Inelastic light scattering by gel modes in semi-dilute polymer solutions and permanent networks at equilibrium swollen state

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Inelastic light scattering by gel modes in semi-dilute polymer solutions and permanent networks at

equilibrium swollen state

J.P. Munch, S. Candau, J. Herz, G. Hild

To cite this version:

J.P. Munch, S. Candau, J. Herz, G. Hild. Inelastic light scattering by gel modes in semi-dilute polymer

solutions and permanent networks at equilibrium swollen state. Journal de Physique, 1977, 38 (8),

pp.971-976. �10.1051/jphys:01977003808097100�. �jpa-00208664�

INELASTIC LIGHT SCATTERING BY GEL MODES

IN SEMI-DILUTE POLYMER SOLUTIONS AND PERMANENT NETWORKS

AT EQUILIBRIUM SWOLLEN STATE

J. P. MUNCH and S. CANDAU Laboratoire

d’Acoustique

Moléculaire

(*)

Université

Louis-Pasteur, 4,

^rue

Blaise-Pascal,

⁶⁷⁰⁷⁰

Strasbourg,

^France

and

J.

HERZ,

^{G. HILD}

Centre de Recherches sur les

Macromolécules, C.N.R.S.,

⁶⁷⁰⁸³

Strasbourg Cedex,

^France

(Reçu

^{le 14 mars}

1977, accepté

^{le 25 avril}

1977)

Résumé. ²⁰¹⁴ Le coefficient de diffusion

coopératif,

mesuré par diffusion de la lumière dans des réseaux

polymériques gonflés

^{en bon solvant, varie avec} la concentration à l’équilibre ^des gels ^selon

une loi d’échelle identique à celle des solutions semi-diluées. Ce résultat

implique

que la distance moyenne ^entre n0153uds adjacents ^{du réseau est} égale ^{à la}

longueur

d’écran. Cette hypothèse ^confirmée

par les ^mesures de compression uniaxiale, permet de reconsidérer la structure réelle des réseaux

polymériques

^{et en}

particulier

^{de mettre en} évidence l’influence des enchevêtrements

piégés

^durant

le processus de réticulation.

Abstract. ²⁰¹⁴ The

cooperative

diffusion constant, determined from light scattering experiments ^on

polymeric

networks swollen in a good solvent, varies with equilibrium concentration

according

to a scaling law similar to that observed in semi-dilute solutions. This result

implies

that the average distance between

adjacent

crosslinks of the network is

equal

^{to the} screening length. This assump- tion, supported ^also by

compressional

^{modulus data, allows us to} reconsider the real structure of polymeric networks and in

particular

^{to consider} the influence of entanglements

trapped during

the

gelation

process.

Classification Physics ^Abstracts

5.660 ^- 7.146 ^- 7.221 ^- 7.610

1. Introduction. ^-

Cooperative

diffusion associated with network deformation has been observed

recently by light scattering

in both semi-dilute

polymer

^solu-

tions and swollen networks

[1-5].

In both _cases, the

cooperative

diffusion coefficient

D, depends

^{on the average}

distance ç

^between

adjacent

cross-links and is

given by [6] :

where kB

is the Boltzmann constant, T the

tempe-

rature, and ilo the

viscosity

of the solvent. For semi- dilute

solutions., ç

^{is the}

dynamical screening length,

identical to the static characteristic correlation

length,

which

depends only

^{on the} concentration.

Using dynamical scaling

^{laws, de Gennes has shown that,}

(*) ^{E.R.A. au} C.N.R.S.

for a

good solvent,

^the concentration

dependence of ç

is

given by [6] :

which leads to the

following

^{law for}

D,

Evidence for this

dynamical

correlation has been observed

recently by

Adam et al.

[1]

from the time

dependence

of the autocorrelation function of scatter- ed

light.

^The

cooperative

modes have also been observed in swollen permanent ^networks

[2-5].

^The

autocorrelation function of scattered

light

^{has been}

found to

decay exponentially.

^For

scattering angles.

ranging typically

^{between 10° and 90°, the}

decay

^{rate r}

follows a k2

dependence (k, scattering

^wave

vector),

within an accuracy

comparable

^to that obtained for dilute solutions of

monodisperse polymers [5].

