Polymer adsorption on heterogeneous

J.

Phys.

ii

France

(1993)

3

JANUARY1993,

121-138

121

PAGE

Classification

Physics

Abstracts

68.45-61.25-82.65

adsorption

Polymer solid

monolayers

surfactant

on

and

heterogeneous

surfaces David

(~) School Tel-Aviv

(~)

of Physics University,

Institut

(Received

Abstract. where

the

adsorbing

29

monolayer

Astronomy,

and

Sadron,

July 1992,

6

Raymond

Boussingault,

rue

accepted

October

6

soluble for

interactions

Exact

of

Sciences,

Cedex,

France

flat

We

a

fixed

1.

is

increase

on

average

of the

neutral

surfactant

for

surface

adsorption

chemical

polymer

between

potential) and

or

surfactant.

insoluble Even

(at

if the

isolated polymer chain adsorbs via a adsorption of a polymer solution can induce phase transit-ions iii an insoluble monolayer that phase separates into dense regions where the polymer adsorbs and dilute regions from which the polymer is depleted. Phase transitions induced by polj,trier adsorption can also for soluble monolayer. For a surface with periodic occur quenclied heterogeneities we calculate the adsorbed of polymer as a function of the amount wavelength of the heterogeneity.

local

polymer

Faculty

but heterogeneous surface, in both cases always find that the heterogeneity of the annealed heterogeneity example of surface with a

surfactant attractive

Strasbourg

67083

adsorption. As an monolayers (at

enhances

either

Sackler

Israel

1992)

lve study polymer adsorption on heterogeneity is quenched or annealed.

surface

Beverly

and

69978, Tel-Aviv,

Ramat-Aviv

concentration)

fixed

a

Charles

consider

we

(~) and Jeaii-Fr.ail&ois Joaiiny (~)

AiideInian

concentration.

an

The

Introduction.

adsorption fi.oin solution plays a major role in a number of industrial problems such lubrication, stabilization, adhesion and and has been the subject of as numerous experinieiital and theoretical Over the last few years a rather studies. detailed description of the and properties of adsorbed polymer layers on "ideal" surfaces (in the of structure sense Polymer

colloidal

an(I cheiuically homogeneous) has been developed [1-5]. adsorption of polymers is a cooperative efsect, the polymers which are attracted adsorbed by the surface strongly and form thick adsorption If the layer. adsorption arc a takes place from semi-dilute polyiuer solution, the thickness of this layer is of the order of a length ( of the bulk polymer solution. the correlation On the other hand, if the adsorption proceeds fi.oiii a dilute solution, the thicl;tress is of the order of the radius of gyration of an isolated cliaiii and can, thus, be as large a~~ a few hundred angstr6ms. In the adsorbed layer, the polymer chains form loops with a broad distribution of sizes and the of the structure

perfectly Since

flat

the

JOURNAL

122

layer

self-siniilar;

is

distance

from

the

the

monomer

surface.

This

PIIYSIQUE

DE

profile decays

concentration

self-similar

N°1

II

structure

[3]

as

inverse

an

well for

accounts

law of the

power

of the

many

observed

polymer layers [2, 5]. neither homogeneous in composition nor Most real surfaces smooth. Although are, however, influence the adsorption of polymer solutions, these the heterogeneity may strongly non-ideal theoretical situations have received relatively little attention [6-12]. The aim of this paper is to study the effect of chemical heterogeneities on polymer adsorption while keeping the adsorbing properties

surface

adsorbed

of

flat.

types of adsorbing

Two

surfaces

[12]. A heterogeneous

faces

quenched

considered:

are

surface

quenched

is

if the

and

disorder

annealed heterogeneous surcharacterizing the adsorbing A typical example is a sense.

