Polyelectrolyte molecule conformation near a charged surface

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surface

O. Borisov, E. Zhulina, T. Birshtein

To cite this version:

O. Borisov, E. Zhulina, T. Birshtein. Polyelectrolyte molecule conformation near a charged surface.

Journal de Physique II, EDP Sciences, 1994, 4 (6), pp.913-929. �10.1051/jp2:1994174�. �jpa-00248016�

Classification Physics Abstracts

61.40K 64.75 68.45 82.45

Polyelectrolyte molecule conformation

_{near a}

charged surface

O. V. Borisov

(*),

E. B. Zhulina and T. M. Birshtein

Institute of Macromolecular

Compounds

of the Russian Academy of Sciences, 199004,

St.

Petersburg,

Russia

(Received 6 November 1993, received in final form 3 January ^1994,

accepted

16 March 1994)

Abstract.-The

scaling theory

of conformation of _a weakly

charged polyelectrolyte

molecule attached _{onto a}

charged

surface is

developed.

It is shown that the

long-range

interaction between

charged

monomer units and

similary

(or

oppositely) charged

surface provides _{many more}

qualitatively

different

regimes

of behaviour of

polyelectrolyte

molecule than short-range

repulsion

(or attraction) ⁱⁿ the case of uncharged polymers near

repulsive

(or

adsorbing)

surface. The

scaling

indices

describing

orientation and

stretching

of

grafted polyion

near

similarly charged

surface _as well _as the

adjorption

^of macromolecule due _to

long-range

electrostatic attraction to the

oppositely charged

surface _are obtained.

1. Introduction.

The behaviour of

charged

macromolecules

(flexible polyions)

near

solid-liquid

interfaces

being

of great ^{interest for}

general polymer

science _as well _as in _numerous

technological

aspects has been

extensively

studied both

theoretically

and

experimentally during

the last years.

The theoretical

analysis

of conformations of

charged

macromolecules attached to inert

(uncharged)

interfaces

(grafted polyelectrolyte monolayers

_or

polymer brushes)

has been made in

[1-7].

The

understanding

of structure and

properties

of

grafted polyelectrolyte monolayers

is of great

importance

for the

problems

of colloid stabilization in aqueous

media, hydrodynamic

friction

reduction,

reversed

phase chromatography

of water-soluble

polymers

and

biopolymers,

etc.

It _was shown that due to the

long-range

character of electrostatic interactions the

cooperative

effects

(chain

orientation and additional

stretching)

in

grafted polyelectrolyte monolayers

immersed into _a salt-free solution become

significant

at

comparatively

low

grafting

densities

(far

below

overlapping threshold),

that

is,

in

pronounced

contrast to the situation in neutral

brushes

[8, 9].

(*) Present address

iaboratoire

_{Ldon Brillouin, CE-Saclay, 91191} Gif-sur-Yvette, France.

At the _same time it is well-known that the surface of colloidal

particles (silica particles,

for

example)

_as well _as surfactant _{mono- or}

bilayers

_at interfaces _are

usually charged

due _to the dissociation of

ionogenic

_groups and _at

comparatively

low

grafting density (in

the

« mus-

hroom

»

regime)

the interaction of

polyions

with the surface

charge predominates

_over the intermolecular _one and determines the

grafted polyion

conformation.

The

adsorption

of

polyelectrolyte

molecules onto an

oppositely charged

surface has been studied

recently [10-12] (see

also

[13-15])

_on the basis of numerical solution of the system ^of

self-consistent field

equations including

the Poisson-Bolzmann

equation

in the

ground

state dominance

approximation

and under the condition of finite

Debye-Huckel screening length (non-zero

ionic

strength

^of the

solution).

Thus the interactions in the system ^{remained short-} range on the scale of

polyion

dimensions.

The aim of this paper is to present a

general picture

of

single polyion

behaviour _near

charged

interface and _to describe the influence of

long-range

electrostatic interaction between

polyion

and

charged

interface _on the

polyion

conformation in

scaling

terms. Both _cases of

similarly

and

oppositely charged

interface

corresponding

to the

polyion repulsion

from or to

adsorption

at the interface will be considered.

It will be shown that

polyion adsorption

caused

by long-range

attraction of

charged

monomers to the surface includes different stages

corresponding

to conformational rearrange-

ment on different scales in contrast to one-stage

adsorption

caused

by short-range

attrac- tion

[16].

2. Model.

We shall consider _one

weakly-charged polyelectrolyte

molecule in _a salt-free dielectric

solution

terminally

attached _{onto an}

impermeable charged planar

surface. Let N be the number

of _monomer units in the chain and every m-th unit carries _an

elementary charge

_{e so} that the

total

charge

of the

polyion Q

_{= e} ^~ Let the backbone of the chain be flexible

(the

Kuhn

m

segment

length

A is

equal

to the _monomer

length a)

and the condition of weak

charging,

m»j~ _projides

^the conservation of chain

flexibility regardless

of its

charging.

Here

u =

~

m

~j

^{ml, is the usual} interaction parameter

including

the electron

charge

_e, the

a a _e

dielectric constant of solvent _e and the temperature in

energetic

units T

i~

is the

Bjerrum length.

