Application of scaling to the study of polymers at interfaces

Progress in Surface Science, Vol. 27(1/2), pp. 5-23, 1988 Printed in the U.S.A. All rights reserved.

0079-6816/88 $0.00 + .50 Copyright © 1988 Pergamon Press plc

APPLICATION OF SCALING TO THE STUDY OF POLYMERS AT INTERFACES E. BOUCHAUD, L. AUVRAY, J.P. COTTON, M. DAOUD, B. FARNOUX and G. JANNINK Laboratoire LemonBri//ouin, (commun CEA-CRNS) CEN-Saclay, 91191 Gif-sur-Yvette Cedex, France

ABSTRACT

The structure of polymer solutions at interfaces has been the object of intensive study : scaling analysis has provided many theoretical results. We present a survey of the work initiated by de Gennes and Alexander, and followed by Eisenriegler, Kremer and Binder using the magnetic analog. We compare the surface multicritical state at absorption edge to the bulk t r i c r i t i c a l state at the Flory temperature . We give some of the detailed results obtained in Saclay. From the experimental point of view, test of scaling is given by the molecular weight dependences of the observables. However, more recently, deeper insight has been gained by use of neutron scattering techniques developed by the Saclay Group.

CONTENTS

I.

Introduction

6

II.

Scaling predictions for polymer interfaces near a surface

7

I l l . Experimental evidence

15

IV.

22

Conclusion

6

E. Bouchaud, L. Auvray, J.P. Cotton, M. Daoud, B. Farnoux & G. Jannink I. Near surfaces,

the bulk s t a t e are,

for

is

the

symmetry in

broken.

instance,

INTRODUCTION the

atomic

force

field

characteristic

The forces a c t i n g on atoms at a l i q u i d - a i r

weaker on the

surface

side

than on the

bulk

of

interface

side,

simply

because of atom d e f i c i e n c y beyond the surface. In many cases the p e r t u r b a t i o n of the atomic s t r u c t u r e due to the i n t e r f a c e extends state

of

double to

only

to

a few atomic

near c r i t i c a l i t y ,

layers,

the

infinity.

In

layers.

or

if

perturbation

flexible

When however the atomic

the

forces

are

long

reaches many atomic

polymer

solutions,

the

system is

in

a

range as in e l e c t r i c a l

layers

and may even tend

perturbation

of

the

polymer

c o n f i g u r a t i o n due to the presence of a s u r f a c e , also extends to a large distance compared

to

the

conditions.

size,

Here the e f f e c t

polymer l a y e r existence

atomic

is

of

long time I ,

even

in

but

laws f o r

scaling

solution

state

far

from

demixion

is due to the large size o f polymer c o i l s .

then formed and t h i s

scaling

a fact

these

is

systems.

was introduced f o r

in

close

Diffuse

relationship

layers

the study o f

A diffuse w i t h the

were studied

for

polymer surfaces o n l y

more r e c e n t l y by de Gennes 2 and Alexander 3. Scaling scale

is

used whenever in a system there

and to which a v a r i a b l e

investigation behaviour. and i t s

of critical

is

ascribed 4.

i s a q u a n t i t y which f i x e s

It

is

systems, where mean f i e l d

The c r i t i c a l

nature

of

the

particularly fails

polymer chain

is

now well

the

bulk

of

"good"

solutions,

the

physical

quantities

a scale are the squared average radius o f g y r a t i o n in the l i m i t concentration

or

the

must now be i n t r o d u c e d . The l a t t e r (Fig.i). This w i l l

which

define

of zero polymer solutions.

r e l e v a n t scales but others

depend on the type of i n t e r f a c e to be studied

For instance a polymer chain may be a t t r a c t e d or r e p e l l e d by a surface. be described

and surface.

below in

For an i n f i n i t e

terms

chain,

of there

interaction is

situation

is

very r e m i n i s c e n t

bulk

solution,

separating

swollen

led to d e f i n e a " h i g h e r order" theta

temperature

A major

is t r a n s l a t i o n

difference

and

of

the

from

critical the

energies

a special

surface i s n e u t r a l and which separates the a t t r a c t i v e This

surface.

established

Edwards screening length f o r more concentrated

When we approach a surface these q u a n t i t i e s are s t i l l

the

in the

to p r e d i c t the c o r r e c t

r e c o g n i t i o n is f o l l o w e d by s p e c t a c u l a r r e s u l t s .

In

for

useful

the

which the

from the r e p u l s i v e regimes.

so c a l l e d

collapsed

between monomer

energy f o r

theta

state.

temperature

transition"

however between them,

i n v a r i a n t at the theta p o i n t , i t

is

the

In both cases one i s

p o i n t 5 , 6 , 7 namely a " t r i c r i t i c a l "special

in

that

for

an

point"

interacting

whereas the

system

is not at the special t r a n s i t i o n .

Structure of Polymer Solutions at Interfaces

7

Thus we expect in the l a t t e r case many properties related to the higher order critical

point, to occur only close to the surface. I t

is the purpose of this

review to examine this question and other topics related to scaling in polymer solutions near and at the surface (section I I ) .

We shall then give experimental

evidence for the large extension of the polymer concentration profile.

