Field theoretic approach to the dissolution of polyelectrolyte complexes

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Field theoretic approach to the dissolution of polyelectrolyte complexes

P. Haronska, Ch. Seidel

To cite this version:

P. Haronska, Ch. Seidel. Field theoretic approach to the dissolution of polyelectrolyte complexes.

Journal de Physique I, EDP Sciences, 1992, 2 (8), pp.1645-1655. �10.1051/jp1:1992232�. �jpa-00246645�

J. Phys. I France 2 (1992) 1645-1655 _AUGUST 1992, PAGE 1645

Classification Physics Abstracts

36.20 64.75

Field theoretic approach to the dissolution of polyelectrolyte complexes

P. Haronska and Ch. Seidel

Max-Planck-Institut fur Kolloid- und

Grenzflichenforschung,

Kantstrafie 55, D/O-1530 Teltow- Seehof,

Gernlany

(Received 19

July

199I, revised 17 March 1992, accepted 6 April 1992)

Abstract. We

study

the dissolution of

fully symmetric polyelectrolyte complexes

that _occurs when the

polymer

concentration reaches a critical value. Edwards' approach is used to derive the

monomer correlation function. Near the critical concentration the correlation function shows _an

oscillatory

behaviour which is _a

typical

feature of _a

mesophase

fornlation.

1. Introduction.

In a

preceding

article

[I]

we discussed the dissolution of

polyelectrolyte complexes

for _a very dilute

polymer

concentration. The dissolution itself iS caused

by

addition of low molecular Salt. In this paper ^we

investigate

_an

abrupt

dissolution of

polyelectrolyte complexes

above _a

critical

polymer

concentration

larger

than the

overlap

concentration

[2, 3].

This kind of dissolution has _some features of _a

phase

transition.

We restrict ourselves _to

fully symmetric polyelectrolyte complexes

and discuss the

transition

by studying

the

stability

^of the

homogeneous phase,

^{I.e. above} the critical

polymer

concentration. Then surface effects _can be

neglected [4].

In _some

respect

^the system ^has similarities with _a

binary polymer

blend.

However,

_a characteristic feature of

polyelectrolytes

is the dominance of Coulombic interaction _over the excluded volume effect. Such _a circumstance

requires

other theoretical

approaches

^than those used _to

study

usual

polymer

blends. Our treatment is different _to that of Brereton _et al.

[5]

who considered the

phase

behaviour in

charged polymer

systems

by using

_a

generalized

copolymer problem.

The statistical mechanics of flexible

polymers

can be used if the

Debye screening length

is much smaller than the radius of

gyration.

Then _a suitable

starting point

for _a theoretical

treatment is

given by

the Edwards Hamiltonian

[6]. By employing

_a combination of

path

integral

and field theoretic

techniques

an

instability

_{at a} finite _{wave vector} q~ can be

predicted

which

implies

that the system

undergoes

a

mesophase

transition

[7].

This article is

organized

_as follows. In section 2 _we recall the

expressions

^of

polymer physical quantities

of interest in the framework of the Edwards Hamiltonian. In section 3 the

JOURNAL DE PHYSIQUEi -T 2, N'S, AUGUST 1992 59

Gaussian model is

introduced,

and its

generalization using

_a Hartree

approximation

is

presented.

In section 4 the concept ^{of the effective} interaction is

introduced,

and its connection _to the

phase

transition is shown. In the final section

concluding

remarks _are

given.

2. Hamiltonian.

Due to the

complexity

of

polyelectrolytes simplifications

_{are necessary} to

develop

a

theoretical

approach.

We shall _use the model of

uniformly charged polyelectrolyte

chains with

charges

smeared

along

the chains. Furthermore _we conf~ne ourselves to effective

two-body

interactions between _monomers. In the

following

units will be used such that

x(s)

=

/r(s)/f _(2.i)

where

f

_{is the Kuhn}

length.

