The theta-point and critical point of polymeric fractals : a step beyond Flory approximation

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The theta-point and critical point of polymeric fractals : a step beyond Flory approximation

Daniel Lhuillier

To cite this version:

Daniel Lhuillier. The theta-point and critical point of polymeric fractals : a step beyond Flory ap- proximation. Journal de Physique II, EDP Sciences, 1993, 3 (4), pp.547-555. �10.1051/jp2:1993150�.

�jpa-00247853�

Classification

Physics

Abstracts

05.40 61.40 82.70

The theta-point and critical point of polymeric fractals

_{: a}

step beyond Flory approximation

Daniel Lhuillier

Universit£ P, et M. Curie, Laboratoire de Mod£lisation _en

M£canique

(*), Case162, 4 Place Jussieu, 75252 Paris Cedex05, France

(Received J6 October J992, accepted in

final form

22 December J992)

Abstract. We propose a new form of free energy for

polymeric

fractals (chain-like, branched _or membranes) based _on self-similarity arguments ^and on the existence of _a maximum

gyration

radius. Results _are obtained

conceming

the size exponents at the

theta-point

and critical

point.

These _two exponents depend on a

single

_parameter

only,

for which _a

simple phenomenological expression

is

provided.

A very

good

fit is obtained with numerical _or

experimental

results

conceming

linear and branched

polymers.

A

comparison

with the

Flory-de

Gennes

free-energy emphasizes

the _non mean-field features of the _new

free-energy-

1. Introduction.

When _a

polymer

is immersed in _a

solvent,

its

equilibrium

size

depends

_on their mutual

affinity

which

(in

most

cases)

is _a

decreasing

function of the temperature ^{T. For} a

given polymer-

solvent

couple,

there is _a

special

temperature ^{9 such that the}

polymer

shrinks into _a compact

globule

for T~ 9 while it

expands

into _a

highly

_porous

panicle

for T~ 9. For _a N-mer molecule, the compact ^{size scales like N '~~ where d is the dimension of the}

embedding

_space,

and the

expanded

size scales like N ". The excluded volume exponent v

depends

not

only

on d but also _on the

polymer

structure

(chain-like

_or

branched).

When T

=

9,

the

polymer

is somewhere between the

compact

^and

expanded

states and the size

exponent

v g lies in the range between I/d and

v. In _{some cases} vg is known

precisely

_: for d

=

3,

_a

polymer

chain at the

theta-point

behaves like _a Brownian walk with _vg

=

1/2

[1]

for d

=

2 the

presumed

exact value is _vg

= 4/7

[2]

_; for d

= I the obvious result vg = I should hold. Can those results be recovered

by

a

phenomenological approach

? The

Flory-de

Gennes model of the

theta-point

of

a

polymer

chain

[I]

is based _{on a}

free-energy

which associates the entropy ^of a Brownian walk with the energy

coming

from

three-body repulsions

between _monomers.

Minimizing

that free- energy leads to vg =

2/

(d

₊ I

),

^{which is} a

good

result for d

=

3 and d

= I but a poor one for

d

=

2. De

Queiroz [3] suggested

that the

three-body

interaction _was overestimated and he

(*) Associ£ _au CNRS.

548 JOURNAL DE

PHYSIQUE

II N° 4

proposed

_a screened interaction _energy with the result

v g ⁼

7/12 for d

=

2. This is close to the

presumed

exact

value,

but when extended to other

d-values,

de

Queiroz's

_argument results in vg =

(5

+

d)/4(d

+

I)

which is

good

for d

= 3 but not for d

=

I. At variance with de

Queiroz's

decrease of interaction energy,

Roy

et al.

[4]

have

developed arguments

which amount to an

entropy

^{increase with the result} _{v g =}

(d

+ 5

)/ (d

+ 2

) (d

+ I

)

which is

good

^for d

=

I,

but leads to _a critical dimension less than three

(d$ w2.4). Conceming

branched

polymers,

the

theta-point

_was located

by

Daoud and

Joanny

at vg =

7/4(d

+

1) [5].

Based _on

an extension to lattice animals

[6]

of the

Flory free-energy,

the above

prediction

in

certainly

true for the _two extreme values d

= 6 and d

=

4/3. For

intermediary

values of

d,

it is

likely

to

give

no better results than the

corresponding prediction

for

polymer

chains. An extension _to branched

polymers

of de

Queiroz's screening

_or

Roy's

et al. extra entropy can be of _some

help

but does not guarantee a real

improvement

_over the range 4/3 _~ d _~ 6

[7].