The solvent is

supposed

to be _or atherrnal solvent with respect to

short-range

interactions of

uncharged

monomer units of the chain.

We suppose the surface to be

uniformly charged

with the

density

_{p so} that the

sign

of surface

charge

_may be the _{same as or}

opposite

_to that of the

polyion.

The dielectric constant of the

surface,

_e~, can differ from that of the

solution,

_e, The condition of

electroneutrality

results in the presence of _a

corresponding

amount of counterions in the solution, These counterions

compensate ^the

charge

^of the surface and _are

supposed

to be monovalent _as well _as

charged

groups on

polyelectrolyte

^chain,

3.

Polyion grafted

onto _a neutral surface.

The conformation of

polyelectrolyte

molecules

sparsely grafted

onto an

uncharged

surface has been discussed in

[7],

It _was shown that if the number of

polyions grafted

_per unit _area is less than

a~

^~~

,

where

Ho

is the characteristic size of isolated

polyion

in the solution, all the

Q~ Ho

cooperative

^effects

(chain

orientation and additional

stretching)

^due to intermolecular interactions _can be

neglected.

Here and below _we suppose the latter condition _to be fulfilled _so

that

only

intramolecular interactions and interaction between the

grafted polyelectrolyte

molecule and the surface will be taken into account.

The conformation and

equilibrium

dimension H

=

Ho

^of _an isolated

polyion

in the bulk of _a salt-free solution is determined

by

the

competition

between the force of intramolecular

Coulomb

repulsion

of

charged

monomer units

fel

*

I _(i)

H

e

and

entropic

force of chain

elasticity

v

H fi

~~°~~ ^~

Na ^~ ~~~

where _v is the usual

Flory

index for neutral chain dimensions _{: v}

= or v m

~

for the _cases of

2 5

and athermal

solvent, respectively.

The balance of these _two forces

gives [17, 18]

1-v

~

Q2

^{~ ~"~~ ~~"}

HomN~~"

_{a u}

j mNam ^~~~

u~~" (3)

e

~

In the framework of the blob

picture,

_a

single polyelectrolyte

molecule _can be

presented

_{as a}

succession of blobs each of size

to

determined

by

the condition

2v ~

f

^~

~2-V ~V-2

~

(~)

~~fel~

so that the average end-to-end distance

(and

the radius of

gyration)

is

given by

Ho

_m

_N~~ iu (5)

NBn

"

~

ii

(6) (iota)

^"

while the dimensions of

polyion

^{in the direction normal} to the end-to-end _vector scale like

Ro

_m ^N

j'~ fo ⁽⁷⁾

Under the conditions of

uncharged

or

weakly charged

surface the

polyion

attached _{at one} end _onto the surface retains in

general

the conformation of _an isolated

polyion

in the solution

and has _a random orientation in the

half-space

above the surface

(Fig. la),

so that its

dimensions both in the direction normal to the

surface, H,,

and

parallel

to

it, H~,

H~,

^coincide

(with

the accuracy of omitted numerical

coefficients)

^with

Ho.

Note that this is the

case when the dielectric constants of the solvent and the surface _are close _to each other,

e e~ we, e~. The

polarization

effects which _are

important

in the

opposite

_case will be discussed later

(see

Sect.

6).

Obviously, equations (3-7)

_are

valij ^only

^{if the fraction of}

^charged

monomer units in the chain is

sufficiently high

to

provide ^~

» T where R~~,j = N " a is the size of the Gaussian _or

Rcoii ~

swollen coil

unperturbed by

electrostatic interactions. Otherwise the intramolecular electro-

a)

Fig.

I.

Polyion

grafted onto a neutral _or

weakly charged

surface _: a) _strong intramolecular Coulomb

repulsion

and chain

stretching

b) weak intramolecular electrostatic interaction.

static

repulsion

is too weak and cannot stretch both free _or

grafted

chain which retains

(with

the accuracy of numerical

coefficients)

Gaussian _or swollen coil statistics and dimensions

2 v

mN" _a

(Fig, 16).

This is the _case at m »N ^~

u"~

4. Electrostatic force between

grafted polyion

and

charged

^surface.

As is well-known the infinite

charged planar

^{surface in the} solution

always

retains its

counterions _near

it,

_even at infinite dilution. The characteristic thickness of the counterion cloud above the surface is

given by [19]

A

=11

^~

₍₈₎

e

This counterion cloud

partially

_screens the surface

charge

at distances _x» A _so that the

resulting

electrostatic field in the solution in

half-space

above the surface is

given by [19]

(we

have chosen the direction of the _{x-axes to} be normal to the

surface).

The

corresponding

electrostatic force

applied

to the

grafted polyion

is

equal

to

~~

^{A »}

^H~

f,

m

QE~(x

=

H~)

_m ^~

(10)

T

~

A _«

H,

eH,

fy=fz=0.

At

sufficiently

low surface

charge density

_we have A »

H~

_so that

screening

effects _on the

scale of

polyion

size _are

week;

the electrostatic field

acting

_on the

polyion

is almost

homogeneous

and

equal

to that of infinite

charged plane

in dielectric medium

(Eq. (10),

A »

H,).

