Dynamical properties of branched polymeric clusters in dilute regime

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Dynamical properties of branched polymeric clusters in dilute regime

M. Delsanti, J. Munch

To cite this version:

M. Delsanti, J. Munch. Dynamical properties of branched polymeric clusters in dilute regime. Journal

de Physique II, EDP Sciences, 1994, 4 (2), pp.265-274. �10.1051/jp2:1994105�. �jpa-00247960�

J. Phys. II France 4 (1994) 265-274 FEBRUARY 1994, PAGE 265

Classification

Physics

Abstracts

61.25H 05.40

Dynamical properties of branched polymeric clusters in dilute

regime

M. Delsanti

I')

and J. P. Munch (~)

1')

Service de Chimie Moldculaire, CEA CE

Saclay,

91191 Gif-sur-Yvette Cedex, France (2) Laboratoire d'Ultrasons _et de Dynamique des Fluides

Complexes

(*), Universitd Louis

Pasteur, 67070

Strasbourg

Cedex, France

(Received 29 June J993, accepted in final form 20 October J9931

Abstract. Static and

dynamical

propenies ^of randomly branched polymers ^{in dilute} solutions

were investigated by light scattering experiments. In the intermediate scattering regime, ^{the main} result is that the time dependence of the

dynamical

structure factor of the polymer is _a function only of the reduced variable Dt, D

being

_a characteristic frequency which varies with the scattering

vector q as q~^~~°~. The

dynamical

behavior is very similar to that observed with linear polymers.

This shows that branching does not affect dynamics, and hydrodynamic interactions between

monomers are dominant _as for linear polymers.

Introduction.

In recent years, much attention has been

paid

to

polymer

solution

physics.

One _reason for this interest is the

discovery

of

analogies

between the

physics

^of

phase

transitions and

polymer physics ill- Quasi-elastic scattering experiments (light

and neutron

scattering) performed

on

linear flexible

monodisperse polymer

^{solutions showed} that the

dynamics

^of a linear

polymer

^is

dominated

by hydrodynamic

interactions between _monomers

[2-61.

In this paper, we are

especially

interested in the influence of

branching

_on

dynamical propenies. Using light scattering techniques,

_we determine the

dynamical

and static structure factors of dilute

solutions of

polydisperse polymeric

clusters obtained _near the

gelation

threshold

by

co-

polymerization

of

polystyrene

with

divinylbenzene.

In the first pan ^{of this} paper, we examine the effect of

polydispersity

_on

stjtic

and

dynamic scattering

measurements. We then report the results obtained

by light scattering experiments performed

in the intermediate

scattering regime

and compare them _to data obtained _on different

systems ^{and in various}

scattering regimes.

(*)

URA C-N-R-S- N° 851.

Preliminaries.

At _a

scattering

vector q,

light scattering experiments

allow the autocorrelation of the electric field scattered

by

_a

polymer solution, (E(q, t)

E+

(q, 0)),

to be measured.

E(q,

t) is the

complex amplitude

of the scattered field _at time _t, and

angular

brackets represent a time

average. When interactions between

polymers

_are

negligible

and all the

polymers

have the

same

degree

of

polymerization

N, the autocorrelation function of the scattered field is

directly proportional

to

S(q,

t) the

dynamical

structure factor of _one

polymer [7]

(E(q, t)E+ (q, o))

_cc

cS(q,

t),

(ii

c

being

the number of _monomers in _a unit voluine.

Explicitly

for

S(q,

t), _we have _:

S(q,

t)

=

) ( _{(exp jiq(r~ (t)}

r~

(0))1) (2)

where

r,(t)

is the

position

of the I-th _{monomer at} time _t. From theoretical arguments ^and

experimental

evidence [1,

81,

one can show that

S(q,

t) has the

following scaling

form

S(q, t)

₌ NP

(qR) g(rt)

with P

(0)

= g

(0)

=

(3)

P(qR)

represents the form factor of the

polymer

where R is the radius of

gyration

of the

polymer

which is linked to N

by R~

cc N, D

being

the fractal dimension.

g(rt)

denotes the time

dependence

of the

dynamical

structure factor where

r

=

q2 D~ H(qR

with

H(oi

=

(4)

Do

the diffusion coefficient of the

polymer

is

equal

_to

kT/f,

where k is the Boltzmann constant, T the temperature, ^and

f

the friction coefficient of the free

polymer.