Our

opinion

^is that the

apparent

deadlock reached

by

the

Flory approach

should not be traced to the _use of _a minimum energy

principle

but to the nature of the statistical distribution

associated with the

free-energy.

The

Flory free-energy

is based _on the distribution of the end to end distance of _a

phantom

^{chain. We}

recently

built _{a concurrent}

free-energy

based _on the distribution of the

gyration

radius for _a

self-avoiding

walk and _we deduced from it the size exponent vg of a

polymer

chain

[8].

It should be

specified

that in _our

approach

_v is _a

given quantity (we

make _no

attempt

to calculate it _or deduce it from _some minimization

principle)

and _we determine _{vg as a} function of

v and d. Here _we extend the

approach

to branched

polymers

and

polymeric membranes,

and _we amend _some of the

predictions

^made in

[8].

In

fact,

the extension will _concem the whole

family

of r-fractals defined _as fractal

objects

made of N elements and with _a maximum

possible gyration

radius

scaling

like N~

[9]. Polymers

are

members of that

family

with _r= I for

chains,

r=3/4 for branched

polymers

and

r = 1/2 for

polymeric

membranes.

The

theta-point

is not the

only problem

we are interested in. We also _{want to} consider the

critical

point.

^{When several}

polymers

are immersed in _a solvent, _a second effect of _a

temperature ^{decrease is} to induce _a

larger

and

larger

attraction between the macromolecules.

For low

enough temperatures,

^the

polymers prefer

to bunch

together

^with _a small amount of solvent and _a demixtion process occurs,

ending

with two separate

phases

of very different

polymer

concentrations. We _are interested in the location of the critical

point,

I-e- the

top

^{of the} coexistence _curve. In

particular,

_{we want to} determine the size exponent _v~ ^{for linear and} branched

polymers, following

^the method initiated in

[8].

The definition of r-fractals and their relevance to the

polymer problem

is

briefly

reminded in section 2. The statistical distributions for the

configurations

and the number of contacts are

presented

in section 3. Sections 4 and 5 deal with the

theta-point

and the critical

point respectively.

Section 6

develops

the

comparison

between the

free-energy

of r-fractals and the

Flory-de

Gennes

free-energy.

2. The r-fractal _{as a}

unifying picture

of

polymeric

fractals.

The structure of _a

polymer

is

usually depicted by

its ideal exponent vo which is the size

exponent ^when any interaction between _monomers becomes irrelevant. This

happens

when the

polymer

is in _a space of dimension d

larger

than _a certain critical dimension d*. For

polymer

chains vo =

1/2 and d*

= 4 while for branched

polymers

_vo

=

I/4 and d*

= 8. Other vo values _are

presumably

^associated with the upper critical dimension d*

=

2/vo.

The

polymer

structure can also be

depicted by

the size reached when the

polymer

is

fully

stretched. For _a

polymer

made of N

elements,

this maximum

gyration

radius is

supposed

to scale like N~

[9].

The exponent r is _an intrinsic property

depending

on the

branching

geometry

only

and not on the dimension of the

embedding

_space. It fixes the lowest space dimension

(d~;~

=

l/r)

which _can

accept

^the

polymer.

It is

quite possible

that I/r also

represents

^the

spectral

dimension of the

polymeric

fractal

[10].

The

gyration

radius of _an r-fractal in _a d- dimensional space spans in the range between

N"~ (collapsed state)

and N~

(fully

stretched

state)

and

consequently

_{« v w r.} Coherent results _are obtained if _{one assumes r}

=

3/4 for d

branched

polymers [9].

It thus appears that _r is linked to vo as

r = v~ +

', _(I)

As _a consequence, vo = 0 for

polymerized

membranes. This strange ^value as well _as the

difficulty

to

interpret

the value vo = I/4 for branched

polymers, prompted

us to delete vo and ^to make _a

systematic

_use of r which has _a much _more intuitive

physical interpretation.

For this _reason the critical dimension of _an r-fractal will henceforth be written _as

d*

=4/(2r-1). (2)

In

fact,

the idea of _a parameter

interpolating

^between

polymer

chains and

polymer

membranes _was

already suggested

in

[I I]

_{as a} trick to suppress the difficulties associated with the infinite critical dimension of membranes. This

parameter

^{is here}

given

_{a very}

simple physical interpretation.

3. The statistical distributions for

configurations

and contacts.