(a)

(b)

-Fig. 1. Polymers at interface. (a) grafted polymers, (b) repulsive surface, (c) adsorbed polymers. The scaling gives a description of the polymer structure reduced to a blob structure. A blob is a part of polymer chain represented by a circle on the figure. Although a surface

is

a strong

local

perturbation

to the structure of

the system, the exact observation of i t s effect is d i f f i c u l t . There are several methods which can be considered as global or detailed. The methods classically used to determine moments of the polymer concentration profile are measurements of

: ellipsometry,

adjacent

hydrodynamic thickness, surface tension and forces between

platelets.

Until

1980, none of

them have produced unambigous

characterization of the structure near the surface, but recent use of the neutron scattering technique has much improved the experimental situation. I f . SCALING PREDICTIONS FOR POLYMERS NEAR A SURFACE

In this

section,

scaling

is

used to describe the various

situations of

polymers near a surface. As an introduction, we f i r s t

recall the properties of the bulk solutions.

Scaling is used to account for the variation of the two body coupling parameter with degree of polymerization and temperature. screening

effects

due to

finite

It

is also used to determine

polymer concentration.

Results obtained

this situation will help us to examine scaling near interfaces.

in

8

E. Bouchaud, L. Auvray, J.P. Cotton, M. Daoud, B. Farnoux & G. Jannink

A. Bulk s o l u t i o n s We c o n s i d e r

the

variation

of the mean squared radius o f g y r a t i o n

R2 w i t h

the number N o f segments of the polymer chain. In absence of i n t e r a c t i o n s , which i s o n l y a conceptual case, the q u a n t i t y R~ i s obtained from the random walk s t a t i s t i c s R~ = N 2 2

(II.1)

where 22 is the average square radius of gyration of a segment. Monomers are in fact

submitted to repulsive and a t t r a c t i v e interactions,

which cause respectively swelling and collapse of interaction T

the

chain.

The two body

T is temperature T dependent. = (T-e)/e

where e i s

the compensation or F l o r y temperature.

The i n t e r a c t i o n

is

repulsive

or a t t r a c t i v e according to the sign of T • The three body interaction W is repulsive and constant. Coupling constants associated

to these interactions are derived from calculation of the polymer

chain swelling4 : the result is

The

z

=

g

= W.

increase

T N1/2

of

the

two

body coupling with

characteristic and the major d i f f i c u l t y

of

N accounts for

polymer physics.

the

essential

Similar scaling

w i l l also be found in the analysis of structures near an interface. The

range of

variation

of

z

is

considerable

and

there

are

several

interesting situations associated to characteristic intervals of z. In the case of swelling (z<< I)

the upper c r i t i c a l dimension is dc = 4

and the scaling prediction is lim z'*~

R2 = z2(2v-1) R2

(If.2)

0

where u is a c r i t i c a l exponent associated the n vector model ( u = .588 in 3d, = .75 in 2d). Relation

(If.2)

reveals

the

"blob"

structure of

the

polymer chain.

If

we define g(T) = T-2 as the number of steps necessary to form a random walk

Structure of Polymer Solutions at Interfaces

9

sequence called blob, then ( I I . 2 ) , ( I I . 1 ) give R2 = [ g _ ~ ] 2 ~ g ( ~ ) ~ 2 If

Izl < 1,

of

the chain configuration.

(II.3)

the three body interaction

is now relevant , secures s t a b i l i t y

The temperature 0

state, with an upper c r i t i c a l

corresponds to a t r i c r i t i c a l

dimension dc = 3. The chain has a quasi random

confi gurati on. If

z <-1,

the polymer configuration shrinks and i t

collapses eventually

to a globule like configuration. At

finite

concentrations,

occupation

of

sites

by polymers screens the

two body interaction. The variation of a physical quantity with the monomer volume fraction ¢ is accounted for by reference to

¢*, the monomer volume fractipn within a chain

in the l i m i t ¢ ÷ 0 ; the quantity ¢*(= N(~-)3 ~ N-4/5)

is

also

called

the

"overlap" volume fraction. There are two noteworthy consequences of screening.

First, the separation

between good or poor solutions given by z = 1 in dilute solutions, is replaced by the condition

T¢"1 --1.

Next,

the

variation

of

a characteristic

length

L with ¢ , in the good solvent domain, may be written L = Nu ~ f(@l@*)

(II.4)

where for strong overlap, x >> 1, the function f(x) has a power law behaviour. In particular,

the

length ~ associated

to the screening of the interaction,

or screening length, obeys to such a law. Screening is a concentration effect which should be independent of N. Therefore, the powerlaw is given by4 ~ ~- (3~- 1)Iv

(II.5)

which can be written in a manner similar to ( I I . 3 ) ~2 =

(g(¢))2u ~2

(II.6)

where g(¢) is the number of monomers associated to the "blob" ~ .

B. Polymer

s o l u t i o n s n e a r an i n t e r f a c e

We consider an impenetrable interface, f l a t

with respect to the polymer

10

E. Bouchaud, L. Auvray, J.P. Cotton, M. Daoud, B. Farnoux & G. Jannink

dimensions. I t is materialized by a solid, a liquid or a gas. We examine two situations. In the f i r s t , there is a short range, attractive potential between monomer and surface, superposed to the i n f i n i t e l y repulsive core. In the second, the polymer is attached to the wall by one end and there is

no other

interaction

between surface

and monomers than impenetrability.