^The

diffusion constant

Dc

determined from r has been

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003808097100

972

found to vary with the _average molecular

weight

between

cross-links,

^the

functionality

^{of the cross}

links and the method of

preparation [3-5].

^{The results}

obtained in these studies have been

interpreted by

Munch et al.

[4]

within the framework of the rubber

elasticity theory,

^{with the}

assumption

^of

ideality

^of

networks,

i.e. the absence of structure defects as

pending

^{chains or}

entanglements.

The purpose of this _{paper is} to show

that, according

to a

suggestion

^{of de}

Gennes,

^{for a network at the}

swelling equilibrium

^{in a}

good ^diluent,

^the average distance between the cross-links and the

dynamical screening length

^are in the first

approximation

^iden-

tical,

^{if one} allows for the _presence of structure defects in the permanent ^networks.

Furthermore,

^we compare the behaviour of permanent ^{networks and semi-} dilute solutions in the case of a

moderately good

solvent.

2.

Experimental.

^-

a)

^Two

samples

^of

polystyrene

have been used in the

experiments reported

^{here. Sam-}

ple

^{1 of}

weight

average molecular

weight MW

^{= 700000}

has been

prepared anionically

^{and has a narrow}

distribution of molecular

weights (MW/Mn 1.04).

Sample

^{2 is a fraction of a crude}

polystyrene;

^its

weight

average molecular

weight

^is

MW

^{= 3.5 x 106.}

The solvents are benzene and

ethyl

^{acetate. Cha-}

racteristics of the systems ^{relative to}

sample

^{1 are}

given

^{in table I. The}

comparison

between values of the radius of

gyration RF

and the second virial coef- ficient

A2

obtained for

benzene, ethylacetate,

^and

cyclohexane

^at the theta temperature,

respectively,

shows that

ethylacetate

^{is a}

moderately good

^{solvent of}

polystyrene.

TABLE I

Polystyrene M,

^{= 700 000}

e) Determined by conventional light scattering.

The solutions of

polystyrene

^{were clarified}

by ultracentrifugation.

^For concentrations

larger

^than

about 6-8 x

10-2 g/cc,

^{some dusts were still} present in the

solutions, giving

^{rise to a}

partial heterodyning.

For this _reason,

light scattering experiments

^were

performed

^{in the}

heterodyne regime

^{with an external}

oscillator, using

^{a Michelson} type interferometer.

For

sample ^l,

^the

scattering angle

^{was 0 = 90°.}

At this

angle,

^the

scattering

wavevector is much smaller than the inverse of the coil radius of

gyration

at zero concentration

RF

and the coil deformation modes do not

give

any

significant

contribution to the spectrum of scattered

light.

^For

sample 2,

^{which has} much

larger dimensions,

^{0 was set at 24°.}

b)

^The characteristics and

light scattering data,

relative to

polystyrene

networks swollen in

benzene,

have been

given

in earlier _papers

[3-5].

^It is worthwhile to

point

^out that several series of networks have been

considered, prepared according

^to different methods of

crosslinking. However,

^{in all cases the}

crosslinking

reaction has been

performed

^{in a}

medium-good

solvent of

polystyrene.

^Some of these networks have been swollen at

equilibrium

ⁱⁿ

ethylacetate.

^The

characteristics of these

gels

^are listed in table II.

TABLE II

Characteristics

of polystyrene

networks swollen

by

.

Ethylacetate (a)

e) The networks have been prepared by anionic block copoly-

merization of styrene ^with divinylbenzene (DVB) ^{in the} proportion

of 4 molecules of DVB per living ^{end of} polystyrene [5].

(’) ^Molecular weight ^{of the} precursor polymer determined by gel permeation chromatography.

(’) Swelling equilibrium concentration measured by weighing samples in swollen state and dry state, respectively.

3.

Experimental

results and discussion. ^-

Figure

¹

shows the concentration

dependence

of the diffusion constant of the solutions of

polystyrene

in benzene.

In the dilute

limit,

^{the diffusion constant of the} individual coils increases

slightly

with concentration.

This increase is in

good

agreement with the decrease of the radius of

gyration

which has been evidenced

by

^neutron

scattering [9].

^{In the} semi-dilute

regime,

^the

observed diffusion coefficient increases

considerably

with concentration and can be identified with the

cooperative

^{diffusion constant}

Dc

of the network.

