Polymers at interfaces: from a quasi self similar adsorbed layer to a quasi brush first order phase tansition

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Submitted on 1 Jan 1993

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adsorbed layer to a quasi brush first order phase tansition

Christian Ligoure

To cite this version:

Christian Ligoure. Polymers at interfaces: from a quasi self similar adsorbed layer to a quasi brush first order phase tansition. Journal de Physique II, EDP Sciences, 1993, 3 (11), pp.1607-1617.

�10.1051/jp2:1993221�. �jpa-00247928�

Classification Physics Abstracts

36.20C-81.205-82.65D

Polymers at interfaces: IFom

_a

quasi self similar adsorbed layer

to

_a

quasi brush first order phase transition

Christian

Ligoure

Groupe de Dynarnique des Phases Condens4es

(Unit4

de Recherches assoc14e _au CNRS

233),

Universit4 des Sciences et Techniques du

Languedoc,

Case Courrier 26, Place Eugbne Bataillon,

F-34035 Montpellier Cedex 5, France

(Received

I April 1993, revised 5 August 1993, accepted 13 August 1993)

R4sum4. Nous discutonsl'adsorption de polymbres, pour1esquels tous les monombres sont faiblement attir4s _par la surface, h l'exception d'un _groupe terminal qui lui est fortement attir4 par cette surface. La structure 1l'4quilibre de la couche adsorb4e est d4crite _en utilisant des

arguments de lois d'4chelle. Cette structure est interm4diaire entre celle d'une couche adsorb4e autosimilaire et celle d'une brosse. Nous pr4disons l'existence d'une transition de phase du premier ordre d'un r4gime de quasi couche adsorb4e autosimilaire h _un r4gime de quasi brosse.

Le modble est confront4 h des exp4riences r4cente3.

Abstract. We discuss the adsorption of polymer chains, where all _{monomers are} weakly

attracted by the surface, except _an endgroup which is strongly attracted by the surface. The equilibrium structure of the adsorbed layer is described using scaling _arguments. This structure is intermediate between _a self similar adsorbed layer and _a brush. A first order phase transition is predicted from _a quasi-self similar adsorbed layer to _a quasi brush regime. The model is

compared with _{some recent} experimental results.

1. Introduction.

Adsorption

of

polymer

chains from _a

good

solvent onto _an interface _{can occur} for two different

reasons. When all _monomers of the

chain, although

still soluble have _a finite

affinity

to the

surface,

the chains _are

essentially uniformly adsorbed, leading

to a self similar form for the concentration

profile iii;

_we shall refer to it _as the self similar adsorbed

layer (SSAL).

On the other

hand,

the surface attraction may be due to _a

specific

segment of the

chains,

which

adsorb

strongly

to the surface:

adsorption

of endfunctionnalized chains _or block

copolymers

examplifies

_a tethered surface

layer.

In this _case, the stickers of the chains

(the

insoluble block for

copolymers

_or the

endgroup

for the endfunctionalized

chain)

anchor to the

surface,

whereas

the other soluble _{monomers are} forced to strech into the solution to form brushes [2]. ^The

properties

of such _a brush _are

quite

different from those of _a SSAL. For

instance,

the thickness of _a SSAL is _on the order of

magnitude

of the radius of

gyation

of _a free chain in

solution,

whereas it is much

higher

in the _case of _a brush. Both structures

(SSAL

and

brush)

have been

extensively, theoretically

_as well _as

experimentally

studied in recent years. References _are

given

^{in three} review articles [3, 4].

So far

relatively

^less attention has been

paid

to the _case of

weakly adsorbing homopolymers carrying strongly adsorbing

terminal _monomers [2, 5].

Ou-Yang

and Gao [5] have studied the

adsorption

of A-B-A triblocks

copolymers (with hydrophobic

small A blocks: the stickers and

an a

weakly adsorbing

backbone: PEO

block)

_on PS

microspheres

in aqueous solution.

They

have measured the

hydrodynamic layer

thickness _{as a} function of the number of

polymers

_per latex

sphere by dynamic ligth scattering

^technic. The

layer

thickness built _up

gradually

at low

concentration,

but _{over a narrow} concentration _range it increases

sharply

and then reaches _a

plateau.

