Crosslink Effects on Equilibrium Polymers

(1)

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Crosslink Effects on Equilibrium Polymers

K. Elleuch, F. Lequeux, P. Pfeuty

To cite this version:

K. Elleuch, F. Lequeux, P. Pfeuty. Crosslink Effects on Equilibrium Polymers. Journal de Physique

I, EDP Sciences, 1995, 5 (4), pp.465-474. �10.1051/jp1:1995140�. �jpa-00247072�

(2)

J. Phys. I Hauce 5

(1995)

465-474 _APRIL 1995, PAGE 465

Classification Physics Abstracts

05.20 64.60 82,35

Crosslink Effects

_on

Equilibrium Polymers

K. Elleucu

(~),

F.

Lequeux (~)

^and ^P.

Pfeuty (2)

(~) Laboratoire d'Ultrasons et de Dynamique des Fluides Complexes (*) Institut Le Bel, 4 _rue Blaise Pascal, 67070 Strasbourg Cedex, France

(~) Laboratoire Léon Brillouin (**), Centre d'Etudes de Saclay, 91191 Gif _sur Yvette, France

(Received

2 November 1994, revised in final form 21 December 1994, accepted 4

January1995)

Abstract. We study trie elfects of three- and fourfold crosslinks _on wormlike micelles. This mortel is worked _using _a correspondence between magnetic mortels and equilibrium polymers.

We obtain the phase diagrams and trie critical behavior of the concentration of extremities and of crosslinks. These crosslinks generate _a demixing for both three- and fourfold connections.

We show that, due to the dilference of parity between the three- and fourfold connections, the behavior _is quite dilferent for each _case.

1. Introduction

Wormlike micelles in

solution, liquid sulfur,

etc. _are known to be

equilibrium polymers. Tuey

are aise called

living polymers. They

_are linear

abjects

whose

length

is net fixed

chemically

but

thermodynamically. They

^reach a

dynamic equilibrium,

where

they

^do not cease ta break and recombine _or grow and _retract. Hence these systems _are dioEerent and

apparently

_more

complex

than usual

polymer

^solutions.

Some surfactant solutions _were observed

by

S.A.N.S. _to form _worm like micelles

[ii.

Rireo-

logical

measurements and their

analysis

_gave _an estimate of trie micelle

length [2,3]. Recently

these micelles showed _some strange behavior:

increasing

sait concentration leads to _an increase of tue

lengtu

_as

usual,

but at

higu

concentrations, an apparent ^decrease ^of ^trie

length

_{[3] fol-}

lowed

by

a

phase separation

_[4] _was observed. Tuis uas been

suspected

to be trie

signature

of trie _appearance of connections due to

charge

_screemng

[3-5].

However, _many doubts remain:

are there

really

connections and is trie

demixing

linked to trie existence of these connections?

It is clear that _we _now need _a theoretical

description

of connected

living polymers (Fig. 1).

Trie _atm of this _paper is ta introduce _a theoretical

analysis

of trie rote of crosslinks in

equilibrium polymers

_more detailed than trie

recalling

trie

magnetic

model used to

investigate

^such

equilibrium polymers,

we describe trie

phase diagram

of trie

(*) U-R-A. 851 CNRS.

(**) Commissariat à l'Energie Atomique, ^C.N.R.S.

(3)

Fig. 1. We _are interested in _a systeIn of linear objects that have _a chemical potential _p _per unit of length. These chains _con close in rings or have extremities of energy El. They _con also form

connections of _energy E3.

connected

equilibrium polymers

for both 3 functional and 4 functional connections.

Finally,

we discuss trie consequences of these 3 and 4 connections _on _worm like micellar systems.

2.

Correspondence

Between

Ising

Model and

Equilibrium Polymers

A

polymer theory taking

excluded volume eoEects into accourt is diflicult _to work. An alterna- tive way is to use trie

equivalence

that exists between trie

polymers

and trie

magnetic

systems since

they

^bave ^been studied _very

deeply

in trie

theory

^of critical

phenomena.

Living polymers

_are ^dioEerent ^from ^dassical

polymers

since

they

do not bave _a fixed

length

and also because

they

_con sometimes

polymerize

into

rings.