For _a

fully symmetric polyelectrolyte system consisting

of the two

components

^I

(poly-anion)

and 2

(poly-cation)

the interaction

potential pV~~

is _a

superposition

of excluded-volume effect and screened Coulombic interaction. We have

Pv~~(x)

_{= u}

(x)

+

(-

i_)@ ⁺ v

A~ f2 ^~~~~ )'~~~~

(2.2)

where

A~

is the

Bjerrum length

A~

=

~~~ (2.3)

4

"80D

and

A~

denotes the

Debye screening length

A

g

² ₌ ^A_~ cj

(2.4)

Here ci is the concentration of the counter-ions and the added low molecular salt, e is the electrical

charge,

_eo is the

permittivity

of _vacuum, D is the dielectric constant of the

solvent,

and

f

denotes the fraction of

charged

_monomers in _a

polyion.

In the paper we shall confine ourselves _to

length

scales

larger

than the radius of

gyration.

As _{soon as}

A~

«

R~ ^(2.5)

where

R~

^{is the radius of}

gyration given by

R(

=

N

(2.6)

and N denotes the

degree

^of

polymerization,

the

potential (2.2)

_can be

approximated by

a 3- function

pv~v(x)

₌ u

(x)

₊

(- i)»

⁺ " v

(x). (2.7)

A

comparison

with

equation (2.2) gives

vm

f2A(. _(2.8)

The limit

(2.5)

allows _us to

apply

^the statistical mechanics of flexible

polymers.

Then _a suitable

starting point

for _our further considerations is the Edwards model

[6].

The

conformation of any chain is

given by

_a vector

position

function

r(s) (0

_{« s w} N

).

The free

N° 8 HELD THEORETIC APPROACH 1647

energy is defined _{as a}

path integral

exp(- pF )

₌

In fl Dxi

;

Dx~

;

exp[- PHI (2.9)

, =1

where

lDx ^represents

^{the summation} over all

possible configuration

of the chains

[8].

Here

pH

is the Edwards-Hamiltonian

given by

pH

=

( ( _l~ ^{s) ~~~' ~}

+

( f

j~

^ds

j~

^ds'

pV~~ [x~,(s) x~~(s')]

+

2~~~i=1

₀ ^~~

~~=li,j=1

0 0

n N N

+

£

dS

ds'~Vj2[Xj;(S)-X2j(S')1 (2.10)

j,_{j =1}

0 0

where _n denotes the number of

polymers,

and

p

is

(k~ T)~

Due _to the

coupling

of chains in Hamiltonian

(2.10)

_a calculation of the

path integral

in

equation (2.9)

is very

complicated. Decoupling

can be

accomplished using

normal random fields

[9].

The

corresponding expression

reads

[10]

exp(- pF)

=

D~ii, D~i~

_exp

i- pJci~ii, ~i~ii (2. ii)

where

~3~[§~1' §i21"

~ ~~

~~~,~

^~

~)~ ^~~~~~~~~

U~ ~~~~~~~~ ~~~~~

~~ ~~~

~=I _~ =l

and

G~

₌

ld~x(0) j _d~x(N

) ~~~~

Dx exp

(- j~

^{ds ~~ ~ i}

j~ ^{dr~Y~ [x(s)] (2.13)}

' x(0) 2

0 a~

0

In the

following capital Y'~

denotes Fourier transformed

quantities

which are convenient to use for

treating

the

problem.

We have

~ ~ ²

pxj§'i, §'~]

= n

z

In

[G~,]

+

j)~ _{Z ~'»~~~ ~'~ ~~ ~~}

,~~ ~=i q

u~

~ ~'~~~~ ~'~~ ~~

~~ ~~~

The short-hand notation stands for

1= _ld~q/(2 ^7r)~. (2.15)

q

3. Gaussian model.

In _some respect our system ^has similarities with _a

binary polymer

blend. The characteristic feature of

polyelectrolytes

is the dominance of Coulombic interaction _over the excluded

volume effect _so that the

sign

of

pV~~

^becomes

important.