The friction coefficient

strongly depends

_on the

hydrodynamic

interactions between _monomers and there _{are two} well

defined limits

iii.

If

hydrodynamic

interactions between _{monomers are} dominant

(Zimm

limit), f

is

proponional

to the radius of

gyration

of the

polymer

:

f

cc ~R where _~ is the

viscosity

of the solvent. Without

hydrodynamic

interactions

(Rouse limit),

the friction is N

times the friction of _{one monomer}

f

cc

N~ao,

where ao is _a

typical

size of the _monomer.

More

generally,

we have

f

Cc

nR~

^~

₍₅₎

where Z is _a

dynamical

exponent. In the _two

limiting

_cases mentioned

above,

Z is

equal

to 3

(dimension

of the

space)

and

equal

to 2 ₊ D

respectively.

The functions

P,

_g and H _are rather

complex

and often unknown except ^{in the} two

limiting regimes, qR

« and

qR

» I. When

qR

« 1,

equation (2)

reduces _to the well known form

S(q, t)

=

N

e~~~

with r

=

Do q~.

^{Under these}

conditions, polymers

^{look like}

points,

and

quasi-elastic scattering probes

the Brownian motions of the

polymers.

In the intermediate

regime qR

» 1, the functions

P,

_g and H _can be

approximated by

a power law function

11, 6,

91

S(q, 0)

cc N

(qR )~~

_{cc q~} ^~

,

(6.a)

g (rt

)

cc e- ^~~~~~~~

,

rt _~ l

,

(6.b)

r _cc

q2 ~)

~

(qR)z-2

_cc

~~'qz (6.c)

~R ~

N° 2 DYNAMICAL PROPERTIES OF BRANCHED POLYMERS 267

In this

regime

static and

quasi-elastic scattering probe

the

properties

of the

polymer

at a space

scale

q~

which is much smaller than the

polymer

size.

Polymers

_seem infinite and all the

quantities

measured become

independent

of the

degree

of

polymerization. Scattering

measurements in the intermediate

q-regime provide

_a

powerful

tool to determine the

dynamical

and structural

properties

of

polymers.

Let _us examine the effects of

polydispersity

on

scattering

measurements. We consider _a

polymer sample

whose

polydispersity

in the

degree

of

polymerization

is characterized

by

its

weight

distribution C

(N

).

Explicitly

C

(N

dN represents ^{the number} of monomers per unit

volume which

belong

to

polymers having

_a

degree

^of

polymerization

between N and

N ₊ dN. The autocorrelation function of the total scattered field is the

superposition

^of the autocorrelation functions of the fields scattered

by

each son of

polymer

and _can be obtained

by integration

of

equations (1)

and

(3)

over the distribution C

(N ). Equation (1)

holds, but _{now c} is

equal

to C

(N

dN and

S(q, t)

represents an effective

dynamical

structure factor which

can be written _as follows

S(q,

t ₌

N~ (P (qR ))~

G

(t) (7)

N~

is _{a mean}

degree

^of

polymerization (weight average)

N~= jC(N)NdN/ jC(N)dN. ⁽⁸⁾

(P (qR))~

is _{a mean} form factor, and the brackets

( )~

denote the

following

_average

(P (qR ))

~ ⁼

lC (N )

NP

(qR

dN/ C

(N

N dN

(9)

G(t)

describes the time

dependence

of the

dynamical

structure factor G

(t

=

(P (qR

) _g

(rt )) ~/ (P ^(qR ))

~

with G

(0

=

lo)

