Collapse of grafted polyelectrolyte layer

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Collapse of grafted polyelectrolyte layer

O. Borisov, T. Birshtein, E. Zhulina

To cite this version:

O. Borisov, T. Birshtein, E. Zhulina. Collapse of grafted polyelectrolyte layer. Journal de Physique

II, EDP Sciences, 1991, 1 (5), pp.521-526. �10.1051/jp2:1991186�. �jpa-00247536�

Classification

Physics

Abstracts

6140K 64.75 68 45 82 45

Coflapse of grafted polyelectrolyte layer

O. V.

Bonsov,

T. M.

Birshtein,

E B. Zhulina

Institute of Macromolecular

Compounds

of the

Academy

of Sciences of the USSR, 199004,

Len~ngrad,

USSR

(Received 5 November 1990,

accepted

31 January

1991)

Absbact. The

theory descnbing

chain conformation _{in a}

planar

layer ^of

grafted polyelectrolyte

~polyampholyte) molecules and the conformational transition related to the collapse of this layer caused

by

the decrease in solvent

strength

_is

developed Depending

on the values of the layer parameters

(grafting

and charge densities) this transition may occur as a continuous (cooperative)

or as a first order

phase

transition

The

layers

of

uncharged polymer

chains

grafted

at one end _onto

impermeable

surface and immersed

(or not)

_{in a} solvent _were the

objects

of intensive theoretical

investigations dunng

the last decade

[1-12]

The interest to these systems was initiated

pnmarily by

_many

practical applications

the

layers

of

grafted polymer

chains used to be the model of superstructures

formed

by block-copolymers

in the strong segregation ^limit

,

the

grafting (or

the

adsorption

at the anchor

end)

of the

polymer

molecules _can be used for the stencal stabilization and

flocculation of colloidal

dispersions,

etc.

At the _{same time} the

grafted polymer layers appeared

to be _very

interesting

in the theoretical aspect. ^It was shown _{in a} number of _papers

[1-6]

that under the conditions of strong

overlapping

the chains _m the

layers

_are

always

stretched in the direction normal _to the

grafting surface,

_so that their dimension in this direction scales with N _as the dimensions of the free chain in the d-dimensional _space, where d

=

3

d~

and

d~

is the effective dimension of the matnx

(d~

= 2

,

0 and d

= 2 3 for the chains

grafted

onto the

planar, cylindrical

and

sphencal surface, respectively)

This effective reduction of the

grafted

chain dlrnensiona-

lity

_is manifested _in the character of the

collapse

of the

layer

caused

by

the decrease of the solvent

strength.

As _is known

[13-14]

for _an individual

chain,

tins

collapse (coil-globe transition)

_{occurs as}

the second-order

phase

transition

(the

first-order

phase

transition for the stiff

chains).

In the

layers

of the chains

grafted

_onto the

spherical

or

cylindncal

matnces

(an

effective

dimensionahty

d

= 3 and 2

respectively)

the transition becomes

broader

_than

in an isolated

macromolecule,

but retains its

phase

character In contrast to

this,

the

collapse

of the

layer

of the chains

grafted

onto a

planar

surface

(effectively

one-dimensional

chains)

_{occurs as a non-}

phase

conformational transition

accompanied by

_a smooth decrease of the

layer height

and _an

exclusion of the solvent from the

layer [6, 12].

To prevent any

misunderstanding,

_we should note that the alternative result obtained

by Halpenn [15] (the possibility

of the first-order

522 JOURNAL DE

PHYSIQUE

II M 5

phase

transition

accompanied by

the jumpwise

change

of the dimensions _in the

planar layer) appeared

to be _an artifact and _was caused

by

the _use of the incorrect

expression

of the elastic free _energy of the

compressed

chains

(See [12]

for the

discussion).

All the results obtained

previously

referred to the

layers

of

uncharged grafted polymer

chains. We will start the theoretical

analysis

of the

grafted polyelectrolyte layers by

the present paper

(some

results

concerning

the structure and interaction of these

layers

_were

recently

obtained

by

Pincus

[16],

the

corresponding

numencal calculations _were

performed

in

[17, 18]).

Up

to the present time _a sufficient progress was achieved _in the theoretical

investigation

of such condensed

polymer

_{systems as}

polyelectrolyte

networks. In _{a senes} of _papers

by

lchokhlov _et al.