degree of freedoiu in the thermodynamic chemical irreversibly attached impurities which laterally diffuse on cannot disordered surface for the the surface. A quenched environment represents a difserent local adsorbing iuodel it by interaction which has local fluctuations about ive surface iuonoiuers, a value. randoiu quenched disorder first calculate the of For properties [13], one must mean a adsorbed polymer layer for a given realization of the surface interaction and then the average these properties ~vith respect to the disorder. On the other hand, if the heterogeneity is annealed, the disorder of the adsorbing surface equilibriuiu. This is, for instance, the of chemical heterogeneities is at thermodynamic case that fi.eely diffuse laterally on a solid surface. Another example which we consider in can details is a surfactant monolayer at an air-water interface. The annealed heterogeneity is then surface

solid

is

"fi.ozen"

a

with

surface

the

surfactant

concentration

surfactaiit-polymer properties

such

coupling its

as

phase

per

unit

affects

transitions

important issue to be adsorption as well as

The

area.

the

polyiuer and

surface

addressed the

is how

surfactant

the

surface

tension.

organized as follows. In the next section, we present general considerations paper polyiuer-surfactant therinodynaiuics of mixed Then, in section the 3, we study systenis. on the adsorption of a single polyiuer chain on a surfactant monolayer. Although this is a rather academic problem, it gives solve physical insight into the problem of surfactant-polymer inSection 4 is devoted adsorption of polymers from semi-dilute solutions teractions. to the on monolayers. The descripannealed heterogeneous surfaces, and iuore specifically on surfactant solution properties field level along the lines of the tion of the polymer reiuains at a mean is

Our

density functional apl)roach of polymer adsorption [14, 15]. Section 5 considers pequenched heterogeneous surfaces both in the case where the surface is neutral on and in the where it is strongly adsorbing. The last for the polj a~, concentration, 6c 0, If t
0), the equivalent homogeneous For negative repel the polyiuer. When i is decreased, there is a first order transition between depletion and adsorption at a,,alue to- We distinguish between three liiuits: For

this

effect

to

-ci~/1(/~ -3a2/81(/~ > 7 > -a2/21(/~

314(-17ci14)~/~li

ci~

"

0 > 7

to"-271(/~-3ci~/4 to

-ci~/21(/~

-7~14/ci~

=

»

»

(3.8)

7.

monolayer undergoes a phase separation and negative, the pure surfactant iuonolayer state used to derive equation (3.8) is that of the denser coexisting phase. As was excluded volume interactions mentioned above, it is possible to include for the polymer chain. Ilowever, since the qualitative picture does not change by including these interactions, ,ve do not treat this effect in the present paper. The adsorption of a single polymer chain on a quenched disordered surface has been studied and the chain is large but finite, there by several authors [6-10]. If the surface is infinite the ,vith a sufficiently large is always a region on infinite surface attractive fluctuation for the chain to adsorb. Thus, the polyiuer chain always adsorbs on a surface with quenched heterogeneities, but this is very specific to adsorption of a single finite chain and will not be that

Note

the

discussed.

further

adsorption

Polyi»ei

4.

In

when i is

reference

this

section

(Gibbs)

discuss

we

adsorption

the

mouolayers:

suifactaut

on

of

polymers

semi-dilute

from

a

semi-dilute

polymer bulk

solutions.

solution

on

soluble

(Langiuuir) iuonolayers. The equilibrium and interfacial properties of the combined from the total free energy of the polymer-surfactant systeiu can be obtained mixed system: Ft discussed in sections 2 and Whereas F~ and Fp~ have been F~ + Fp + (,s. 3, the interfacial fi.ee energy (, for a seiui-dilute polymer solution has to be specified. Within field approach [3, 15] that ignores excluded volume correlations, it can be written (in a mean and

insoluble

"

units to

the

of kBT

and

local

per

unit

surface

(4.I)

concentration

describes cb

"

1b/

a

"

a

functional

polymer

#~(z

oc). -

profile in The

polymer

of the

cp(z) by 1~2(z)

=

contact

profile

order

parameter

1~(z)

related

cp(z)

~(l~~

/~~ (~lb)~ ~

~P Equation

as

concentration

iuononier

~l~)~) ~Z. with

obtained

a

from

(~.i)

polymer reservoir at infinity of equation (4.I) decays towards

JOURNAL

128

the imposed bulk polymer chains. excluded

value

The

volume

c~

co + bc and

=

the

It is

convenient

~a~("bC

z/(

e

u

/

fl The

#~ = with

e

+

2

free

energy

#(v

0),

=

67(

(V#)~

=

fl

is

and

t, 14 and

ci,

2

We

then

polymer

mininiize

one

(#~

l)~) of

surfactant

the

#s and

value

surface

fi.ee

fl

thus

can

du

the

#(

& 6c 2

with

fi.ee

fl is is

where

At this

free

separately 4.I

iiuplicit and

energy insoluble for

ADSORPTION

insoluble, the

an

theriuodynaiuic

the

surfactant

ON

function

and

AN

concentrat.ion

+

as

separates

and

d~fl W When

6c

=

0, the

monolayer

u

polymer

the free

profile 0, #(u)