The function of interest both theoretically and experimentally, is the polymer concentration p r o f i l e , which builds up in the v i c i n i t y of the interface. ( I ) interactive surfaces. This situation is of great interest, because i t

provides many developments

from the point of view of c r i t i c a l phenomena. The single polymer chain We consider a single polymer chain in a good solvent, close to the wall. We define the parameter B where Wb,

related to the adsorption edge by the relation 8

AWc - (Ws - wb)

Ws are

interaction

(11.7) energies

and at the surface ; the q u a n t i t y for

an i n f i n i t e

chain

for

monomers r e s p e c t i v e l y

in

the

bulk

AWc is the s m a l l e s t energy (Ws - Wb) necessary

to be adsorbed. Thus by d e f i n i t i o n

the i n t e r v a l s

6 > O,

< 0 correspond r e s p e c t i v e l y to the a d s o r p t i o n and d e p l e t i o n phases o f i n f i n i t e chains. Because poljaner are l i n e a r f l e x i b l e which w i l l the

bulk

depend on the

degree o f

chains,

the a d s o r p t i o n is a phenomena

p o l y m e r i z a t i o n N,

s w e l l i n g depends on N. There e x i s t s

in

the

namely a cross

same way as over,

in

defined by

the c o n d i t i o n aN¢ where

=

i

(11.8)

¢ is a c h a r a c t e r i s t i c exponent. Thus f o r a chain o f degree N, a d s o r p t i o n

occurs i f

6 > N-¢ , and d e p l e t i o n i f -N "¢

6 < -N -¢ . In between, i . e .

for

< ~ < N-¢

(11.9)

we f i n d a cross over r e g i o n , in t h i s region the polJaner chain remains e s s e n t i a l l y isotropic. as • = O,

The c o n d i t i o n ~ = 0 d e f i n e s a m u l t i c r i t i c a l is,

in

shows t h a t the l i m i t

the

bulk,

a tricritical

point.

point,

in

the same way

The analogy with

~ ÷ 0 d e f i n e s the special t r a n s i t i o n .

magnetism

Structure of Polymer Solutions at Interfaces

11

A lower bound for the exponent 0 can be found easily, comparing the number of monomers on the surface to that in the volume of an isotropic configuration N0 - N~(d-1)__ N Nvd Here 0 = 1-~ , but for an impenetrable barrier

(II.10) 0 > 1-v. In fact, calculations9

based on renormalization give 0 = .67. In a

the

blob

situation

structure.

(g(~))0 ~ = 1.

6N¢ > 0,

Let

the polymer chain adsorbed to

namely g(a)

The sequences of

be

size

the

number of

D : (g(~))v~

the wall

has

segments such that

are

isotropic

and form

the surface blob. The chain is then seen as a succession of such blobs. The number Ns of monomers in contact with the wall is a function of the scaling variable

Ns

aN0

=

N0 f(~N 0)

11.11)

Scaling gives I0 Ns ~ N~(I-O)/0 ]~]-I

, ,

NO6 >> i NOa << - i

(11.12)

Adsorption in d i l u t e pol~/mer solutions As the polymer concentration

becomes f i n i t e ,

excluded volume e f f e c t s

are

important and a polymer concentration p r o f i l e builds up near the wall. There exists the number of called

a domain of

polymer d i l u t e

adsorbed monomer per unit

the plateau regime.

has a c h a r a c t e r i s t i c

In this

behaviour

observed experimentally.

of

surface, r ,

regime, great

The competition

(bulk)

concentration, is

for

a constant.

This

the polymer concentration

theoretical

interest,

which is

profile

which

can be

between confinment, surface and volume

forces yields the following results obtained by scaling. The

concentration

p r o f i l e O(z)

includes

three

regions

with

different

behaviours I I , as shown of Fig. 2.b. ( i ) In the distal region (z>>~), the monomer f r a c t i o n is the bulk f r a c t i o n O(z) ~ 0 (ii)

In

the

(11.13)

proximal

region

(a<
and one expects a new behaviour, O(z) where 0s is

:

%

with

is most important

of weak adsorption.

(~)-m

identified

atomic distance).

the surface e f f e c t

characteristic

We assume (11.14)

~z=a)

(which is a distance of the order of the

In order to determine the exponent m, l e t

us consider the

12

E. Bouchaud, L. Auvray, J.P. Cotton, M. Daoud, B. Farnoux & G. Jannink

~(z}

~(z)

(a)

e(z)

Ib)

0.2/3

~8"~ _ ~

O'

(c)

I

a

D

a

Rl

Fig. 2. Variation of the concentration p r o f i l e ¢(z) of of the distance z from the surface, predicted by the concentration, a the monomer size, D the mean distance the blob size and R the chain size. These curves are case f o r (2a) grafted polymers, (2b) adsorption case one.

13

monomers, as a function scaling. ~ is the bulk between grafted points, given in the semi-dilute and (2c) the depletion

cross-over value for ~ : a ~ N-~. Then D~ ~ and the proximal region extends from the wail to the distal region. Continuity of the profile for z=~ provides a condition

for

the determination of m. Using ( I I . 1 4 ) ,

(II.13),

and ( I I . 5 ) ,

we find ¢(Z) (iii) This

:

The central region

is

¢s ( ~ ) - i / 3

(a << z << D)

(11.15)

region (D<
expected to

a screening length ~(z)

be semi-dilute,

varying with

with

a monomer f r a c t i o n

the depth z.

