Interaction between two polymer brushes in binary solvent mixture

(1)

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Interaction between two polymer brushes in binary solvent mixture

Tuan-Anh Tran, Andrea Liu, P. Pincus

To cite this version:

Tuan-Anh Tran, Andrea Liu, P. Pincus. Interaction between two polymer brushes in binary solvent mixture. Journal de Physique II, EDP Sciences, 1994, 4 (8), pp.1417-1426. �10.1051/jp2:1994207�.

�jpa-00248050�

(2)

J. Phys. II France 4 (1994) 1417-1426 _AuGusT 1994, PAGE 1417

Classification Ph_v.vie.I Abstiact.I

36.20 82.70 82.65

Interaction between two polymer brushes in binary solvent mixture

Tuan-Anh C. Tran

(~),

Andrea J. Liu (2.

*)

and P. Pincus (~. 3)

(') Department

of

Physics, University

^of Cal)fornia, Santa Barbara, CA 93106, U.S.A.

(2) Department of Chemical and Nuclear

Engineering, University

of California, Santa Barbara, CA 93106~ U-S-A-

l')

_Department of Materials,

University

of California, Santa Barbara, CA 93106, U-S-A-

(Received 27 Jaiiuaiy J994, receii'ed _in final farm /9 April /994, aiiepted 22 April /994)

Abstract. The system ^of two

polymer

brushes in _a near-critical binary mixture of good and _poor

wlvents is studied within the framework of the Alexander-de Gennes model. When the brushes do not overlap, the problem is shown to be

analogous

to the system ^of two hard wall in _a _near critical binary mixture. As in the two-wall _case, it iq found that the dis

joining

_pressure between the brushes is negative. This attractive interaction, which arises from the preferential attraction of the good

solvent to the brushes, has _a range comparable to the correlation length of the mixture. When the brushes do overlap~ the interaction between the bruqhes i~

repulsive.

This leads _to _a minimum in the

disjoining

_pressure at contact.

1. Introduction.

A

polymer

^brush [11 is an interfacial structure

consisting

of

polymeric

chains

end-grafted

to a

solid surface. At

high grafting densities, strong

interactions between the chains

give

rise _to _a brush

height

that is

linearly proportional

to the

degree

of

polymerization

N. The brush structure and

properties

thus

depend sensitively

on the free energy

required

to stretch the chains, and _on the solvent

quality.

As the solvent

quality decreases,

the brush thickness shrinks

continuously

however, the

height

is still

linearly proportional

to N

[2].

Polymer

brushes _are used in _a

variety

of

applications, including

colloid stabilization in

nonpolar

~olvents. Colloidal

suspensions

tend _to flocculate due to van der Waals interactions.

When the colloids _are coated with

polymer

brushes, the brushes resist

compression,

so the colloids

repel

each other _at short distances, thus

preventing

flocculation. This

problem

has been

investigated

in detail both

experimentally

and

theoretically

for

polymer

brushes in _a

single

solvent

[I].

Less attention, however, has been

paid

to the

problem

of brushes in _a

(*) Peini. adch.e.I.I Dept. of

Chemi~try~

405 Hilgard Ave., UCLA. CA 90024, U.S.A.

(3)

solvent mixture. In

applications

such _as

lubricants,

colloids _are often

suspended

in _a mixture of different

liquids. Recently, Beysens

and coworkers

[3]

and

Gallagher

and Maher

[4]

have done

experiments

_on colloids in

binary liquid

mixtures.

They

^find that the colloids tend _to flocculate, _even in the

single-phase region

of the mixture,

possibly

due to

preferential

attraction of _one component to the colloids. While their

experiments

_were

performed

with

charge-stabilized

colloids, the

interpretation

of their results suggests ^that brush-stabilized colloids

might

also flocculate in

liquid

mixtures.

Previous theoretical work _on brushes in solvent mixtures has focused _on the behavior of _a

single

brush. Lai and

Halperin [5j employed

the Alexander-de Gennes model and Marko

[6]

used _a self-consistent _mean field

theory

to examine the _structure of _a

single

brush in _a

binary

solvent mixture. In both

analyses,

the

binary

solvent mixture _was assumed _to be in the

single- phase region.

The behavior of _a brush in _a

binary

solvent in the _two

phase region

has been studied

by

Johner and coworkers

[7].

In this paper, ^we carry out a theoretical

study

of the

interaction between _two brushes in _a

binary liquid

mixture of

good

and poor solvents. As in most

grafted

on

flat, parallel

surfaces. We have

examined the system ⁱⁿ the

single-phase region

of the

binary mixture,

_near its critical

point.