^It ^uas ^been ^shown ^that ^such a

fluctuating

_system

corresponds

to _an

Ising (one component) magnetic

model [7].

Liquid

sulfur is _a

representative example

of _a system ^that ^bas ^been treated this _way. Since trie

giant

^micelles _are _very similar to sulfur

polymers,

trie former could also be descnbed

by

an

Ising

^model.

Dur system ^will consist of _a _monomer ensemble

(chemical potential p)

that _can assemble to form

rings

^and chains. Their ends bave _a

fugacity

Ki _"

exp(-Ei/kT),

where Ei _is trie end energy. These chains _are

placed

in _a

good

solvent _so that there is _an excluded volume

interaction between them.

Let _us establish trie

correspondence

between trie chains and _rings of

living polymers

and trie

Ising

model.

Trie ensemble of cuains and _rings uas the

following partition

function [8]:

~g ~g

(Î)

z =

~ n(Ne,l~b)lÙl~~P~

~

~~~~~

Ki

and

Kp being

trie

fugacity

_per end and _per bond

respectively.

Ne

,Nb

_are trie number of chain ends and of bonds in chains and

rings, respectively.

n is the number of

configurations

for fixed N~ and Nb. It _must be noticed that it also takes into

account the exduded volume effect

by giving

less

weight

to

configurations

where _two

polymers

cross.

(4)

N°4 CROSSLINK EFFECTS ON EQUILIBRIUM POLYMERS 467

2.1. THREE-FOLD CROSSLINKS.

Following

trie _same

approach,

_we _can write trie

partition

function for _a system ^with ^3-fold crosslinks _:

Zp

₌

£ _n(N~,

_Nb,

Nc)KfeKfbKf~ ₍₂₎

N~,N~,N~

where Kc is trie

fugacity

_per crosslink and Nc trie number of these crosslinks.

This enables _us to write trie free _energy _per unit

partiale (we

_suppose kT ₌

Ii, fp

₌

-1IN

Ln

Zp

^and ^deduce ^trie concentrations of

ends,

_monomers and crosslinks that look like

~~ ô

Î~KI' ^~

ô

Î~K2' ^~~

ô

Î~K3

^~~~

Let _us _now seek for trie

"magnetic"

Hamiltonian that will

correspond

to this

polymer

system.

Trie

typical

_way to do it is to make _a

high

temperature _expansion and then

identify

both theories.

We

rapidly

review trie main fines for trie unlinked

polymer theory.

Let _us start with trie Hamiltonian:

H

=

£ _JS~Sj £

_hS,

₍₄₎

<;,j> _<

where

Si

is trie

spin variable;

trie second _sum is _over ail sites of trie lattice. Trie first is _over the _1,

j pairs

of nearest

neighbors.

Trie

partition

function Z _e

£e~~~

could then be

expanded

in

fl (supposed

_to be

small).

A schematic

representation

of the different _terms of _our

expansion

will

surprisingly

^look like

polymer

and

ring configurations

_[7]. Trie excluded volume is also taken into account in this

one component

model,

_even if it is not _as strong as in trie limit of _zero

spin

components [9].

Considering universality,

^this

repulsion

is in fact suilicient.

In order to generate crosslinked

polymers,

it is easy ^to prove that trie

following

Hamiltonian will do it

nicely:

H ₌

£

_KS

_Sjsk ~ _J$Sj £

_hi

₍₅₎

<1,j,k> <i,J> <

The first _sum is _over trie

triplets of1, j,

^k with 1,k two nearest

neighbors

of

j.

The

partition

function _can then be

expanded

in

fl

units and is in fact the

partition

function

given

in equation

(2):

Z =

£ n(N~, Nb, N~)h~eJ~bK~~ (6)

Hence the

polymer

and the

"magnetic"

models

belong

to the _same

universality

Mass. We

deliberately

write the two

partition

functions in the _same form _as in

equations (2)

and

(6)

and the two systems are

equivalent

in the _sense that

Ki

~

h, K2

_~ ^J ^and

K~

m~

K.

As it is somewhat diilicult to work _on trie discrete

"magnetic" models, especially

^when

including fluctuations,

_we will then work _on the continuous formulation of trie

Ising

^model. ^Of

course, it is also

possible

to do _a direct

diagrammatic correspondence

between trie continuous

models, using

trie

Feynman diagrams

for trie correlation functions of trie

"magnetic"

model and

identifying

them _to trie continuous

polymers [loi.

H =

1(?~9)~

+ u2~9~] + uo~9~ h~9

(7)

(5)

For trie

Ising

model _we will bave in trie continuous limit _a Hamiltonian of trie form: where _u2 is

proportional

to

(J J~),

h is trie _same _as the _one introduced

before;

_vo _is

responsible

^for the excluded volume interaction and does not appear

explicitly

m the discrete Hamiltonian since it is

intrinsically

contained in the discreteness of the lattice.

Adding

^the ^3-fold crosslink is

equivalent

to add _a

-u3~~

_term, with _u3

~ K. The Hamilto-

nian will become _:

~ ~~~~~~ ~ "~~~~

"~~~

_~

"° _