[19, 20]

the chain conformations and the conformation transitions in

polyelectrolyte

networks _were

investigated.

As _was

shown,

the

collapse

of _« dilute _» networks caused

by

the decrease in the solvent

strength

_{occurs as} the first order

phase

transition.

The purpose of the present _paper is the

investigation

of the

collapse

of the

layer

of

polyelectrolyte

chains

grafted

onto a

planar

surface caused

by

the decrease _in the solvent

strength.

The

dependence

of the character of this conformational transition _on the main parameters ^{of the}

system (grafting

and

charge densities)

and the

scaling

relations for

layer

dimensions _in different _regimes will be

analyzed.

Model.

Let _us consider _a

layer

formed

by long (N

»

I) polyelectrolyte ~polyampholyte)

chains

grafted

at _one end onto an

impermeable planar

surface with _a

density I/«

_ensunng the

overlapping

of

polymer

coils and inunersed _{m a} dielectnc solvent. A chain part of

length equal

to chain thickness _{a is} chosen _{as a} unit. For

simplicity

let the backbone of the

polyelectrolyte

chain be

flexible,

_i-e its Khun segment

length

_is A _ma The nonelectrostatic volume interactions between the _monomer units will be described

by

the

second,

_v, and the

third,

_w, vinal coefficients. The former

depends

on

temperature (solvent strength)

_near the b-

point as vmvor, ^where r =

(T-0)/T,

and _we will consider the

changes

_in

layer

charactenstics accompanying the variations of _r, whereas the latter _is

approximately independent

of temperature

(solvent strength),

_{w m} const

(r).

Let _every m-th _unit of each chain carries

positively

_or

negatively charging

_{group, so} that the total number of

charging

groups in the chain _is

N/m.

Let m'mm be the _mean distance

along

the chain between

uncompensated charges

of the _same

sign,

_so that the total

charge

of the macromolecule is

Q

=

Ne/m'.

We have

Q

=

0,

m'mm _in the isoelectnc _case and m'mm for _«pure»

polyelectrolytes,

when the chains carry the groups of _one sign. We will restnct our

consideration

by weakly charged polyelectrolytes,

for which

m ~'~

» u ₌

e~/(eaTl (1)

where _{e is} the dielectric constant of the solvent. In this _case the intrachain Coulomb interactions do not lead to the

changes

of chain stiffness

[21].

We will consider the situation when _no salt _is added into the solution The condition of the total

electroneutrahty

of the system ^results in the presence of S

I _N/m'

_mobile counterions

~per chain)

_in the solution. These counterions _are retained

by

infinite

planar layer

of

grafted polyions

_near it

just

as near an infinite

charged

flat surface

[22]

^{Under the condition}

KH »

I,

where

I/K

is the

Debye

_screening

length

_{in a}

layer

and His the

layer thickness,

all countenons are located in _a

layer.

In _our consideration we will _assume this condition to be fulfilled.

Free energy.

For the

analysis

of the conformations of

grafted polyelectrolyte

chains _{m a}

layer

we will _use the standard

approach

based _on the minimization of free energy which _is

presented

_{as a surn}

of _terms of different

physical origin.

AF

=

Afdci

₊

Afconc

₊

AFm

+

Afco« (2)

We will restrict ourselves

by

the

simplest approximation

_in which these terms _are

expressed

as functions of the total

layer thickness, H, only. (A

_precise

analysis

of _a

planar layer

of

uncharged grafted

chains shows

[11, 12]

that this approximation allow to obtain all

scaling dependences

of

layer

charactenstics _on its parameters ^and to

analyze

the character of the transition related to the

layer collapse).