However,

_as the surface

charge density

increases the counterion cloud becomes _more dense and is located closer to the surface thus

screening

the interaction between

charged

surface and

grafted polyion (Eq. (10), «H~).

The force

given by equation (10)

is the force of

polyion repulsion

from _or attraction to the

surface in the _cases of the silimar or

opposite sign

of

charges

of

polyion

and -interface

respectively.

S.

Polyion

near

similarly charged (repulsive)

surface.

We shall start the

analysis

from the _case of

similarly charged polyion

and surface when the force between

polyion

and the surface

given by equation (10)

has the character of

long-range repulsion.

Let _us consider first the conformation of _a

sufficiently strongly charged polyion,

which is stretched

by

intramolecular Coulomb

repulsion

in the absence of the

charge

_on the surface

(Eq. (3)).

At

comparatively

low surface

charge density,

when A »

Ho, f~

«

f~j

_or that is the _same, p «

~l~,

^{the extemal force is weak in}

comparison

with intramolecular

repulsion

and _causes

Ho

only

the orientation of the

grafted polyion

in the direction normal _to the surface

(Fig. 2a).

The average

angle

between the

polyion

axis

(end,to-end vector)

and the direction normal _to the surface _can be estimated from the condition

f,Ho(I

_-cos

@)mT. (ll)

The orientation effect becomes

significant

at

f,

_m ^~ _or at p m

~~

N~~

Ho QHO

An increase in the surface

charge density

and,

consequently,

ⁱⁿ

f~,

results in decrease in and at p ^»

~~ the

polyion

becomes

strongly

^oriented

perpendiculary

to the surface

QHO

@~m

^~~ Ml

(12)

QHOP

and its lateral dimensions become

equal

_to

_Ro, equation (7).

As the surface

charge

and the

repulsive

force

applied

to

grafted polyion

increase _more

they

become stronger ^than the intramolecular force

equation (I),

and _cause noticeable additional

stretching

of

polyion

in x-direction

(Fig.

2bl.

--~T~

',

H~

I

9 /

aj hi

Fig. 2. -Polyion grafted onto a similarly charged ^surface al orientation in the x-direction cl

stretching

in the x-direction.

At

f,

»

f~j,

or that is the _same, at p ^» N~ _~, the chain

stretching

is determined

by

the

Ho

balance between

f~

^and f~~~~,

equation (2)

and is

given by

i-v

j

~

i-v

H~

_m N ~~ ~ ~

a =

~ _~~ ~ ~~ "~ ~ ^~

~'

~ ~

(l

3)

T

Nam"~~ H~»A.

As _{we can see} from

equations (13)

and

(8)

_an increase in the surface

charge density

_causes

simultaneously

a monotonic decrease in A and _a monotonic increase in the chain

stretching

untill

H,

« A.

However,

_at

H,

_m A the

polyion

becomes

strongly

screened from the surface

by

the counterion cloud _so that further increase in _p does not lead _to additional

stretching (Fig. 3).

InH~

V

In

In p

Fig. ^3. The dependence of the polyion x-dimensions _on the surface charge density similarly charged

(repulsive)

^surface.

The

polyion

stretched in the x-direction due _to the

repulsion

from the surface _can be

~

^-iiv

presented

as a succesion of

N~

= N Pincus

stretching

blobs of size

a

mN~~u~~())ammA/N ^H,«A

~~~-

~ ~

(~~)

~f~

_m"a

_H,»A,

Note that _at

H,

WA every Pincus

stretching

blob of size f ₌ ^m" a contains _one

charged

group.

It is worth

noting

that the

polyion

^dimensions at

saturating (at H,

» Aj coincides with those of the

polyion

in the

polytelectrolyte

brush in so-called _« osmotic _»

regime [3, 5].

The lateral

dimensions of the stretched chain _are

given by

2v-1

~ 2V

N ^"

mu~ )

a

H~

«

~ j~ _j~~^lf2

~

^{U p} ~~~~

v~ z~ B ~~ i

N~'~m

^~

a

H~»A.

As _can be _seen from

equation (15)

the

polyion

lateral dimensions decrease

monotonically

with

p till

H~

WA or remain constant as well _as

H,

at

H,

WA, In the _case of solvent

(v

₌

1/2)

the lateral dimension of

polyion

does not

depend

_on the normal

stretching

and

remains

equal

to R~~,j =

N ^"~ _a.

All the

regimes

described above _occur for

polyions sufficiently strongly charged

and stretched

by

intramolecular Coulomb force, If the _own

charge

^of

polyion Q

is not

large enough

to

provide

chain

stretching,

the orientation effect _near the

charged

surface does not take

place

_;

the chain remains

unperturbed by

the electrostatic interaction conformation

(Fig.

lb) _up to the value of

f,

»

~~

m

~

At

higher

surface

charge density,

^{~~ ~} » mu~ N

",

the chain is

e R~~,i ^e

stretched in the x-direction

by

^{the force of}

repulsion

from the surface and this

stretching

_as well

as lateral chain dimensions _are described

by equations (13, 15).

6.

Polyion

near

oppositely charged

surface i

adsorption.

Let us start

again

from the case of

weakly,

but

oppositely charged

surface and

sufficiently

strongly charged (stretched by

intramolecular Coulomb

repulsionl polyion.

The external force

applied

to

polyion

is

given (in

absolute

value) by equation (lO)

but attracts the

polyion

to the surface.

Though

this attraction is weak, the

polyion

attached to the surface at one end has the random orientation in the half space above the surface _{as was}

described above

(Fig, la),

However, _at

Ho f,

_m T the attraction of

polyion

to the surface becomes sufficient _to

provide

its orientation

mostly parallel

to the surface thus

reducing

the

height

of the free

polyion

end above the surface and

decreasing polyion

_energy in the field of the surface

charge

without any rearragement ^{of intemal} conformational structure

(Fig, 4a),

We _can characterize this

regime

as

pre-adsorption.

As _a result, the average dimensions of

polyion

in the x-direction decrease with

an increase in _p like

~' jr j~

'~~

~"~

a~p

^{'~~ ~~ ~~~~}

where the second

equality

is

provided by

A

»H~,

while

polyion

lateral

dimension

_remain

equal

to

Ho,

This character of decrease in the

polyion height

^above the

surfaci, H~

,

occurs until

H~ mRo given by equation (7),

P

Further increase in _p

and,

_{as a}

result,

in attractive force between

polyion

and the surface leads to transversal

compression

of

polyion by

electrostatic force. This deformation is related to rearrangement ^of conformational structure of

polyion

_on the scale smaller than

Ro

^but

larger

than

fo,

The

elasticity

of the chain of

fo-blobs

with respect to the

compression

in

the ~-direction is

given by

NB~

f/

~c°n~ ~ ~

~f3