~~~

We stress here that in the Hubbard

transformation,

_u2 _is identified with J J~. Hence _u2 in

trie continuous model is trie chemical

potential

_per unit of

length

while h and _u3 _are

fugacities.

Calculating

the

density

of the

magnetic

^free _energy

f

=

-1IN

Ln

Z,

_we _can deduce the

end,

the _monomer and the crosslink concentration.

~~ ô

Î~KI

ô

ÎÎ

h' ^~~~~

~

ô

Î~K2 ÎÎ2

^~~~~

~~

ô

Î~K3

_ô

/Îu3

~~~~

For similar _reasons, _we could also calculate trie monomer-monomer correlation function [11]:

Io(z,z')

_oc

j< ç9(z)~ç9(z')~

> < ç9(0)~

>~i (10)

Remark _: In order ta describe

networks, Lubensky

et ai. [12] ^used a Hamiltonian of _a similar form but with _a tensorial field. Trie upper critical dimension _was then 6 and

generated

_a ricin

topological variety.

With _Dur scalar field trie critical dimension _is 4 but the

topology

_is fixeà and does not allow _us to

distinguish

between the different structures, ^1-e- ^for instance, _we

cannot separate the rote of

loops

m the clusters.

Hence,

the information _we get has

nothing

to

do with cluster

topology

and

percolation transition,

wuich in fact has _no

thermodynamical

rote.

Tue

complete

tensorial model is _more

complicated

and _uses _a

development

in _e

= 6 d

= 3,

which is _a very crude

approximation. Unfortunately,

Dur scalar model does net allow _us ta get

an

insight

into the

topological

structure of the clusters.

2.2. FouR-FOLD CROSSLINKS

[13].

In the _same way as

previously,

ta mclude 4-fold _con-

nections with _a

fugacity K4,

_we

only

have _ta add _a

u4~~

term ta _Dur

Harniltonian,

with U4'~1Ù4.

H = [(T7çg)~ +

u2~~Î

+

(vo u4)~~

₊

u6~~

_hçg

(II)

2

vo

engendering

trie excluded volume

through

the solvent

quality,

the addition of trie _u4 term _is in fact

equivalent

ta _a

change

of the solvent

quality.

We also added _a

u6~~

term with _u6 _> o,

smce trie _çg~ and _çg~ coefficients could vamsh.

We _can also deduce the 4-fold crosslink concentration:

~~

ô

/~K4

_ô

/Îu4

~~~~

(6)

Then

using

the field

theory

and critical

phenomenon techniques,

_we _can calculate the _previous

quantities

and

principally

find the

phase diagram.

For the sake of

simplicity,

let _us first show the _main mean-field results.

3. Mean Field

3.1. THREE FOLD CROSSLINK. In the Landau mean-field

approximation,

the free energy

cari be written _as

At the transition point h

= o, _u2 _" o, _u3 _" o and _vo _> o, ^the concentrations of ends

Xi,

monomers

#

and crosslinks X3 _vary hke

(using Eq. (9)):

Xl'~lll, É'~l7l~, X3'~l7l~

# being

^the

polymer

parameter that _can be controlled

(supposing

that ail the surfactant _con- tribute ta the

polymerization),

_we _get:

~i

+~

#~/~

and X3 _+~

~3/2

These results _were

already

derived

by

Cates and

Drye

_[6] within the frame of the Gaussian

approximation.

