Phase separation of weakly cross-linked polymer blends

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Phase separation of weakly cross-linked polymer blends

A. Bettachy, A. Derouiche, M. Benhamou, M. Daoud

To cite this version:

A. Bettachy, A. Derouiche, M. Benhamou, M. Daoud. Phase separation of weakly cross-linked polymer

blends. Journal de Physique I, EDP Sciences, 1991, 1 (2), pp.153-158. �10.1051/jp1:1991121�. �jpa-

00246309�

Classification

PhysicsAbsmacts

8270G M.75

Sho~ Communication

Phase separation of weakly cross-linked polymer blends

A~

Bettachy (~),

A~ Derouiche

(I),

M. Benhamou

(I)

and M. Daoud

(2)

(1) Groupe

de

Physique Thdorique,

Facultd des

Sciences,

Ben

M'sick, Casablanca,

Marocco

(2)

Laboratoire Ldon Brillouin

(*),

C-E-N-

Saclay,

91191 GiffYvette Cedex, France

(Received

22 November199fl

accepted

4 December 1990

)

Abstract. We consider the

microphase separation

that _occurs when _a mixture of _two different

polymers

^is

brought

to

phase separation

after

being

cross-linked. For

deep quenching, microphase separation

^takes

place,

related to the

competition

that exists between demixion and

elasticity.

We consider more

carefully

_very ^weak

cross-linking

ⁱⁿ two cases. If the

gel

is assumed to be

regular,

the effect is present above the

gelation threshold,

when _a

gel

is present, ^and

disappears abruptly

at the threshold. The characteristic size of the

microphase

is related to the distance between

(trapped) entanglements.

When the sol is present, the effect exists both above and below the

gel point.

The characteristic size of the

microphase

is of the order of the chain

radius,

and is much smaller than the mesh size of the

gel.

1. Introduction.

Phase

separation

in linear

polymer

blends

ill

is studied both for

practical

and fundamental _rea-

sons. Branched

systems

^and

gels

were also considered _{to some} extend

[2, 3]

^An

interesting

case

was

recently

considered

theoretically by

de Gennes [4] ^and

experimentally by

^Briber and Bauer [5j ^{This is the} case when two linear

polymers

A and B _are first cross-linked in their coexistence

region,

and then cooled down _{at a}

temperature

^where

phase separation

should take

place.

The

previous

authors considered the case of

strong cross-linking,

when the average ^{distance between} cross-links is much smaller than the radius of _a chain. As _a

resul~

there is _a

competition

between the

tendency

of the

system

^to

phase separate,

^{and the}

elasticity

of the network that resists such

separation. Agreement

between the theoretical

predictions

and the

experimental

results is

quite satisfactory, except

for very ^low _q. ^The theoretical

approach

however assumed the existence of _an

ideal

network,

with _no dead ends. It also assumed that every ^linear

polymer belongs

to the

gel.

This may ^{indeed be realized}

by

a

strong cross-linking

^and

subsequent washing

of the

gel

in order

to remove the sol and the unreacted

parts.

^{It is also}

interesting

to consider the _case when those

two

parts

are still

present. Moreover,

it is also

interesting

to consider the _case when the

gel

is very

poorly cross-linked,

that is when _one is close to the

gelation threshold,

so that the mesh size is very (* Laboratoire commun C-N-R-S--C-E-A-

154 JOURNAL DE PHYSIQUE I N°2

large.

^{In what}

follows, _wq

will consider the latter _case, when the

gel

is "fractal" _{on a}

large

distance scale

f.

two cases may ^{then be}

considered, depending

_on whether _one

keeps

the sol _{or one} washes it out of the container before

cooling

down. We win restrict this

study

to the former _case. Thus

we consider the

following

case: a mixture of two linear

polymers,

both made of the _same number

Z of monomers is first cross-linked in the coexistence

region

of both

components.

^We assume

that each of A and B may ^react

only

^with the other

species.

This

step

^is

stopped above,

and in

the

vicinity

of the

gelation

threshold. Thus _one has a mixture of _a sol made of finite branched

polymers

and a very ^tenuous

gel.

This is _now

quenched

^{below the} coexistence

temperature

^of the

mixture,

and _we would like _to

study

the behavior of such mixture. In order _to do

this,

_{we are}

going

to extend the de Gennes

approach

of

strongly

cross-linked

systems

to this _case. For _a

deep quenching, microphase separation

^{of the}

gel

becomes dominanL As _we shall see, the

typical

size of the instabilities is related _to the radius of the linear chains rather than to the mesh size

f

of the

gel.