Our aim is to understand how the mixture affects the

disjoining

_pressure between _two _non-

overlapping

brushes. We will show that there is _an attractive interaction driven

by

^the solvent concentration

gradient

that arises from

preferential adsorption

of the

good

solvent to the brush.

In section

2,

_we review the _case of _a

single

brush in _a

binary

mixture. We compare the

results of Lai and

Halperin's approach [5],

based _on the Alexander-de Gennes model

[8]

of _a

polymer

brush, _to Marko's self-consistent field calculation

[6]

and the

experimental findings

of

Auroy

^and

Auvray [10].

Near the critical

composition

of the

mixture,

where the concentration

of each solvent is

comparable,

the Alexander-de Gennes model agrees well with the self-

consistent field calculation. This

justifies

_our _use of the

simpler

Alexander-de Gennes

approach

to

study

the interactions between brushes. In section 3, _we present our model and the form of the free energy for

nonoverlapping

brushes in _a near-critical

binary

mixture. Within _a

reasonable

approximation,

we show that there is _an

analogy

to the

problem

of critical

adsorption

in _a

binary liquid

mixture confined between _two walls, first studied

by

Fisher and de

Gennes [I1,

12].

We also present a model for the case of

overlapping brushes,

where the

problem

reduces _to the Alexander-de Gennes model.

Finally,

_we discuss the solvent

concentration

profiles

and the form of the pressure for both the

overlapping

and non-

overlapping

_cases in section 4, and compare the results _to the

scaling theory

for _a system ^of a

binary

fluid confined between two hard walls.

2.

Single

brush in

binary liquid

mixtures.

Here _we review the work of Lai and

Halperin [5]

on the _structure of _a

polymer

^brush

grafted

_on

a flat surface in the presence of _a reservoir

containing

a

binary

mixture of _two

simple

solvents A+B. The

end-grafted polymer

chains _are assumed to be neutral,

monodispersed,

and

uniformly

stretched. Within this framework, _we _assume that the solvent concentration _can be

approximated

_as _a _step function. We denote the volume fractions of the

polymer, good

solvent A and poor solvent B

by

~b, ~b~ and ~b_~,

respectively.

The

assumption

that the

tertiary

system is

incompressible yields

the condition _~b ₊ _~b

~ + ~b~ =

I. The _excess free energy per unit _area

of the system ^is

given by

h~

~

(~Tla~) _~lfmix

~bA /~<e, +

"<e,

+

~elJ,i,c (~'~~

The _term in brackets is the free energy

required

to create a unit volume of uniform solvent

composition

^from a solvent reservoir of

exchange potential

_v,~, and osmotic pressure

(4)

N° 8 TWO POLYMER BRUSHES IN BINARY SOLVENT MIXTURE 1419

n,~,.

The volume fraction of solvent A in the reservoir far away from the brush is denoted

by

~fi. The free energy

density

of

mixing

is

~fm,~=~b«'og~bA+(I-~b-~bA)log(i-~b-~b~)+

+XA~b~bA+XB~b(I-~b-~bA)+XAB~bA('-~b-~bA). (2.2)

Here, _XA characterizes the interaction between the

polymer

and solvent A, _x~ the interaction between the

polymer

and solvent

B,

and _XAB the interaction between the two solvents. The second term in

(2,

ii is the elastic free energy per unit _area that arises from

stretching

^the

polymer

chains _:

~~~~~"~ ~~

~~~

~~ ~~

where the brush thickness L is related _to _~b

by

N «a/L

by assuming

a constant monomer

density profile

within the brush

[8].

The dimensionless

grafting density

is defined

by

_« ₌

(a/d)~

where d is the average distance between

grafted

ends and _a is the _monomer size.

Minimization of AF with respect to ~b and ~b~

yields

two

equations

of _state

relating

the

exchange potential

_v and osmotic pressure ⁿ in the brush to v,~, and

n,~,.

~fm,, _~f<e,

~

^~'~' _~~ ~

(2.4)

J

f

_~

~2

~b

$

₊ _{WA v<e,}

fmix ~

^~

^fire,

^"

^4~v

re,

f<e,

The

equations (2.4) completely specify

the

equilibrium properties

of the brush for _a

given

set of

«, ~fi, and _x parameters. ^In

general,

these two

coupled equations

_are

complicated

and _can be solved

analytically only

when the solvent reservoir is

nearly

_pure A or B.