The

phase diagram

_is shown _m

Figure

2. For h

= o there is a critical

point

at u~ = u~ = o and three- first-order transition fines

emerging

from it.

U~

)

~/

/

",

, /

u~

Fig. ^2. ^The mean field phase diagram for threefold crosshnks. _u2 is the chemical potential anà

u3 the crosslinks fugacity. The discontinuous fines _are for h

= 0

(no extremities).

There _are three first-order bues coming ^out of the critical point located at _u2 = u3 " 0. Trie continuous fines _are for h _> 0

(with

extremities). The critical point _moves from trie origin leaving a gap and creating _a

dissymmetry in the phase diagram.

If h

#

0, then trie

phase diagram

^looks

^quite

^different ^with a region where _no first-order transition _can

happen

for weak _u3

(Fig. 2).

(7)

u~

u

~

Fig. 3. The mean-field phase diagram for fourfold crosslinks, _u2 is trie chemical potential and _u4 the crosshnk fugacity. h

= 0

(no extremities)

is represented by trie discontinuous fines, A critical fine appears for _u2

" 0 and _u4 _< _uo. The point at _u2

# 0 and _u4

" uo is tricritical followed by _a parabohc

first-order fine for _u4 > _uo.

3.2. FouR-FOLD CROSSLINKS. Trie Landau free energy is written

The

phase diagram

is shown in

Figure

3. If h

=

o,

trie

point

where _u4

" ~o and _u2

" 0

corresponds

ta _a tricritical

point,

a critical fine _appears _at _u2

" 0 and _u4 _< _vo and trie

parabolic

fine for _u4 _> _vo is of first order. When h is non-zero, trie tricritical point transforms into _a critical _one and the first-order fine remains. If h grows but remains close to zero, these

critical effects will still be felt.

In the vicinity of the tricritical and critical parts of the

phase diagrarn,

the end and crosslink concentration will vary like:

Xi

~

#~/~

^and X4

~ #~

4. Renormalization

Group

While for the fourfold crosslink _case the

phase diagrarn

will not show _any

qualitative

^difference when

induding fluctuations,

_m the _case of threefold

crosslink,

trie

phase diagram

will be affected

by

fluctuations and will be

topologically different,

_as

argued by

^Alexander and Amit [14].

For trie threefold crosslink _case, _we _can do trie renormalization _group calculation. In

fact,

as shown

by

Alexander and

Amit,

this

problem

_can be transformed into _an easier _one. We _are

going

ta take _up this demonstration

again:

If _we start with:

H =

(T7çg)~

+

ju2 ^u3

⁺

^uo~~

^hçg

⁽¹⁵⁾

with trie

following change

of variable _çg

= ~b + "~

trie Harniltonian becomes:

4uo

~

~~~~~

~

~Îo

^~"~

Îo ~~~

~

~~Î ÎÎ

^~~~ ^~

^"°~~

~~~~

We _see

that,

_since trie order parameter ^is

scalar,

_we _cari transform trie

original

Hamiltonian with _çg3 into _a classical Hamiltonian _~b~ with

magnetic

field.

(8)