The

adsorption

_{curves are very} different from those of both the

homopolymers

and

grafted polymers

with surface

repelling

^backbones [6].

They

have

interpreted

their results _as

a first order

phase

transition from _a

pancake

to a brush

configuration

_[2].

Alexander considered the

"pancake-cigar"

transition in his

pioneering

_{paper on} tethered chains [2]. ^{It is}

expected

to occur in brushes

comprising

of chains

capable

of uniform

adsorption.

The Alexander

analysis

utilizes _a

highly simplified

model. The adsorbed

layer

is modeled

as a concentration step function and the fraction of _monomers is identified with the ratio

(monomer size/layer thickness).

This

oversimplification

is

only

to be

expected,

since this paper

predates

the modern notions of self-similar

adsorption layers.

A clear summary of his

analysis

_can be found in _a recent review article

iii.

Alexander

predicts

the existence of three

regimes

^of

adsorption:

_a two-dimensional

(dilute

_or

semi-dilute) regime,

_a three-dimensional semi-dilute

regime

of unstretched

chains,

and

finally,

_a three-dimensional semidilute

regime

of stretched chains

(the

brush

state). Using

_a

grand

canonical

approach,

^{Alexander shows} that the chemical

potential

of adsorbed chains is _{not an}

increasing

monotonic function of the surface coverage. It exhibits a maximum in the twc-dimensional

regime

and _a minimum at the _crossover between the unstretched three-dimensional

regime

^and the brush

regime. Consequently

the

three-dimensional

regime

of semidilute unstretched chains is unstable and _a first order

phase

transition is

expected

to occur around _~ Gt

Rj~.

_~ denotes the number of adsorbed chains per unit area, and RF is the three-dimensional radius of

gyration

of the chains.

Indeed,

the transition _{occurs as soon as} the external chemical

potential equals

the minimal value of the chemical

potential

of adsorbed chains: it should correspond to a very low surface _coverage in the

vicinity

of the two-dimensional

regime.

The _purpose of this paper is to

give

_a theoretical

description

of the structure of adsorbed

lay-

ers of such

weakly adsorbing homopolymers carrying

_a

strongly adsorbing

terminal monomer, in order _to

investigate

the

adsorption

isotherm of these

polymers

and the

possible

existence of _a transition from _a SSAL

configuration

to _a brush

configuration.

To

perform

this

study

_we

use a very

elegant approach

due to Guiselin [8].

Using scaling

arguments like Alexander's [2], Guiselin has shown that _a SSAL _can be viewed _{as a} very

polydisperse

brush [9]. ^This

analogy

allows _us to

give

_a unified

description

of _a SSAL and _a brush and is _a convenient framework for

describing

the intermediate

expected

structure.

The _paper is

organized

_as ^follows: in section 2 _we review the

approach

^{of Guiselin} _[8]

and

apply

it to the _case of

weakly adsorbing polymers

with _a

strongly adsorbing endgroup.

Section 3 is devoted to the

study

of the equilibrium structure of these adsorbed

layers

both in the canonical and

grand

canonical

approaches.

The existence of _a first order phase transition from _a

quasi

^SSAL to _a

quasi

^brush structure will be shown. In section 4 _we compare the

model with _some

experimental

results of reference [5].

2 The model.

Let _us consider chains with

polymerization

index N

terminally

anchored to _a flat surface. The chains _are constituted

by (N- I)

_monomers which have _a weak interaction

(6

kT per

monomer)

with the surface. The last _monomer

(the sticker)

has _a

high

attractive interaction

(AkT)

with the surface. In the limit A _» 6 _{we can suppose} that all stickers of the adsorbed chains _are

grafted

to the surface. We denote

by

_~ the number of adsorbed chains _per unit _{area: ~ =} a~

/Z,

where Z is the _average surface _per adsorbed chain and _a is the Kuhn statistical

length;

_4ls

denotes the volume fraction of

polymers

at the

adsorbing

^surface. It is _very

important

to notice that the thickness _on which _Ala is defined is not the _monomer

length

_a, but the correlation

length

at the surface

(; (

is

always larger

than the _monomer

length

a. For

simplicity

_{we put}

a = I and kT

= I in the

following.