In analogy with

¢(z) (11.5)

and we

may then write ~(z) ~ {¢(Z)}-3/4a In this

(11.16)

region, we expect a self similar behaviour and thus a power law

decrease for @(z). Using continuity with

(II.13)

for z ~ ~, and with

(II.15)

for z ~ D, we find ¢(z) ~ (~)-4/3, D << z << ~

(11.17)

Inserting this result into ( I I . 2 7 ) , we obtain ~(z)

~ z

which is a d i r e c t expression of s e l f s i m i l a r i t y 5. Note that scaling is not able to determine the parameter a -Depletion. concentration

For a repulsive

profile

to

include

interaction three

regions

with

as above,

compl i cations. ( i ) In the d i s t a l

region (z>>(), the p r o f i l e is f l a t

¢(z>> ~) ~

the

wall,

we expect

however with

the

added

Structure of Polymer Solutions at Interfaces (ii)

In order to determine the p r o f i l e

to consider the l i m i t

of

strong

in the central

repulsion, a ~1,

to be discussed below reduces to the f i r s t

region, i t

13 is convenient

so that the proximal

layer,

region

D ~ a. Then, we expect the

p r o f i l e to decrease when z goes from ~ to a. In t h i s system, the osmotic pressure n(z)

is

conserved at every depth, and is equal to the bulk osmotic pressure 4

~b. Using ( I I . 5 ) and ( I I . 6 ) we get ~b ~ ca3'~ +g/4 g We assume a power law variation f o r ¢(z)

i(z)

:

(II.18)

Cs (a)a

(II.19)

where the surface fraction Cs is determined by noting that for a~1, the surface energy per unit surface is ~(z=a) ~ Cs ~ ~b The exponent a in

(II.19)

(II.20) is

determined by continuity with

(II.18)

for

z ~ C . We find ¢(z) ~ ¢9/4 (~)5/3

(II.21)

in the weakly repulsive case, 1 >> ~ >> N-3/5, one expects depletion

only for

distances larger than D. (iii)

In the proximal region, a<
D to a, with the same power law as in the adsorption case, relation (II.15). The reason for such increase is related to the existence of a multicritical point, the adsorption threshold, for an infinite chain. There is a multicritical region for

I~I < N"3/5, corresponding to a neutral

behaviour of the surface,

i.e. neither attractive nor repulsive. At the scale of surface blobs the overall repulsion

is weak and the surface is neutral,

z"1/3 dependence for

the profile,

leading to the characteristic

equation (II.15).

In the repulsive case,

the profile has the same form : ¢(z)

= Cs(Z) (~)-1/3

(II.15')

where Cs is determined by matching ( I I . 1 5 ' ) and (II.21) for z ~ D. We find 11 @(z) ~ ¢(-D)5/3 (~)-I/3 The resulting

concentration

(II.22) profile

in

the depletion

case is

given in

Fig. 2.c. (2) Grafted pol~nners We consider the solution of polymers attached by one end to the wall. The chains are constrained to a half space, as shown in f i g . l a .

Leta be the

14

E. Bouchaud, L. Auvray, J.P. Cotton, M. Daoud, B. Farnoux & G. Jannink

graft density on the wall corresponding to an average distance D between graft points D ~ o-1/2 a)

For a low density, oR2 << 1, every chain occupies a half sphere of

radius R (where R is

given by I I . 1 ) .

The average concentration p r o f i l e at a

distance z from the wall is then written

~(z)

=

where ~ (R) i s

~(R) f ( z / r )

the monomer volume f r a c t i o n

by the wall area f r a c t i o n

a

within

small

independent of

@(z) that

a3N/R 3, m u l t i p l i e d

R2 occupied by c o i l s .

arguments

the c o i l

(11.24) x of

the

function

~ oz 2/3,

dR) ~ ~ as

f(x),

we expect ¢ (z)

to

be

number of monomers N. Assuming a power law behaviour

f o r f ( x ) and taking i n t o account r e l a t i o n ( I I . 2 4 ) ,

Note

the c o i l ,

= ~o N ~ N2/5

¢(R) For

(11.23)

we get

z << R expected.

(II.25) Finally,

for

z >> R,

the

profile

decreases

very f a s t . b)

For

large

d e n s i t i e s , oR 2 >> I ,

Alexander 3 showed t h a t

every

grafted

polymer may be subdivised in blobs with size D made of gD monomers, with D ~ gD a

(11.26)

Because of the i n t e r c h a i n i n t e r a c t i o n s ,

the polymers adopt a more elongated

structure

and may be considered as a l i n e a r

direction

normal

to

the

surface.

In

string

the d i r e c t i o n s

of

blobs stretched

parallel

to

the

in the surface,

the s t r u c t u r e i s random. Assuming L ~ N3/5 a h (oR 2)

(11.27)

We f i n d R~ ~ N oI/3 a

(11.28)

RH ~ NI / 2 o -I/12 a

(II.29)

F i n a l l y , the p r o f i l e is f l a t , goes to

o as in

last

except in the f i r s t

subsection, relation ( I I . 2 5 ) ,

blob, z < D, where i t

and for

z > R , where i t

drops to zero very fast. Thus scaling provides a direct way to a set of simple power laws behaviour, in

the

results

different

cases we considered

are

in

not

contradiction

(adsorption,

with

the

depletion,

conclusions

of

grafted).