Then trie eoEective

magnetic

field and trie eoEective _u2 Parameter _are _given

by

~~~ ~

Îo

~"~

~lo

^~~~~~

and

~

3~2

"~~~ "~

4

uÎ

^~~~~~

This kind of Hamiltonian is well-known and allows _us _to _use trie results

already computed.

For trie

following

Hamiltonian:

H = (T7~b)~ + ~a21b~ + a41b~ h'~b

(18)

2 2

there is _a critical

point

at a2 " a2c and h'

= 0 and _a first-order fine for _a2 _< _a2c and h'

= 0.

This enables _us to find trie

phase diagram (Fig. 4) by mapping

trie

following phase diagrarn

on ~12, u3 usmg u2ea and h~R. There _are three critical

points

and three first-order

fines,

one

coming

out of each critical

point.

^Trie important ^dioEerence with trie _mean field is trie appearance of trie gap for _~2 and _~3 weak. Thus this intermediate

region

between trie two critical

points

is _a _crossover _one.

>

é'

Fig. 4. The threefold crosslink phase diagram including fluctuations. For h

= 0 (no extremities)

in discontinuous fines there _are three first-order hnes coming ^out of three critical points. ^When ^h

(putting

extremities), represented here _in continuous fines _grows, _a dissymmetry _is ^created. ^The polymer physical _region _is for e~~3/~~ _> _0. _The _three _vertical _fines _represent _trie trajectories followed

when the concentration # is varied. The critical point at the left of hue _one corresponds to the

overlapping transition, the _one _near hne 2 is the beginning of _a phase separation.

Trie

scaling

part of trie free _energy is written

~ ~ ~~'~~~~

~~ AÎÎ2)~

'

~

_~'~ _~~~~

f(z) being

_a

regular

^function ^of z and _a,

fl

and _~/ trie _one component

magnetic

critical _expo- nents, ^with ^trie ^usual ^notation.

(9)

We _can estimate trie variation of ends and connections with trie concentration in trie

vicinity

of trie critical

points. Using equation (9),

_we get:

#

m~

(A~2)~~"

Xi

~

(A~2)~

~ #~~~~~"~

(20)

With trie _sonne

scaling

^form ^{of trie} ^free _energy as in

equation (19),

_we derive X3.

Including

trie

~3 term is

just equivalent

to

replacing

_~2

by

_~2efl and h

by

h~~

(17a), (17b),

in trie

previous equation.

Near trie first critical point we bave _~3 _- 0. Hence A~2

" ~2 a2c and h

m~ ~3.

Using

equation

(9c),

_we _get

easily:

X3 _~

(A~2)"

=

(A~2)~

~

#~/l~~"~ (21)

which is trie _same variation in

#

_as for Xi

Near trie second critical

point,

trie situation is

completely

dioEerent and _more

complicated.

X3 and

# varying

with trie _same

exfionents,

_we bave trie very dioEerent result: X3

~

#.

Hence, trie eoEect of fluctuations _is to

modify

trie

topology

of trie

phase diagram.

Two critical

points

appear with _a very dioEerent behavior of X3.

Remark: We could also think of _a mixture of three- and fourfold crosslinks. In ibis _case, trie critical eoEects will be dominated

by

trie threefold connections in trie critical

region,

_~3 like h

removing

trie system ^from ^trie tricritical

point.

5.

Consequences

for

Equilibrium Polymers

5.1. THREEFOLD CROSSLINK. Let us

just

discuss trie

meaning

of _our parameters. ^Since _~13

and h _are

fugacities,

trie

physical region corresponds

to _~3 > o and h _> o; _~2 ^is ^trie chemical

potential

_per _monomer and

then,

trie

conjugate

of concentration. Then _a decrease of _~~ is

equivalent

to _an increase of concentration

#.

On trie other

hard,

h contrais trie concentration of extremity.

Similarly

_a decrease of

h,

1-e- _an increase of trie extremity _energy, decreases trie

extremity

concentration.

Varying #,

_we _are

going

to

explore

three

regions

in trie

phase diagrarn (Fig. 4). Increasing

trie chemical

potential

_~2 is

equivalent

to decrease trie concentration.

Along

fine 1

(Fig. 4),

we go near trie critical point ^located at h

= o, _~3

= o and _~2

" ~2c

(actually,

_we _can reach this point

only

for _a solution without any end _nor

crosslink).

This critical transition

corresponds

to a

merged polymerization

and

overlapping

transition.

Adding

extremities will lower trie critical eoEect and trie

signature

of trie

overlapping

transition will be _a maximum for trie correlation

length

and for trie

compressibility adding

extremities will increase trie number of

polymers,

then shorten their

length

and soften trie critical behavior.

Putting

connections between _our chains will also _move trie system _away ^from ^that

criticality.

Trie concentration _near trie critical point ^is trie

overlapping

one,

Increasing

h _moves trie critical

point

to trie

non-physical region

of trie

phase diagrarn.

Trie distance from this critical point is h~R m~ h

~~~~c/4~o

(~~c <

o).

We _see that h and _~3

play

a similar rote.

This

explains

that Xi

(the extremity concentration)

and

X3 (the

crosslink

concentration)

have the _same critical exponent

(#°/~~") (20), (21).

Let _us _note that this exponent for Xi bas also been derived

by

Cates

il

_Si

using scaling

arguments. It is suspected that trie

tadpole

terms

(Fig. Sa)

_are

controlling

trie eoEect of _~3

near this first critical

point

because

they engender

eoEective ends.

