Experimental structure factor of solutions of randomly branched polymers

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Submitted on 1 Jan 1990

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Experimental structure factor of solutions of randomly

branched polymers

F. Schosseler, M. Daoud, L. Leibler

To cite this version:

Experimental

structure

factor of solutions of

_randomly

branched

_polymers

F. Schosseler

_(1),

M. Daoud

₍₂₎

and L. Leibler

₍₃₎

(1)

Institut Charles

_Sadron,

6 rue

_{Boussingault,}

67083

_Strasbourg

Cedex, France

(2)

Laboratoire Léon Brillouin

(*),

CEN

_Saclay,

91191 Gif-sur-Yvette, France

(3)

Laboratoire de

_{Physicochimie Théorique,}

E.S.P.C.I., 10 rue

_Vauquelin,

75231 Paris Cedex

05, France

(Received

Il

_April

1990,

accepted

in

_{final form}

27

_May

₁₉₉₀₎

Résumé. 2014 Nous avons effectué des

_expériences

de diffusion de rayonnement sur des solutions diluées de

_phases

sol

_produites

lors d’une réaction de

_{gélification.}

La

_comparaison

des facteurs de

structure mesurés sur des échantillons fractionnés ou non fractionnés _permet de

_séparer

les effets de

_dispersion

en tailles et les

_propriétés

conformationnelles des

_espèces

ramifiées. Les résultats

montrent _que, dans une solution

_diluée,

le _comportement de ces

_objets

est

_plutôt

en accord avec celui attendu pour des amas de

percolation gonflés.

Abstract. 2014 We have

_{performed scattering experiments}

on dilute solutions of

_polydisperse

and fractionated sol

_phases

obtained in a

_gelation

process. The

comparison

of the structure factors allows one to _separate size

polydispersity

effects and conformational

_properties

of the branched

species.

Our results show that their behavior in solution agrees rather well with the one

_expected

for swollen

_percolation

clusters.

Classification

Physics

Abstracts

82.70G - 36.20C - 64.60F

Introduction.

During

the last decade the structure of branched

_growing

_aggregates

has been the

_subject

of a

large

number of

_experimental

and theoretical studies

_[1].

The

large

aggregates

that are built in

a

_{given growth}

_process are characterized

_by

the same

_decay

of the

_pair

correlation function.

Scattering experiments

measure the Fourier transform of the

pair

correlation function and

thus are a convenient method to

_{study growth}

_processes.

Gelation is a

_particular

class of

_growth

_{processes in} which the

growing

clusters

interpen-etrate each other and

_finally

constitute a

_{macroscopic object}

which _{spans the whole reaction}

bath

_{[1, 2].}

In this

_process,

the interactions between the

_growing

clusters are

_important

and

screen the intramolecular interactions

_beyond

a distance that

_depends

on the concentration in

the reaction bath. Thus the branched molecules

_constituting

the sol

_phase

are more

_compact

than the branched

_objects

obtained from

_aggregation

in a dilute solution. One

_striking

feature of the sol

_phase

is the

_huge

size

_{polydispersity}

of the

_growing

clusters. At the

_{gel point,}

all sizes of clusters are

_present

in the reaction

bath,

from the constitutive units to the

_macroscopic

infinite cluster. The size distribution function of these

_{objects obey}

a

_scaling

form and the

physical

properties

are those of individual branched molecules

_averaged

over this size

distribution function

_[2-12].

This paper

reports

on the conformational

_properties

of

_randomly

branched molecules

synthesized by crosslinking

of

_pre-existing

linear chains in solution. This reaction allows one

to obtain

_gels

but,

for all the

_samples

studied

here,

the reaction has been

_stopped

before the

gel point

is reached. Fractions with low

_{polydispersities}

have been

_{prepared by precipitation}

fractionation from a

_pregel

_sample.

We have measured

_by

small

_angle

neutron

_scattering

(SANS)

and

_{light scattering (LS) techniques}

the

_intensity

scattered from

_polydisperse

and fractionated

_{samples. By}

_comparison

of their structure factors measured on

_length

scales between the monomer size and the radius of

_gyration,

we can

_separate

_{polydispersity}

effects

and

_gain

information about the conformation in solution of both the branched molecules and the linear precursor chains. The same method has

_already

been used

_by

Bouchaud et al.