=

(4A)

2

order

#s and

energy, for the

coth(u+[),

=

by #~

parameter

obtain

surface

relation

a

value

equation (4A) adsorption of =

where coth

I

the

between

6c

of the

)16c~

+

+

surfactant

polymer

concentration

~14 6c~ 6c

surfactant

(4.6)

given by equation (4.5).

concentration

adsorption.

This

becomes

fi.ee

When

MONOLAYER.

is is

obtained now

from

discussed

the

monolayer

is

and monolayer remains constant The stability of the monolayer is fl; the monolayer becomes per unit area negative. This defines a spinodal becomes

in

molecules

conserved of the

iuetastal)le

=

)il.

~14 6c~

(4.5)

LANG&iuiR

the

parameter.

order

by studying the convexity of the free

which

z

of the

concentration

the

if the

line

at

function

a

is + 4)

unstable

derivative

0)

=

3tfa~2~fip~,

%

=

obtained

second

with

inonolayers.

surfactant a

=

~[4s

~ &

determines

INSOLUBLE

of

6af, I

e

4

the

parameter

to

value of the

the

soluble

is

&

+

2

concentration

surfactant

of the

equilibrium, completely

number

total

I)(#j

ibc~

+

Miniiuizing

6c.

respect

fl

energy

(is

=

~fi)

~fi2(z

lengths:

and

polymer profile (#(u)),

concentration

energy surfactant the

total

the

express

surfactant

(4.2)

Ft/(a2(~fi(/3kBT),

=

6c +

We

the

concentration

(4.3)

the

to

between

tb/tbb.

+

profile (#>(u)) yields the usual order polymer solution on an ideal solid surface seiui-dilute a I is an integration constant related to the surface value respect

a

obtains:

functional

a

the

of the

positive

z/(a~/3u1~()~/2

=

7:

with

7)#~.

+

concentrations

4

By properly rescaling Ft, 14 + 314(a~~~fip~, and f

elasticity

the

interaction

with

surface

the

dimensionless

introduce

to

for

accounts

volume

interaction

direct at

"

term

conditions.

nionoiuers

FPS

N°1

II

excluded

the

in

u

Fps

term

gradient

The

describes

term

good solvent includes only the

parameter

coupling

The

concentration

infinity.

at

~fib

second

PIIYSIQUE

DE

energy

energy

regions

unstable

I ~ ~ j ~~2 ~ unstable

WI

/~2

at

a

@

~

positive

value

~~ ~~

I =

is

jj ~~

~~ ~

i +

jj

~~'~~

N°1

IiDSORI~'rION

POLYAfER

polyiuer

the

where

order

paraiueter

the

on

Is

I

ON

surface

j(I +

=

129

AIONOLA"ERS

SURFACTANT

#~ is

@i).

(4.9)

phase by the construction of the free value of bc, the standard tangent energy comiuon a (4.7). If I is io value higher than the spinodal value given by equation transition at a occurs larger than io, the monolayer retrains homogeneous and the polymer does not adsorb. If I is phase onto smaller surfactant than io> the monolayer phase separates into two phases: a dense and a dilute surfactant which the polymer adsorbs phase from which the polymer is depleted. with a of a interaction In the strong adsorption liniit, &~ » 14> the energy monomer molecule is of order kBT, and a first order transition takes place at surfact.ant When

by changing the precise location of

decreased

is

transition.

The

transit-ion, to is positive

the

The

surfactant

independent In

the

polymer

of the

dense

the

obtained

(4.10) concentration bc

in

&3 /314

=

the

bulk.

which

is also

surface

order

concentration.

polymer

the

adsorption

is

enhanced

and

the

is

parameter

#~ The

order

first

&~/1814.