2. I "GUISELIN MODEL". The

polymer layer

_can ^be characterized

by

its

loops

and tails distribution [8]. ^Let us call

Vi (n)

and

Vi (n)

the number _per unit area of

loops

and tails made

of _{n monomers}

respectively,

and

D'(n)

=

D((n)

+

2D[(2n)

is the number _per unit _area of

pseudotails

made of _{n monomers.} Guiselin [8] defines

Din)

=

f$ _{D'(n)dn, Din)}

_{is then the}

number _per unit _area of

pseudotails

made of _more than _{n monomers.}

Guiselin _{[8] assumes} that the adsorbed

layer

is

analogous

to _a

polydisperse

brush and that the

pseudotails

_are all stretched in _a similar _way. This

approximation corresponds

to the step

profile approximation

of Alexander for the _case of _a

polymer

brush [2].

Using

this

analogy,

he

obtains the

following

relations:

~~

j~(~)I/3

dn

(1)

4l(z)

=

D(n)~/~

where _z denotes the distance from the surface

(the

n-th _monomers of all

pseudotails larger

than _{n are} at the _same distance from the

surface),

and

4l(z)

is the volume fraction of

polymer

at distance _z

perpendicular

to the surface. The Guiselin model refers to three-dimensional

states of adsorbed

layers,

since the correlation

length ((z)

and the volume fraction of

polymer

4l(z)

_are related

through

the relation

((z)

=

4l(z)~3R

We have the

following

conservation relation:

/nD'(n)dn ~

₌ ^N~

(2)

The thickness of the

layer

is

given by:

N

h =

/ _{D(n)~/~dn (3)}

1

All these equations are valid in the

overlapping regime,

I.e.

D(n)

>

N~6/5

_The

validity

of

equations (1-3)

_can be verified for the _case of _a brush and _a SSAL. In the brush _case

4l(z)

= ~

and h

= N

«~/~

_[2];

equations (1-3) give

the _same result. In the SSAL _case, the volume fraction

profile 4l(z)

varies _as

z~4/~,

and the thickness h scales _as

N~/~ ill. D(n)

has been calculated in reference [10]:

D(n)

+~

n~6/~

_and

corresponds

to the limit of the

overlapping regime. Putting

this

expression

for

D(n)

in equations

(1,3)

leads to the correct result for h and 4l.

The free _energy of the

layer

_can be

expressed

_{as a} functional of the

D(n)

distribution. In

principle,

the

equilibrium

structure _can be obtained

by

the functional minimization of the free _energy, which

gives

_a

Euler-Lagrange equation. However,

this method _supposes that the

distribution of

pseudotails D(n)

is _an

analytical

^{function of} _n. We shall show that it is

certainly

not the _case. Our _purpose is to

postulate

_a

given

form of the

pseudotails

distribution and to

verify

that it leads _{to a} smaller

equilibrium

value of the free _energy of the

layer,

than for both

a brush _{or a} SSAL.

2. 2 POSTULATED PBEUDO-TAILS DISTIIIBUTION. We

postulate

the

following

distribution of

pseudotails D(n)

D(n)

=

4l(/~

l < _n < n*

D(n)

₌

^4l(/~

^~ n* < _n < N ~~~

n

where _a

=

6/5,

and n* is _a parameter

characterizing

the distribution

D(n).

D(n) obeys

^all

physical requirements:

it is _a continous and

decreasing

function of _n.

By substituting

n*

= I _or n*

= N

respectively

in

equation (4),

_one gets the structure of _a SSAL

or a brush

respectively.

The

expected

structure of the

layer

is _a combination of the SSAL and the brush

(I

< n* <

N).

The

proximal region (n

<

n*)

has the inner _structure of _a brush with _a

grafting density #(/~

The external

region (n

>

n*)

has the structure of _a SSAL. This choice for the distribution of

pseudotails D(n)

is not

arbitrary:

it mimics the _scenario

proposed by Ou-Yang

and Gao [5]: ^at low surface coverage ~, because backbones _can

absorb, polymers

assume low

profiles

and the structure of the

layer

is _a SSAL.