The concentration

dependence of Dc obeys

the follow-

ing scaling

^law

FIG. 1. ^- Diffusion constant versus concentration for PS-benzene system in dilute solutions, semi-dilute solutions and swollen net-

works.. Solutions of sample 1. ) Swollen ^networks.

The value 0.68 obtained

experimentally

^{for the}

critical

exponent

is smaller than the theoretical value 0.75 but agrees very well with that obtained

by

^Adam

et al.

[1] on

^{the same} system.

Furthermore,

^Delsanti and Adam

[10]

^have obtained the

following

^molecular

weight dependence

^{of the zero} concentration diffusion coefficient

The cross-over between the dilute and the semi- dilute

regimes

^{occurs at a} concentration c* which

corresponds

^to the situation where the _average dis- tance between

neighbouring

^{coils is}

equal

^to

RF.

By combining

eqs.

(4)

^and

(5),

^one obtains the follow-

ing

^molecular

weight dependence

^{of c*}

This molecular

weight dependence

^is

quite

^close

to the

M-0.8

_power law

dependence predicted by

the

theory [11].

^It should be noted that the concen-

tration c* cannot be determined

accurately

^{since the}

diffusion constant varies with concentration is the dilute

regime. However,

^{in the}

following,

^{we will}

consider networks

prepared

^from precursor

polymers

of molecular

weight

^{M 50 000. In that} range, ^we have not observed a

significant

variation of D with concentration in dilute solution.

Let us examine now the

properties

of swollen permanent networks. If one assumes

that,

^{at the}

swelling equilibrium,

the elastic chains of the networks have an average dimension

RF,

the situation is

quite

similar to that of a solution of macromolecules of dimension

RF,

^at the concentration c* where the coils

just begin

^to

overlap.

^{One would} expect,

then,

^{for a} permanent

network,

^an

equilibrium

concentration _ce

equal

^{to c* and}

depending only

^on the molecular

weight

of the elastic chains. As a matter of

fact,

^{it has} been shown

previously

that ce

depends

^{also on the}

experimental

conditions of the

crosslinking

^reaction

and on the

functionality

^{of the}

crosslinkages [4, 5].

In

figure

^{2 we have}

plotted

^the

experimental

^values

of _Ce as a function of the values of c* calculated from eq.

(6),

for three series of networks.

Although

^the

uncertainty

ⁱⁿ the determination of c* is

quite large,

one can observe

that c.

c* for trifunctional

gels,

ce N c* for 3 DVB

gels and ce

> c* for 10 DVB

gels.

This behaviour can be

explained by

^the

nonideality

of

gels

^{and the} presence of structure defects in the networks.

The

probability

^of

trapping physical entanglements during

^network formation increases with the number of _{DVB per}

living

^end

(and

also with the initial concentration in styrene

prior

^to

crosslinking).

^{As a}

result,

^the average dimension of the

elastically

^effec-

tive chains

joining

^two

junction points (physical entanglements

^{or chemical}

crosslinkages)

^{would be}

smaller than the radius of

gyration RF

^{of the} precursor

polymer,

^and

equal

^{to the}

screening length ç.

FIG. 2. ^- Polystyrene ^{networks swollen} by ^{benzene :} equilibrium concentration versus concentration c* calculated from _eq. (6)

as indicated in the text. + f3, ^{0 3 DVB, a 5 DVB, 0 10 DVB.}

The full line represents Ce ⁼ c*. Dashed lines are guides ^for the eye.

For

f3 gels, trapping

^of

entanglements

^{is less}

probable,

^{but on} the other hand

pending

^{chains would}

drastically

affect the structure of the

networks,

^since each

pending

chain results in an

elastically

ineffective

crosslinkage. Then,

^the average number of monomeric units between two

crosslinkages

increases and the average dimension of the elastic chain of the network is

larger

than the value of

RF corresponding

^{to the}

precursor

polymer.

^{From a}

general point

^of

view,

the final structure of the network will

depend

^{on the}

amount of both

entanglements

^and

pending

^chains.

The

preceding

conclusion relative to the

apparent

variation of the dimension of the elastic chains with the nature of the

gel

^{is also}

supported by

^the

light scattering

^data.