In the intermediate range of ~fi, where the

composition

of the reservoir is

nearly

critical, _a numerical solution of

(2.4)

_was obtained

by

Lai and

Halperin [5]

for _a mixture of

good

and poor solvents.

They

found that _at

high grafting density,

the

height

of the brush decreases

abruptly

as the volume fraction of

good

solvent decreases below ~fi

~ 0.75. This effect has been observed

by Auroy

and

Auvray [10] experimentally,

and is attributed _to

preferential

solvation of the

good

solvent. Moreover, Marko

[6]

showed that

depending

on the

miscibility

of the solvents, the brush

height

can vary

non-monotonically

with ~fi. As XAB decreases, the two solvents become more miscible with each other, and the free energy cost of the solvent

concentration

gradient

in the reservoir

eventually

exceeds the _sum of the

stretching

_energy and the

repulsion

between the

polymer

and the poor solvent. In this _case, the brush

height actually

increases with

decreasing

^~fi.

For ~fi S 0.75, MarLo showed that the self-consistent field calculation

yields

_monomer

density profiles

that have a step at the

layer tip.

Thus, the Alexander-de Gennes model, which

assumes that the _monomer

density

has a step

profile,

is _a

good approximation

in this

concentration

regime.

Since _our calculation is concerned with two brushes in _a near-critical

binary liquid

mixture (~fi~ = 0.51, the _use of the Alexander-de Gennes model for the brushes,

as

opposed

to the

computationally

more

complex

self-consistent field

approach,

is well

justified.

3. Two brushes in _a near~critical

binary

mixture.

We _now consider _two

non-overlapping

^brushes in _a

binary

mixture. As discussed in the

safely

assume that the free ends of the

polymers

_are all located _at the

JOURNAL DE PHYSIQUE it T 4 N' x AUGU~T ,994 54

(5)

same distance L from the

grafting

surface. The geometry ^is ^shown ⁱⁿ

figure

I, where 2 h is the

separation

between the

grafting

surfaces and d is the average

separation

between

grafts.

The volumes

containing

the brushes _are referred _to _as

region

I the volume

containing only

solvents is called

region

II. The _excess free energy per unit _area is

given by

~~

=

lF[,~

_(~b

[

_~fi

_v,~,j

L ₊ ~

)

_~~~

+

)

j~ ~'[~

^~,x

^(~b~

^~

⁾

Vres ⁺ ~~(d~b~/l~X

~j

(~. l)

L

The first _two terms _are identical to those for _a

single

brush in _a

binary

mixture

(see (2.2)

and

(2.3)).

The free energy

F$,~

describes the free energy of

mixing

^of the

binary liquid mixture,

and is

given by equation

(2.2) with _~b set to zero. The

length

scale b is the correlation

length amplitude,

which is _on the order of the molecular size. Note that

by assuming

the Alexander-de

Gennes model for the brush and

using

_~b

=

~ _~~

,

we have

implicitly

assumed that the solvent L

concentration

profiles

are also flat inside the brushes

(region I).

As in the

single-brush

_case,

~b_~ is the volume fraction of

good

solvent A, and the

three-component

system ^is ^assumed to be

incompressible.

As usual, the solvent is considered

good

^when the

polymer-solvent

interaction is attractive

(XA

_~ 0.5 and is considered poor when the interaction is

repulsive (x~

_~ ^0.5 ).

The

equation

for the free energy

(3,I)

has _a

simple physical interpretation.

The first two terms, which describe the free energy in

region

I, _can be

regarded

_as _a _« surface free energy »,

where the _« surface

» is located at the

tip

of the brush, at x = L. The last _term, which describes the free energy in

region

II, is

just

the Cahn-Hilliard free energy that describes the

binary liquid

mixture confined between the two _« surfaces

».

it I

d

« »

L

« w

2h

Fig.

I. Two

opposing

neutral brushes with separation 2 h

> 2 L and _mean

grafting spacing

d.