~~~~

,

while the

compressing

electrostatic force is still

given by equation (10),

The balance between these forces results in

~'

~ ~

~

" u

~~"~~

_j e i13

t ^a

~ ~

(j~

~ 3(1-vj 5-4v

Note that the

point _H~

_m

Ro

_or ^{~ ~} _m N ^~'~ _m ^{~ "} _u ^~ " ~

corresponds

to the

beginning

of

e

the chain

adsorption

onto the surface due to electrostatic attraction _{: at}

higher

surface

charge

densities

H~

becomes

independent

of N

(Fig.

4b). It is also

interesting

that in this

regime

the

adsorption layer

thickness

H~ independently

of the solvent

strength (of v).

P ^"~

Ho H~

al

~

H~ ^~

/

~j cj

Fig.

4.

-Polyion grafted

onto an

oppositely charged

surface a)

polyion

orientation parallel to the surface without conformational rearrangement b) weak

compression

in the x-direction

beginning

of

adsorption

c) strong compression in the x-direction

adsorption.

The thickness of the

adsorption layer

formed

by polyelectrolyte

molecule _near the surface decreases with _an increase in the surface

charge density according

to

equation (18)

until

H~

^{exceeds the} electrostatic

bloj~size~ _f~.~

At

H,

m

fo

or at

~~~

mm "~~

u

"~~ the chain of

fo-blobs acquires

2,dimensional

e

conformation

completely lying

_on the surface. As _no extemal force is

acting

in the lateral direction, the

polyion

stretched

by

intramolecular

repulsion

^has a random orientation _on the surface. If the z-axis coincides with end-to-end vector, we have

H=

_m

Ho, _H~

_m

Ro (Eqs. (3, 7)).

Further increase in the surface

charge density (and

in the attraction

force)

leads _to the deformation of

fo,blobs

so that the chain _can be

presented (in analogy

to

[16])

_{as a} succession of

adsorption

blobs of size

H,

the chain part ^{inside each blob remains}

unperturbed by

intramolecular electrostatic interaction and retains Gaussian _or excluded-volume statistics in

the _cases of _or athermal

solvent, respectively (Fig. 4c).

The energy of interaction of

charges

within each blob with the

charged

surface is

m T _so that the blob size

(the adsorption layer thickness)

is

given by

v

H~m m~~u~~~ ~~~a. ₍₁₉₎

e

The decrease in the thickness of adsorbed

layer

with _an

incjease

^{in p}

^according

^to

equation (19)

continues still

H~

remains

larger

than A that _{occurs at} ^{~ ~}

w u~ m~ ^" At

higher

e

surface

density

the counterion cloud thickness becomes smaller than the size of _one

adsorption

blob

(adsorption layer thick~less)

^{and the}

^charges

^on

^polyion

appear ^to be screened from the

surface

charge.

Thus _at ~'~

»u~~

_{m~" the thickness of the}

adsorption layer

becomes

e

independent

of _p and

equal

to

H~

_m m ^" a

(20)

I,e, _one

adsorption

blob contains _one

charged

_monomer,

The schematic

scaling dependence

of the

polyion

dimensions in the direction normal _to the

oppositely charged

surface _on the surface

charge density

is shown in

figure

5a. It is

interesting

that in the _case of solvent Gaussian

blobs)

the

slopes

^of this

dependence

in the

regime

of transversal

compression

of chain of

fo-blobs

and in that of deformation of

fo-blobs

coincide, This is _not the _case for the chain with excluded volume interactions.