The

dependences

of the tennis in

equation (2)

_on H

(through

the

swelling

coefficient with respect to the chain Gaussian dimensions _a

=

H/N~'~

_{a or}

through

the _{mean unit}

density

in _a

layer

_p

=

Na~/( «Hi)

have the

following

forms

~per chain)

_:

elastic

(conformational)

term

~~~

~a~~+

_In

_a~,

a ml ~~~

free _energy of volume interactions

AF~~~~ m NV _{p +}

Nwp

^~

(4)

translational

entropy

^{of mobile}

$ountenons

AFm

_m s in

w/m') (5)

electrostatic

(correlation)

_term, related to the correlations of local

charge density

AF~~~

_m ^N

(uim

j~'~ p ~'~

(6j

Here and below all numencal coefficients _are omitted and all

energetic

values _are

expressed

in kT _units.

Note that the correlation

(electrostatic)

contnbution

AF~~~, equation (6),

_is written _in the

Debye-Huckel approxbnation,

^I-e as for

weakly

unideal

plasma [23]

In the most

typical

_cases

uml and this

approximation

is valid for the solutions of

weakly charged (m

ml

polyelectrolytes

_{m a} wide range of concentrations p «1.

Equafion

for

swelling

coefficient.

The minimization of the free energy, equations

(2)-(6),

with respect to _a

(or pi

leads to the equation for the

equihbnum layer

tluckness Tlus equation may be

presented

_in the

following

form _:

a

(S

₊ I _{a +} Ga ^~'~

= Br ₊

Cla

, a m1

-a~~-,(S-I)a~Ga~'~=Br+Cla,

awl ~~~

where

B m vo

N~'~a~/« _(8.I)

C m wN

~(a~/«)~ (8.2)

524 JOURNAL DE

PHYSIQUE

II M 5

are the renormahzed

parameters

^of pair and

temary

nonelectrostatic interactions _{in a}

layer, respectively,

and

G m N

~'~(u/m

_)~'~

(a~/«)~'~ (9)

is the renormahzed paranJeter ^of electrostatic interactions.

CoUapse

of the

layer.

At

S,

G

=

0 equations

(7)

describe the

dependence

of the thickness of

uncharged grafted layer_on

_r and _give the

following scaling

asymptotes for it.

Na(vo ^ra~/«)~'~,

^{r » r *}

^(10.1)

~"

Na(w~'~a~/«)~'~, _jr

_«

_{(r* (10.2)}

Na(wa~/(vo jr

_«

)),

_r « r ^*

(10.3)

in the regions of

strong,

b- and poor

solvent, respectively (comp [5, 6]).

The temperatures

± r *

m ± C

~'~/B

₌ ±

(w~'~/vo)(a~/«)~'~ (l I)

correspond

to the _crossover between different _regions The decrease _{in r} leads to the continuous

(without

_a

jump)

decrease _in the

layer thickness,

the transition of the

layer

into _a

collapsed (globular)

state _is not of the

phase

character

(see [6, 12]

^for

details).

In the

general

_case of _a

grafted polyelectrolyte layer,

the

analysis

of

equations (7)

shows that the character of the

dependence

of the

equihbnum layer

thickness H

(swelling

coefficient

a)

_on the solvent

strength,

_r, is determined

by

the relation between the parameters

C,

S and G and three different _regimes of the

layer

behavior _can be

distinguished.

I)

The regime

of

_«

uncharged» layer

Cm

S~, G~'~

_{In this case the}

polyelectrolyte

effects _{in a}

layer

_are weak companng with

strong

volume interactions. The thickness of the

layer

_{is given}

by

_equations

(10), (11)

The

collapse

^of the

layer

with decrease _{in r occurs}

smoothly _just

_as the

collapse

of the

layer

of

grafted uncharged

chains

(Fig. I,

_curve

I).

It's easy ^to see that the condition C »

S~

_means that under the conditions of strong and o-solvent there _is less than _one mobile _ion

(or

_one

uncompensated charge)

_{per one} blob

[1, 2, 5]

^{of the}

stretched chain _{in a}

layer

2)

The

«polyelectrolyte»

_regime Cm

S~, G«S~'~C~'~(m'mm

_~) In this _case the

equllibnum layer

dimensions under the conditions of strong ^{and b-solvent} are determined

by

the

competition

between the translational

entropy

^{of mobile countenons}

(Eq (5))

and the

c£

2

5

qf

Fig.

I Different types of

dependence

of the swelling coefficient, _a, of the

grafted polyelectrolyte

~polyampholyte) layer

on the solvent

strength,

r

(all

comments to the _{curves are in} the

text)

elastic free _energy of chain

stretching (Eq. (3))

and

slightly

decrease with the

decreasing

in the solvent