The behaviour of r-fractals in _a

good

solvent

(high temperature)

_was described in

[9] by

the statistical distribution of the

gyration

radius. If the maximum number of

configurations

_occurs

for _a

gyration

radius

N",

the

self-similarity

of the fractal

compels

the number

A'(R)

of

configurations

with

gyration

radius R to follow the

scaling

law

j

~ r-v fi~V ^v-I/d

logA'(R)~RN- (-) (-)

N" ^R

where _R is _some

positive

constant. Note that the powers of R/N ~ are not

arbitrary. They give

a

vanishingly

small number of

configurations

for the _{two extreme} values

R~N~~~

_and R

~

N~

[9].

We deleted all numerical coefficients but _R to insist _on the

scaling

features.

To describe the r-fractals in _a poor solvent

(low temperature)

_one needs _a second statistical distribution which is the number JL

(R )

of contacts between non-consecutive elements in _{an r-}

fractal of size R. Each of those intemal _contacts lowers the energy

by

_an amount

2

kTK,

where _K has the _same

meaning

_as the

Flory

interaction

parameter

^{and the}

prefactor

2 is used for convenience

only.

The free energy of the r-fractal _can be written in adimensional units

as

~(R)=-logA'(R)-2KJL(R).

The

equilibrium

size

R(K)

is the _one

minimizing

~ and the behaviour of

R(K)

from the

expanded

to compact ^states

gives

the value of _v

g.

The distribution JL

(R

that _was

proposed

in

[8]

and

partly

checked

numerically

in

[12]

for

polymer chains,

can be extended _to other

polymeric

fractals in the form

~ ⁱ

NV v-iid R r-v

JL(R)~

aN + +

JL*(N, R)

R N"

550 JOURNAL DE

PHYSIQUE

II N° 4

where _a is _some

positive

constant. If _we discard JL* for _a

while,

the above distribution _means that the number of contacts has _a maximum of order

(a

₊ I

)

N for

compact configurations

and that it decreases

steadily

for

increasing

^R. It is

noteworthy

that the number of _contacts is still of order N around the most

probable

size R N ^"

[13]

^and that it is

vanishingly

small for the

fully

stretched state. When JL* is

discarded,

the

resulting free-energy ~(R)

leads to _an

abrupt

transition from the

expanded

to the

collapsed

_state,

occurring

for _«

~

l/2. To get a true theta-

point,

_we must take JL* into _account. It appears that it is _a function of N and

R/N ^" which cannot be deduced from the

arguments

^of

self-similarity

used _to obtain the other

contributions. For this _{reason, we} henceforth

speak

^{of JL*} as the _« anomalous _» number of contacts. With the above two statistical

distributions,

the

free-energy

of _an r-fractal of N elements _can be written _as

j

~ r-v fi~V ^v-I/d

~~

(l +2K)(-)

₊

(1-2«)(- -2«JL*(N,R/N") (3)

N" R

where _we deleted contributions

depending

on N

only

_: albeit

important numerically,

these contributions _are irrelevant in the

forthcoming

_energy minimization.

4. The

theta-point.

We define the relative size of the r-fractal _as

j X =

(R/N")~~"

This

quantity

becomes

(R/N"°)~

for any space dimension

larger

than the critical dimension.

With X instead of R the

free-energy (3)

becomes

~=

(l +2«)X+ (1-2«)X~~ -2«JL*(N,X).

The exponent ~ is

positive

and defined _as

r- v

~

v

I/d'

^~~~

~ locates the

position

of _{v as}

compared

to its extreme values _r and I/d. When

d

=

d*,

then _v

= vo = r 1/2

= 2/d* and

consequently

_~

=

d*/2. In the

Flory approxi- mation,

the excluded volume exponent ^is _{v~ =}

(1+

2

r)/(d

₊

2)

and _~ has _a

remarkably simple expression

_~~

=

d/2. In what

follows,

_we do _not make any

special assumption conceming

_v, and _~ will be

given by

its

general expression (4). Minimizing

~ leads to a

relation between the interaction parameter « and the relative size

~ j

_~~+l

~ ~

i

+x~~~(1-

^~~'~~

Two consequences are

noteworthy

:

I)

the relative size is

expected

to decrease when _« increases and any reasonable

expression

for JL* should

satisfy

I (X~

⁺

$

~ 0

,

and

(6)

?X 3X

it)

the size _at the

theta-point

is deduced from

~ J~~

~ ^(~

~~ ⁼

i.