In what

follows,

we will first remind the case of

strongly

cross-linked

polymers

that _was

considered

by

de Gennes in section 2. Section 3

generalizes

it to the case of _a weak

regular gel

where the average ^{distance between} cross-links is assumed to be constant and _no sol is

present.

In section

4,

_we consider the _case of random

cross-linking

in the

presence

^of a sol.

2.

Micmphase separation

in

strong gels.

Consider _a

strongly

cross-linked mixture of two

polymers

A and B made of the _same number Z

of monomers. Let _n be the average ^{number of} monomers between consecutive cross-links. We

assume that _n is much smaller than Z and that the network is ideal. Thus _we

neglect completely

the

possible

_presence of dead ends.

Similarly,

we also

neglect

the

presence

^{of finite}

polymers

that do not

belong

to the

gel.

The latter

approximation maybe easily justified

because if

present,

these may

easily

be washed out of the

gel.

The former

approximation

on the other hand needs _a very ^{skilful chemist. Let} us now

change

the

temperature

so that the mixture is

brought

to the two

phase region

where the

polymers

would

phase separate.

^{Because of the}

competition

between the

phase separation

^{that would}

normally

occur in the A-B

mixture,

and the elastic

properties

of the

gel,

a

microphase separation

takes

place.

This was described

by

de Gennes [4]

by considering

the

following

free energy

per

^site:

F

=

(

Ln @ +

~

~

Ln(I

_@) +

x@(I

_@) +

a~(V@)~

+

CP~ (I)

where @,

Z,

_{x are}

respectively

the fraction of A monomers, the

length

^of both A and B

polymers,

and the

Flory

_[6] interaction

parameter

^{between A and B} monomers. The first three _terms in relation

(I) correspond

to the

Flory-Huggins _{[6, 7j}

free

energy

^{of an} uncross-linked mixture of A and B chains. The

gradient

term takes into _account the presence ^{of interfaces between A-rich and}

B-rich

phases. linking only

these terms into account

corresponds

to the classical

phase separation

between

polymers

A and B. P is the average

separation

between the _centers of _mass of A and B-rich

phases,

and C _an elastic constant, to be evaluated below. The last term takes into account the

elasticity

of the

gel

that is

resisting phase separation.

In order to estimate this term, de Gennes first solves the uncross-linked

problem.

One finds

@c =

1/2 (2a)

xo -~

2/N (2b)

It is then

possible

to

expand

the free

energy

^{in the}

vicinity

of the critical

point, assuming

₌

1/2

_{+ p :}

Fo

=

(Xo X)

P~ +

a~(?P)~ (3)

The

last, elastic,

term in relation

(I)

is estimated

by introducing

an

analogy

between the

present problem

and the

polarisation

of _an

electrically

neutral medium. In thin

analogy,

the

displacement

P of the center of _masses of A and B-rich

phases

is the

polarisation

_{and p the} ^local

charge.

^Thus

we have

div(P)

_{= -p}

(4)

Fourier

transforming

relation

(I)

and

taking

into account

equation (4),

we find

F"~jlXo-X+q~+ §)P( ⁽⁵⁾

q

~

The scattered

intensity

in _a

light

of neutron

scattering experiment

is

S(q)

~'

Xo

X + q~ +

(6)

q It has _a maximum for

q* -~

CU' (7)

with value

S

(q*)

-~

(xo

^{X +}

2q*~) (8)

This

diverges

for _Xc such that

Xc Xo ~' q~~ ~'

C~/~ (9)

Usually,

the

Flory

interaction

parameter

is

inversely proportional

to

temperature

^{T. This}

implies

that the

spinodal temperature

^{for such} cross-linked material is

changed

and the coexistence

region

is

larger

than

(or

the uncross-linked mixture.

It is

interesting

to note that the scattered

intensity,

relation

(6),

is the _{same as} for a

polymer

mixture for

large scattering

vector: for q > q* ^and _q » qi, with

qi~

_-~

Cl (xo xi

,

the elastic contribution _to the scattered

intensity

in relation

(6)

_may be

neglected,

and

S(q)

is

comparable

to the scattered

intensity by

a

polymer

mixture close to its

phase separation:

S(q)

_m

So(q)

-~

(Xo

_X +

q~)

^~~

(10)

We also note the existence of _a second

length,

_qi

~~,

which coincides with q*~~ for _x

= xc and

corresponds

to _a

larger stability

limit. We do _not know however whether its

keeps

_any

significance beyond

the

present

^{first order}

expansion.