=

dense

bulk

phase,

surfactant

monolayer undergoes a function of 6c) is as a #. For fixed

independent of the polymer phase at the transition is

and

in

concentration

(io

transition

io At

the

temperature,

this

experimental

relevant

cb((4s -1).

quantity

is

=

the

+ &~ /314

(4. II)

polymer surface independent

r

excess

f/~(cp(z)

=

cb)dz

=

polymer bulk concentration ot~~ corresponding to a very strong adsorption. and it is of the order of r Although we have discussed here possible phase transitions in Langmuir monolayers inadsorption, ,,,e have not phase diagram of duced by polymer constructed the global surface polyiuer-surfactant iuixture. Sortie general be found in reference [17]. the considerations can calculated using the full free energy of the surfactant In principle, the phase diagrams can be monolayer valid in the whole range of and not only a Landau expansion. The concentrations phase diagram of the monolayer contains two lines, the "special transition" line transition between adsorption and depletion for the polyiuer and a binodal line representing the phase In

the

surfactaiit

phase

§

dense

it is

of the

=

of the

surfactaiit.

concentration

co

transitions factant

The

"special

monolayer

of the

transition" and

has

only depends on the approximated here by

nominal

line been

a

vertical

sur-

line

temperature-surfactaiit concentration binodal line is shifted by plane. The surfactant coupling t-o t-he adsorption of the polymer. The spinodal line of this transition is given by equation (4.7) ,vith 6c 0 ,vhere t is the inverse osiuotic compressibility of the monolayer calculated derivat.I,>e of the surfactant froiu the second free energy F~. Using the Bragg-Williams form of the fi.ee energy, find that the spinodal line is always shifted to higher temperature by ,ve polymer and that the critical corresponding maximum the adsorption of concentration to the than for the pure the of the spinodal line has a higher value surfactant. general, In two lines of the phase diagram surfactant concentration which be either smaller larger meet at a or can concentration of the surfactant than the critical phase transition.

in

the

the

=

4.2

ADSORPTION

surfactant

imizing attractive

ON

the

surface for

the

soLuBi~E

A

concentration fi.ee

is

a

energy,

polymer, I-e-,

GIBBS

non-conserved 7 is

parameter

order

equation (4A). ,vhen

In

NIONOLAYER.

large,

In

the

the

case

surface

a

and

soluble its

where free

monolayer, the by minmonolayer is strongly has a single minimum

value the

energy

Gibbs is

obtained

130

JOURNAL

adsorption

the

and

creased free

belo,v

these

small

is

first-order

When is 6c

and

This

the

result

of r that

Note

surfactants in

the

on

polymer

be

can

values

As 7 is dethe

concentration),

bulk

exchange of stability change in surfactant

an

the

oft

surfactant.

and

of the

between concen-

at

polymer

the

[12].

lo,,,er

oft, the transition

adsorption

concentration

is r

excess

order

oft, the

values

to-

values

surface

first

The


e

life do not take influence of the

the

is

but

solution

aqueous

an

is

virtually

of

surfactant

no

from

polymer

of the

interaction

interface.

adsorption

Polyuier

5.

and

and

solution

surfactant

there

the preferred state for t adsorption vanishes, at large

reference

condition

a

and

in

in

=

solution

such a

in

(independent

high

At

N°1

adsorption is weak. For a critical value to> there is a discontinuous and a much the iuonolayer towards a higher concentration stronger

cia2cb/t

=

obtained order

oft

function

a

decreased.

II

polymer

between

is also

equation (4.10) the

coupling

the

&~ /(4.33/214),

=

As

i is

the

average

&/(2i)

=

§c

value

when

transition

adsorption.

by

enhanced minima.

t,vo

niiniiua

t,vo

tration

a

has

energy

is

critical

PIIYSIQUE

DE

heterogeneous

queuclied

on

surfaces.