Then it

is,

for

instance,

not

possible

to use an

incompressibility

constraint

[11].

Such _a circumstance

requires

other theoretical

approaches

than those used _to

study

usual

polymer

blends.

An exact calculation of the free energy

(2. II)

is

beyond

_our abilities. Hence it is necessary to carry ^out calculations within _a

perturbational

treatment. The Gaussian model is used _{as a}

starting point

for

constructing

_a

perturbative expansion. Neglecting higher

order fluctuations of the random fields In

[G]

reads

[10]

n in

jG~,i

₌

j

p

j ro(q) Y~~(q) Y~~(- q)

₊

o(v~() _(3.i)

q

where _p is the concentration of the

polymers given by

p =

2 n/V

(3.2)

V is the volume and

ro

is the

unperturbed scattering

function defined _as

[10]

r~(q)

₌ N

2(2/q2

N

(2/q2

N )2

ii exp(- q2

N/2

)1) (3.3) Using equation (3.I)

_one

gets

a Hamiltonian which is

quadratic

in

Y'~.

^{In this} case the

correlation function

(Y'~(q) Y'~(- q))

=

exp(pF) j

D~Yi

D~Y~

Y'~(q) Y'~(- q) exp[- p3C

[~Yi, ~Y~]]

(3.4)

can be calculated

rigorously.

We have

~~"~~~ ~'~~~ ~~~°

~

ii

+ 2

pure(q)ji

+ 2

pure(q)1 ^~~'~~

and

(Y'j(q) Y'j(- q))~

₌

⁽ Y'i (q) Y'~(- q))~

+

~ _~

(3.6)

+ 2

pure(q)

The

unperturbed scattering

function

ro

is well

approximated by [9]

ro(q)= p~2

~ ~.

(3.7)

2 ₊

(q( _Rg

To get ^{the correction} to the Gaussian model _we shall _use here _an

approach

which is similar to the Hartree

approximation [12].

Then _a modified Gaussian model is obtained. We shall _see that such _an

approximation

holds

only

when VW _u.

The

expansion

of

equation (3. I)

in terms of random fields suggests ^the

following expression

for the correlation function

(see

Ref.

[10])

(9'~(q) 9'v(- q))

=

( 9'~(q) 9'v(- q))o

₊

+

z 1Y/~(q) ~l~v,(- q))ozv,~<(q)i~l~~,(q)~l~v(-q)). (3.8)

From

equations (3.5)

and

(3.6)

it follows

immediately

that in the limit _{VW u} the

identity l~l~i(q) ~l~i(- q))o

=

l9'i(q) _~l~~(- q))~ (3.9)

N° 8 HELD THEORETIC APPROACH 1649

holds. Then

equation (3.8)

is reduced to

(9'~(q) 9'v(- q))

=

( 9'~(q) 9'v(- q))o

+

+

(- 1)»

^{+ v}

(v~~(q) v~~(- q))~ z(q)(v~~(q) v~~(- q)) (3.io)

Dimensional arguments suggest ^that a

perturbative

treatment of

equation (3.10)

is _an

expansion

in terms of UN ^~'~ when

vpN

^~ is fixed

[9, 10].

In other

words,

the effective

coupling strength

is very

large

_so that _a conventional

perturbation theory

breaks down

[9].

To

overcome this

difficulty

let _us first examine the

physical meaning

of

vpN

~.

Starting

from the

definition

(2.6)

of the radius of

gyration R~

we find

vpN~mR)f~ (3.ll)

where

f

is the Edwards

screening length [5] given by f~

=

(3.12)

4

vpN

Then

equation (3.9)

takes the form

lj~ ^)~2

^-1

l'l'i(q) 9'i(- q))o

=

l'l'i(q) 'l'2(- q))o

= V +

~ ~

(3.13)

2 ₊

(q( _Rg

If _{we assume} that

R/f

^{is variable and the condition}

R/f

» I is fulfilled _a

perturbative

expansion

in terms of UN

~i(R/f)

is obtained. This is due to the fact that _a concentrated solution _can be treated

by

_a

simple

_mean field

theory [6].