The elastic _constant

C,

which controls the size

f*

of the

microphase

was estimated

by

de Gennes for _a

perfect network,

with _{n monomers} between consecutive cross-links. The constant per ^{link is}

Cl -'n~~a~~ (11)

156 JOURNAL DE

PHYSiQUE

_{I N°2}

Assuming

that no dead ends _are

present,

^the

rigidity

constant

per

^{site is} C

-~

(naj~~ (12)

Thus the characteris tic size of the microdomains is

f*

-~

C~~/~

-~

n~/~ (13)

and is of the order of the mesh size of the

gel.

It is then

interesting

_to look at the limit when the

gel

is close to the

gelation

^threshold.

3. Weak

regular gels.

As a first extension of the

previous work,

let _us consider _a

gel

that is less cross-linked. In this

section,

we make the

assumption

that the

gel

is

regular:

we assume that the distance between

cross-links is _a constant, ^{and that there is} no sol. Thus _we extend the

previous

model to situations where the

cross-linking density

is smaller. It will be

interesting

to compare ^{the results of this} sec- tion to those of the

following

_one, where _we will consider the

vicinity

of the

gelation threshold,

and thus take into account the

presence

of the sol and of the defects in the structure of the

gel.

The main difference between this _case and what _was considered above _comes from the presence of

trapped entanglements

that act as

permanent

cross-links.

Indeed,

because we are

assuming

a

regular

chemical structure for the

gel,

^all the

entanglements

that _are

present

are

trapped.

This

implies

that _as

long

as the distance between chemical cross-links is smaller than the distance

Ne

between

entanglements,

the

previous

^results are valid. On the

contrary,

^when n becomes

larger

than

N~,

_we are led to consider the "effective"

cross-links, including

both chemical ones and the

trapped entanglements.

This in _turn

implies

that the average ^{distance saturates} to

Ne.

The rest of de Gennes'discussion remains unaltered. As _a consequence, ^the

microphase separation

^{is dom-}

inated

by

^the

entanglements:

whatever the

cross-linking density,

_we

have,

as

long

as a

(regular) gel

is

present

q* -~

C~~l'-~ _N~~~/~ (14)

Similarly,

the

temperature

^shift saturates

and,

from relation

(9),

we

get

,yc Xo ~' q~~ ~'

Ne~~ (15)

With this

assumption,

the

gel disappears

when _n becomes

larger

than the

length

Z of the linear

chains,

and _so does the

microphase separation.

For _{n >}

Z,

"normal"

phase separation

betwecn

linear chains _occurs for _x

" So ^at q = 0. Thus in this

approximation,

we

expect

sudden

drops

both

in q* ^{and in} Xc for _n

= Z.

4. Weak random

gels.

Let _{us now} relax the constraint of

regularity

in the chemical

cross-linking procedurc,

and consider the differences with

previous

section. When the

cross-linking density

is

low,

and the

gel

is in the

vicinity

of the

gelation threshold,

two additional

points

have to he taken into account when

compared

to the

regular gel assumption

_we considered above:

first,

the

gel

_may no

longer

be considered _{as a}

perfect

network. On the

contrary,

we know that it is _a

fractal,

with many ^dead ends and other

imperfections [8,

9]

Second,

_we also know

that,

at least in thc reaction

bath,

a

large

amount of sol is

present.

^{This is made of}

finite, eventually

_very

large

branched

polymers.

Among these,

unreacted linear A and B chains _are

present,

^that

might _separate freely

because

they

are not

subject

to any ^{direct elastic} constraint. Note that the finite

polymers

of the sol are

also

subject

to the

restoring

elastic

force,

as the

gel.

Thus _we may

partition

the mixture into two

parts, namely

those

polymers

^{that have} reacted and _are

cross-linked,

and those that did not, and

are still linear. In what

follows,

we win

neglect

^the

polydispersity

of the

sol,

and consider the distribution of _masses

only through

^the

partitioning

that _we

just

mentioned. It is then

important

to realize that the elastic contribution that is

resisting normal, macroscopic, phase separation

is

on the scale of the

polymer chains,

and is

completely

^{different from the elastic modulus of the}

gel.

The latter vanishes at the

gelation threshold,

while the former is still

present

^{and finite. As}

a matter of

fact,

^it is

present

even in the

sol,

below the threshold. The free energy ^{of this}

system

is very ^{similar to what} was considered

previously

for _a

strong gel.