_[10]

to

_study

the clusters obtained

_by

the

_{polycondensation}

of small molecules.

Theoretical

_background.

In recent _{years fractals have become} a familiar

_concept

in a wide range of

_{physical phenomena}

[1,

13,

14].

Fractal

_aggregates

have a self-similar structure in a

_large

_{range of}

_length

scales.

They

are invariant

_by

dilation :

_pictures

of an

_aggregate

taken with different

_magnifying

lenses look the same. This can be

_{expressed mathematically by}

a _{power law}

_{decay [5, 15]}

for

the

_pair

correlation function

_{g ( r )}

defined as the monomer concentration at a distance

r from a

_given

monomer. For a finite

_fractal,

this

_{algebraic decay}

is valid for distances

between the size a of the

_constituting

units and a

_typical

size

_{roughly equal}

to the radius of

gyration R

of the

_aggregate.

Therefore we have :

where D is the fractal

dimension,

d the space dimension and

_{h (x)}

a crossover function : for

x 1, h (x) ~ 1

and for

_large

x,

h ( x) decays

faster than any power law.

Upon spatial

integration

of

_{equation (1),}

one obtains a _very

_simple

_relationship

between the mass

M and the radius of

_gyration

R of a fractal

aggregate :

The fractal dimension D is

_always

smaller _{than the space} dimension and

_{equations (1)}

and

(2)

reflect the _{presence in the}

_aggregate

of self-similar defects that are

_responsible

for its

smaller

_density.

In

_highly

diluted solutions one can

approximate

the total scattered

intensity by

the sum of

1

the intensities scattered from each

_aggregate.

Then the

_intensity

_{SM(q )}

scattered from a set of statistical fractals with the same mass Mis

_proportional

to the Fourier transform of

_equation

(1) :

crossover function

_{h (x),}

one defines the intermediate

_{scattering regime}

_{(R-1 q a -1)}

where the scattered

_{intensity decays}

as a

_simple

_power law :

where C is the concentration

_{and q}

denotes the

_scattering

wave vector.

Equations (1), (2)

and

₍₄₎

are three

_widely

used definitions of the fractal dimension

D. A

_simple

well-known fractal is a

_{long polymer}

chain in a

_good

solvent whose fractal dimension is very close to

_5/3.

For a set of fractal clusters with different molecular

_weights

the

_preceding

relations are

modified

_{by polydispersity}

effects as

_emphasized

earlier

_by

Daoud et al.

_[4]

and Martin et al.

[5].

The size distribution function of the

_largest

clusters obtained in a

_gelation

reaction is

usually

written in the

_vicinity

of the

_{gel point}

as a

_{scaling equation [3] :}

where

_{P (M, e)}

dM is the number of branched molecules with a molecular

_weight

between

M and

_{M + dM,}

when the relative distance to the

_{gel point}

is e =

_{( 1 - p /pc).}

Here

p is the extent _{of the reaction and pc}

_{the p}

value at the

_{gel point.}

The function

f (x)

is a crossover function with usual behaviour

(f (x) -

1 for x « 1 and

_{f (x) «}

1 for

x >

_1).

In the

_{scaling equation (5)}

the distribution function

_{P (M, c)}

_depends

on the extent of the reaction

_through

the

_{single quantity}

M*. The cut-off molecular

_weight

M* characterizes the

_spreading

of the size distribution function and

_diverges

at the

gel

point

where the

polydispersity

is infinité :

Thus,

at the

_{gel point,}

P

_{(M, s}

=

0 ) decays

as a

_simple

_{power law :}

The

_{exponents o-}

and T that appear in

equations (5)

to

₍₇₎

are critical

exponents,

whose values can be calculated in the

_{Flory-Stockmayer approximation [ 16] (u}

=

1/2,

_T =

5/2)

or in

the

_percolation

model

_{[3] (a m}

_0.46,

T -

2.2).