The

more detailed

Structure of Polymer Solutions at Interfaces

15

calculations 1. I I I . EXPERI~NTAL EVIDENCE Tests for scaling laws in polymer physics are commonly derived by observing physical

quantities as a function of molecular weight in the samples. However,

a given observation

of molecular weight variation can correspond to different

models. Therefore

the deeper insight

by the structure

function,

the

is

necessary.

This

polymer form function

is

naturally

given

and the concentration

profile.

A. Variation of moments of polymer concentration profiles ~ t h mlecular Right. Let 4¢(z)

= ~(z) - ~

(III.I)

be the polymer concentration profiles near a solid liquid interface, where ¢ is the bulk volume fraction. Measurement of profile

moments are made in several

different ways12,13,

which are b r i e f l y described below. (1) Ellipsometr~ In this experiment, the ratio between reflectance associated to polarization states respectively orthogonal and parallel to the surface plane, is measured. Let F(q) be the Fourier transform of the concentration profile F(q) = 5® eiq z A¢(z) dz o and l e t r i (i = 0,1,n) be the moments. For a profile of the type ~¢(z)

(~)m

,

z < Zmax

(111.2)

(III.3)

a simple calculation shows rl/r ° : (_~)m-1

= ee

(Ill.4)

This is called the apparent thickness. Inserting the value m = 4/3, we have ee

~ R2/3 = M2/5

This relation was tested for polystyrene in a good solvent. (2) Hydrodynamic thickness A weak shear flow near the wall creates a velocity gradient

(111.5)

16

E. Bouchaud, L. Auvray, J.P. Cotton, M. Daoud, B. Farnoux & G. Jannink dv n ~-~ (z)

where is

:

o

(111.6)

o i s the s t r e s s , n the v i s c o s i t y and f a r away from the w a l l , the v e l o c i t y

proportional

a singular

to

z,

function

but

of

z.

within

the

polymer

layer,

The cross over occurs at

the

velocity

profile

is

the hydrodynamic thickness

e H. Dynamical s c a l i n g shows t h a t e H ~ R ~ N3/5

(111.7)

Measurements 14 c o n f i r m t h i s r e l a t i o n s h i p . (3) Contact thickness The force are of

adsorbed

distance has

relationship

been

a characteristic

measured

thickness

e c,

between two mica sheets on which polymers

very at

a c c u r a t e l y 15. which

It

repulsive

reveals

the

interaction

existence

between the

sheets begins to be detected. This thickness scales as e

= N0.43 c

(III.8)

in agreement w i t h ( I I . 1 7 ) . (4) Surface t e n s i o n This surface interest tension

is

a

tension to of

test for

scale pure

appropriate a given

the

liquid-air

interfaces.

polymer volume f r a c t i o n

surface

solvent.

for

¢

Let y(¢)

in the bulk.

tension y - Yo with~!¢ , where Yo i s

For an adsorbing

interface

and w i t h i n

be the It

is

of

the surface the

plateau

regime (see s e c t i o n 2), the p r e d i c t i o n i s

B(Y-Yo)

= -62 -

al/3 N In(¢/¢2)

(III.9)

where 02 is a cross over volume fraction separating semi-dilute from concentrated regimes close to the surface. Until

now, experiments16 have indicated linear relationship between Y-Yo

and ¢ . However, a more detailed experiment should reveal ( I I I . 9 ) .

B. Detailed observations of the concentration p r o f i l e .

Scattering experiments

The predictions of the scaling theory presented above have not yet been tested thoroughly by experiments, because the global techniques used to study adsorbed layers do not have the required spatial resolution. In this section we present the r e l a t i v e l y new techniques to study adsorbed layers. We express for these techniques the predictions of the scaling theory and we discuss the experimental facts obtained so far, in relation with these predictions.

Structure of Polymer Solutions at Interfaces

17

The essential parameter of a scattering experiment is the momentum transfer q (in A-I) that an incident particle exchanges by interacting with the scattering centers of a sample. The intensity scattered at a momentum transfer q provides informations

on the structure of the sample at the scale q-l.

case, this scale must be in the range 10-1000 A and

In the polymer

t h u s corresponds to

the

range of small angle scattering of neutrons or X-rays. There are then two strategies to study polymer at interfaces. The f i r s t one

uses classical

small angle scattering

by

suspensions or porous media presenting a large by polymer, in

bulk

samples of

amount of

colloidal

interfaces

coated

the q-range where the intensity only depends on the local

structure of the interfaces. The second one uses neutron or X-rays r e f l e c t i v i t y of a single, oriented

interface,

the momentum transfer being provided by the

reflection process. In both cases, neutrons present a definite advantage because the scattering amplitude of the sample constituants may be monitored by isotopic substitution 17 and the

signal

of

the

polymer can thus be greatly

enhanced. Also,

as a

consequence of this substitution, linear and non linear "optical" effects are made conspicuous. (1) Small angle neutron scatterin 9 by interfaces The interest

of the small angle neutron scattering

adsorbed polymer layers

coating

colloTdal

grains

technique to

has been f i r s t

study

emphasized

and demonstrated by the group of Vincent18-21 and rediscussed in ref.