A typical example of quenched heterogeneities of the adsorbing surface. to the case contaminations which heterogeneities due to chemical terminally attached to a are solid surface. If the amplitude of the heterogeneity is small, it can be expressed as a sum of semi-dilute Fourier modes. discuss the adsorption of a polymer solution onto a solid We first surface ,vith one single q-iuode periodic heterogeneity. Two considered: (I) a strongly cases are adsorbing surface ,vith a small periodic iuodulation; and (it) a neutral surface whose average transition line ("special bet,veen adsorption and desorption. In the is just on the transition" latter extend results randoiu heterogeneity which is a sum of all q-modes. true to our case ,ve a We

turn

now

would

be

STRONG

5.I

ADSORPTION

polymer solution approximation by the

of

ON

the

~/~BI~ first

The be

teriu,

noted,

and

the

Fp,

ho,vever,

gradient

"

in

/ ((

is the that

terra

,vith

contact

functional

F

Fp

=

(~~/~)~ ~

bulk

since

(V~fi)~

HETEROGENEOUS

PERIODIC

A

a

Jl~)~j ~~

l~(~~~

both

given

is

~~

in

(perpendicular to the surface). The second term, Fp~, is the interaction between the polymer and the (assui»ed attractive) 7 coupled to the polymer the surface, concentration ~fi). on As our example of a periodic heterogeneous surface, we choose a coupling varies periodically along the x-direction with a wavevector q.

In

the

weak

strong adsorption heterogeneities for

liniit

1/71

which

67

is of
ii

+

+

671f> and

heterogeneity liniit the polymer concentration profile can be found perturbatively concentration profile order in the heterogeneity strength, the in 6f. At zeroth I), coth(u where the #o(u) surface, ideal homogeneous for + an same as 21ii b/( is such that sinh 2i extrapolation length I perturbation to this profile is periodic with a wavevector q, the order first order =

=

=

the

parameter

be

can

written

as

# Substitution

lo + 6f #1(u)

=

equation (5.3) gives

in

~

d#i

where Far

«(

a,vay

correction

fi.oin

the

the

order

to

#1(u) where sionless

A is

an

~

j~(o)

~

(5 5)

surface, the order parameter is # parameter profile is then given by

solid

=

A

e~~~("+~~ (coth~(u

integration

decay length K[

decay length is of the (q( » I) the decay length this

integration

constant

distances

order

of the

is of the

A

can

e

r/cbf.

a

'~~

surface.

At

(5.6)

small

with

the

in the

«ii

limits:

two

i~

«ii

«

£e~~~

Kii

»

1.

r.

We

I.

In

«

I),

wavevectors

«ii

» I and

dimen-

(q(

wavevectors

length f, whereas at large period of the heterogeneity q~~

of the

The

~

correlation

estimated

«

I

I

(5.7)

bKi

relevance

For

+

vanish.

must

exponentially

decreases

solid

the

bulk

order

be

=

ofexperimental f

from

I)

+

#i

and

I

=

coth(u

+ Ki

=

A

quantity

far

I)

+

correction

This

constant. at

A

excess

j)

k~.

4 +

e

+

#il

for

lllil~~~+

ill (0)

+

$

(5A)

kw.

cos

following equation

the

~~

? ~~~~

The

obtained

(5.3)

expansion as an is exactly the

surface

equations

smooth,

667f.

In the

The

dimensionless

the

0

fi

=

qf, and

+

0

=

f

is

on

=

+I(w)#s

parameter

and

as

same

k

and

131

#s.

flat

otherwise

are

)) 6f

z/(,

%

u

# + ~fil~bb with a surface surface is only heterogeneous but is

MONOLAYERS

SURFACTANT

is the

surface

homogeneous

excess

to

surface

"

define the

here presence

a

dimensionless of

a

periodic

heterogeneity +o~

r

In

the

limit

of

small

,vavevector

=

ro

26f

+

(«ii

qb < I,

to

t =

#o41 du.

coskw

« +

(5.8)

I)

&I

cos

kw.