(7>

As

suggested

in

[8],

JL* is

generally

the _sum of two

contributions,

_one

being

dominant when X _« I. Such _an

inequality prevails

in the

neighborhood

of the

theta-point

and

consequently,

we

will _not write the

complete expression,

but

only

that part ^{of JL* relevant} to theta-conditions.

For _a

given

monomer of the

polymeric fractal,

the anomalous number of contacts is

presumably

_{some power} of the average monomer volume fraction

N/R~

_and if the contributions

of the N _monomers add

together,

we are led _to propose

JL *

=

N

(N/R~)

^~ "

(8)

where _n is _some

positive _exponent

to be determined later _on. Note the minus

sign

which is essential if condition

(6)

is _to be satisfied. The above

expression

_can also be rewritten in terms of the relative size and the free energy of _an r-fractal of N elements

finally

_{appears as}

~

~=

(l+2«)X+(1-2K)X~~+2KN

~

X~~~ ₍₉₎

The

o-point

defined in

(7) gives Xg

or R

~N"~

_with

~

rd

° ~

d ₊ i/n

(lo)

The next task is _to propose ^a definite

expression

for _n.

Any

such

expression

should

satisfy

the

following requirements

:

n=1/2 for

d=d*,

n =

I for d

=

(d*, ₍₁₁₎

(2r-1)d-2~~~~

for

d~~d*

n ~ 4

The first condition holds because JL* should _not differ from the mean-field number of _contacts

N~/R~

_{when the}

space dimension is

equal

to

(or larger than)

the critical dimension. The second condition must be satisfied if vg

equals

its ideal value _vo

= r at _some

special

_space

2 dimension

d$

with

di

~ d*. This

special (critical)

dimension is three for linear

polymers,

^six

for branched _ones and is

di

=

~

=

~ d*

(12)

for _an r-fractal. The upper bound of the third condition is necessary for

(6)

to be fulfilled with

expression (8)

for JL*. The lower bound is necessary for

vg to be

larger

than its ideal value for any d~

di.

The

partial knowledge

of exponent n

represented by (11)

is _not sufficient _to

deduce it

unequivocally

^and we cannot but _{resort to}

phenomenology.

It is desirable _to express n as a function of _r, d and _v

(or ~)

since these _are the

only quantities already appearing

in the free energy

(9).

We have found _a rather

simple expression

that leads to

predictions

for _ve which fit

remarkably

well with all known results

conceming

linear and

JOURNAL DE PHYSIQUE II T 3, N'4, APRIL 1993 22

552 JOURNAL DE PHYSIQUE II N° 4

branched

polymers.

This

expression

is

1/n

=

d/d$ (2

_r

1)

_~

(l d/d$). (13)

One _can check that

requirements (I la)

and

(I16)

are satisfied. As to

(I lc)

^{it is satisfies in all}

cases where _v

(hence ~)

is known.

Replacing

the above value of I/n into

(10),

_{we get}

ve as on

explicite

function of _r, d and _v that appears as

3

(rd

~~~~

"° ~~~

2(1+r)d+ (2r-1)~ i(2r-1)d-31'

The

predicted

values for linear and branched

polymers

in _two and three dimensions are

given

in table I. For two-dimensional linear

polymers equation (14) gives

the value obtained

by Duplantier

and

Saleur[2].

For two-dimensional branched

polymers,

the

prediction

is

remarkably

close to the value _vg

= 0.5095 found

by

Derrida and

Herrmann[16].

For

polymeric

membranes _r

= 1/2 and

equation (14) predicts

_vg

=

I

Id,

I-e- _a

theta-point occurring

for

collapsing

membranes.

Table I. Predictions

for

_ve and v~. The

only ingredient

is the value

of

_v

Jkom

the

quoted references. Concerning

_v~, when

predictions of (17)

and

(20)

_are

different,

that

given by (17)

is between brackets.

d _{v ~}

l/n

_{vg v~}

from

(4)

from

(13)

from

(10)

from

(20)

or

(17)

Chains

2 ~

j18]

^{~ 13}

~. ₌ j 4 3 7 24

3 0.588

[14]

1.616 ' 0.461

2

(o_471)

Branched

2 0.64075

[15]

0.776 0.0746 0.5090 0.50525

r = 3/4

~

(0.50407)

~ ~~~~

2

10 ^~

5. The critical

point.

As

suggested

in

[8],

the interaction parameter K ^* between two

polymers

is related to the interaction parameter _K between _{two monomers,} and _to the anomalous number of _{contacts as}

K*=K(I+X(). ₍₁₅₎

When _K * reaches the critical value Km =

2,

the

suspension

suffers _a demixtion process. Since

K is of order one-half when demixtion _occurs, the size of the

polymer

at the critical

point

is obtained _as the solution of

~ J~~

x-

(~=~

= i.

(16)

The above

equation

is _to be

compared

with

(7).

Notice that both the

theta-point

and the critical

point

are defined in _terms of JL*

only.

With

expression (8)

for

JL*,

_one finds X~ ^and R N ^"~ with

"~~~~~nd~