(6)

Minimization of

(3,I)

with respect to ~b and ~b~

yields

four

coupled equations

_:

~1 ^~~~

=

l)i~

^« _<e~

^(3.2)

~

~~~~~

~

=

(L/b)

^~~~"~~ _v<e~

'

d~; d~b

( _~[

~AL

d~b((x

=

2h -L)

d~b((J.

=L)

b =-b

dJ ~r ,

~i~

^#

_~ll'~

+

FL~ FS~ i _b~ ^li~ ^j,

~

-

°

~3.4~

In the limit h

~ o~, the

equations

^above reduce to those for _a

single

brush in _a

binary

mixture.

The concentration

profile

of the solvent in

region

II between the brushes is

given by

equation (3.2).

The

slope

of the solvent concentration at the _« wall

» at x =

L is

given by equation (3.3). Finally, equation (3.4)

determines the

position

of the wall, or

equivalently

the brush

height,

with respect to the

grafting

surface _at _J.

=

0. The

equations

_are _more

illuminating

when _we

expand ~b(

and ~fil around ~fi( =

(l

_~b), and

expand ~b(

and ~fi'~ around the 2

critical concentration ~fi(' = ~fi~ m 0.5 _to _recast them in

Landau-Ginzburg

form. If _we define

Mi

"

16(

_~fi), and

Mjj

=

~b(

_~fi)~, then

equation (3.2)

becomes

~

d~mjj

~ ~

~r~ ~~"

_~~~ _~

"~

_"

~" '

~~ ~~

where

t ₌

2(x~~ xci, (3-6)

for Xc =

2,

u =

16/3, (3.7)

and

v~~, =

log (~fil(I

~fi )) +

XAB(1

²

~fi). (3.8)

Note that when the fluid is at critical

composition

4~

=

4~)'

₌ 0.5, the chemical

potential

v~~~ vanishes. The differential

equation equation (3.5)

must

satisfy

the

boundary

^conditions

b~~'~j ~~=-jij-GMj(,<=L)-v,~,L/b,

dmjj

(x

=

2

~-

L )

~

'

b

~

=Hj+GMj(J.=2h-L)+v~~~L/

,

where

hi

=

(L/b) (x~ x~)

~b

,

(3. lo)

and

G

=

2(L/b)I1

^~ _~

^ABl ^(3.1')

(7)

Note that

Mj

₌

Mjj

+ ~b, so

equation (3.9)

_can be rewritten _as 2

dmjj (,i

=

L b

~_

~-Hi-Gmii(,I=L),

dmjj(J.

=

2 h L

'~

b

=Hj+Gmjj(,1=2h-L),

where

Hi

=

hi

₊

_G~b

₊ _v,~, _L/b.

_(3.13)

These

equations

_are

equivalent

to those

describing

_an

Ising

model confined between _two walls, _one at.i ₌ L and the other at,<

=

2 h L [I1,

12]. By comparing equations (3.5-3.12)

to the

equations

of Nakanishi and Fisher

[12],

_we

interpret Hi

as a surface field. As

expected, Hi

increases as ~b increases. From

equation (3. lo),

_we _see that

hi

_is

_always _positive

_because

x~ ~ x~. Thus,

hi _always _prefers

_the _better solvent A, _as

expected,

and

changes sign

if the labels A and B _are

exchanged.

The actual surface field

Hi,

however, is not

always positive,

due to the _extra term

G~b

in

equation (3,13).

This term arises from the presence of the

2

polymer

in the brush. The volume fraction of A

preferred

in the brush may be lower than that between the brushes because the

polymer occupies

_some of the volume. Thus, the surface field

Hi

_may be

negative

even when the

polymer prefers

A to B.

In the formulation of Nakanishi and Fisher for

Ising

models between

plates,

G is the surface enhancement field, that _measures the deviation of the effective

coupling J,

at the surface from its bulk value J

[13].

In this _case, the effective interaction between A and B is reduced in brush because of the presence of the

polymer.

This reduction is reflected in

negative

value of G. As

expected,

the

magnitude

of G increases _as the

polymer

volume fraction _~b increases. In the

single-phase region

of the

binary

mixture,

equations (3.5-3.12) yield adsorption profiles

that

decay

_away ^from the walls with the correlation

length fmb/,/~.

_The

_quantity

t measures the difference between _XAB and its critical value, and is

analogous

to the reduced

temperature

(T T~)/T~

ⁱⁿ the

Ising

model.

By invoking

the

analogy

_to _a near-critical

binary liquid

mixture confined between _two

plates,

we have shown that the _two brushes

provide

surface fields at their

tips

that act on the

binary

mixture. We discuss the consequences of this

analogy

ⁱⁿ ^the next section. The

analogy

fails, however, when the brushes

overlap.

In that case

(2

h

~ 2 L

),

_we have _a

tertiary

mixture with _a different critical