The

shape

of the

dynamical

structure factor is

complex

and may be

dependent

_on the

scattering

vector. The

simplest

means to

analyze

the time

dependence

^of

G(t )

is _to determine its

slope

at

time

origin

and its

integral

_:

=

~(~~~

⁼

^D~ q~, (11.a)

rH ^t i-o

r~ =

j~ ^G(t)

^dt =

1/D~

q~

(11.b)

o

D~

^and

D~

_are two apparent

q-dependent

diffusion coefficients _:

D~

=

(P (qR Dn

(R H

(qR ))

/

(P (qR ))

~

(12.a)

and

1/D~

₌

(P (qR

)/

[D~(R

H

(qR II

/

(P (qR ))~

^l

^2.b)

In order to determine the

q-dependence

of the structure factor and of the _mean characteristic times _r~ and _r~, we assume that the

weight

distribution has the

following

form _:

C(N)ccN'~~~f(N/N*) _{fo(No/N) (13)}

where N* and

No

are the

typical degrees

of

polymerization

of the

largest

and smallest

polymers, respectively.

Functions

f

and

fo

are cut-off functions

~f(0)

=

fo(0)

=

ii,

The

distribution

(Eq, (13))

is the

general

form encountered in any

polymer growth

_process

[8, 101.

For

instance,

in the _case of

percolation

clusters r~ is

equal

to

2,2,

and it is smaller than for Shultz distribution. The results obtained after

integration

of

equations (7), (11)

and

(12) using equation (13)

are

reponed

in table I. In the

regime

q~ «

Ro,

where

Ro

^{is the}

typical

size of the smallest cluster

(R(

_cc

No ),

the behavior of the _structure factor is identical with that

found in the _case of

monodisperse polymer

when q~ « R. At _a space scale q~ «

Ro

all the

polymers

_seem infinite and

polydispersity

becomes

unimponant.

If the exponent _r~ ^is

larger

than

2,

in the

regime Row q~~ «R*,

where R* is the

typical

size of the

largest

cluster

(R

^*~ cc N

*),

the

polydispersity

_{can smear some exponents}

(see

Tab.

I).

For

instance,

if the

exponent _r~ is

larger

than 2, the static _structure factor is _no

longer proportional

to

I/q~

but _to

I/q~

with D

= D

(3

_r~

).

This

point

_was shown

by

small

angle

neutron

scattering experiments

_on

polyurethane

and cross-linked

polystyrene

clusters

[12, 131.

When

hydrodyn-

amic interactions between _{monomers are}

dominant,

for whatever kind of

polydispersity

the _q-

dependence

of the different characteristic times _r~ and r~ is the _same

they

are

inversely proportional

to

q~,

Therefore,

if the diffusion coefficient of flexible

polymer

is

inversely proportional

to its

geometrical

size then in the intermediate

scattering regime,

I,e, when q~ ^is smaller than the size of the

largest polymer,

whatever the

polydispersity,

the

dynamical

structure factor should

be _a function

only

of _a dimensionless variable

fit,

fl

being

_a characteristic

frequency

proportional

to

q~,

From

quasi-elastic light scattering experiments

the establishment of

Table I,

Dependence of

static and

dynamic

structure

factors

_{as a}

function of

the

scattering

i>ector q.

S(q)

is the static structure

factor.

_r~ and _{r~ are mean} characteristic times w>hich

character"I=e the time

dependence of

the

dj~namic

_structure

factor

_{(see tea-t,}

Eqs, (11)

and

(12)),

R* represents ^the i"adius

of gyration of

^the

largest polymers

_present in the

sample

which is linked _to the

degree ofpolymerization of

the

largest polymers

N ^* and _to the

weight

_average

of degree of polymerization Nw of

the

polymer sample by

the

following

relations:

R ^*~ N * and R ^*~

NW-

The

polydispei"sity of

the

polymers

is characterized

by

a

w>eight

distribution

of

molecular

w,eights

w.hich is assumed to

obey

_{a pow>er} law>

: C

(N

) N ^{' ~~}

[8,

10,

ii

I. If

_{rD <}

2,

the exponent D is identical with D the

fractal

dimension

ofthe polymer,

and

if

_r~

~ 2, D is

equal

_to

D(3 r~),

Q-Regime

q~ W

Ro Ro

" Q~ " R ^* R ^* W q~

Static

structure

S(q)

cc

q~~ _S(q)

cc

q~~ _S(q)

_cc R ^*~

factor

Hydrodynamic

~~~~~~~~~~~ ~H CC ~A ^CC Q ^~ ~H CC ~A ^{CC ~} TH ^CC T~ CC R

*/q~

dominant

Without

~~ ~ ~~ ^c~

q~

^{l~+ ~J} _{r~ x} R

*~/q~

~li[lllslllll~~

^{~~ °~ ~~ °~ ~ ~} _{r~ cc q} ^{~2 +D~} _{TA ~c} R

*~/q~

N° 2 DYNAMICAL PROPERTIES OF BRANCHED POLYMERS 269

relationships

between different characteristic times and the

scattering

vector, in the inter- mediate

q-regime,

is of great ^interest to determine the

dynamical properties

of

polydisperse

branched

polymers

obtained

through gelation

_process,

Experimental procedure.

The

investigated

system was made of clusters

synthesized by

free radical

co-polymerization

of styrene ^and

divinylbenzene

in benzene solution _{at a} temperature ^{of 60 °C, The}

growth

_process

during

this

sol-gel

transition has been

intensively

studied

[141.

The main conclusion _was that free radical

co-polymerization

leads to the formation of branched clusters that _can be described

by percolation.

The

product

obtained at different times of

co-polymerization

_{was «}

quenched

_», dried and then redissolved in benzene for

light scattering

measurements, In the present work, the studied

sample

resulted from _a chemical reaction which _was

stopped

at a time, t, before the

gelling

time t~

(t~ t)/t~

^So-1.

The

samples prepared

at different dilutions _were transferred into the

scattering

cells aid

centrifuged just

before the

scattering

measurements.

Scattering experiments

were

performed

_{on a} spectrometer

fully

described elsewhere

[151.

For static measurements, we measured the

intensity

scattered

by

the pure solvent

(benzene),

(I~),

and the

intensity

scattered

by

the solution,

(I~(q)).

The

intensity

scattered

by

the

polymer

was obtained from the difference :

iI(qi)

=

ilE(q)12)

=

lI~(q)) (I~)

(proponional

to the static _structure factor S

(q,

⁰

)

of the

polymer

ⁱⁿ the limit of infinite

dilution,

see

Eqs.(1)

and

(7)).

For

dynamic

measurements, at a

given scattering

vector, the autocorrelation of the scattered

intensity, (I,(q,

t

) I~(q,

0

)),

and the _mean scattered

intensity, (I,(q)),

_were measured. The time

dependence

of the

dynamical

_structure factor _was obtained

by

renormalization [71

1(1,(q, t)I~(q, o)) (I,(q))~l/(I,(q))~

=

AG~(t).

A is _a numerical factor which

depends

on the geometry ^{of the}

experiment.

The correlator used

was a Malvern correlator K7025 with _a linear time scale which extends from At to

128 At, At

being

the time per channel. To

investigate

all the

profiles

of

G~(t),

the time per channel _was varied

by

a continuous increase. The

experimental

function

G~ (t)

was obtained

by superimposing

the data collected at different times per channel.

Results and comments.

Scattering experiments

were

performed

on dilute solutions

(c

<

10~~ g/g)

and for _a q range

lying

between 8 _x

10~~ i~

_{and 3}

x

10~~ l~

_{From static} measurements, we found that it

was not

possible

to find a

q-domain

where the scattered

intensity

_can be fitted _to the Zimm

approximation

(1 (q ) )

=

(I (q

- 0)

)

/

ii

_{+ q~}

R(

where

Rz

is _{a mean} radius of

gyration.

In fact, the data fitted

quite

well a power law _{as a} function of q. This result _means that the

degree

of

polymerization

of the

sample

is _so

large

that