As

above,

we assume that the free energy ^{b made of two}

parts.

^The energy ^of

mixing

is assumed to be identical to the

Flory- Huggins part

^of an uncross-linked mixture of A and

B,

_as in relation

(I).

The elastic contribution is

analogous

to the

previous

one.

But,

in the estimation of the elastic constant

C,

one has _to take

into account

only

those

polymers

that have reacted: the unreacted linear chains do not contribute to thin elastic

part.

^Let p be the

probability

that _a

polymer

is cross-linked. In the

vicinity

of the

gelation threshold,

we assume

that,

at

equilibrium,

few

entanglements

are

trapped,

^{and that the}

average ^{distance between} "effective" cross-links is of the order of the radius

R(Z)

-~

Zl/2

_{of a} linear chain. In _a

sphere

of radius

R(Z)~ ^Zl/2

chains _are

present. Among these, only pZ~/2

are

reacted. The modulus

Cl

of _one A-B

pair

is of the order of the one for _an elastic chain:

Cl

-'

I/Z

Thus the modulus

per

^{site is}

C

-~

jpZ~/22~~/~

-~

§ ₍₁₆₎

Thus the total free energy per ^{site is} identical _to relation

(Ii,

with C

given by

relation

(16)

^in-

stead of

(12). ^Therefore,

^{in the}

^vicinity

^{of the}

threshold,

a

microphase separation

_occurs, with _a characterhtic distance

q*~~

q*~~ -~

R(Zl _P~~~~ (17)

Note that p is smaller than

unity

and that this

corresponds

to some

swelling

of the size of the linear

chains,

because of the presence of the unreacted linear chains. Close _to the threshold

however,

this effect is rather

limited,

because _p is of the order of10~ and of the small value of the

exponent

in

(17).

Thus the characteristic size

q*~~

^of the

microphase separation

is much smaller than the

diverging

mesh size

f

of the

gel. Finally, nothing special happens

at the

gel point.

5.

Concluding

remarks.

Our main conclusion is that

microphase separation

^takes

place

both above and below the

per-

colation threshold.

Indeed,

it is clear from the above considerations that _one does not need the presence ^{of the}

gel

to

get

^the

microphase separation.

The _reason for this is that the elastic restor-

ing

force that resists normal

phase separation

h

present

at scales of the order of the radius of the linear chains.

Thus,

_even below the

threshold,

the elastic

restoring

force between cross-linked A and B

parts

^{is still}

present,

^and we still

expect

^the same

phenomenon

to take

place.

This may ^be

158 JOURNAL DE PHYSIQUE I N°2

seen for instance for small values of the

probability

_{p of a} chain

being

cross-linked. One is then

dealing

with a mixture of free A and B linear chains and A-B block

copolymers:

the

probability

of

finding

such diblocks is of order p, and for smafl _p, dominates the

p2

terms

corresponding

to

larger ~branched) polymers.

^{In this} case, one may still obtain micellar behavior of the

copoly-

mers, with characteristic

length q*~~,

^relation

(17).

^{Note also} that surface effects

might

become

important,

with the

copolymers

and _more

generally

the branched

polymers, sitting

at interfaces between A and B-rich

phases possibly leading

to microemulsions. A detailed

study

of a mixture of A-B

copolymers

in A + B was

given recently by

Leibler _et al.

[10]

We discuss

briefly

the influence of the

trapped entanglements

in the random

cross-linking

case.

The first effect h to

change

the location of the sol-

gel

transition. Below this

threshold,

we assumed above that

they

relax. Thin is

probably

correct for

large

times. For shorter

times, they

_may affect the effective distance between "effective" cross-links.

They

would then tend _{to saturate} it at a

value of the order of

Nel/2.

_{As a} consequence, q* ^{would remain}

roughly

constant over a wide range ^{of the}

probability

_p ^around the

gelation point.

Recent studies

by

Rubinstein _et al.

ill]

indicate that the relaxation times in these

systems

are very

long

in the

vicinity

of _pc. It would

certainly

be

interesting

to test

experimentally

whether such

entanglement

effect is observable _or not.

Acknowledgements.

This work _was made

during

_a visit of A.

B.,

h

D.,

and M. B. to

Saday,

and

they

wish to thank The L. L. B. for its

hospitality.

M. D. wishes to thank B. Hammouda for _a related discussion.

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Avril 1990".