The total

_intensity

scattered from a

_{polydisperse sample}

in dilute solution is obtained

_by

a

straightforward generalization [5]

of

_{equation (3) :}

In the

_{small q}

_range,

_{equation (8)}

reduces to the usual Guinier

_expansion

of the scattered

intensity :

where

_C,

_Mw

and

_R2Z

are

respectively

the

concentration,

the

_weight

averaged

molecular

Using equation (5)

into

₍₁₁₎

and

_(12),

it is easy to obtain the

_scaling

relation between

Mw

and

_{R z 2[3-5] :}

In the intermediate

_{scattering regime}

defined now

_by

Rz

1 « q « a - 1,

the scattered

intensity

obtained from

₍₈₎

still

_decays

as a _{power law but with} a different

_{exponent :}

The

_comparison

of relations

₍₁₃₎

and

₍₁₄₎

with

_{equations (2)}

and

₍₄₎

shows that the effect of the

_{polydispersity}

is to

_replace

the true fractal dimension D

_by

a smaller

_apparent

fractal

dimension

_Dapp

=

D (3 - T ) .

Expérimental

part.

Pregel samples

are

_{prepared by y-irradiation}

induced

_crosslinking

of

_polystyrene

chains in

cyclopentane.

The

temperature

during

irradiation is

_kept

to 22 ± 1 °C which is

_slightly

above the theta

_temperature

of the

_PS-C5Ho

_system.

The concentration in the reaction bath

(0.1

gjcm3)

is close to the

_overlap

concentration of the chains.

The

_{sample preparation}

follows the

_{procedure already}

described

_{[6, 9].}

The extent of the reaction is controlled

_by

the irradiation dose. Free radicals are induced

_by

y-rays on the chains

and on the solvent molecules. Then

_crosslinking

comes about

_by

a combination of

neighbouring

radicals on the chains. Direct

_radiolysis

of

_polystyrene

chains in bulk is known

to induce

_{approximately}

1 chain scission for 10

_crosslinking

reactions. But here solvent

radiolysis

is the dominant process in the

polymer

solution because the

_{polymer weight}

fraction is

_only

about 0.1. Therefore most of the radicals created on

polystyrene

chains are obtained

by

transfer from the free radicals induced on solvent molecules. This process reduces to a

large

extent the chain scission that is

_essentially

due to the direct

_radiolysis

of the chains. In fact no chain scission is observed in size exclusion

_{chromatography experiments.}

Small

_angle

neutron

_{scattering experiments}

have been

_performed

on _{the PACE}

spec-trometer in Laboratoire Léon Brillouin

_during

two different runs.

During

the first one the

_intensity

scattered from dilute solutions of

_polydisperse

branched

samples

has been measured as a function of the extent of the reaction. The

_samples

used in this

_experiment

have

_already

been studied

_{by light}

_{scattering [9]}

and size exclusion

chromatography coupled

with

_{light scattering [6] (SEC-LALLS) techniques. They}

are

prepared

from linear

_hydrogenated

PS chains with

_{weight averaged}

molecular

_weight

Mw =

55 000 and

_{polydispersity}

index 1.2. Their characteristics are listed in table I.

Small values of the momentum _{transfer range, 4.71} x

10 - 3 -«--- q (Â - 1’) -«--- 5

x

10 - 2,

have

Table 1. - Characteristics

of

the

_{polydisperse samples.}

The

_pregels

with no concentration C indicated have not been studied

_by

SANS

_{technique. Sample}

450 contains

_only

the linear

precursor chains. Error bars

correspond

to a 68 %

confidence

interval

for

the

experimental

values.

neutrons and a distance between the

_sample

and the detector

_equal

to 2.5 m. The

_samples

have been diluted in

_C6D6

at concentrations around

10- 3

_g/cm3

_(see

Tab.

_I).

As a

_sample

background

we have used _pure

C6D6,

thus

neglecting

the incoherent

scattering

from

hydrogenated polystyrene.

With the small

_polymer

concentration of the

_samples,

this extra

incoherent

_scattering

is

_only

about 4 % of the total incoherent

_scattering

of the

_sample

background.

The data treatment has been made

according

to a standard

procedure.

All

_spectra

have

been first corrected from the electronic noise measured with a Cd

_plate

in

place

of the

_sample

absolute scale

_{by introducing}

the differential cross section per unit volume of

_H20.