point.

The presence of the brushes suppresses the critical

point

of the

binary

mixture _; thus, even when the system ^is near the critical temperature ^of ^the

binary

mixture it is still well inside the

single-phase regime

inside the brush. The _excess free energy p,er unit _area for this

tertiary

mixture is

where _~b

=

N«a/h. Note that this

equation

is similar to

equation (2.I).

The _extra term

(3 «la~)(Na~/h~),

_prevents

_complete _collapse

of the brush in _a poor solvent. At

large elongations (3 «la~)(h~/Na~)

is the dominant term in the elastic free energy, Minimization of

equation (3,14) yields equation (2.4)

^with h

replacing

L. When the _term

(3 «la~)(Na~/h~)

is

negligible,

^the ^results are identical _to those of the

single

brush

problem.

Hence the

interesting

physics

^lies ⁱⁿ ^the

nonoverlapping

_case.

(8)

4. Discussion.

In the last

section,

_we demonstrated

that,

under certain

conditions,

the

equations

for _a

binary liquid

mixture confined between _two

polymer

brushes _can be

mapped

onto the

equations

for _a

binary liquid

confined between _two

plates.

This

mapping

leads _to the

prediction

of

scaling

behavior, It is convenient _to rescale the parameters

m =

Ml

,/t hi

,

=

Hi

b/t,

g =

Gb/

,j,

fi

<e, ^~ v,e, ^t~ ^~~ (4. )

z =

,t/f

,

(

=

h/f,

=

L/f

Far from the critical

point

ⁱⁿ the

single-phase regime (x~~

_~

x~),

we can

neglect,

for

sufficiently

small surface field

hi,

the nonlinear term in m(z) in

equation

(3.5). Then

equations

(3.5-3,12) have the solution

cosh

((

z)

m

(z)

=

(mL fire,

₊

Ji,~, (4.2)

cosh

(( I)

where the reduced

composition

_m _at the

tip

of the brush is

hi

₊

Ji,~,

b tanh

(( I)

m~ =

(4.3)

g + h tanh

(( I

₎

In the infinite molecular

weight

limit of the

polymer

(N

~ o~ ), and for the

special

_case of critical

composition (fi,~,

=

0), equation (4,3)

reduces _to

m~ =

Ail (g

,

(4.4)

which is

independent

of L. In that

limit,

the

slope

at the

tip

of the brush reduces _to

d~b('

(,

_{_~} =

M~/f. (4.5)

d-r

Near the critical

point

of the

binary liquid

mixture t

= 0, when the

spacing

between the brushes is small _or

comparable

to the correlation

length,

we must include the nonlinear _term in

m(z)

in

equation (3.5).

In all of the

following equations,

we have considered

only

^the

special

case of critical

composition, fi,~,

=

0. We find _a power ^law

decay

^of the order parameter away from the surface at x = L at distances,r smaller than the correlation

length

f

M(.r)

= sgn

(Hi ~'~~~~

,

(4.6)

where

=

b,~(G(/(Hj _(4.7)

(9)

At _x

= h,

midway

between the two

brushes,

_we find

jbK(

^II

^/)

₍₄

₈₎

M~

= sgn

(Hj p

_~ _~ ,

where

K(

II

,fi)

_is _the

_complete _elliptic _integral

_of _the _first _kind _evaluated

at I/

/,

_which _is

of order

unity.

This

expression

holds when the

spacing

h-L is

sufficiently large

that

M~

«

M~,

where

M~

^is

given by equation (4.6)

_as

M~

=

Hj/(

G

(.

^In ^the

opposite limit,

where

M~

S

M~,

we find

M~

w

M~/ /1

+

M)

_u

(h L)~/b2 (4.9)

2

The volume fraction of

polymer

in the brush _~b is obtained

by solving equations (3.2-3.4) simultaneously.

In

general,

this _must be done

numerically.

^In the limit L/b

~ o~, however,

~b(

^and ~b

satisfy dF$,~/d~b(

=

0 when 4~

= 0.5, _so we can solve for

~b(

ⁱⁿ terms of

~b. Moreover, when L/b is

large,

_~b

depends only weakly

_on h, _so _we _may take the

single

brush limit h

~ o~. If _we further _assume that _~b « I and that t #

0,

we find

~b = [3

«~/(l

_x~ ₊

a2)]~~~,

(4,

lo)

where

~

_j/2~ ~-x~+XAB-2). ~~~~~

With _~b and

m(z)

in hand, we can calculate the pressure between the two brushes

i

d(AF F,~~)

(4.12)

P

"

f

_dh '

where AF is

given by equation (3,I). Upon differentiating

AF, we find that the pressure

between the brushes is

simply

the _excess free energy relative _to the reservoir per unit volume evaluated _at the

midplane

_.r

=

h

~

~ m