First-order

perturbation theory yields (see

Ref.

[10])

3(q)

= P

In (q, P)l'l'i _(P) 'l'i(- P))o (3.14)

P

where

D

(q, P)

=

i~(q, _P) ro(q ) ro(P ) (3.15)

and

4(q, _P)

=

8(P, 0, q)

+

8~P,

_{P + q,}

q)

+

8(P,

_{P + q,}

P)

+

&(q,

_P + q,

q)

₊

+

8(q,

_{P + q,}

P)

₊

8(q, 0, p). (3.16)

The function

&(q,

_p,

k)

is

given by [10]

N

rj r2 r~

&(q,

_p,

k)

=

dri dr~ dr~ dr~

_x

o

o o o

x

expi- ^{~ (ri r~) ~ (r~ r~)} i~ ^(T~ 4~i

~3.1?~

Note that

3(q) corresponds formally

to the field-theoretic _mass operator

[12].

Thus

equation (3.10)

_may be considered _as

Dyson equation

and the

following expression

is obtained

l'l~p(q) ll'~(- q))

=

(- 1)~

^{+ "}

'/~ ₊ 2

Pro(q) 3(q) ^(3.18)

For _our further

investigations

an

analytic expression

of

3(q)

is needed.

Unfortunately equation (3.14)

cannot be calculated

analytically.

For

(q

- 0 _an exact

expression

^is derived in reference

[10].

The result reads

3(q)

=

2N~) (q(~R( (3.19)

where the dimensionless

quantity

_a is

given by

a=

~

S.

(3.20)

37r

pN

Note that up ^to an irrelevant numeric

prefactor

the

relationship

a w UN

~/~/(R/f) (3.21)

holds. In the limit _q

- co we obtain after _a tedious

algebra

the

following

closed

expression

for

3(q)

z(q)

= 2

pN

² _a

(

_q ²

Rj)- _(3.22)

To get a formula which is valid both for q - 0 and q - co we use here the

following

Pad6

approximation

1(q)

=

2

pN2 "(q(2R2

~ _~ ~ ^~

~~~ _(3.23)

This

yields

Pro(q) li(q)

=

~~

~ ~

PN

^{~ " ~}

(~~

~

(3.24)

2 ₊

q(~

Rg

⁶ +

)q~ g)

4. Phase transition and effective

potential.

The interaction

potential

between _monomers is modified

by

the presence of other chains

[13].

To obtain the effective

potential

between _monomers of the

component

^I we have to

integrate

out the system ²

[1Ii.

Then the free energy reads

exp(- pF)

=

D§ki expj- p3Ciii§kijj (4.1)

where

p3Cii

[~Yi] ^{becomes in the limit} v » _u

pJciii§Yii

=

§Yi(q)[pro(q) ^z(q)

⁺

) _{§Yi(- q). (4.2)}

The effective

potential Wii(q)

is

given by [13]

Wii(q)

=

l~i(q) ~i(- q))

=

~~~

(4.3)

Pro(q) j 3(q)

+

1/(2 v)

N° 8 FIELD THEORETIC APPROACH 1651

From

symmetry

^{it follows}

immediately w(q)

=

wit(q)

₌

w~~(q) (4.4)

The

generalization

to

oppositely charged

_monomers is

straightforward

and

yields W12(q)

= W

(q) (4.5)

Equation (4.2)

_may be considered _as the

quadratic

term of _a Landau

expansion.

As _{soon as}

Pro(q) j _li(q)

+

~

= 0

(4.6)

the

system

^{reaches its}

stability boundary

and the fluctuations of the order

parameter

~Yi become

anomalously large.

Therefore

equation (4.6)

determines the condition for which the transition takes

place.