Indeed,

it has been found that

Dc Dop (Dop,

^{diffusion constant of the precursor}

polymer)

^for

f3 gels, Dc - Dop

^{for 3 DVB}

^gels

^and

Dc

>

Dop

^{for 10 DVB}

^{gels [4].}

Furthermore,

^the

assumption

^{that the} average distance between

junction points

^is

equal

^{to the}

screening length implies

^{that for permanent}

^networks,

the variation of

Dc

^with

equilibrium

concentration ce

obeys

^the

scaling

^law

given by

eq.

(4).

The results relative to different series of networks are

given

ⁱⁿ

figure

^{1. The}

experimental points

^{lie on, or}

slightly

above the

straight

^line

representing

the semi-dilute solutions. Therefore _eq.

(4)

^is also satisfied for _per- manent networks with

only

^{a small}

change

^{in the} pre- factor of the

right

side of the

equation.

974

We have observed similar results for the system

polystyrene-ethylacetate. ^Figure

3 shows the concen-

tration

dependence

of the diffusion constant relative to

samples

^{1 and 2.} One observes

again

^{in the semi-}

dilute domain an increase of the diffusion constant with concentration

according

^the

following

^law

Furthermore

Dr

^is

independent

of molecular

weight.

At this stage, ^{we must}

point

^{out two}

experimental

observations that we do not understand.

i)

^{For solutions of}

polystyrene

ⁱⁿ

ethylacetate

^at

concentrations

larger

than about

10 - 2 g/cc,

^{the auto-}

correlation function of scattered

light

^{exhibits an}

additional

component

^of

decay

^{rate much}

larger (about

⁵⁰

times)

^{than the} component

arising

^from

diffusive motion of the

polymer.

^This component, of unknown

origin,

^{is observed even} in solutions of

polystyrene

of low molecular

weight (- 20 000).

ii)

The values of c* determined from

figures

¹

and 3 for

sample

¹ in benzene and

ethylacetate

respec-

tively

^{do not} follow the

RF ’ dependence predicted by

^the

theory [11].

The ratio

is

equal

^{to 1.4 whereas}

On the other hand the variation of c* with M relative to the two

samples investigated

ⁱⁿ

ethylacetate obeys approximately

^{the - 0.8 power law}

dependence.

It seems

then,

that the chains need to be

overlapped

more in

ethylacetate

^{than in}

^benzene,

^{in order to}

give

rise to

cooperative

^modes.

On

figure

^{3 we have also}

reported

^{the data relative}

to some networks swollen in

ethylacetate.

^The coope- rative diffusion constant of the permanent ^networks

FIG. 3. ^- Diffusion constant versus concentration for PS-ethyla-

cetate system ^{in dilute} solutions, semi-dilute solutions and swollen networks.. Solutions of sample ^1. 0 Solutions of sample ^2.

0 4 DVB networks.

obeys

^{the same}

scaling

^{law as} the semi-dilute

solutions,

but the difference between

prefactors

^{is more} pro- nounced that in benzene. This last

point

^{could be}

related to the fact that the networks have been _pre-

pared

^{in a} solvent of better

quality

^{that the}

swelling

agent.

Let us consider now the structure of the permanent networks. As discussed

above,

it is reasonable to assume

that,

^for permanent networks swollen in benzene the

equilibrium

concentration _Ce can be identified in the first

approximation

^{with the concen-}

tration c* relative to chains

joining

^two

elastically

effective

junction points. Then,

^{one can} estimate the average molecular

weight Meff

of such chains from eq.

(6).

^The values obtained for

Meff

^are

compared

to the molecular

weight Mp

^{of the precursor}

^polymer

in table III.

In the

light

^{of the}

preceding results,

^we have also

reanalyzed

^{the data}

reported by

Belkebir-Mrani

et al.

[12]

^{of the}

compressional

modulus of

elasticity

^E

which are included in table III.

According

^{to the basic}

assumption, i.e.,

^the average dimension between ^two

crosslinkages

^is

equal

^{to the}

screening length,

the modulus should also

obey

^the

same

scaling

^{law in} swollen networks and in semi- dilute solution. De Gennes has shown that for semi- dilute solutions E oc

C2.2S.