<1>(a.u.)

i o

~ ^o

i o i

Fig. I. Variation of the intensity scattered by the polymer _{as a} function of the scattering vector in _a log-log scale. Points and _crosses

correspond

to measurements

performed

on samples

having

_a

concentration of 9.39 x10~~ g/g and 4.85 _x 10~~ g/g

respectively.

In the q-range and concentration regime investigated, the intensity scattered _is proponional to the concentration and varies _as

q~~~~~~°~~

the condition

qR*

< cannot be reached in the q-range

investigated

₁₁

61.

For concentrations

smaller than 10~ ^~

g/g,

within

experimental

_error lo

fl),

the scattered

intensity

is

proponional

to the concentration. The

figure1

shows the variation of the

intensity

_{as a} function of the

scattering

vector for

samples having

a concentration of

9.39x10~~g/g

and 4.85 _x

10~~ g/g.

The fact that the scattered

intensity

is

proponional

to the concentration _means that, for the concentration and _q range

investigated,

interactions between clusters do _not affect the

scattering

and

consequently

the

theory

of

scattering

from dilute solutions _can be

legitimately applied

in

analyzing

the data.

The

representation

used in

figure1

is _a

log-log

scale and the observed linear behavior

indicates that the static _structure factor _can be described

by

a power law

S(q, 0)oc

q~ ^{~ from the best fits} we find D

=

1.35 ₌ 0.05.

Following experimental

work

performed

_on

similar systems

[12-15,

₁₇₁

accepted

values of D and r~ are D

=

2.1= o-i and _r~

=

2.3 ₌

0.1,

_so the

intensity

is

expected

to vary as

q~~

with

b

= 1.5 ₌

0.3,

which agrees with

the observed

q-dependence.

Whatever the exact values of the exponents and the size

distribution of the

clusters,

the results of the present

scattering experiment

_are consistent with the fact that _we have _a collection of fractal branched clusters and show that _{we are} in the

intermediate

scattering regime qR

* » 1.

Dynamic scattering experiments

_were

performed

_{on a}

sample having

_a concentration of 9_39 x

10~~ g/g.

An

example

^of the

profile

of the autocorrelation function of the

intensity

fluctuations is

given

in

figure

2 where

G~(t)

is

plotted

_{as a} function of time in _a semi-

logarithmic

scale. At short times,

G~

_(t

) exponentially decays

whereas, at

long

times, the

major

part ^of

G~(t)

is _non

exponential.

Let _us consider the behavior at short times. To

analyze

the time

dependence

^of

AG~(t)

_at shon times, the

logarithm

of

AG~(t)

is fitted _{to a}

quadratic polynomial

of time which allows _us to determine the

amplitude

factor A and the initial

slope

of

G2(t)

_:

_1/ro

=

(d

log G~(t)/dt)j

~o, in fact

1/rjj

₌

2/r~

(see

Eq. (ll)),

The

major

part ^of

N° 2 DYNAMICAL PROPERTIES OF BRANCHED POLYMERS 271

1o

i o

o

i i

Fig.

2. -A typical example ^{of the} autocorrelation function of the

intensity

fluctuations

G~(t)

as a

function of the reduced time t/To in _a

semi-logarithmic

scale, where the characteristic time To is deduced from the initial slope of

G~(t)

_{(see text), In} this

representation

_a linear behavior is only

obtained at shon times. At long times, the continuous _curvature shows that the major part ^of

G~(t)

_{cannot be described by}

an exponential function, The straight line represents an exponential behavior _:

G~(t)

= exp(- t/To). The present example corresponds to the results obtained _{on a} sample

having

_a concentration of 9.39 _x 10~~ g/g at a

scattering

vector q = 1.49 _x 10~~

fi~'

1o

i o

Fig. 3. -Typical variation of the autocorrelation function of the intensity fluctuations

G~(t)

as a

function of

(t/To)~

with p

=

0.676. The data _are the _{same as} in figure 2. The ordinate being _a log scale, in this representation the linear behavior at long times clearly shows that the profile is well described by _a