From

equation (3.24)

it follows that the

instability

_occurs at a finite

wave vector

q~R~

^{and at} a critical

coupling strength

_a~. Furthermore both

quantities

_are

functions of the dimensionless parameter

ilR~.

Figure

I shows the critical

coupling strength

_{a~ as a} function of

f/R~.

^The

quantity

a~ reaches its lowest value for

f/R~

_-

0,

_so that the

theory

has its best

validity

in this limit for which

approximately

a~(f/R~

-

0)

= 0.85

(4.7)

holds.

cx~

3

2

O-O

$/R~

Fig,

a~

_mz

From

(3.20)

a

critical

concentration p~. Up

to

an

irrelevant

numeric prefactor

_{we have} in the

f/R~

- 0

p~

_m

UN

Relationship (4.8) predicts

that the critical concentration of _monomers, I.e.

p~N,

is _a

function of _v

only

^and

fully independent

of the

degree

of

polymerization.

This is _a

characteristic feature of _a concentrated solution

[9, 14].

Figure

2

presents

^{the critical} wave vector q~.

Again q~R~

^is

^nearly

constant when

f/R~

^becomes zero. One finds

q~

R~

= 2.6

(4.9)

Combining figure

I with

figure

2 the

dependence

of the critical _wave vector q~ ^{to the critical}

coupling strength

_a~ is obtained

(see Fig. 3). Using equations (2.8)

and

(3.20)

_we find that q~

R~

^is a function of both the

charge

_per unit

length

_on the chain and the salt concentration.

q~Rg

3

2

1

O

O.O O,I O.2 O.3 O.4 O.5

(/R~

Fig.

2. Critical _wave vector q~ as a function of

$/R~.

^{The value for which} q~ has its maximum is q~ =

2,62/R~.

q~R~

3.O

2.5

2.O

.5

1.O

O-B O.9 1.O I.I 1.2 1.3

CKc

Fig.

3. Critical _wave vector q~ as a function of the effective

coupling strength

_a~. The _curve ends at a~ = 0.85.

N° 8 HELD THEORETIC APPROACH 1653

Unfortunately

the

validity

of the

theory

is restricted to such values of a~ for which condition

(4.7)

holds. Therefore _our results _can

only

be discussed

qualitatively

and _a

comparison

with the results of reference

[5]

will not be made here.

If the concentration _p tends toward its critical value the effective interaction

diverges

at

(q

= q~

(see Fig. 4).

At

higher

_p the

peak disappears.

The

corresponding

effective

potential

in r-space ^is obtained

by

the Fourier transform

W(r)

=

W(q)e~~~ (4.10)

The

graphical representation

is

given

in

figure

5. When the

polymer

concentration reaches its

~

W(q)/v

7

6

5i

4

3

2

,_---

O

O 1 2 3 4 5 6

qR~

Fig.

4.- Plot of

W(q)

for two values of the dimensional parameter Y =

(P

-Pc)/p~:

(-)

y = 0.02, (--,--) _y

= 0.2, where

$/R~

₌ 0.I has been used.

W(r)/V

O.5 O.4

i

O.3 O.2

,

O.I

,

O.O ',

__-

-o, i -O.2

-o.3 -O.4

-O.5

O 2 3 4 5 6 7 8 9 O

r/R~

Fig.

5. Effective potential _near the critical

point.

The

(-)

_curve

corresponds

to (p

p~)/p~

=

0.02 and the

(-=--I)

_{curve belongs to} (p

p~)/p~

₌ 0.2, where

$/R~

=

0.I has been used.

critical value p~ the effective

potential

oscillates with the

period

2

7r/q~

and becomes

long- ranged.

The

density

correlation between

oppositely charged

_monomers results from intermolecular interactions. This enables _one to introduce _a radial distribution function

gi~(r)

like that of the usual

liquid theory [15]. Using equation (4.5)

_we find

gi~([r~-r~[) =exp[+w([r~-r~[)]. (4.ll)

The radial distribution function

(4.