(5.9)

JOURNAL

132

DE

PIIYSIQUE

N°1

II

(b7/71) cos qx. As soon as the wavechange in the surface is br/ro excess heterogeneity is larger than the extrapolation length b, qb > I, the fluctuation of the surface follows the heterogeneity of the solid surface. In the strong adsorption limit excess considered here, the extrapolation length b is of the order of the size a. In practice, monomer all heterogeneities of the soli(I surface will thus fluctuations of the surface create excess. of an adsorbed polymer layer to a heterogeneity is different from the Note that the response of a thin wetting film fluctuations for which it has been shown [21, 22] that with a response wavelength smaller than a finite healing length (in general larger than the film thickness) are damped. For the liquid fiIni, the daiuping of the surface corrugations is due to surface tension between the film and the there is no Iii the polymer adsorption problem, interfacial vapor. tension bet,veen the adsorbed layer and the bulk solution. This result can easily be understood looks at the local adsorbed layer. Locally, the adsorbed layer has the if one of an structure seiui-dilute polymer solution, it thus responds of perturbations with a wavelength to structure a length. The heterogeneity produces a perturbation on the correlation larger than the local ,vhere the local correlation length is the extrapolation length b. Hence, it perturbs the surface profile as soon as the wavelength is larger than b. concentration relative

The

length

"

of the

within the Although this result has been obtained mean-field approximation, the same physpicture prevails ,vhen excluded voluiue interactions taken into This has been account. are de Gennes free energy checked quantitatively using the Cahn appropriate for polymers in a

ical

good solvent. the

In

liniit

is

excess

small

,,.a,,elengtli heterogeneities (qb

of very small and in the

approximation

field

mean

»

I),

the

fluctuation

decay length is the

its

in

bulk

the

surface

correlation

length f

~ ~~'~~~

4+~~(2 5.2

NEUTR,iL

PERioDic

AND

A

suRFAcEs.

DISORDERED

neutral

surface

with

chemical

het-

preference to the polymer adsorption. a absence of heterogeneity, the concentration profile is flat, the reduced order parameter In the vanishes ro between the polymer is # I and the surface 0. The coupling constant excess here only the limit and the surface is given by equation (5.I) ,vith 71 ii consider 0. We where the aiuplitude 67 of the heterogeneity is small. The or(ler parameter profile is also obtained froiu equation (5. I ). In order to find this profile, expand # in Fourier series we

erogeneities

is

surface

wltose

averaged

I)ehavior

=

has

no

=

"

"

#

~j

+

=

#n(u)

(5.ll)

nkw.

cos

n

We

then

expan(I

in

If only'

order

the

polymer profile

t-he first

in

ii

harnionic

po,vers

does

not

where The

K(

%

first

4 + k~

order

as

in

the

correction

previous to the

~

ii

+

cos

reduced

vanish.

$)41

("1

)

of the

kW

"

=

It

heterogeneity 6§. Up the equation

to

first

satisfies

0

0

(5.12)

section.

polymer profile

41(u)

=

((

is

thus

exp(-«iu).

(5.13)

POLYMER

N°1

To

first

order in

order

6§

6f,

in

in order

to

the

ADSORPTION

surface

obtain

lo and #2 have

components

requires only the

surface

lo

of

to

3&12

d~4i~~

4j(2)

of the

surface

excess

given by

It is

eXp(-2Kiu)

~j

d2u

°

calculation

#i~

6f,

in

expand up to second only the first two Fourier

reasons

The

133

thus

must

one

(6f)2.

order

second

MONOLAYERS

symmetry

order

to

and

For

excess.

contributions

contribution

vanishes

still

excess

the

SURFACTANT

ON

1

This

leads

to

~j? 4i~~

Therefore,

w>e

can

deterniine

bi~

~

2Ki

(5.14)

3bi~

j2 ~~

$

"

d#[~~

~

~

3 + k2

surface

i-educed

the

4(3

~-2~iu

k2)

+

+OJ

f

excess

(5.15)

K(

+

du[-#(

=

~j2 r

Thus, and If

although the average and ro surface is neutral heterogeneity enhances adsorption, as in the the of heterogeneity is small, q( < I, ,vavevector

The

surface

correlation

roughly

adsorption length (. If the

becomes

excess

length of the independent

the

thus

oft

heterogeneities

=

/ dp (b7(0)b7(p ii of the

surface

P

G'(q correlation

correlat.ion

The r is

surface of the

length is of length (, is excess

various

can

the

be

w,avevectors

«

r > 0

section

3.