By increasing

the surface _coverage

~, the stronger

adsorbing endgroups (the stickers)

will _{start to}

displace

the

weakly

adsorbed backbones from the

surface,

because backbones cannot

overlap

_very much in _a

good

solvent.

This

competition

between the stickers and the backbones will continue to

"pop"

the

polymers

backbones into the

solution, by anchoring

_more chains _onto the surface.

Consequently

small

loops

and tails _are

destroyed by increasing

_~.

It is convenient to define _an order parameter _{q =}

~/4l(/~

From

equations (2, 4)

_one gets:

~a-i

n =

a/(a i)x 1 ~) ₍₅₎

where _x

= n*

IN (I IN

< _x <

I).

As _a matter of

fact,

the

physical

parameter _q describes the

degree

of "disorder" of the

layer.

The ordered state

corresponds

to the brush

(~

=

4l(/~)

^{[2] and q = I}

^(putting

^{x = I in}

^{Eq. (5)}

leads to q =

I).

The disordered state

corresponds

to the SSAL

(putting

_{x =}

I/N

in

Eq. (5)

leads _{to q}

=

6/N;

in the

thermodynamical

limit N

= cc, q

equals strictly

0 in the disordered

state).

2.3 FREE ENERGY OF THE LAYER. In order to evaluate the free _energy of the

layer,

_we

use

scaling

arguments, ^{which do} not

give

precise estimate of

prefactors

in the free energy of adsorbed chains.

Consequently,

_we omit all these

prefactors.

Still the

scaling

arguments

predict

a correct order of

magnitude

and correct

scaling laws,

_we expect that the model describes well the essential features of the adsorbed

layer. However,

it should be noticed than _{an exact} but

quite complex

_mean field calculation in the strong

segregation

limit could be

performed using

the model of reference _[9] and the

analogy

^of Guiselin.

For _a

given

surface coverage ~ and _a

given pseudotails

distribution

D(n),

the free _energy of the adsorbed

layer

_per unit _area F reads:

F # Fu + Fs + Fconf ₊ Fads + Ftr

(~)

Fv denotes the excluded volume interaction _energy of the

layer

_per unit _area:

F~

₌

/~ ^§~(z)@dz

=

/~ D(n)%dn ₍₇₎

2 0 2

1

The second and third terms of

equation (6)

_are

entropic

contributions _to the Helmoltz energy.

l~ is the

stretching

_energy of chains per unit _area.

Using

the trick of MiIner _et al.

Ill],

it is evaluated _as the

stretching

_energy of _a dense

(and

therefore

ideal)

melt of chains of blobs of dimension

((z)

₌

D(n)~~/~

_[8]:

K ₌

/~ _{D'(n)dn /~ D(n')~/~}

_dn'

= F~

(8)

1 1

The

equality Fa

₌

Fv

in

equation (8)

is obtained

by

a

simple integration

_per part; it _seems to be

surprising,

but it is _a

noteworthy

consequence of the

equality ((z)

₌

D(n)~~/~. However,

Guiselin [12] ^{has shown that this relation between the local} correlation

length

and the local

stretching

^{of the adsorbed} chains minimizes the free _energy of the

layer.

F~onf takes into account the variation of the number of

configurations

of _an adsorbed chains

as a function of the

pseudotails

distribution. It has been shown in the _case of _a SSAL [12] ^that this contribution cannot be omitted. F~onf

=

-Log(Q) IA,

where Q is the number of

configu-

rations of tails and A is the _area of the

adsorbing

surface. There _are

AD(1)

₌

£$~~

AD

'(n) pseudotails,

but all tails of _same

length

_are

indiscemable;

_one deduces the number of

possible configurations _Q[12]:

(AD(i))1

(9)

~ "

flj~i _{(AD '(n))!}

Using

the

Stirling approximation,

the contribution F~onf reads:

F~onf ₌

-4l)/~ _Log (4l)/~)

⁺

/ i~

_D

'(n)Log(D '(n))dn (10)

The fourth term in

equation (6)

is the _energy

gained by adsorption

of the

weakly adsorbing

monomers and the

strongly adsorbing

stickers.