~~~~

InH~

InH~

I InR~~,j

for _v lf2 v

~ + l

v ~

fi

^~

in P Inp

aj hi

Fig.

5.-The dependence of the

polyion

x-dimensions _on the surface

charge

density:

oppositely

charged (attractive) surface;

a)polyion

sufficiently strongly

charged,

stretched by intramolecular repulsion dashed line corresponds to the _case of Gaussian blobs (o solvent) b) weakly charged

polyion,

The

compression

of

polyion

in the x-direction also affects

(in

the _case of athermal

solvent)

its lateral dimensions,

Actually,

at

H, wRo

the

polyion

is

lying

on the surface and stretched

by intramolecuiar repulsion parallel

to it. In the

region

of weak transversal

compression

^of

polyion by

the force of

attraction _to the surface,

H,

»

fo,

the dimension of

polyion

ⁱⁿ the lateral direction remains

equal

to

Ho. However,

at stronger

adsorption

when the chain _can be

presented

_{as a} system ^of

adsorption

blobs of size

H~

«

fo

^{the chain}

elasticity

with respect to

stretching

in the lateral

direction decreases _as the

compression

^increases.

In order _to derive the

dependence

of the lateral dimensions of adsorbed

polyion

_on the surface

charge density

we should

equilibrate

the elastic force

acting

ⁱⁿ stretched 2-dimensional chain of blobs, each of size

H~

v~

j ^{"2 "}

~~~~

~

~j

^~

_~~~~

^{~~~ ~~~}

~ ~~

JOUR~AL DE PHYS,QUE Ii T 4 N' 5 JUNE >q94

Q2

with the force of intramolecular electrostatic

repulsion f~j

_{m ~} ^that

gives H~

2 v~ 2 v~ ^v v2

H, _m N _am ^{~ "~} _u ^{~ "~}

(H~la

^{"~~ "~~}

⁽²²⁾

where _v~ is the 2-dimensional

Flory

exponent

(v~

_m ^3/4 in athermal

solvent). Equation (22)

describes the lateral dimensions of

polyion

confined into _a slit of width

H~

«

fo. Substituting expression (19)

and that for the adsorbed

layer

thickness

H,

we get ^the

dependence

of the lateral dimensions of adsorbed

polyion

_on the surface

charge density

_p at m" _a «

H,

«

fo.

"2 "

H~~N(a~ pie)~"~~~~~~"~~ (23)

As _{we can see} from

equation (23)

the

compression

of

polyion by

the electrostatic force in the _x- direction causes at

H~

«

fo

a decrease in the chain

elasticity

with

respect

to

stretching

in the _z- directions

and,

as a

result,

additional

stretching

of the

polyion

_on the surface.

Equations (21, 22)

_are

analogous

to

equations (2, 3)

and describe the

stretching

of 2-

dimensional chain of

H,-blobs by

intramolecular Coulomb

repulsion. Correspondingly,

^2-

dimensional

planar stretching

blob («

pancake

») of size

2 v~ v~ 2(v v2)

f~

_m m

~ "~

u

"~ ~

a

(H~la

^{"~~ "~~}

⁽²⁴⁾

can be introduced

(compare

with

Eq. (4)).

It is easy to see that at

H~«fo

we have

f~

»

H,.

The

last'ratio

_{means that} the adsorbed

polyelectrolyte

chain is unstretched _on the scale of

H~-blobs.

The number

N~

of

f~-blobs

in the chain is

given by

N~~ ^» N

(i~/H~ ^)-

^~'"2

(Hja )-

^1'"

(25)

so that for the lateral dimensions of adsorbed

polyion

_we have _:

H~

_m

Ns~ f~ (26)

~ p~ ^lf2

~ (~~j

y B~ ²

As it follows from

equations (24,

25,

27)

the transversal

dimension, H~

^of

polyion

_on the surface decreases with _a decrease in

H~

or, that is the _same, with _an increase in the surface

charge density.

Of _course additonal

stretching

of adsorbed

polyion

takes

place

in the _case of _a

good

solvent

while the

polyion

with Gaussian

elasticity

retains its lateral dimensions

equal

to

Ho

independently

of

H~ (and p). Strictly speaking

it is _not the _case for adsorbed

polyion

under the condition of the solvent because temary ^interblob interactions in _two dimensions may affect the chain

elasticity.

However, the

analysis

of this delicate

problem

is _out of the framework of

our paper. Let us

only

note that for _a 2-dimensional chain under the b-conditions

v~, _~ m 4/7.

The

dependence

of the x-dimensions of

weakly charged (unperturbed by

intramolecular Coulomb

interactions) polyion

_on the surface

charge density (oppositely charged surface)

is

presented

ⁱⁿ

figure

5b. In this _case, the

beginning

of

adsorption

is related _to the deformation of the 3-dimensional Gaussian _or the swollen coil into