II

)

determines the

density

correlation between any

pair

of

monomers a and b in

dependence

^of their relative distance

(r~ r~[.

Up

to now we have considered dimensionless

quantities.

^In order to obtain _a

graphical representation

from

equation (4.

II

)

the

knowledge

of the

coupling parameter

v is needed. It

can be _seen from

equation (4.

II that the behaviour of

gi~(r )

is

already

well understood when the effective interaction is known. This

implies

that the correlation function shows _an

oscillatory

behaviour like that of the effective

interaction,

which is _a

typical

feature of _a

mesophase

formation.

5.

Concluding

remarks.

We have

presented

_a theoretical treatment of the dissolution of

fully symmetric polyelectro- lyte complexes.

Due _to the dominance of the Coulombic interaction _over the excluded

volume effect _an

incompressibility

constraint cannot be used _{as a}

starting point

of theoretical

investigations.

Therefore

I

modified Gaussian

model

_{has been used}

wliich _predicts au instability

at a finite _wave vector q~. This

implies

that the system

undergoes

a

mesophase separation

transition.

In the paper we have considered

charged polymer systems

^{in the} limit of

strong

Edwards

screening.

This is because _our

theory

breaks down for

f

=

R~. Density

fluctuations which _are much

larger

than that of

f

should be therefore _not affected

by

the interaction

strength

v.

Using equations (2.8)

and

(4.8)

the

theory

derived here is

only

valid when

f~N

_WI.

_(5.1)

Then the

period

of the

mesophase q~R~

^is

independent

of the

charge

_per unit

length

_on the chain. For very small

f

the critical

wavelength

_{q~ is} influenced

only by

structure when the chains _are very

long.

It _must be

pointed

out that the results obtained here _are

only

correct in _a

qualitative

_sense.

So the effective

coupling

_{parameter a} is

approximately

I when the system ^{reaches its}

stability boundary.

It will be the

topic

of _a further _paper to include the

higher

order _terms.

Acknowledgment.

The authors thank Dr. T. A.

Vilgis

Max-Planck-Institut fur

Polymerforschung

Mainz for

helpful

discussions.

N° 8 HELD THEORETIC APPROACH 1655

References

[I] HARONSKA P., J.

Phys.

France 50

(1989)

1827-1841.

[2] BUNGENBBRG DE JONG H. G., DEKKER W. A. L.,

Kolloidhefte

43 (1936) 143-212.

[3] KOTz J.,

private

communication.

[4]

BORUE V. Yu., ERUKHIMOVICH I. Ya., Macromolecules 23

(1990)

3625-3632.

[5]

BRERETON M. G., VILGIS T., Macromolecules 23

(1990)

^2044-2049.

[6]

EDWARDS S. F., Proc.

Phys.

Soc. London 88

(1966)

265.

[7] LEIBLER L., Macromolecules 13

(1980)

1602-1617.

[8] WIEGEL F. W., Introduction to

Path-Integral

Methods in

Physics

and

Polymer

Science (World Scientific

Publishing

Co Pte Ltd.,

Singapore,

1986).

[9] OoNo Y., Adv. Chem. Phys.

61(1985)

353.

[10]

OHTA T., NAKANISHI A., J.

Phys.

A _: Math. Gen. 16

(1983)

4155-4170.

[ll] BRERETON M. G., VILGIS T., J.

Phys.

France 50 (1989) 245-253.

[12] RAMOND P., Field

Theory

_: A Modem Primer (MA _:

Benjamin, Reading,

1981).

[13] MUTHUKUMAR M., EDWARDS S. F., J. Chem.

Phys.

75 (1982) 2720-2730.

[14] DES CLOIzEAUX J. and JANNINK G., Los

Polymbres

_en Solution (les

llditions

_de

physique,

Paris, 1987).

[15]

FRIEDMAN L., A Course in Statistical Mechanics

(Prentice-Hall,

Inc.,

Englewood

Cliffs, New

Jersey,

1985).