strength

HmNam'~~'~ ₍₁₂₎

1-e the

polyelectrolyte

^effects

predominate

_over the volume interactions. This situation retains in the range of r~~ _w

r « 0 where

r~~m-S~'~C~'~/Bm-m'~~'~w~'~. ₍₁₃₎

Note that the condition C _«

S~ provides r~'

_« r*. At the

point

_r

= r~~, the attractive

pair

interactions become strong

enough

to compete ^with countenon entropy ^and the

layer

thickness suffers _{a jumpwlse} decrease _up to the value of the thickness of

uncharged layer

at the _{same r}

(Fig. I,

_curve

2)

3)

Isoelectric

regime.

Cm

G~'(

_{G »}

S~'~C~'~(m~ mm').

In this _case the number of

oppositely charged

_groups in the chains _is

large

and their attraction

(correlation

_term,

Eq. (6))

_{gives a} sufficient contribution _m the total free energy of the

layer.

If m' _{mm ~,} this

electrostatic attraction results _m the

collapsed

state of the

layer

_even under the conditions of

strong ^solvent

(Fig.1, curve3),

where the

layer

dimensions _are determined

by

the

competition between electrostatic attraction and pair

repulsive

volume interactions :

H _m

Na(vo r)~ (miu)~ (a~ia-) (14)

In the intermediate _case,

m~

_« m' _{« m}

~, under the conditions of _a strong ^{solvent the}

layer

is m a swollen state,

equation (10.I),

but the decrease _in

r leads to the

collapse

of the

layer

into the state described above

(Eq (14))

Thls transition _occurs

smoothly

at _rm

m~ ~'~(

«la

_{~)~'~ if G} »

S~'~

_or with _a jump in

layer

thickness _{at r}

m

m'/m~

if G «

S~'~ (Fig. I,

curves

4, 5, respectively).

The further decrease in the solvent

strength,

_r, transforms the

system

into the

b-conditions,

where the

layer

thickness _is determined

by

the

competition

between

repulsive

_temary volume

interactions and electrostatic attraction

H m Naw

~'~(miu (a~ia-) (15)

The

b-region

_occupies the range of

r,*

« _r «

(r,*(,

where

* -1 113

(16)

T, m-m W'

~Note,

that at C _«

S~,

m~

« m' _we have

r,*

_{» r} *

).

At r «

r,*

the

layer

_{passes into} the

ordinary globular

_{state, equation}

(10 3)

Discussion.

The jumpwise

dependence

of the dimensions of the

planar layer

of

grafted polyelectrolyte

chains _on the solvent

strength

_in contrast to that of the

layer

of

uncharged polymer

chains indicates that the

collapse

of the

planar grafted polyelectrolyte lajer

_{may occur as} the first order

phase

transition. This result _{is not} surpnsmg, because the translational entropy ^of mobile countenons, which _is

responsible

for this

effect,

contributes _an affective

long-range

interaction into the system.

The _main

relationships

of the

collapse

of

grafted polyelectrolyte layer

coincide with that of the

collapse

^of

polyelectrolyte

networks

[19, 20].

^The main difference _is the appearance of the

526 JOURNAL DE

PHYSIQUE

II M 5

regime in which strong volume interactions _in

densely grafted layer

dominate _over all

polyelectrolyte

effects

Note

that,

_{as was} shown

above,

the

layer

of

strongly charged

and

moderately densely grafted polyelectrolyte

chains _{remains in a} swollen state far below the

b-point.

^This is

why

this kind of

protecting layers provides

_a stencal stabilization of colloidal

particles

below the b- point ⁱⁿ contrast to

stabilizing layers

of

uncharged

chains

[24]

Note addkd _m

proof

Recently

Y. Rabin and S. Alexander

(Europhys

Lett. 13

(1990) 49)

have shown that if the chains _in the

layer

_are stretched

by

the extemal force the chain

elasticity

_must be descnbed

by

the Pincus law

(AF~j

_m

(H/(N~'~ar~'~))~'~)

_i-e differs from that of Gaussian chains. An

analogous

situation takes

place

in the _«

polyelectrolyte

» regime when the osmotic _pressure of countenons acts as an extemal

stretching

force.

Analysis

shows that if _we take this effect into account the factor m' ^~'~ _in

equation (12)

obtained _in the _mean field

approximation

should be

replaced by

the

scaling

factor m' ^~'~

r

~'~ at _r » m' ~'~.

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