22 and

23. (i)

The method. The coherent intensity

of an isotropic

(in cm-1)

incompressible sample containing

scattered

per unit volume

a polymer (p)

the solid surface of a porous medium or colloTdal

adsorbed on

grains (g) in the presence

of pure solvent (s) is the sum of three terms resulting from the interferences between the amplitudes scattered by the solid and by the polymer :

i(q) = (Bg-Bs)2 Sgg(q) - 2(Bg-Bs) (Bp-Bs) Spg(q) + (Bp-Bs)2 Spp(q)

(III.lO)

The Bi are the scattering length densities (in cm-2) (i=g, p, s) and Sij the partial structure factors, q, the scattering vector is scattering angle B and the wavelength ~ by : q = (4~/~) sin ~/2. Provided that

q-1 is

smaller than the grains

related

to the

radius Rg, the scattered

intensity only depends on the specific area (A/V) of the solid surface and

18 Eo Bouchaud, L. Auvray, J.P. Cotton, M. Daoud, B. Farnoux & G. Jannink on the structure of the adsorbed layer. a) The intrinsic surface structure factor of the pol~nner. A f i r s t direct method of studying the adsorbed layers is to choose by isotopic substitution Bs = Bg (contrast matching ip(q) = (Bp-Bs)2Spp(q). In principle,

conditions).

Then

i(q)

is

reduced

Spp(q) depends both on the average concentration

< ¢(z)>,

and on the concentration

profile)

in the layer 26,27.

fluctuations

(deviations

to

profile

from the average

In the simple case where the adsorbed layer can

be assumed homogeneous (without

fluctuations),

Spp(q) is

simply related

to

the square of the Fourier transform of the profile

Spp(q)

= Spp(q) : 2~ (A/V) q-2 [f~dz <0 (z)> eiqZl2 o

(III.11)

= 2~ (A/V) q-2 a6 Ir(q)I2 Until the

now, this

underlying

investigation the

expression,

hypothesis of

polymer

concentration

of

which

concentration

fluctuations

can be inverted

homogeneity, in

have

p r o f i l e s 18.

an adsorbed

to y i e l d

b e e n the The e f f e c t

layer

is

<@(z)>, and

basic

tool

on Spp(q)

certainly

of of

small 23,24

but should deserve more attention in the future.

b) The pol~nner-solid cross structure factor.

A second method of getting

information on the average polymer profile is to measure the cross-term Spg(q). This term cannot be measured directly, contrast variation experiment (varying Spg(q) yields < ~ (z)>

but must be deduced from a

Bs at constant Bp and Bg). I f qRg>>1,

independently on fluctuations,

the Sine Fourier

transform of

:

Spg(q)

:

2x(A/V) q-3 fo
= 2x(A/V) q-3 a3 Im (?(q)) Expression ( I l l . 1 2 )

is

somewhat simpler than relation

(Ill.12) (III.11).

In many

cases, i t may be advantageous to perform a contrast variation experiment. With this

method, which yields

both Spg(q) and Spp(q), the small angle neutron

scattering is a powerful tool to achieve the measurement of density profiles. ( i i ) Predictions of the scaling theory. The evaluation of the partial structure factors Spp(q) and Spg(q) has been carried out in ref. 9, including fluctuations

Structure of Polymer Solutions at Interfaces effects, for the singular profile

19

¢(z) ~ z-4/3 predicted by the scaling theory

in the case of strong adsorption (6~ >> 1). We focus here on the cross structure factor,

Spg(q) which is the most sensitive to the singular structure of the

layer. In the q-range a
:

2~ (A/V){2a4(qa) -8/3}

(III.13)

Experimental results. A f i r s t experiment to test the predictions ( I l l . 1 3 )

has been made at the Laboratoire L~on Brillouin. The sample was a large molecular weight

monodisperse poly(dimethylsiloxane)

on porous silica (pore radius 3000 A)

in

(Mw = 270000)

presence of

strongly

adsorbed

p u r e cyclohexane ;

the

q-range 10-2 - I0-1A-1 was chosen to probe the inner structure of the adsorbed layer. The signal of the polymer for different contrasts was extracted from the difference between the signal of the polymer coated silica the bare s i l i c a .

and the signal

of

One major problem of the experiment is the smallness of the

polymer signal. The experimental on Fig.

results obtained for the cross-term Spg(q) are displayed

3 in a log-log representation.

where the

signal

to noise

ratio

is

For the f i r s t

sufficiently

20 experimental

points

high, we observe that the

agreement between the experiment and the prediction of the scaling theory ( I I I . 4 ) is very good : Sexp Pg

= c q-b

b : 2.65 ± 0.1 [W~(ql)]

,

l

0 -1 -2

f

-s

l -~

L °°l~ -3 -2

-

I IOgl(ql-1

0

Fig. 3. Log-Log plot of the Fourier transform Spg(q) of the concentration profile as a function of the scattering vector q. The data are obtained with polydimethylsiloxane polymers adsorbed on s i l i c a . The slope 2.65 ± 0.1 confirms the scaling predictions ( I I I . 4 ) .