~~~,x ~<e, (~b~

~

) V<e,j,

h

(4. l~)

At ~fi

= ~fi~ = 0.5, this reduces _to P

=

~j tM(

₊

uM()

,

(4,14)

a 2 4

where t and _u _are defined

by equations (3.6, 3.7).

In the

single-phase region,

the first term dominates _so the pressure

decays exponentially

with h _as

expected

P

=

~§ ^tM( ^sech~ (h L)/f

(4.15)

2 _a

The

negative sign

indicates that the brushes attract each other. Near the critical

point,

the second term in

equation (4,14)

dominates. When the

separation

distance h L is smaller than the correlation

length f

^the _pressure ^varies as

p

=

~~ K4

^b ⁴

ua~ ,fi

h L

(4.16)

(10)

Equations (4.15, 4,16) imply

that the interaction between _two brushes in _a critical mixture is shorter in range than the _van der Waals at~action.

However,

_our estimate of the pressure is

flawed because _we have used the mean-field

approximation

to describe the

binary liquid

mixture _near

criticality.

We may obtain the actual form of the pressure

by using

finite-size

scaling

_arguments. In the limit f » h L, we

employ hyperscaling

to obtain

j2 «~/~

~

h

~ L

kT

(h L)3

^{~~ ~~~}

This

expression

has the _same

sign

and is similar in form to the _van der Waals attraction. If _we

replace

the walls and brushes at x =

0 and _x

= 2 h with walls _at _x

= L and _x

=

2 h L

characterized

by

_an effective Hamaker _constant

A,

the

expression

for the pressure

arising

from

van der Waals interactions is

P~~w

=

~

~.

(4,18)

6 ar

(h L)

Typical

values for the Hamaker _constant _are of order A

=

10~ ^~~ ergs, so the pressure

arising

from van der Waals interactions is

comparable

in

magnitude

to the pressure

arising

from

preferential

attraction of the

good

solvent _to the brush _near the solvent-solvent critical

point.

Once the brushes

overlap,

the pressure

changes sign

and increases

by

^several ^orders ^of

magnitude.

A

plot

of dimensionless pressure

(Po

=

a~ P/kT),

where the correlation

length amplitude

b is

arbitrarily

set

equal

to the _monomer size _a, is

plotted

in

figure

2 _as a function of

h/L. The pressure appears discontinuous _at L because we have used a step

p?ofile

for the

polymer

concentration and have

neglected gradient

terms in A-solvent concentration within the brush _a _more realistic model would smooth _out the

discontinuity

at h

=

L _on the scale of the correlation

length.

o.oo2

P~

o

/

h/L

Fig. 2. Normalized pressure, Pow a~ _P/kT, _is plotted _as _a function of h. The plots _are done for N

= 30 000, _«

=

0.01, _xA

= 0, _xu

=

I, and xAu 1.9999(dotted line), 1.99999(dashed line), 1.999999(solid line). Note that the minimum _occurs _at the overlap distance.

(11)

The results in

equations

(4.4,

4,17) apply only

in the

special

_case of critical

composition.

When ~fi is

off-critical,

the results _are

qualitatively similar,

and are contained in the

analysis

of Nakanishi and Fisher

[12].

Finally,

we note that _our results _are

applicable

_even when the brush

profile

is

parabolic

as

long

as f » L. When the correlation

length

is

large,

the interaction between brushes should be insensitive _to the brush

profile shape.

5. Conclusion.

We have

investigated

the interaction between _two brushes in _a

binary

^mixture near

criticality.

Within the Alexander-de Gennes

approximation,

the brushes _can be

replaced by

two

imaginary

surfaces located _at the

tips

of the brushes when the brushes do not

overlap. Thus,

the

interaction between the brushes _can be obtained from

binary liquid

mixtures confined between two walls [I1,

12].

Acknowledgments.

This

study

was stimulated

by experiments

carried _out _on latex colloids in mixed solvents at the

University

^of

Pittsburgh by

the group of J. V. Maher. We thank F. Stiemelof, ^S. T. Milner and D. C. Morse for useful discussions. The work _was

supported by

a California

Biotechnology

Training

Grant and

by

the U-S.

Department

^of

Energy

under grant No. DE-FG03-87ER45288

(TAT and

PP),

and

by

^the ^National Science Foundation,

through

the Materials Research

Laboratory

at the

University

of Califomia, Santa Barbara, under grant. No. DMR-9123048

(AIL).

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