Fa~~ ₌ -I

(~~ ~)

/~~

(ii)

The last term in

equation (6)

^denotes the translational entropy of the

polymers

in the

layer:

Ftr =

~Log(~) (12)

All the contributions of the free energy of the

layer

_can be _now

expressed

as a function of the internal variable _x and the extensive _{one ~.}

3.

Thermodynamical equilibrium.

3, I CANONICAL ENSEMBLE: FIXED SURFACE COVERAGE. We _assume that the

averaged

surface _{coverage ~} and therefore the _{area per}

polar

head is fixed.

Using

the

postulated

distri- bution

D(n) (Eq. (4)),

the free _{energy per} unit _area of _a

homogeneous layer

can be

explicitly

calculated from

equations (6,

7,

10-12)

_{as a} function of _x and _~. It reads:

F =

~iN4l(~/~x

_{+ i} ^~

~a~

_{6 (4l~}

~) 4l(/~ Log (4l(/~)

⁺

4l(/~ ((i x") _(Log (a4l(/~) ^(Log(N)

^{+ i + i}

la)] ⁽ⁱ

^{+ ax"}

^Log(x)

^{A~ +}

^~Log(~)

(13)

where 4l~ is

given by equation (6)

and _a

=

6/5,

_{~i =} 11

/6.

The

equilibrium

state is obtained

by

the minimization of F with respect to x = n*

/N(i IN

<

x <

i).

The minimum

corresponds

to the value _x

= xeq

(~).

The order parameter

q(~)

and the thickness of the

layer h(~)

_can be then calculated

by putting

^the value _x

= xeq

(~)

in

equations

(3

and

5) respectively.

~~

~chaln

°~~'~~~~°

200

150

3= 2

xli~

50

-4 -3

~

lo lo lo

~~ ioo

a=io

iso

Fig. I. Dependence of the free energy per adsorbed chain fchain _on the internal parameter _x ^char- acterizing the pseudotails distribution

D(n)

for the following set of parameters: polymerization index N = 10000, adsorbing _{energy per monomer} 6

= 1, sticking _energy A

= 30, and different values of the surface coverage ~.

The

expression (13)

of the free _energy F does _not allow for _an

analytical

_treatment of the

thermodynamical equilibrium;

nevertheless the numerical calculations _{are easy} to do. The variation of the free _{energy per} adsorbed chain f~ha;n

"

F/~

is illustrated in

figure

I where

f~ha;n ^is

plotted

_on the parameter _x

characterizing

the distribution

D(n),

for the

following

set of parameters:

(N

= 10000; ⁶ = 1; ^A = 30; and three values of _{~ ~}

=

10~~;

_~

= l.13 _x

10~~;

_{~ =} 2 _x 10~~

).

Notice that these three values of _{~ are}

larger

than the

overlapping

surface coverage ~overi. "

N~~/~

= l.6 _x

10~~,

which _means that the adsorbed

layer

is continuous

(there

_{are no} individual adsorbed

polymers

referred to _as

"pancakes").

Each _curve presents

a minimum xeq. The value xeq ^increases

by

increasing ^the surface coverage. For small values

of _~, xeq ^is very small

(for

instance for _~

=

10~~,

_{xeq =} 9 _x

10~4),

and the structure of the

homogeneous layer

is very close to _a SSAL:

only

small

loops

_are forbidden

(n

<

10).

At

higher

values of _~

(for

_{~ =} 2 _x

10~~,

_{xeq =}

0.175),

the structure of the

layer

is intermediate between

a SSAL and _a brush

(quasibrush).

The transition from _a

quasi

^SSAL to a quasi brush is rather

sharp.

In

figure

2 the order parameter _q

corresponding

to the

equilibrium

value

xeq(~)

is

plotted

_{as a} function of _~ for different _sets of parameters

(N, 6).

The transition is

abrupt

for

high

values of N _or small values of 6, whereas it is smoother for smaller values of N and

larger

^values of 6. The

position

^of the transition _~t

depends strongly

_on the

polymerization

index N and _on the _energy

gained by adsorption

of _{a monomer} 6. One finds from numerical calculations

(see Fig. 3),

for 100 < N < 10~ and 0 _< 6 _< 8, that _~t

~

6N~°.~~

o.6

0.5 ,,.~

~

^0.4

/

i

0.3

/

o-1

-4 -3 -2 -1

lo lo lo lo

~i

Fig. 2. Dependence _on the order parameter of the adsorbed layer _{q =}

~/ (lbs)~/~,

_on the surface

coverage a for different set of parameters

(- -)

N

= 10~, 6

= 1;

(-)

N

= 10~, 6

= 1;

(--)

N

=104,

6

= 4;

(...)