~~~~

Introducing

X~ ⁱⁿ

(5)

_one deduces

(1-~)

2 K~

Xl

N

(18)

For linear

polymers

in three dimensions _n is

equal

to one, and with _v

= 0.588

[14]

one gets

v~ = 0.471 and 2 _K~ I

>N~°.~~'.

_These

predictions

_are in conflict with

experimental

findings

which _are v~ = 0.461 and 2 K~ ~

N~°.~ _[17].

_{However, with}

a rather

simple

modification of

(15)

into

I-i

~~M~*

K*=K

I+N~~

_X-

₍₁₉₎

~X

One gets

vd

(20)

"C ~

j

^~ _{2 nd}

and

j

2 _K~ I

= N ^~~

(21)

The

predictions

for linear

polymers

in three dimensions _now coincide

perfectly

with the

experimental results,

but we have _no

convincing

_argument which could

explain

the presence of the

screening

factor

N~'~

~ in

(19).

^{For this} reason, we will consider

(15)

and

(19)

_as two

distinct

possibilities

with consequences

(17)

and

(20) respectively.

^Just notice that the

screening

factor

disappears

if _v

equals

the

Flory

value

(~~

=

d/2

).

The

predictions

of

(17)

and

(20)

for linear and branched

polymers

in _two and three dimensions _are

given

in table I. For

polymeric

membranes the critical

point

is

predicted

to occur under

collapse

conditions

(v~

₌

I/d),

_{as was} the _case for the

theta-point.

6. About the

Flory approximation.

Though free-energy

minimization is _not the

only possible phenomenological approach

to the

polymer problem [20],

_we here

adopted

that

procedure

to deduce the size exponents at both the

theta-point

and critical

point.

At variance with

Flory's theory

_{we never} tried to deduce the

excluded volume exponent ^{from such} a minimization.

Instead,

_{v was a}

supposedly

known

quantity

which

helped

us to build the

free-energy

of r-fractals with the

scaling

variable R/N~. As a consequence we deduced _vg and v~ as a function of _{v, r} and

d,

with results

(10)

and

(20) completed by (13).

Predictions

gathered

in table I _are

pretty

^close to exact or

numerical results when

they

exist.

Conceming

the

free-energy

of _a self-similar

r-fractal,

its differences with the

Flory-de

Gennes

free-energy

~FDO

=

N

~~

~

+

(l

2 _K

) ~j

+ N

~

^~

₍₂₂₎

N R R

554 JOURNAL DE PHYSIQUE II N° 4

are best exhibited if _we rewrite

(9)

in the form

j i _n

R r-v N ^vd-i N r-v

~=

(l +2K)N(-) +(1-2K)N(~) +2KN(~) (23)

N~ R R

The

exponents appearing

in

(22)

_are mean-field _ones indeed since

they

_can be obtained from the

corresponding exponents

ⁱⁿ

(23) assuming

_v

= vo and d

=

d*.

Moreover,

it appears that in the FDG

free-energy,

the

exponent

n is constant and

equal

to one, which is its actual value

when d

=

d$ only.

We also want to stress the different

interpretation

of _our

free-energy

_as

compared

to the usual

interpretation

of

(22).

What is called the entropy ^{of the random walk in} the

Flory-de

Gennes

free-energy (the

first term of

(22)

_can be rewritten _as

(R/N"°)~) corresponds

to a term in

(23)

which is not of pure

entropic origin

_as

suggested by

the factor 1+ 2 _K. What is called the

two-body

interaction in

(22) corresponds

to a mixed entropy- energy ^term in

(23)

while the

three-body

interaction is

replaced by

a «

(1

^{+ ~}

^-body

^»

r v

interaction. It is noticeable that _n increases

steadily

above _one when d decreases below

d$.

An increase of _{n means a} decrease of the effective number of _{contacts at} the

theta-point

and

a

consequent

move of the

theta-point configuration

towards _{a more} and _more

compact shape.