Let _us sweep fine 2,

decreasing

trie chemical

potential

_~2 hence

increasing

the concentration.

Near trie second critical point, ^the êoEects ôf _~2 ând _~3 are mixed in trie heR and _~~eR. As

(10)

N°4 CROSSLINK EFFECTS ON EQUILIBRIUM'POLYMERS ₄₇₃

a)

b)

Fig. 5.

a)

Tadpole diagram.

b)

Bubble diagram.

calculated

before,

X3

~

#

while Xi remains in

#~/(~~")

_near the critical point. Since the ends _vary

slowly

^and ^the connections grow

proportional

to trie _monomer

quantity,

_we suspect that _we do not bave _a stretched network at ail

(in

contrast with the arguments of Cates and

Drye

_{[6] of}

"saturated/unsaturated regime"

in order to locate the

phase separation).

We guess that the system ^could be made of bubble structures

(Fig. 5b).

This would _mean

that,

_m this _case, the connections _are

uniformly

distributed _on the

polymers

and not

attracting

two chains.

Following

fine 3, we cross a first-order fine.

Crossing

this

fine,

the concentration _is discontinuous. This _means that _a

phase

separation occurs there between _a dilute

phase (up)

and _a dense

phase (down).

5.2. FOURFOLD CROSSLINK. We _saw that the addition of the _~14 term _is

equivalent

to a

change

in trie solvent

quality.

When trie attraction due to ~4 compensates trie exduded volume eoEect _~o, _a

phase separation

will

happen

and that is what _we _can check

by plotting

trie

phase

diagram (Fig. 3).

The concentration could be varied

by changing

the chemical

potential

_as in the _case of the threefold crosslink and _we also _see that _we bave three

typical regions.

Since, in _our

physical

system, ends _are

always

present, we must bave h _> o. We _saw that when h is small

enough,

we still feel trie tricritical point if _~4

" ~o and _~2

" o and trie critical

fine _{for ~4} _< _~o.

Trie critical fine will generate

polymerization

when _~4 is weak. A first-order fine _starts from trie tricritical

point

and represents trie

demixing.

This

demixing

is in fact similar to that due to the

change

of solvent

quality

in classical

polymers.

With _a variation of trie crosshnk _energy, _we have _a _continuous _passage from linear

polymers

to a compact reticulated system ûntil ^trie êxcluded ^volume êoEect îs

compensated by

trie attraction due to trie crosslinks. Then _we get a

demixing

between two

phases,

_one compact ^and one dilute.

Around this tricritical point,

X4

~ #~ ^which means that trie

density

of

entanglements

is

proportional

to the

density

of trie contact points ^and ^that ^trie

phase separation

must be

quite

dioEerent from the threefold crosslink _case.

We notice that

apparently

trie three- and fourfold crosslinks bave _a similar

phase diagram,

(11)

with _a critical point ^followed

by

_a first-order

demixing

fine. However, trie nature of trie critical

point

and that of trie transition _are _very dioEerent.

6. Conclusion

In this _paper, _we have studied the effects of crosslinks _on

equilibrium polymers.

We bave used field

theory techniques,

which _are well

adapted

to treat fluctuations. We observed that fluctuations

play

_a

predominant

rote for this kind of system

(mostly

for threefold

crosslinks).

The

physical

^results are the

following:

Both three- and fourfold crosslinks induce

phase separations

and these

phase

separations _are of _a dioEerent _nature. The fourfold crosslink is

equivalent

to a solvent

change

and thus leads _to

a tricritical behavior, wuile tue threefold crosslink leads _to _a critical beuavior.

Surprisingly,

for the threefold crosslink _case,

X3

~

#

_near the

demixing

critical

point

_so that there _is

apparently

_no

stretching

^of the

polymer

iietwork close to trie

phase separation.

Trie drawback of _our

simple

scalar model is that _we _are trot able to characterize trie

topologies

of _our dioEerent

phases.

It is

certainly interesting

to better understand the

phase

separation and the structure of such

phases by performing

_more experiments and also numerical simulations.

Acknowledgments

We _are

grateful

to L. Scuàfer for

extremely stimulating

discussions and useful remarks.

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