N

= 104, 6

= 8;

(- -)

N

= 10~, 6

= 1.

In order to determine the

thermodynamical equilibrium

in the canonical

ensemble,

the free energy

F(xeq, «)

_must be _now

minimized, subject

to the constraint that the total amount of adsorbed

polymers

is fixed

(xeq

^{has been}

numerically

calculated for each value of _~: it is the value of _x which

gives

the absolute minimum of

F(x, ~)).

In

figure 4,

the free _energy F

(xeq, ~)

is

plotted

_{as a} function of _~ for the

following

set of parameters:

(N

₌

10000;

⁶ ₌ 1; A

=

10;

A

= 250; A

=

500).

In all _cases, the _curves present a first minimum at low values of _~ and _an inflexion point.

Increasing

the value of the

sticking

_energy A leads to the existence of _a second minimum for

high

values of _a. Two

phases

_can coexist if it

possible

to

construct a

tangent

line that touches the free _{energy curve at two} points _~i ^and _~2

(~i

_<

~2),

but otherwise lies

everywhere

below it

(common

tangent

construction).

Common tangency

at each of the two

points

of the free _{energy curve} is then found

by demanding

that at

point

~i each of the

following

two

quantities

is

equal

to its value _at point _~2

~~~~~~'

~~

and

~~~~

~~

b~

~~~~~'

~~ ~

ii~

At very low surface coverage

(but

still

larger

than the

overlapping

surface

coverage),

I-e-

a < al, the adsorbed

layer

is

monophasic;

it is _a

quasi

SSAL with _a small order parameter

10'~

_~ a b

~° a~N"°'~~

o- ^o-

l0~~

N~io~

S=1 10'~

1o2 1o3 1o4 1os ^{° 2 4}

~

6 8 1°

Fig. 3. The surface _coverage at the transition

(inflexion

point of the

curves in Fig. 2) _at is plotted

on the polymerization index N

(for

6

= 1) ^and on the adsorbing _energy

6(for

N

=

10000).

~ ^A=io

N=ioooo b=1

4 ^~"

F

2

0.01 0.015 0.02

O

-2

Fig. 4. Dependence _on the free energy per unit _area of the adsorbed layer F

(xeq., a)

_on the surface coverage a for the following set of parameters

(N

= 10000; 6

= 1) ^{and three different values of the} sticking _energy A.

q referred _{to as} the "disordered state". At _a

= al, a first order

phase

transition _occurs: the adsorbed

layer

separates ⁱⁿ two

phases:

_a quasi SSAL

(a

_{< al} coexists with _a quasi brush

(a

>

a2)

the

quasi

^brush is the ordered _state with _a finite order parameter _q

(it

is not

strictly speaking

_a

brush,

because _{q is} smaller than

I).

The

proportion

of each

phase

follows the rule

of _momenta. For _{a > a2,} the

layer

is

monophasic again:

the

single phase being

_a

quasi

brush.

The _range domain of coexistence of _two

phases (al

< a <

a2) depends

_on the

polymerization

index N _as well _as the energy 6, ^{but it}

depends only

_very

weakly

_on the

sticking

_energy

A,

as can be shown in

figure

4 whereas the proportion ^{of each}

phase depends evidently strongly

on A.

isoo

1000 _N=10000 $=1

soo

~

0.001 0.002 0.003 0.004 0.005

-soo ~i

-iooo

isoo

Fig. 5. Dependence _on the effective chemical potential of the adsorbed chains

p'

on the surface coverage a for the following set of parameters:

(N

= 10000; 6

= 1).

3,2 GRAND _{CANONICAL ENSEMBLE:} EQUILIBIIIUM WITH A RESERVOIR.

Generally,

the

total _amount of adsorbed

polymers

is not fixed. The

adsorbing

surface is in contact with _a solution of free chains in _a

good solvent,

and the surface _{coverage a}

depends°on

the chemical

potential

of free chains in solution _pext, and _on the osmotic pressure in the reservoir _arext. We restrict _our attention to the _case of _a dilute solution of

polymers

in the reservoir; in this _case, the osmotic pressure arext

acting

on the adsorbed

layer

_can be