case

decreases

the

with

in bulk

concentration

(5.18) the

larger

is

wavevector

is

neutral,

length on the wavelength larger

the than

a

random,

but

,vhich

for

siiuplicity

equal

is

bulk

the

it

characterize

we

is

than

taken

to be

to

profile (, and

correlation

by

its

power

Lorentzian

A2 =

bulk

the

concentration

surface

(5.19)

+11

heterogeneity

surface

surface

discussed

correlation

q~

The

heterogeneous

,vith

~

bulk

r

surface

periodic

is not

e~q

the

on

annealed

when

average

local

the

0,

"

~~~~~~~~

If the

and

(5.16)

=

siuall

very

solution.

layer follo,,>s

heterogeneity

spect.ruin G(q

tions

k2).

+

4«1

the

r

is

»(2

=

2#(~~]

+

2

o

~.

is

supposed

to

be

much

smaller

than

the

(. obtained

in

For

q.

a

each

first

approximation vector

wave

the

by summing contribution

to

over

the

the

surface

contribuexcess

given by equation (5.lfi).

/ )( ~i' )/ ~

r

=

G(q ).

(5.20)

134

JOURNAL

In the

limit

is

where

(


energy with a non vanishing cubic teriu in bc. The coefficient 13 of this cubic term is positive if co > ck and negative if co < ck. lIere, ,ve discuss single chain adsorption on such a monolayer in the neutral where 7 thickness adsorbed polymer chain D is a 0. The of the case To + ciao function of the local concentration in the iuonolayer. coefficient 13 It is independent of the and is given by the values found at the critical point. The total free energy function of a as the surfactant is obtained along the same lines as equation (3A), concentration

When

the

reference

separation

concentration

Landau

the

=

adsorbs is

,vhen

continuous:

6c

=

surfactant

t < t~

=

the

not

the

of

critical

concentration

free

surfactant

~~)

j6c~(t

+

)13 6c~

larger

concentrations

a~ and the change in the

1 ic)

=

is

for

is

=

Ft If13 is positive, (for

co

expansion

/13.

thickness

The

as

)i

D* ~-

the

than

tc

critical

value),

the

due

to

concentration

monomer

of the

(Al

(14 6c~.

+

adsorbed

layer diverges

at

polymer chain adsorption

the

the

transition

/13]~~

(A2)

ci

with

a

different

po,,>er

If13 is negative, ([or

corresponding

to

the

la,v

divergence

adsorption

than

when

concentrations

surfactant

of

the

polyiuer

13

"

smaller chain

0, equation (3.5). than is

the

critical

discontinuous

value), and

the occurs

transition at

to

"

JOURNAL

136

a~+21(/(914). is 6c

=

The

juiup

in the

The

thickness

-213/(314).

PHYSIQUE II

DE

concentration

surfactant of the

finite

Appendix

expansion

surface zero

long

as

13

as

quenched

for

I'>lx)) adsorption,

polyiuer

the

polymer r

where

A is the

~vhereas

for

surface

is

area,

annealed

the

quenched

(tJ)h

e

=

f (~b2jr)

A

rq

case

P(h),

Hamiltonian

degree

simple

where

case

distribution

about

of

quenched

for

and

disorders

annealed

llre~~~ioih iie~~~ioih

[13].