This would

explain why

the actual vg is

always

below the

Flory prediction (except

for

d

=

d$

where _n has the _same value _n

= I in both

approaches).

The present

attempt

to go

beyond

the

Flory approximation

is based _on

phenomenology

and

consequently gives

more hints than final _answers.

First,

it would be nice if guesses like

(13)

or

(19)

could be

given

_some theoretical

justification.

^For

instance,

renormalization group results

on the number of _contacts between two

interpenetrating

SAW

[21]

is

part

^of a more

general approach

to K ^*

Second,

if the value _we obtained for v~ is in close agreement ^with

experiment (for

3-d linear

polymers)

_a modified

Flory-Huggins

model based _on the

present approach (see [8]

for _more

details)

leads to a coexistence _curve with mean-field exponents. ^This means that

we have

correctly

^{described the} correlations between

spatial positions

of _monomers in _{a r-} fractal but _not the correlations between

density

fluctuations. This is at variance with the models

developed by

Sanchez

[22]

and

Cherayil [23]

who took critical fluctuations into account, obtained a

satisfactory description

of the coexistence _curve but could _not locate the critical

point correctly.

It is _our

hope

that _a combination of both kinds of

approaches

will be

performed

in _{a near} future.

References

ill _DE GENNES P. G., J.

Phys.

France Leit. 36 (1975) L. 55.

[2] DUPLANTIER B, and SALEUR H.,

Phys.

Rev. Len. 58 (1987) 2235.

[3] DE QUEIROz S. L. A., Phys. Rev. A 39 (1989) 430.

[4] ROY A. K., CHAKRABARTI B. K. and BLUMEN A., J. Stat.

Phys.

61 (1190) 903.

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

Phys.

France 42 (1981) 1359.

[6] ISAACSON J, and LUBENSKY T. C., J.

Phys.

France Leit. 41 (1980) L 469.

[7] CHANG I. and AHARONY A., J.

Phys.

I France 1(1991) 313 _;

DE

QUEIROz

S. L. A., SEND F, and STELLA A. L., J. Phys. I France 1 (1991) 339.

[8] LHUILLIER D., J. Phys. II France 2 (1992) 1411.

[9] LHUiLLIER D., J.

Phys.

France 49 (1988) 705.

[10] CATES M. E., Phys. ^{Rev. Len. 53} (1984) 926 V1LGis T. A., J. Phys. France 49 (1988) 1481.

II Ii ARONOVITz J. A, and LUBENSKY T. C.,

Europhys.

Leit. 4 (1987) 395

KARDAR M. and NELSON D. R., Phys. Rev. Lett. 58 (1987) 1289 and erratum 58 (1987) 2280.

[12] VICTOR J. M. and LHUILLIER D., Submitted for

publication.

[13] _DES CLOizEAUX J., J. Phys. France

41(1980)

223 SCHAFER L., J.

Phys.

A. Math. _gen. 9 (1976) L 71.

[14] LE GUILLOU J. C. and ZINN-JUSTW J.,

Phys.

Rev. Lett. 39 (1977) 95.

[15] DERRIDA B, and STAURER D., J.

Phys.

France 46

(1985)

1623.

II 6] DERRIDA B. and HERRMANN H. J., ^J.

Phys.

France 44 (1983) 1365.

[17] DOBASHIT., NAKATA M. and KANEKO M., J. Chem.

Phys.

72 (1980) 6685 _; SCHINOzAKI K., VAN TAN T., SAITO Y, and NOSE T., Polymer 23 (1982) 728 _; SANCHEZ I. C., J. Appl. Phys. 58 (1985) 2871.

PERzYNSKI R., DELSANTI M. and ADAM M., J.

Phys.

France 48 (1987) lls _; MEL'NICHENKO Yu. B., KLEPKO V. V. and SHILOV V. V.,

Polymer

29 (1988) 1010.

[18] NIENHUIS B., Phys. Rev. Lett. 49 (1982) 1062.

[19] PARisi G. and SOURLAS N.,

Phys.

Rev. Lett. 46 (1981) 871.

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

Phys.

Rev. B 39 (1989) 2846.

[21] KRUGER B. and SCHAFER L.,

unpublished

results.

[22] SANCHEZ I. C., J.

Phys.

Chem. 93

(1989)

6983.

[23] CHERAYiL B. J., J. Chem.

Phys.

95 (1991) 2135.