th<

2-dimensional coil. The swollen coil

consists of

adsorption

blobs of size

given by equations ^(19,

^{201. This} deformation is caused

by

the force of electrostatic attraction of

charged

_monomers to the surface and

begins

_at

f,

_{R~~,j m} T _or

~~ _~

m mu~ N~ " The

corresponding

increase in the lateral dimensions

(in

e

the _case of athermal

solvent) according

to

equations (22, 23) begins

later when the chain

elasticity

^with respect to

stretching

in the lateral direction decreases

sufficiently

and becomes too weak to oppose the intramolecular Coulomb

repulsion.

7.

Polyelectrolyte globule

_near

charged

interface.

Up

to _now we considered the situation when the solvent is

good

_or b-solvent with

respect

to the interaction of

uncharged

_monomers of the

polyion. Correspondingly,

it _was assumed that the

chain parts inside electrostatic blobs

obey

excluded volume

(vm3/5)

_or Gaussian

(v

=

1/2) statistics, respectively. However,

in many

practically important

_cases

uncharged

backbone of the

polyion

is insoluble in _water. This _means that water is _a poor solvent for

uncharged

monomers and the interaction between them has _a character of

short-range

attraction. If this attraction is

comparatively

weak

(close

_to the

b-conditions)

_or intrachain electrostatic

repulsion

^is strong, the

fo-blobs

^remain Gaussian and all the

picture

of the

polyion

behaviour _near

charged

interface described above remains valid. However, if the

short-range

attraction between

uncharged

_monomers becomes

sufficiently

_strong, the

polyelectrolyte

chain

collapses forming polyelectrolyte globule.

7.I POLYELECTROLYTE _GLOBULE. Let _us start from the brief review of

single polyelec~o-

lyte globule

conformation in the bulk of the salt-free solution.

As _was shown in

[20],

the local structure of the

polyelectrolyte globule

coincides with that of

the neutral

polymer globule

and is determined

by

the balance between

s(ort-range binary

attraction and temary

repulsion

between

uncharged

monomers. It is characterized

by

a local

monomer

density proportional

to _{r =} (@

T)/T

and _a thermal correlation

length

f~ _m r a :

every chain part ^of a size smaller than

f~

^{remains Gaussian and the}

globule

_as a whole _can be

presented

_as a system ^of close

packed f~-blobs.

Parameter _r describes

binwy

attraction between

uncharged

_monomers thus

characterizing

the solvent

strength

_; it increases when the solvent

strength

decreases.

On the

large

scales the

polyion

remains stretched

by

unscreened Coulomb

repulsion proportionally

to the

degree

of

polymerization, _{H~ ~N,}

^and can be

approximated by

the

tangled cylinder

of

length H~

^{and thickness D} m

(N/(H~

r

))~'~

The

equilibrium

dimensions of

polyelectrolyte globule

_are determined

by

the balance between the energy of intramolecular electrostatic

repulsion, F~~~j~~~

m

Q~/ (H~

^e

),

and the interfacial energy,

F~

_{m y}

_(H~

D where the _« surface tension _{», y,} at the

globule/solvent

interface _can be

presented

as y m

T/f/.

The

minimization of the free energy,

F~~~j~~~

+

F~

with respect to

H~ gives

~

it

(28)

4f3 2f3

~ l

NU~ _~t

_~0

*

~0 £

H~

m Nam _U

0

D _m

fo

where

fomm~'~u~~'~a

is the electrostatic blob size in the b-solvent, _r =0,

(Eq. (4)

_at

v

=1/2).

The

polyelectrolyte globule

_can also be

presented

_as a stretched chain of blobs of size D : the

local _structure of these blobs coincides with that of neutral

globule

and the energy of

electrostatic

repulsion

of

charges

ⁱⁿ every blob is

equilibrated by

the interfacial energy

m

D~

_y. The transversal dimension of

polyelectrolyte globule

is

given by

~g

^fit~~~ Um~ ^~'3

~l/6

~ ij~

Nf~

^ij2

fo

^{~. ~~~~}

As it follows from

equation (28),

the

short-range

attraction between

uncharged

_monomers

causes the

collapse

of the

polyelectrolyte

chain into

polyelectrolyte globule

at _{r m} m~ ^~'~ u~'~

or at f~ w

fo.

At smaller _r electrostatic

repulsion predominates

over

short-range

attraction and the

polyion