20

E. Bouchaud, L. Auvray, J.P. Cotton, M. Daoud, B. Farnoux & G. Jannink (2) Reflection from pol~mer solution surfaces The second technique for the investigation of the near surface structure

of a polymer interface is the measurement of the r e f l e c t i v i t y curve with X-rays or

s l o w neutrons. This

horizontal surface.

technique is

well

suited

for

the study of

a free

I t is based on the fact that, in the range of the radiation

wavelength ~, the refractive index of a medium can be written as25 n = 1-

(~2/2~)B

The r e f r a c t i v e

(111.14)

index i s e s s e n t i a l l y

C through the s c a t t e r i n g

dependent on the atomic c o n c e n t r a t i o n

length d e n s i t y B. For neutrons B=C b, where b i s

the

coherent s c a t t e r i n g l e n g t h . The d e v i a t i o n Therefore

total

of

n from

external

unity

reflection

is

in

can

both

cases

be observed

of

the

order

for

the

two

of

10-6 .

radiations

at a l l grazing angles less or equal to a c r i t i c a l angle ec given by cos ec=n, or ec

=

~(B/~)1/2

(Ill.15)

A similar relation 26 defines e.

the c r i t i c a l

wave length ~c for a given angle

In this situation an exponentially decaying radiation beam penetrates at

distances t y p i c a l l y of the order of several tens of X. If

the

medium is

homogeneous,

the

reflectivity

function

decreases

monotonously as e increases above e c, or as ~ decreases below ~cThe variation is given by the Fresnel law25 R =

Irl 2

where the r e f l e c t a n c e i s r

=

(1-p)/(l+p)

p

=

(1-

and

We examine profile.

a

(ec/O)2)1/2 non

homogeneous

The r e f l e c t i v i t y

associated

medium

characterized

by

to such a medium d i f f e r s

a

concentration

from the Fresnel

law, and the observed d i f f e r e n c e i s used to determine the c o n c e n t r a t i o n p r o f i l e . The s c a t t e r i n g l e n g t h d e n s i t y associated to the bulk volume f r a c t i o n

B(~) Introducing

=

is

Bp(¢) + Bs(l-¢)

the isotopic

compositions :

deuterated polymer -

non deuterated

solvent and vice versa, we obtain different scattering length densities B(¢). For a given concentration

profile,

the optical

properties of

the interface

w i l l depend to a great extent, on the value of B(¢). This is the isotopic effect.

Structure of Polymer Solutions at Interfaces

21

We examine this r e f l e c t i v i t y as a function of the "mo~ntum transfer ''30 q =

(~i~)1/2 (n:2- cos2B)1/2

(III.16)

The correspondence between r e f l e c t i v i t y

function

and concentration

profile

is explicitely given in two extre~ situations. a) In the limits of

the

refraction

~ ÷ ~c (q÷ O) and ~ ÷ ~, and for positive increments

index An = n(z) - n

solvent), the r e f l e c t i v i t y

(deuterated polymers, non deuterated

is a universal

function of q, characterized

by a

singularity and the exponent m (m = 4/3) (Dietrich and Schack28). b)

In

function

the

range V ~ << 1 (q>>O), the

Im (2q) (see equation ( I . 5 ) ) .

reflectivity

Bouchaud et

is

related

to

the

a129, gave an explicit

fo~ula, based on the 1st Born approximation AR(q) where ~(q)

is

= a3 (4~/q)(Bp-B s (R(q))I/2 (l-R(q))Imr(2q)) the

change in

reflectivity

with

respect

(III.17) to

pure solvent

r e f l e c t i v i t y R(q). In q-1/3.

particular, First

(II.16)

experi~ntal

implies

that AR(q) decreases as~ptotically

results 29 are shown in

are displayed as a function o f ~

Fig.

4. Reflectivities

c for three samples : i)

Polydi~thylsiloxane in deuterated toluene, i i i )

as Ri

pure solvent, i i )

Deuterated Polydimethylsiloxane

in toluene.

0.0

~

0.6 ~~.~.~.. O.L •, ,

0.0 0.0

I

_ .....

I 0.2

t

I O.L

I,

I 0.5

i

I O.O

m I0 "z

Fig.4. Reflectivities ~asured as a function of incident wave length. The data are represented as a function of the "momentum transfer" q. o data sample 1 error bars are given by vertical lines • data sample 2 1st Born approximations, samples 1 and 2 !

Fresnel function associated with an idealized equivalent bulk solution.

22

E. Bouchaud, L. Auvray, J.P. Cotton, M. Daoud, B. Farnoux & G. Jannink Comparison between experiment and theory in case ( i i )

of a concentration profile with

(Ill.14).

experimental

In

reveals the existence

due to a strongly adsorbing surface, in agreement

case ( i i i )

a

significant

data and 1st Born approximation.

discrepancy It

appears30 between

is related to the repulsive

interaction between the incident neutron and the perturbation created by the concentration reflectivity

profile.

Shape and depth of

function

near

total

this

reflection,

perturbation

but

the

determine the

effect

is

not

fully

understood. IV. CONCLUSION We have examined the application

of

scaling

to

polymer solutions

near

the interfaces. Scaling predicts a great variety of structures, all characterized by

a concentration

variation

of this

profile.

The most spectacular

structure with the adsorption

predictions

concern the

energy near adsorption

edge.