We have used the values listed in table II as

_they

are

_{provided by}

the

_utilitary

program WATWET

(ILL, Grenoble),

that

_gives

a

polynomial

interpolation

_{through experimental}

values

measured

_by

R. C. Oberthür. We thus obtain the absolute scattered

_{intensity I (q ) (in bams),}

which is

_proportional

to the structure factor

_{,S(q ).}

The

_{proportionality}

constant K is

_given

_by

the contrast between the

_polymer

and the solvent. K has been measured

_{by using}

the linear precursor

sample

as a standard

_since,

for this low molecular

_{weight sample,}

the

_scattering

function is measured in the Guinier _range.

_By

_{extrapolating}

the data to the zero wavevector

value in Zimm

_{representation}

we

_{get :}

where C is the

_polymer

concentration and

_{A 2}

the second virial coefficient of PS chains with molecular

_{weight Mw}

= 55 000 in

C6D6.

In fact we have used for

_A2

the

_{corresponding}

value

in

_{C6H6, A 2}

= 6.44 x

10- 4

cm3/g.,

which should be _{very close} to the true value. Thus we are

able to measure the absolute structure factor

_{S(q )}

for all the

_samples.

Figure

1 shows the SANS results for some of the

_samples

together

with

light scattering

data

obtained from

_{previous experiments}

on the same

_samples

diluted in toluene

_[9].

The

_light

scattering

data

_points

are obtained from

_{extrapolation}

to zero concentration whereas neutron

scattering

data are obtained

_by

_{simply dividing}

the scattered

_intensity

_by

the

_polymer

concentration. However the

_agreement

between the two data sets is

_{quite good.}

In fact

neglecting

the virial terms for

_high

molecular

_weight

branched

_samples

introduces

_only

a small error since the

_polymer

concentration

_(see

Tab.

_I)

and second virial coefficient

_(around

10- 5

_{CM3 /Î)}

are small for these

_samples.

Thus the virial term in

_{equation (15)}

is

_only

about 1 % of the term

Mw 1.

In the second

_experiment

the

_intensity

scattered from différent fractions of the same

branched

_sample

has been measured as a function of molecular

_weight.

A

_polydisperse

sample

has been

_prepared

as

_usually

from the same _{precursor chains and then}

_separated

into

fractions

_{by using}

_{précipitation}

fractionation. These fractions have been characterized in toluene

_by

a

_{light scattering technique.}

The

_apparatus

is an AMTEC

_{photogoniometer}

with

incident

_{light supplied by}

a He-Ne laser

_(A

= 6

328 À )

and

scattering angles comprised

between 10° and 160°. Values of

_{MW and R,}

are listed in table III and

_plotted

in

_figure

2.

Table III also

_gives

for each fractionated

_sample

its

_{approximate corresponding weight}

fraction in the

_{polydisperse sample.}

As in the first

_experiment

the

_samples

have been diluted in

_C6D6

for SANS

_experiments

(see

concentrations in Tab.

_III).

In order to

_improve

the

_counting

rates,

scattering

cells with a

10 mm

_{optical path (instead}

of 5 mm in the first

_experiment)

have been used. Two différent incident neutron

_wavelengths

have been

used,

A = 7.46

Á

and À = 15.5

Â,

with a

_sample

detector distance

equal

to 2.5 m. The

corresponding q

_{range for the} two

_{configurations}

is 1.01 x

10- 2 -

q (Â- 1)

1.07 x

10-1 and

4.86 x

10-3 --

q(Â- 1)

5.16 x

10- 2,

respectively.

Scattered

_{intensity data,}

corrected as in the first

experiment,

are shown in

_figure

3

_together

with the

_light

_scattering

results.

Results and discussion.

Figures

1 and 3 show that there is a q range where the scattered intensities tend to be insensitive to the molecular

_weight

of the

_samples

as the molecular

_weight

increases.

Fig.

1. - Intensities scattered from

polydisperse pregels

with different

weight averaged

molecular

weight Mw. Sample

450-0 contains

_only

the linear precursor chains. The

extrapolation

to zero

q value

corresponds

to

_Mw

for each

_sample.

Data

_{points for q}

5 x

10- 3

have been obtained

_by

the

light scattering technique.