stretched exponential,

G~(t)

_(t

~

ro/2)

is well described

by

_a stretched

exponential

function

A~

^{e~~~~~~~. An}

example

of the

profile

of

G~(t

₎is

given

in

figure

3 where the

logarithm

of

G~(t)

_is

_plotted

as a

function of

(t/ro)~

with

p

₌ 0,676. In this

representation,

_{over a} decrease of

G~(t) _by

a

factor100, _a linear behavior is observed which

clearly

shows that the stretched

exponential

behavior is the dominant feature of the time

dependence

^of

G~(t), Practically,

in order to

analyze

the

long

^time tail, the

quantity G~(t)

_is fitted _{to a} stretched

exponential

function

by minimizing

the

following quantity z [Log G~(t~ Log

A

~

(t~/r~)~ l~

with respect to the

,

free parameters

A~,

_r~ ^and

p.

We find that the

experimental

data _are well described

by

a

stretched

exponential

^with an exponent

p

=0.67= 0.05 which is

q-independent

for

8 _x 10~ ^~

i~'

< q < 3 _x 10~ ^~

i~

_To characterize this

long

time tail, we have calculated the

following

_average correlation time _:

ri =

l~

^A_~ ^{e~ ~~~~~~~ dt} =

A~

r(1/p r~/p

o

1«

where r is the gamma function. In fact within 2 fl _r, is

equal

to

G~(t)

_dt,

o

The values of _r, and _To, obtained _at different

scattering angles,

are

reported

in

figure

4 _{as a} function of the

scattering

vector q. An

important point

is that within 4 9b the two characteristic times _ri and r~ are

proportional

:

r,/r~

₌ ^1,55 = 0,06. The present ^results

(see Fig. 4) clearly

show that the

q-dependence

of the _two characteristic times is close _{to a}

q~~-dependence

:

r, oc r~ ^oc q~ ^~~

~~°'~,

_{All these results}

are in agreement ^{with that}

expected

when

hydrodyn-

amic interactions between monomers are dominant (see Tab, Ii, Indeed if

hydrodynamic

interactions _were weak, _we would get ^different

q-dependence

for _rj and _r~ and exponents ^that characterize the

q-dependences

should be

larger

than 3. It is

interesting

to note that the results obtained with

polydisperse

branched

polymers

are similar to those observed

experimentally

for

monodisperse

linear

polymers [2-61.

In both _cases, the time

dependence

of G(t is _a function

only

of

tq~,

Z

being

close _to

3,

and _at

long

times G _t has _a stretched

exponential

behavior with

an exponent

p

=

2/Z. In the present

study

_we find

pZ

=

1.9 ₌ 0.2. These similarities with

monodisperse

linear

polymers

tend to support the theoretical results obtained _on

randomly polydisperse

branched

polymers

₁₁81 which show that the line

shape

of the

dynamical

structure

factor is

mainly

dominated

by

a stretched

exponential

behavior with _an exponent 2/3.

Unfortunately,

numerical

prefactors

were not evaluated and thus _a

quantitative comparison

with _a theoretical

shape

cannot

presently

be

performed.

Tjf~~~

_{To ~IS}

i o

i o

~~

~~3

^o

i

Fig. 4. Variation of characteristic times _Tn (crosses) and Tj (upper

points)

deduced from the initial

slope

and the

integral

of

G~(t)

_(see text) _{as a} function of the scattering vector in

log-log

^scales

(c 9.39 _x lo~~ g/g). The two mean characteristic times _are proportional and _{vary as} q~ ^~~~~~'''

N° 2 DYNAMICAL PROPERTIES OF BRANCHED POLYMERS 273

Dynamic properties

of _some flexible branched

polymers

in various

q-regimes

have

already

been studied

[17, 19, 201.

Let _{us now} compare all these different results with the present

findings.

From

quasi-elastic light scattering experiments performed

_on

polyester

^clusters in the small

q-region (qR

^* « I

),

the apparent diffusion coefficient of

percolating

clusters

[171

_was found to be

approximately proportional

to the

reciprocal

