behaviour is

actually

absent

[27, 34].

One could

hope that,

for trails

(where

bonds may not be visited _more than _once, while sites

may), geometry

would allow the existence of _a fK like transition _{on a} 2D

Sierpinski gasket,

with _an

exponent

to be

compared

to the above. At

least,

this is

expected

_on Euclidean

lattices,

where trails and

§AWS

seem to be in the _same

universality class, although

the situation is not

entirely

clear

(see

_e-g- Ref.

[35]

and references

Table II. End-to-end distance exponent _v~

for

SA Ws at the

fKpoint

_on non-random

fracta?s,

as

given by equations (16)

and

(21).

In those

formulas,

we have used the _exact values

for / _d~,

_{vs~w. For}

definitions of

the

fracta?s,

_see the

respective references quoted

in the column

« Exact _».

(Eq.

Screened

(Eq. (21))

Exact

3 D

Sierpinski

gasket

0.629 (~) 0.533 (~) ^0.529

(Ref. [27])

Modified 3D

Sierpinski gasket

0.607 (b) 0.516

(b)

0.507

(Ref. [28])

Modified

Rectangular

0.625 _(C) 0.528 _(C) 0.536

(d) (Ref.

(J) Data for

f d~,

_vsAw from reference

[29].

(b) ^{Data for}

f _d~,

_vsAw from reference [28].

(C) Data for

f _d~,

_vsAw from reference [27].

(d) See text.

M 3 POLYMERS AT THE &-POINT ON FRACTALS 347

therein). Indeed, recently Chang

and

Shapir [35]

discussed trails _on the 2D

Sierpinski gasket

with attractive interactions _;

they

found tricritical

points,

however of _a nature which is not

easily comparable

to that

expected

for _a standard

fKpoint.

For

instance,

the tricritical

point

for trails

disappears

_{as soon as} self-intersections become favoured _on the other

hand,

the

exponent

v at that

point

is

0.632,

_very close to that obtained in our

approximation.

This

might

add to

arguments

in favour of SAW and trails

being

in the _same

universality class,

also at the

B-point

_on fractals.

4. Branched

polymers.

For branched

polymers,

the ideal behaviour is

given by [36]

J§

N ^~H

(22)

Assuming

_an

entropic

term

S

R~/Rj

=

R~

_N ~/~

(23)

together

with _a

repulsive

_energy

~v

N~R~~ (swollen polymer)

_or

N~R~~~ (fKpoint),

^{it is}

easy ^to obtain the

Flory

results _:

v~ = 5

/2(d

₊ 2

,

d _« 8

(24)

and

V8 "

7/4(d

₊ '

)

,

d _« 6

(unscreened) (25)

Equations (24)

and

(25)

_were derived

respectively by

Isaacson and

Lubensky _[6]

and Daoud and

Joanny [10].

In what follows _we restrict ourselves to

spade

dimensions d

= 2 and

3,

where there _{are more} data available.

Equation (24)

works well for d

=

2

(where

the

predicted v~(2d)=

0.625 is

just

2.5§b off _v =0.64075±0.00015

[37])

and for d=3

(where v,(3 d)

₌

1/2

is

presumed

to be exact

[38]).

For the

B-point,

the extension of the ideas

presented

in section 3 above is immediate and

gives

v e ⁼

(7

2 _a

)/4(d

₊

1) (26)

with

a ₌

d(v~ 1/4) (27)

where v~ is the

exponent

^for the swollen branched

polymer

in dimension d. The results _are summarized in table III. In two

dimensions,

the introduction of

screening changes

the _error from ₊ 14 §b _to it

§b, suggesting

that

screening

is overestimated in this _case.

Unfortunately,

there _seems to be _no data available for the three-dimensional _case for _an evaluation of the

present

status of the matter, we refer to the recent work

by

Lam

[40],

where the _cross-over

exponent

_~b

= 0.814 is extracted. Since this is the ratio between the thermal and

geometrical exponent, nothing

_can done without further

data,

in order to find _v»

Thus,

our

prediction

_v~

= 0.344 for branched

polymers

_at the

8~point

in three dimensions

(an experimentally feasible, though perhaps

hard to

analyse, transition)

stands _{as a}

conjecture

to be tested. In view of the above results for branched

polymers

in d

=

2,

^{this is}

probably

to be

regarded

_{as a} lower bound

(the

_upper bound

being

the unscreened value

0.4375).

For branched

polymers

_on

fractals,

the first

question

is what is the

equivalent

of the

Zimm-Stockmayer result, equation (22)

? If _a

proposal

could be forwarded for

this,

_we would

348 JOURNAL DE

PHYSIQUE

I N 3

Table III. End-to-end distance exponent ve

for

branched

polymers

at the

fKpoint,

_as

given by equations (25)

and

(26).

Euclidean lattices.

(Eq.

Screened

(Eq. (26))

Other

d

=

2 0.583 0.453 (~) 0.5095 ± 0.0030 _(C)

d

= 3 0.4375 0.344 (b)

(a) With v~ from reference

[37].

(b) ^With v~ =1/2 exact.

(C) Transfer-matrix calculation, reference [39].

be able to extend the present

approach

to branched

polymers

_at the

B-point

_on the 2D and 3D

Sierpinski gasket,

where exact results _are available

[41].

5. Conclusions.

In this _{paper, we} have discussed the

Flory approach

for the

description

of linear and branched chains _on

fractals,

both in the

good

solvent linfit and at the

fKpoint. Through

the examinafion of several fractals where exact results _are

known,

_we have shown that the

screening

of three-

body

interactions at short distances

plays

_a fundamental role in the confonnational

properties

of chains at the

fKpoint.