Table II. -

Effective differential

incoherent

_scattering

cross section

_{of H20 for different}

Fig.

2. - Variation of the

_{weight averaged}

molecular

_weight

of the fractions as a function of their radius of

_{gyration (Tab. III).}

It is

_interesting

to _{compare the intensities}

_{S(q ) / C}

scattered from

_polydisperse

and fractionated

_samples

with the same

_{weight averaged}

molecular

_{weight (Fig. 4).}

At small

q

values,

their scattered intensities are the same since

(

S(q) C )q-O

=

Mw.

Then,

in the

Guinier _{range, the decrease in the scattered intensities is}

_{given by}

the

_z-averaged

radius of

gyration (Eq. (9)),

which is

_larger

in the

_{polydisperse sample}

than in the fractionated

sample.

Thus the scattered

_intensity

decreases faster for the

_{polydisperse sample.}

On the other

hand,

in the intermediate

_{scattering regime,}

the

_intensity

scattered from the fractionated

_sample

decreases faster since D is

_larger

than

_Dapp.

_Finally

in the

_{high q}

range,

corresponding

to the intermediate

_{scattering regime}

for the _{linear precursor chains}

_{(Ri) 1 - q - a -}

_l,

a

_being

the

statistical length

of the

_chains),

the scattered intensities become the same for the two

_samples.

In

_figure

4 the difference between the two

_samples

in

_{this q}

_{range is} an artefact due to the fact that the

_intensity

scattered from

_sample

450-55 has been measured in the first

_experiment

(small signal-to-noise ratio).

In the second

_experiment

however

_{(Fig. 3),}

no difference is

detectable in the

_{high q}

_{range between the}

_{polydisperse sample}

450-214 and the fractions. Thus three different

_{scattering regimes separated by}

crossover

_regions

can be

_{distinguished}

in the scattered intensities of the

_{samples :}

the Guinier range, the intermediate

scattering

regime

for the branched

_species

and the intermediate

_scattering

_regime

for the linear precursor chains. Each

regime

can be observed

only

over a

_{limited q}

_range,

_typically

less than

one decade. Therefore the measurement of accurate values for the

_exponents

D and

Dapp

is not _easy. This is

_particularly

true for the

_{polydisperse samples}

because the transition between the different

_regimes

is smoothed off

_by

the

_{polydispersity (Fig. 4).}

Table 1 lists the values of

_Dapp

determined for the

_{polydisperse samples.}

These values have been obtained from a

weighted

least squares fit of the scattered intensities in the

q range 4.7 x

10-3 - q (Á-I) - 2 X 10-2.

For the fractionated

_samples,

the values of

D listed in table III have been determined in the same _{way in}

_nearly

the same

q range

(5

x

10-3 - q

(Â-’)«2

x

10- 2).

Considering

that the radius of

gyration

of the

Table III. - Characteristics

_of

the

_{polydisperse sample}

450-214 and its

_fractions.

Same remarks

as

for

table 1.

(*)

This value

_corresponds

in fact to

_Dapp.

samples

studied here. In the

_{high q}

range, 0.03 q

(Á - 1) -

0.08,

corresponding

to the

intermediate

_{scattering regime}

for the linear

_chains,

the same

fitting procedure

has

_given

the

D t

values listed in table III.

A close

_inspection

of the values of

_Dapp,

D and

_Dl

shows some

_scattering

in the

results,

especially

in the first

_experiment

where the

_{signal-to-noise}

ratio was small. To decrease the statistical errors, we have used in the second

_experiment

a

_longer

_optical

_path

for the

scattering

cells and

_{slightly higher polymer}

concentrations. In

_previous

similar

_experiments

[10],

some

_dependence

of the

_exponent

values on the

polymer

concentration has been

reported. Clearly

the accuracy in the data

(Tabs.

1 and

_III)

is not

_{high enough}

to detect such

an effect here : there is no

_systematic

correlation between the variation of the

_exponents

and

the concentration of the

_samples.

Therefore the values in tables 1 and III have been

_averaged

to obtain the mean values

_Dapp

= 1.53 ±

_0.15,

D

= 2.06 _± 0.10 and

Dt

= 1.70 ±

0.05,

where

Fig.