of the apparent ^{radius of}

gyration

of

the clusters.

Following

the notation used here, these results _mean that, for

qR*

ml,

D~

oc

R*~~~

_{with Z}

= 2.98 = 0.15. On

polydimethylsiloxane

clusters

synthesized by

end

cross-linking,

neutron

spin

^echo measurements were

performed

in _a very different

q-regime qi»1, I being

the minimal distance between crosslinks

[191.

The data show that S

(q,

t

)/S(q,

0

)

could be fitted with _a stretched

exponential

law with _an exponent

2/3,

and is _a

function

only

of the reduced variable fit. The characteristic

frequency

fl is

proportional

to

q~. fl/q~w2 x10~'~cm~/s,

_{and the numerical value is close to that obtained in the}

corresponding

uncross-linked system

[201.

The observed

q-dependence

of the

dynamical

structure

fa/tor

_{is also in}

qualitative

_agreement with _some

quasi-elastic light scattering

results

obtained _on branched silica clusters

[2 II-

In the dilute

regime,

S

(q,

t

)/S (q,

0 _was found _to be

a function of the dimensionless parameter fit, fl

being proportional

to

q~

with Z

> 2.76. All these observations _are consistent with the present ^{results and tend} to indicate that, ^whatever the

space ^{scale where}

dynamics

^is

probed, hydrodynamic

interactions between _monomers of

branched _{structure are} dominant _as for linear

polymers.

Summary

and conclusions.

From

simple calculations,

it appears that in the intermediate

q-regime (qR*

hi where R* is the size of the

largest polymers)

whatever the

polydispersity

of the

polymer,

the

dynamical

structure factor,

S(q, t)

is _a function

only

of the dimensionless parameter ilt with Hoc

q~,

if

hydrodynamic

interactions between _{monomers are} dominant. This

situation is very different from the _case of static measurements, where the value of static exponents

describing

the power law

q-dependence

^{of the static} structure factor _can be affected

by

^the _presence of _an

extremely large polydispersity.

In _{contrast to} static measurements, it is

possible

with _a

single dynamic scattering

measurement to obtain information _on

dynamic properties.

In the present ^work

light scattering

measurements carried _{out on}

polymer synthesized by co-polymerization

of

polystyrene

and

divinylbenzene (before

the

gel point)

show that, in the intermediate

q-regime,

the static _structure factor _can be described

by

a power

law S

(q

oc q~ ^{~' ~~ ~ °°~} As the value of the static exponent ^{is close} to the value 1.6 112,

131,

we expect ^that we have swollen

polydisperse

branched clusters

having

a distribution of the

percolation

type.

Dynamic

measurements

performed

in the _same

q-regime clearly

show that the time

dependence

of the

dynamical

structure factor is _a function

only

of the dimensionless

quantity fit,

fl

being

_a characteristic

frequency

which varies with the

scattering

vector as

q~~~°~

The

q-dependence being

close to a

q~-dependence

_as for linear

polymers,

this

indicates that

hydrodynamic

interactions between _monomers remain dominant _even in the

presence of

branching.

Moreover the

q~-dependence

_seems to be _a universal behavior in

polymer dynamics

whatever the structure is.

Acknowledgment.

We

gratefully

thank Dr. G6rard Hild of the Institut Charles

Sadron, Strasbourg,

for

synthesizing

the

polystyrene divinylbenzene sample.

We also thank Mohamed Daoud for

helpful

discussions.

References

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University

Press, Ithaca, N-Y., 1979).

[2] Adam M., Delsanti M., J. Phys. Lett. Franc-e 38 (1977) L-271.

[3] Han C. C., Akcasu A. Z., Macromolecules14 (1981) 1080.

[4] Wiltzius P., Cannell D. S., Phys. ^{Rev. Lett.} 56 (1986) 61.

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