In

figure

^{4 we have} pre-

FIG. 4. ^- Log-log plot ^{of the} compressional ^{modulus versus} equili-

brium concentration, PS networks swollen by ^{benzene : +} f3,

0 3 DVB, 0 5 DVB, 0 10 DVB, ^PDMS networks swollen by hep-

tane : A. The straight ^{line has a} slope ^{of 2.25.}

TABLE III

Polystyrene

networks swollen

by

^benzene

(Q)

(°) All ^the samples listed here have been prepared ^{with an} initial concentration of styrene ¹⁰ %.

(b) Calculated from _eq. (6).

(C) Calculated from _eq. (6) by letting Ce ^{= c*.}

(d) Calculated from the values of Meff .

(e) Data of Belkebir-Mrani et al. [ 12] (E is related to the parameter G * of the authors through ^the relationship ^{E = G *}

qi- 0 113

^where qiO is the swelling equilibnum ratio).

(f ) f3 samples ^{refer to} trifunctional networks prepared by chemical reaction of a living polystyrene ^{and a} trifunctional electrophilic

deactivator [13].

(9) Samples containing ^one ferrocene unit _per chain end [12].

sented in a

logarithmic plot

the data of Belkebir- Mrani et al.

[12]

of different series of networks swollen

by

benzene and a series of

polydimethylsiloxanes

swollen

by heptane

^{as a} function of ce. One observes

a very

good

agreement with the theoretical

prediction.

Furthermore, according

^{to the rubber}

elasticity theory

^{E is}

given by [14] :

where A is a numerical constant, vo the number of elastic chains _per unit volume of swollen

network,

r2i > and ro2>

^{are the mean square end-to-end}

distances of network chains in the swollen and the swollen-reference states,

respectively.

The calculation of

Meff

^{was based on the}

assumption

^{that no defor-}

mation of the chains occurs in the

crosslinking

process.

This

implies

^that

« rf > / r2 >) _

^1. Then the modu- lus E should be

proportional

^to vo. It should be noted that the

proportionality

^{of E to} vo also results from

the

scaling

^{law E oc}

C;2.25. Combining

^this power law with eq.

(6)

^{one obtains}

In

figure 4,

^{we have}

plotted

^{E as a} function of the effective number of elastic chains _Veff calculated from

Meff (cf.

^Table

III).

^The

experimental points

^cluster

around a

unique straight

line. From the

slope

^{of this}

straight

^{line one can} estimate the value of the _pre- factor A - 1. It should be noted that this

analysis

of the data relative to the modulus is valid

only

^{if the}

number of

pending

chains attached to one

elastically

effective

junction point

^{is not too}

large.

^The presence of such

pending

chains would

drastically

affect the elastic

properties

^without

significantly changing

^the

equilibrium

concentration. Recent

experiments

^of

uniaxial

compression

^on

gels containing

^{a controlled}

proportion

^of

pending

^chains support ^{this last}

point (1).

(’) Bastide, J., Private communication.

976

FIG. 5. ^- Compressional ^{modulus versus} number of elastically effective chains for PS networks swollen by ^{benzene +} f3,3 ^DVB,

o 5 DVB, 0 10 DVB.

4. Conclusion. ^- In this paper, ^we have shown

that,

^for permanent ^networks

synthetized

^{in a}

good

solvent and swollen in a

good

^or

moderately good solvent,

^the

cooperative

^{diffusion constant}

obeys

a

scaling

^{law with}

equilibrium concentration,

^similar

to that obtained for semi-dilute solution. This result

implies

^{that the} average distance between two elas-

tically

^effective

junction points

^is

equal

^{to the}

screening length.

This

assumption

^{is also}

supported by

^the compres- sional modulus data which

obey

^{the same}

scaling

law with the

equilibrium

concentration as the semi- dilute solutions. This

approach provides

^{us with a}

new

insight

^{into the structure} of the networks. In

particular

^the fact that for some

samples

the modulus fall below the curve E ⁼

f (ce)

indicates the _presence of

loops

^or

pending

chains attached to crosslinks.

Quantitative

determination of the number of elasti-

cally

effective chains of the networks can be

attempted,

but the

validity

of the results obtained rests on that of the determination of c*.

Acknowledgments.

^- The authors wish to thank Professor P. G. de Gennes for

having

^{initiated this}

work.

They

^also

gratefully acknowledge

the contribution of Dr. F. Candau to the

experimental

^work.

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