Guided by the magnetic analog, theoreticians identified the surface transition and in the limit of i n f i n i t e degree of polymerization, the special transition. Most experiments depletion.

In

this

are

carried

situation,

in

scaling

situations

of

strong

laws are tested,

adsorption

and the

or

results are

found to be in agreement with predictions. However the problems raised by the cut

off

in

reflectivity

the

concentration

experiment

profiles

indicates

for

are not

solved.

The result

example a strong

effect

of

the

due to

the

structure very close to the surface. Should further from universality

efforts in

be devoted to explore all

pol~nner solutions

the weak adsorption case be examined ? I t greater

attention

at

facets

derivable

particular,

should

seems that other situations deserve

such as pol~n~er-pol~nner interfaces,

of scaling is less straight forward.

the

surfaces ? In

where the

application

The same applies to the kinetics of the

concentration p r o f i l e , which is of fundamental interest. REFERENCES I. 2. 3. 4.

5. 6.

C . A . Hoeve, A. di Marzio, P. Peyer, J. Chem. Phys., 42, 2558 (1965) J. Scheutjens, Macromolecules at Interfaces, PhD Thesis (1985), Wageningen, The Netherlands P.G. de Gennes, J. Physique, 37, 1445 (1976) S. Alexander, J. Physique, 38, 983 (1977) ; 3_8_8,977 (1977) P.G. de Gennes "Scaling concepts in polymer physics", Cornell University Press (1979) J. des Cloizeaux, G. Jannink "Les pol3nn~res en solution : leur mod~lisation et leur structure". Editions de Physique P.G. de Gennes, Macromolecules, 13, 1069 (1980) K. Binder, "Phase transitions and c r i t i c a l phenomena", F. Domb, J. Lebowitz Ed. Academic press (1983)

Structure of Polymer Solutions at Interfaces

23

7. T.C. Lubenski, M.H. Rubin Phys. Rev. B, 11, 4533 (1975) and 12, 3885 (1975) 8." E. Bouchaud, Th~se, Universit~ Paris-Sud, Orsay (1988) 9. H. Diehl, S. Dietrich, Phys. Rev. B, 24, 2878 (1981) K. Kremer, J. Phys. A, 16, 4333 (19837 T. Ishinabe, J. Chem. P-h~s., 76, 5589 (1982) and J. Chem. Phys., 7J_7, 3171-~983) 10. E. Bouchaud, M. Daoud, J. Physics A, 20, 1463 (1987) 11. P.G. de Gennes, P. Pincus, J. Physique Lett., 44, 241 (1983) 12. M. Cohen-Stuart, T. Cosgrove, B. Vincent, Adv. Colloid Inteface Sci., 24, 143 (1986) 13. R__Varoqui, P. Dejardin, J. Chem. Phys., 6_~6,4395 (1977) 14. M. Kawogushi, H. Hayakawa, A. Takahashi, Macromolecules, 16, 631 (1983) 15. J. Klein "Colloldes et Interfaces", A.M. Cazebat, M. Veyssie Ed. Editions de Physique, Les Ulis, France, p. 289. 16. R. Ober, L. Paz, C. Taupin, P. Pincus, S. Boileau, Macromolecules, 16, 50 (1983) 17. Due to the large difference between the neutron scattering lenqth of hydrogen (b H = -0.374110-12cm) and deuterium (b D = 0.6676 lO-I2cm), the deuteriation is in particular a very convenient and powerful method. 18. K.G. Barnett, T. Cosgrove, B. Vincent, A.W. Burgess, T.L. Crowley, T. King, J.D. Turner, Th.F. Tadros, Polymer Commun., 2_22,283 (1981) 19. K. Barnett, T. Cosgrove, T.L. Crowely, Th. F. Tadros, B. Vincent in "The Effect of Polymer on Dispersion Properties", Th. F. Tadros Ed., Academic Press (1982) p. 183 20. T. Cosgrove, T.L. Crowley, B. Vincent, in "Adsorption from Solutions" R.H. Ottewill, C.H. Rochester, A.L. Smith Ed., Academic Press (1983) 21. T. Cosgrove, T.G. Heath, K. Ryoin, B. Van Lent., Pol~mer Commun., 2__88, 64 (1987) 22. L. Auvray, C.R. Acad. Sci. Serie 2, 302, 859 (1986) 23. L. Auvray, P.G. de Gennes, Eur. Phys. Lett., 2, 647 (1986) 24. L. Auvray, J.P. Cotton, Macromolecules, 2__00,~ 2 (1987) 25. M. Born and E. Wolf, Principles of Optics, Pergamon, Oxford (1975) 26. B. Farnoux, proceedings of AIEA, JUlich meeting, Jan. 14-18 (1985) pp. 205209. 27. J.C. Charmet, P.G. de Gennes, J. Opt. Soc. Am., 73, 1977 (1983) 28. S. Dietrich and R. Schack, Phys. Rev. Letters, 5._88--~-140 (1987) 29. E. Bouchaud, B. Farnoux, X. Sun. M. Daoud and G. Jannink, Europhys. Lett., 2 (4), 315 (1986) 30. X. Sun. B. Farnoux, E. Bouchaud, A. Lapp, M. Daoud and G. Jannink, Submitted to Europhysics Letters.