Our results thus confirm and extend those obtained in two

dimensions in

reference[19].

The concepts ^of

screening

have been extended to the

description

of the

fKpoint

_on branched

polymers

_on Euclidean lattices: the estimate

ve=0.344

for the three~dimensional _case has been

forwarded, although

this

probably

constitutes _a lower bound for the actual value.

Acknowledgements.

SLAdQ

would like to thank

Dipartimento

di Fisica at Padova for the

hospitality during

the

course of this work An1os Maritan is _to be thanked for

helpful

discussions. Tiffs work _was

made

possible through

_a bilateral

agreement CNPq (Brazil)-CNR (Italy).

Brazilian Govem- ment

agencies CNPq,

CAPES and FINEP support ^the research of

SLAdQ.

ALS _was

supported by

Centro Interuniversitario di Struttura ddla

Materia, Padova, Italy.

References

[1] FLORY P. J.,

Principles

of

Polymer Chemistry (Comell University

Press, Ithaca, New York, 1971).

[2] FISHER M. E., J. Phys. Soc. Jpn 26

(1969)

44.

[3] DE GENNES, P. G.,

Scaling

Concepts in

Polymer Physics (Comell University

Press, Ithaca, New

York, 1979).

[4] NIENHUIS B.,

Phys.

Rev. Lett. 49

(1982)

1062.

[5] See, e-g- PUREzA J. M., ARAGIO _DE CARVALHO C. A., DE QUEIROz S. L. A., J.

Phys.

A20 (1987) 4409 and references therein.

[6] ^{ISAACSON J. and LUBENSKY T.} C., J.

Phys.

Lett. France 41 (1980) L469.

[7] LUBENSKY T. C., VANNIMENUS J., J. Phys. Lett. France 43

(1982)

L377.

M 3 POLYMERS AT THE fJ-POINT ON FRACTALS 349

[8] DE

QUEIROz

S. L. A., J.

Phys.

A17

(1984)

L585.

[9] REDNER S. and CoNiGLio A., J.

Phys.

A15

(1982)

L273.

[10] DAOUD M. and JOANNY J. F., J.

Phys.

France 42

(1981)

1359.

[ll] NAKANISHI H., J.

Phys.

France 48

(1987)

979.

[12] KREMER K., Z.

Phys.

845

(1981)

149.

[13] RAMMAL R., TOULOUSE G., VANNIMENUS J., J.

Phys.

France 45

(1984)

389.

[14] DEKEYSER R., MARITAN A., STELLA A. L.,

Phys.

Rev. Lett. 58

(1987)

1758.

[15] HAVLIN S. and BEN-AVRAHAM D., Adv.

Phys.

36

(1987)

695.

[16] AHARONY A. and HARRIS A. B., J. Stat.

Phys.

54 (1989) 1091.

[17] BOUCHAUD J. P. and GEORGES A.,

Phys.

Rev. 839

(1989)

2846.

[18] ROY A. K. and BLUMEN A., J. Stat. Phys. ⁵⁹ (1990) 1581.

[19] DE QUEIROz, S. L. A.,

Phys.

Rev. A39

(1989)

430.

[20] ALEXANDER S., ORBACH R., J.

Phys.

Lett. France 43

(1982)

L625.

[21] RAMMAL R., TOULOUSE G., J.

Phys.

Lett. France 44

(1983)

L13.

[22] MANDELBROT B. B., The Fractal

Geometry of

Nature

(Freeman,

San Francisco,

1982).

[23] MARTIAN A., Phys. Rev. Lett. 62

(1989)

2845.

[24] DE

QUEIROz

S. L. A., SEND F, and STELLA A. L.

(unpublished).

[25] DHAR D., J. Math.

Phys.

18

(1977)

577.

[26] DHAR D., J. Math.

Phys.

19

(1978)

5.

[27] DHAR D. and VANNIMENUS J., J.

Phys.

A 20

(1987)

199.

[28] KNEzEVIC M. and VANNIMENUS J., J.

Phys.

A 20

(1987)

L969.

[29] BOUCHAUD E. and VANNIMENUS J., J.

Phys.

France 50

(1989)

2931.

[30] See, _e-g-, KREMER, K., BAUMGARTNER A., BINDER K., J.

Phys.

A15

(1981)

2879 and references therein.

[31] VILANOVE R., POUPINET D. and RONDELEz F., Macromolecules 21 (1988) 2880 re-evaluate earlier

experimental

data and propose

ve(d

=

2)

~ 0.53.

[32] DUPLANTIER B. and SALEUR H.,

Phys.

Rev. Lett. 58

(1987)

2235 _;

SEND F, and STELLA, A. L., J.

Phys.

France 49

(1988)

739 and references therein.

[33] DEKEYSER R., MARTIAN A., STELLA A. L., Phys. Rev. A 36 (1987) 2338.

[34] KLEIN D. J. and SEITz W. A., J.

Phys.

Lett. France 45

(1984)

L241.

[35] CHANG I. S. and SHAPIR Y., J.

Phys.

A21

(1988)

L903.

[36] ZIMM B. H. and STOCKMAYER W. H., J. Chem. Phys. 17

(1949)

1301.

[37] DERRIDA B. and STAUFFER D., J.

Phys.

France M

(1985)

1623.

[38] PARisi G. and SOURLAS N., Phys. ^Rev. Lett. 46

(1981)

871.

[39] DERRIDA B. and HERRMANN H. J., J.

Phys.

France 44

(1983)

1365.

[40]

LAM P. M.,

Phys.

Rev. B 38

(1988)

2813.

[41] KNEzEVIC M. and VANNIMENUS J., Phys. ^{Rev. Lett. 56}

(1986)

1591.