retains the conformation of stretched chain of Gaussian

fo-blobs

described in section 3. In _our further consideration _{we assume} the condition f~ «

fo

to be fulfilled.

7.2 POLYELECTROLYTE _GLOBULE NEAR SIMILARLY CHARGED INTERFACE _: ORIENTATION AND

STRETCHING. If the

polyelectrolyte globule

is attached at one end _onto the

similarly charged interface,

the

repulsive force, f~, given by equation (10)

is

applied

to the

globule.

As the

polyelectrolyte globule

has strong asymmetry ^{of the}

shape,

this force _causes first its orientation in the x-direction

just

like in the _case of

grafted polyion

stretched

by

intramolecular Coulomb

repulsion

under the conditions of

good

or b-solvent

(Sect. 5).

However, the

polyion

ⁱⁿ the

globule

state is lest

elongated

than in the latter _ones,

H~

«

Ho,

so that noticeable orientation is reached at

higher

surface

charge density

: p ^»

~~

The lateral dimensions of

polyelectrolyte QH~

globule

under the conditions of

strong

orientation _are

given by equation (29).

Further increase in the surface

charge density,

_p, and

correspondingly,

in the

repulsive

force,

f,,

_causes

only

weak

perturbation

of the

globule shape

^stabilized

by

^{the surface} _energy

until

f~/T

remains smaller than the inversed thermal blob size

f~. However,

at

f~m

T/f~

_{or p m}

Te/(Qft)

_an

abrupt

transition from

polyelectrolyte globular

to stretched chain of blobs of size _max

(f~,

m~'~ al _occurs

(for

neutral

globules

this

stretching

transition _was

considered in

[21, 22] ).

If f~ »

m~'~

_a,

a further increase in surface

charge density

leads to an

additional

stretching

^of the

polyion

in accordance to

equations (13-15).

In the

opposite

_case, f~ « m~'~ _a, the

polyion

becomes stretched _to the

height H~

_m Nm~ ^~'~ _a further

stretching

is

prevented

due to the

screening

of the surface

charge by

^{the counterion cloud}

(Fig. 6).

lnH~

;.

lnH~

InH~ ^'

Inp

Fig.

6.

Stretching

of the polyelectrolyte globule (f~ « fo) due _to repulsion from the

similarly charged

surface f, » m"~ _{a, solid} line, and f, « m"~

a, dashed line.

Corresponding

dependence for the polyion in the o-solvent (at f, »

fo)

is shown

by

dotted line.

Note that the _same

picture

of

stretching

takes

place

for

weakly charged globule unperturbed by

intramolecular Coulomb

repulsion,

_f~ « N ^"~ _a «

fo,

if m~'~ _{a «} N

f~.

If the last

inequality

is

violated,

the

globule charge

is _too small and the

screening

of the surface

charge

at a distance of the order of the

globule

size,

R~,

^{prevents the}

globule

from

stretching

at any p.

7.3 POLYELECTROLYTE GLOBULE NEAR OPPOSITELY CHARGED INTERFACE ADSORPTION.

The attractive force

applied

to a

polyelectrolyte globule grafted

onto

oppositely charged surface,

_as it increases with _an increase in the surface

charge,

_causes the orientation of the

polyion

to be

parallel

to the surface without any rearrangement ^of

polyelectrolyte globule

conformation _{as a} whole

(pre-adsorption)

and then _causes the

adsorption

of

polyelectrolyte globule

onto the surface.

The

pre-adsorption

starts at p m

~~ i-e- _at

higher

surface

charge density

than for

polyion

in

QH~

good

or

b-solvent,

and the

height

of the free chain end above the surface is

given by equation (16).

The

pre-adsorption regime

_occurs at

H~

»

R~ jhere

^the transversal dimension of

polyelectrolyte globule, _R~,

is

given by equation (29)

_or at ^~ ~

« N ^~'~ m~'~ _u~ ^~'~

r

"~ Thus,

e

pre-adsorption

_range becomes narrower as the solvent

quality

decreases

(r increases).

The weak

adsorption

_range is determined

by

the condition D «

H,

«

R~

^{and is} characterized

by

transversal

compression

of

polyelectrolyte globule by

the force of electrostatic attraction to the surface without conformational rearrangement on a scale smaller than D _m

fo

and without any

change

in the

globule/solvent

interface _area. The elastic force

arising

ⁱⁿ the

transversally

compressed globule

is

equal

to

f~~~~ m

TR(/H(

m

~~

)

(30) fo H,

Combining equations (30)

and

(10)

_{we get}

H

~ u

~~ _~ m~ ^'