neglected.

When the chains in the reservoir do not form micelles; _{pext *} I +

Log (#o/N)

+

Nu#o,

where

#o

denotes the

polymer

volume fraction in the reservoir and _u, the excluded volume parameter. ^At

equilibrium,

the surface coverage can be determined from the

equality

of chemical

potentials

of free _and adsorbed chains:

pext " p "

~~)

^~~

₍₁₄₎

~=~~~_

It is _more convenient to include the

sticking

_energy A in _an effective external

potential:

Equation (14)

_can be rewritten: p[~~ ₌

p'

where p[~~ = pext + A and

p'

= p + A

In

practice,

one needs _to calculate

numerically

the

dependence

of the chemical

potential p'(a). Figure

5 illustrates the

dependence

of the chemical

potential p'(a

_on the surface _coverage

a for the

typical

set of parameters

(N

=

10000;

6

=

1).

It is

important

to notice that

p'

is not a monotonic function of _a. It has _a maximum in the

quasi-SSAL regime

and _a minimum in the

quasi

brush

regime. Therefore,

there _{are no} stable solutions in the intermediate

regime

and _a first order

phase

transition is

predicted.

For p[~~ ^<

p[;~,

_a

single quasi-SSAL phase

is

expected.

For

p[~~

< p[~~ <

p[~x,

two

phases

should coexist: _a

quasi-SSAL

and _a

quasi-brush. Finally,

for

p[~~

_<

_p[~~,

_a

single quasi-brush

is

expected.

It should be

noticed, that,

after

passing

the

point

of

transition, polymers

should

quickly

saturate the interface.

Experimentally,

this

corresponds

to the _case that after

reaching

the

quasi-brush phase, adding

_more

polymers

to solution will

sligthly

increase the number of surface chains and _most of chains will _{go to} the solution. In

figure

6 the

dependence

of the value of the external chemical potential p[~~

corresponding

to the

change

in the

majoritary phase (from quasi-SSAL

to

quasi-brush)

is

plotted

_{as a} function of N and 6. It appears that: p[~~ _+~ 6N~°.~4

1oo0 ~i ~ ~

exi

~~ f~~^~~

CA' ~

~

W ~

~ ^~

loo

$- i N=10~

lo

1o2 1o3 1o4 1os _{o 2 4 6 8 lo}

N 6

Fig. 6. The value of the external effective chemical potential

p(~t

corresponding to the change of

the majoritary phase

(quasi-SSAI

to quasi

brush)

is plotted _on N

(for

6

= 1) ^and

6(N

=

10000)

respectively.

4. Discussion.

Alexander

predicts

the existence of _a first order

phase

transition for

weakly adsorbing polymers carrying

_a

polar

head [2]. However, he

predicts

that the transition should _occur from _a low

density regime

to _a

high density regime.

The low

density regime corresponds

to a structure of

"pancakes",

I-e- the adsorbed chains _are

twc-dimensional;

the

high density regime

is the brush state. This transition is referred _{to as}

"pancake

to brush" transition. It should _occur around

a £i

Rj~ iii.

The model

developed

in this paper

disagreei

with this

interpretation.

It suggests that the transition should _occur from _a

quasi-SSAL

to _a

quasi-brush regime.

It _means that

even at surface _coverage still

larger

than the

overlapping regime,

the

layer

thickness remains rather thin

(% Rg),

but stable.

Moreover,

the ordered _state is not a

perfect brush,

but _an

intermediate structure between _a brush and SSAL

(the

order parameter _{q +~} ^0.6 <

1).

Ou-Yang

and Gao [5] have

interpreted

their

experimental

results in terms of this Alexander's

"pancake

to brush" transition. But there is _no

experimental

evidence that the transition _occurs around _a £i

Rj~.

For instance,

they

have shown that the inflexion point of the transition _at decreases with the molecular

weigth

and goes like _at

~

N~~. _This

implies

that the

pancakes

should be formed