~~~~

~

~~~ =

is

the

~~~

taken

average

theriual

~~ ~~

)~

~~~0

average

with

the

on

order

~fi

bet,veen

the

and

the

7ii

"'IIainiltonian"

interaction

coupling

paraiueter

As in

appearing surface

monomer

the

polymer

surface

Gaussian

the

over

taken

fll~fiexp(-7io)tJ/ JD~fiexp(-7io).

+

polymer-surface bilinear

jB2)

dr

r~ is

(CJ)o is the

and

7io° (CJ)o 7io depends

the

as

is

#~(r). represents

~bi)

averaged differently

cure

fllh P(h)L?/ fllh P(h)

distribution

"Hamiltonian"

The

a

x')

defined

excess

~~

where

in

Gaussian

(Bl)

surface

~~

riLy

possible

0.

~~

the

is

excess.

uncorrelated

an

A~61x

=

=

~

for

disorder

surface

have

surface

mean

l'ilx)'i(x'))

For

0.


To and rq > To- We quenched disorder enhance and below we outline proof that r~ > rq > To within this is a general result a of the cuniulant expansion. the the impurity degrees of freedom (h) in equations (85, 86), out averages over

Carrying for

one

ro, ra

suggesting disorder,

are

Annealed surface

than

excess

the

surface

quenched

the

This

case.

surface

in

excess

itself, yields rearrange enhancement is based

can

an on

even

the

presence

higher fact

that

polymer reaches its bulk condistance ( ft.oni the surface. correlation centration within a Due to the polymer elasticity, the length f is independent of the surface disorder. In addition, for semi-dilute polymer solutions, the polymer concentration epic) is not the order parameter ~fi(r) but the square of it [14, 15]. The inequality ,ve found is strongly based this fact since the disorder like a surface acts on parameter). random interaction and not as a randoiu field [13] (coupled linearly to the order One should, ho,vever, keep in mind that the results obtained here heavily rely on the mean field approxiniation which ignores excluded volume correlations and thus does not take fully the

disorder

into

account

assumed

is

the

only

polymer

of the

enhanceiuent

an

,vhere

lie

to

the

on

surface

and

the

that

fluctuations.

concent.ration

References

ill Napper

D.

II., Polymeric hI. A.,

Cohen-Stuart

[2] [3] de

Geiiies

P.

Auvray L.,

[G]

Ilawaguclii AI.,

[6]

Cotton

Baumgirtner

A.,

M. E.,

[7]

Cates

[8]

Ed,vards

[9]

llone

S.

Cosgrove T.,

G., Adu.

[4]

Ball

F.,

D., Ji II.,

J.

Stabilization

Gall.

P.,

fitolecules

&iuiliukitmar

J.

Prigs.

Aliitltukiiuar Pincus

P. A.,

B., Adu.

(1987)

27

20

J.

Fiance

hi.,

J.

Ghem. 49

Phys.

fifaciomolecules

Phys. 20

Int.

Sci.

202.

(1983)

(1988)

Chem.

Gall.

189.

(1988)

filaciomolecules16

AI.,

Dispersions (Academic

Colloidal

Vincent

Sci.

Itlt.

A.,

Takahmhi

R.,

of

87

1465.

(1987)

3082.

2009. 89

(1988)

2435.

(1987) 2543;

24

Press:

(1986)

143.

London, 1983).

JOURNAL

138

Ji H.,

[10]

D.,

Hone

M.,

Blunt

[11] Odijk T., Andelman

[13]

Lubensky

[14] [16]

[19] [20] [21]

[22] [23] [24]

S.

Gennes

Cabane

F., P. G.,

23

F.,

Matter, Les Amsterdam, 1979).

P/>ys. Soc.

B., Duplessix R.,

J.

24

in

Phys.

(1989)

(1991)

Houches

(1965) 613;

85

Scaling Concepts

22

1458.

1875.

Maciomolecvles

Ill-Condensed

Pioc.

N°1

2600.

Macromolecules

(1990)

II

6040.

1978, R. Balian, R. Maynard

ibid 88

(1966)

Fiance

48

(1987)

Gennes

P.

G., J. Phys.

Ghem.

and

G.

Toulouse

265.

Polymer Physics (Cornell University

Press,

Ithaca, 1979).

651.

(1990) 8407. Chari Il., Hossain T., J. Phys. Ghem. 95 (1991) 3302. Lionti-Addad S., Dimeglio J. M., Ltingmuir 8 (1992) 324. Chemistry of Surfaces (Wiley Interscience, Adamson A. W., Physical D., Joanny J.-F., Robbiiis M. O., Europhys. Left. 3 (1988) Andelman Andelman D., Joamiy J.-F., Phys. Rev. A 43 (1991) Robbins AI. O., Mcconnell H., J. Phys. Ghem. 91 (1987) 1715. Subramaiiiam S., Sammon J., Phys. Rev. Lett. 64 (1990) 1903. Seul M.,

[17] de [18]

C.,

T.

J.

(1988)

21

R.,

Ball

Macromolecules

(North-Holland,

Edwards

[15] de

W.,

D., Joanny

[12]

Eds.

Macromolecules

Barford

PHYSIQUE

DE

94

New 729. 4344.

York, 1990).