3. - Same as

_figure

1 for the

_pregel

450-214 and its fractions.

From the

_{light scattering}

results

_plotted

in

_figure

2 and

_{equation (2),}

we obtain the value

DLS

= 2.04 ± 0.15 which is in

agreement

with

D

measured in the SANS

_experiments.

On the other

hand,

the value

_{D-1app, LS}

= 0.58 ±

_0.06,

deduced earlier

_[9]

from

_{light scattering}

experiments

on the

_{polydisperse samples,}

appears to be not well consistent with

_Dapp.

However in reference

_[9]

the fit of the data was done with a

simple

least squares

_adjustment

that does not take into account the error bars associated with the

_experimental

values. Therefore we have

_replotted

in

_figure

5 these

_previous

results

_together

with some new data

(corresponding

to

_samples

450-178 and 450-61 in Tabl.

_I).

A

_weighted

least _squares fit across

the whole set of data

_gives

_Dapp,

_LS = 1.67 ±

_0.04,

_slightly

different from the

_previous

determination. If we do not consider the two

_{points corresponding}

to the

_higher

molecular

weights

(Mw>

107),

we obtain a smaller value

_{Dapp,Ls=I.61:t0.04}

that is in better

agreement

with

_Dapp.

The latter

_procedure

could be

_{justified by}

_{the fact that the accuracy} on

the values

_{of Mw}

and

_Rz

measured

by light scattering techniques

becomes rather poor for very

large

molecules. Nevertheless in the

_following

we shall use the first value of

_Dapp,

_{Ls ( 1.67 ),}

Fig.

4. -

Comparison

of the intensities scattered from

polydisperse

and fractionated

_samples

with the same

_weight

_{average molecular}

_weight.

Using

the values of

_Dapp

and D measured

_by

the LS and SANS

_techniques,

one can

calculate the

_{corresponding}

r

values,

_tLS = 2.18 ± 0.08 and f = 2.26 ± 0.1.

They

are

consistent,

although smaller,

with the value _{T SEC} = 2.3 ± 0.1

directly

measured _on some of these

_{samples by}

the SEC-LALLS

_{technique [6].}

Table IV summarizes the results.

Previous work has shown that the size distribution functions of these

_{samples obey}

the

scaling

form

_{(5) [6].}

Indirect

_arguments

have also

_suggested

that these

_pregels,

assumed to be

fractals,

should have a fractal dimension close to 2 in dilute solutions

_{[6, 9].}

The results

presented

here sustain further the idea that these

_samples

contain fractals

_objects.

Moreover the values in table IV show that the behavior of these

_objects

in solution is in

_good

_agreement

with that

_expected

for swollen

_percolation

clusters.

Some evidence for the same behavior has been obtained on different

systems

[7,

8,

10-12,

17].

The values of the

_exponents

measured here are in

agreement

within the

_experimental

errors with those measured on

_pregels

obtained

_by

the

_end-linking

of functionalized

Fig.

5. - Same as

_figure

2 for the

_{polydisperse pregels}

listed in table 1.

Table IV. - Mean

_experimental

values

_of

the _exponents.

[7,

10-12, 17].

All these

_systems

are characterized

_by

a small ratio of branch

_points

to

monomer molar concentrations. This feature may

explain

that the

_simple

static

_percolation

model allows one to describe a

_variety

of different

_gelling

_systems.

However once the

_gelling

reaction is

_completed,

the mechanical

_properties

of the

_{resulting gels}

are

_quite

different. For

example

it is well known that _{the upper limit of the} stress before fracture is much

smaller

for

gels

obtained

_{by end-linking}

than for

_{gels synthesized by}

random

_{crosslinking along}

the linear precursor chains. A

simple

model is still

_lacking

to

_explain

these différences.

Acknowledgments.

We are

_grateful

to J.

Teixeira,

our local contact on the

_spectrometer

_Pace,

for his

_{help during}

the

_experiments.

We thank him as well as J.

_Bastide,

J. P. Cotton and M. Rawiso for

_sharing

with us their

_experience

in the SANS

_technique.

We thank F. Candau for

_{kindly giving}

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