by

ideal chains at the

surface;

but there is _no

physical

_reason for which chains in

good

solvent should become ideal _{near an} interface.

The present ^model

predicts

that _at » aoveriap, that _{at goes} like bN~°.~~ and the adsorbed chains remain swollen _as the free _ones do in

good

solvent. The exponent ^{0.94 is} very close to the

experimental

one -I.

Ou-Yang

and Gao have also shown that the transition

regions

_are almost

overlapping

for both C12 and C16

endgroups.

It _means that the transition

region depends

_very

weakly

_on the

sticking

_energy A. Our model _agrees with this

experimental

^fact

(see Fig. 4).

The _same authors have noticed that in the brush

configuration,

the

polymers

_are streched to about four _or five times that of

Rg

in solution.

Using

_a

large

set of parameters

(10~

^{< N < 10~}

0 _< 6 <

8),

the present ^model

predicts

that in the

quasi-brush regime,

the chains _are stretched to about two times that of

Rg.

^This

discrepancy

indicates

probably

that _our model is too

crude: the absence of precise ^{estimates of the}

prefactors

in the free energy of the

layer,

_as

well _as the

rough

calculation of the entropy of the

pseudotails configurations

does not allow to

predict quantitavely

the n* of the

quasi-brush

structure

(see Eq. (4)).

The calculated value xeq ^at the transition

(xeq

_+~

0.3)

is

certainly

underestimated. A _more

sophisticated

model [9]

could

perhaps

be used.

5. Conclusion.

In this _{paper we} have

theoretically

discussed the

equilibrium

of _a

layer

formed

by weakly adsorbing polymers carrying

_a

strongly adsorbing

sticker. We have shown that the structure of the adsorbed

layer

is intermediate between _a SSAL

iii

and _a brush [2]. ^{The model}

predicts

the existence of _a first order

phase

transition from _a

quasi-SSAL

to a

quasi-brush regime.

The transition _occurs at surface _coverage

larger

than the

overlapping

_one. This

theory

is in

good

agreement with the

experiments

^of

Ou-Yang

and Gao [5]. This ^{transition is}

quite

^{different from} this _one

predicted by

Alexander _more than fifteen _{years ago.} Recent

experiments

_[13] could be

possibly reinterpreted by using

^the present ^model. However small

angle

neutron

scattering techniques

^should be

performed

in order _to

investigate

the inner structure of the adsorbed

layers

and the lateral

heterogeneities

in the two

phases

coexistence domain.

Acknowledgements.

1thank J-F

Joanny,

A. Johner and G. Porte for

helpful

comments. I _am

grateful

to A.

Ajdari,

S. Milner and S. Nechaev for

stimulating

discussions.

References

[ii

de Gennes P-G-, Macromolecules 14 (1981) 1637.

[2] Alexander S., J. Pllys. IFance 38

(1977)

983.

[3] Halperin A., Tirrell M., Lodge T.P., Adv. Polym. Sci. loo

(1991);

MiIner S-T-, Sciences 251

(1991)

905.

[4] Cohen Stuart M., Cosgrove T., Vincent B., Adv. Colloid. Interface Sci. 24

(1986)

143.

[5] Ou-Yang D., Gao Z., ^J. Pllys. II IFance 1

(1991)

1375.

[6] Auroy P., Auvray L., L4ger L., Pllys. Rev. Lent. 66

(1991)

719.

[7] Halperin A. Macromol. Rep.

(suppl.)

A29

(1992)

107.

[8] Guiselin O., Europhys. Lent. 17

(1992)

225.

[9] MiIner S.T., Witten T-A-, Cates M-E-, Macromolecules 22

(1989)

853.

[10] ^{de Gennes} P.G., C.R. Acad. Sci. Paris II183

(1988)

739.

[11] MiIner S., Witten T., Cates M., Macromolecules 21

(1988)

2610.

[12] ^Guiselin O., Structures de polymbres adsorb4s. Boucles et Miroirs, PHD-Thesis University

Paris VI

(1992)

unpublished.

[13] Field J-B., Toprakcioglu C., Dai L., Hadziioannou G., Smith G., Hamilton W., J. Phys. II IFance 2

(1992)

2221.