~~~ ~~ ^~~~

a

(31)

' e

fo

that is smaller than the x-dimension of

weakly

adsorbed

polyion

under the conditions of the @- solvent

by

the factor

(f~/fo)"~

The dimensions of the

polyelectrolyte globule

in the lateral direction under the conditions of weak

adsorption

remain constant and _are

given by equations (28,

29). The

validity

of

equation

(31) is restricted

by

the condition

H~»D

_m

fo

or

(a~

pie « (m/u

)(f~/f().

Further increase in the surface

charge density

in the range of _p values (m/u

)(f~/f()

w

a~ pie

_w

(m/u 11

^~

f/

^'

only weakly

affects the

polyelectrolyte globule

conformation because the surface tension in this range is stronger than the electrostatic attraction to the surface.

However,

as the surface

charge density

^increases _up to the value

(a~

pie

)

_m (m/u

(11

^~

f/

^'), the force of electrostatic attraction

(Eq. (10))

becomes

sufficiently

strong to compete ^{with the force of the surface tension}

f~

_m

fi

»

~~~ ⁽³²¹

~~

<

H,

)

so that a further increase in the surface

charge density

leads _to the deformation of the

polyelectrolyte globule.

This deformation is

accompanied by

_a decrease in the adsorbed

globule

thickness,

H,, according

to the

equation

~~y~ lf2

H,

_{m a}

(33)

"aP~t

which works until

H,

»

f~

or

(a~ pie)« (m/u) f/'

At

higher

surface

charge density

an

ordinary regime

of

adsorption

under a-conditions _occurs.

8.

Image charge

^effect.

In the

previous

sections _we considered the influence of the interaction between _a

grafted polyion

and «frozen» surface

charge

of fixed

(and uniform) density

_p. However, if the dielectric _constants of the solvent, _e, and of the surface, _e~, ^differ

greatly,

the presence of the

polyion

_near the interface _causes different

polarization

of this _two media that is

equivalent

to the appearance of the

image charge

in the

symmetrical point

with respect to the interface and its

value

Q'

is

given by

e e~

Q'" Q

e+e~

In most

important

_cases water

plays

the role of the solvent for

polyelectrolyte

molecules while the surface is made of the

unpolar

dielectric

(silica, polymer

latex, etc. with much lower

dielectric constant so that _we have

Q'

_m

Q.

Here _we restrict our consideration

only

to a short summary of the results.

The interaction of the

polyion

with this

image charge

results in the appearance of the additional force

repelling

the

grafted polyion

from the surface. When the dimensions of

polyion

in the

direction

perpendicular

to the surface _are much

larger

than lateral

dimensions, H~

»

H,,

this force is

given by

f,l'~~(H,

m

Q~/ (H[

e

j (341

As it follows from

equation (34)

this force _causes noticeable

polyion

orientation in the _x-

direction _even in the _case of

uncharged

interface

[23]

when

H~

_m

Ho

~

and the force of

repulsion

between

polyion

and its

image

is

equal

to the force of intramolecular

repulsion,

(we suppose

fo

« N ^"~ _a, indices _« 0

» and _« g »

correspond

to the _cases of

polyion

stretched

by

intramolecular Coulomb

repulsion, fo

«

f~,

or

polyelectrolyte globule,

_f~ «

fo,

respec-

tively)

and the average

angle

between the

polyion

axis and the x-direction is

given by

~

m T/

~f['~l Ho

~ ^m

THO

~

e/Q~

«

(36)

In the _case of

similarly charged

surface the _« frozen _»

charge density

almost does _not affect

polyion

orientation in the range of _p values _p «

Q/H(

~.

Under the conditions of

good

or @- solvent the

stretching

of the

polyion

in the x-direction _occurs in accordance with

equations (13- l5)

at surface

charge density

_{p m}

Q/H(.

The

stretching

of

polyelectrolyte globule

starts at a

much

higher

^surface

charge,

_{p m}

Te(Qft)

»

Q/H(,

see section 7.2.

The effect of

image charge

is _more

significant

in the _case of

oppositely charged

surface.

Here the

competition

between the

repulsion

of

grafted polyion

from its

image

and the attraction to

oppositely charged

surface leads _{to a} noticeable modification of the

polyion

behaviour. At

small surface

charge density

the effects of

repulsion

between real and

image polyions

dominates and

polyelectrolyte

chain is oriented

perpendicularly

to the surface up to

2 pie _m

Q/H(

~.

Under this conditions its

height

above the surface is of order of the end-to-end distance of the

polyion

in the

solution, H,

_m

Ho

~.

At

relatively high

surface

charge density

when the attraction of

polyion

to the surface flattens

polyion

so that

H,

«

H~,

^the

repulsion

force

f)'~~

between

polyion

and its

image

is

given by

fl'~

m

Q~/ (H, _{H~,e )} (37)

Analysis

shows that this

repulsive

force does not affect the

adsorption

mechanism and characteristics of

polyion (its

normal dimensions

H,)

under the conditions of

high

surface

charge

densities _p,

corresponding

to the

rearrangement

^of electrostatic

fo,blobs

_:

a~ pie

» m~ ^{"'~~ "~}

u~'

^~ "~' _" ~ for Gaussian or swollen

to-blobs

and

a~

_{pie » m~} ^"~ _u~ ^"~

r for

collapsed to-blobs-

In the intermediate range of

p-values,

^the conformation of _a

polyion

is determined

by

the balance of attractive and

repulsive

electrostatic forces, _so that

H~

_m

~

(38) Ho,

g P

Thus, in the intermediate range of _p

values,

the

image charge

^effect leads to a noticeable increase in normal dimensions of

polyion.

Note also that due to

image charge

^{effect the}

plateau

of the

dependence H,(p

for

collapsed polyion (Fig. 7), disappears.

InH,

-I

Inp

Fig.

7.

Adsorption

of

polyelectrolyte globule

_at the

oppositely charged

surface the

dependence

of the normal dimension of the

globule, H,,

_on the surface

charge density.

9. Conclusion.

In this paper we have

presented

a

scaling description

of the behaviour of _a

single

polyelectrolyte

molecule

grafted

onto a

charged solid-liquid

interface and

analyzed

the

dependences

of the

polyion's

conformational characteristics _on the

sign

and absolute value of the surface

charge density.

Phenomena such _as

polyion orientation,

additional

stretching

at the

similarly charged

surface _as well _{as an}

adsorption

onto an

oppositely charged

surface _were considered. This

picture

remains

adequate

for

polyions

in the

grafted polyelectrolyte layer

if

the surface

charge density

exceeds

by

far the average

charge density

related _to

grafted

polyelectrolyte.

It is shown, that in the _case of _a

large

difference between the dielectric

permeability

of the surface and that of the solution the interaction of

polyion

with the

image