Phase separation of grafted polymers under strong demixing interaction

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Phase separation of grafted polymers under strong demixing interaction

Hui Dong

To cite this version:

Hui Dong. Phase separation of grafted polymers under strong demixing interaction. Journal de

Physique II, EDP Sciences, 1993, 3 (7), pp.999-1020. �10.1051/jp2:1993178�. �jpa-00247893�

Classification

Physics

Al~sn.acts

36.20 36.40

Phase separation of grafted polymers under strong demixing

interaction

Hui

Dong

The James Franck Instiiute, The

University

of

Chicago, Department

of

Physics,

5640 South Ellis Av.,

Chicago,

Illinois 60637, U-S-A-

(Received 15 Januaiy 1993, revised 30 March 1993, aicepted 13 April 1993)

Abstract. We studied the

phase separation

of

grafted homopolymers

of two different kinds

(A

and B) under strong

demixing

interaction, with demixing energy far greater ^{than kT} per chain. We

successfully

constructed _a stable self-consistent model for the

phase-separated

state will all the chains end at the _same

height.

The

configuration

contains pure A and B _monomer regions and inevitable A-B mixed regions. We studied the composition and magntitude of the free energy of the system, ^{calculated the size} of the

periodic

regions and its

dependence

_on the

demixing

interaction strength. The system ^with distributed end density

profile

is further studied.

Comparison

is made with results obtained from the

previous

weak interaction model using second order correlation

function under _mean field

approximation.

1. Introduction.

Phase

separation

of

grafted polymers

under

demixing

interaction is _a

subject

^of

general

interest. It has theoretical _as well _as

practical significance. Many previous

works have

already

been done

[1-4]. Generally

the

polymer

chains _are

grafted uniformly

on an x-y surface. These chains _can be

composed

of the _{same or} different _monomers of various

configurations.

Since

monomers

repel

each other in _a crowded environment, there exists _a strong

repulsive

interaction _among the

polymer

chains. This

parts

^{of the} interaction _{energy can} be described

by

a pressure

profile

under _{a mean} field

approximation. Furthermore,

the pressure causes the chains to stretch, which in tum store a

comparable

amount of elastic energy. if the

polymers

are

composed

of _two different kinds of _monomers

(A

and

B),

there is also

demixing

interaction between them. When the

demixing

interaction reaches sufficient

strength,

the

grafted polymers

start to

phase

_separate.

By

now the

general

method for

treating grafted polymers composed

of

A and B _monomers under weak

demixing

interaction has been studied. Witten and Marko have

obtained many results

using

second order correlation function under _mean field

approximation

[5]. They

studied the

phase separation

^of

homopolymer

chains, and obtained the

density

distribution of the

phase-separated

monomers.

They

also calculated the minimum

demixing

interaction

strength

and its

dependence

_on the system parameters.

Recently, Dong,

^{Marko and}

Witten have

generalized

the

problem

to diblock

copolymer

_systems of various

configurations [6].

We studied various

properties

of their

phase separation

patterns, as well _as the

dependence

of minimum

demixing

interaction

strength

on various system

configurations.

However,

when under much stronger

demixing

interaction

(greater

than kT per

chain),

the

perturbations

_on

polymer

chains _{are no}

longer

small, therefore the

previous

models based _on

perturbations theory

are no

longer

valid. When the

demixing

interaction reaches sufficient

strength,

the

phase separation

can be _so

significant

that the system ^will form

nearly

_pure A _or B

monomer

regions

_to reduce the strong

demixing

interactions. The overall distribution of A and

B chains could be

completely reorganized.

Thus _we need _{a new}

approach

_to the

study

of the

phase separation

in this situation. Unlike the _case of _a blend where the chains _{can move}

freely,

here the A and B chains _are

grafted

onto a surface where

they

_can

only

stretch _or bend. Thus _an

extra constraint is added which makes the

phase separation

_more

complicated.

A self-

consistent solution must first

satisfy

the melt

condition,

that is the overall _monomer

density

must be constant

throughout

the entire

configuration.

The

microscopic

balance of all the

homopolymer

chains _must also be satisfied. Under these

constraints,

the

phase-separated configuration

will have A and B

polymer

far away from each other with minimum additional

bending

and

stretching.

So it _can have low

demixing

_energy while

avoiding

a

big

increase in the

stretching

_energy. In other words, the

segregated configuration

will have the lowest total free energy under the constraints.

One self-consistent state of strong

phase separation

_was

proposed by

Witten and MiIner

[7].

They

assumed that the chains

segregated

^into two

layers

an A-rich

layer

near the

grafting

surface and _a B-rich

layer beyond,

_or the _same pattern ^{with A and B reversed. The}

resulting saving

of

mixing

_energy is

only

_a fraction of the

mixing

_energy of the

un-phase-separated

state.

But this work did _not consider the

possibility

that the

phase-separated regions might

not be

layers,

but

might

instead show

periodic

variation

along

the

surface,

_as in weak

phase separation [5].

In this paper, we consider such states, and show that

they

have

substantially

lower energy than the

layered

state considered in

[7].

The

phase separation problem

becomes

excessively complicated

if _a realistic system ^is

considered,

in which different chains end _at different

heights

with _a definite

probability [5, 6].

The chains with different

heights

behave very

differently

and interact with each other under

demixing

interaction. To

simplify

the

problem

and obtain _a

general picture

of the

phase separation

_{process, we} first

study

_a

simplified system

^{in which} all the chains stretch to the

same

height h,

in section 2. We _{construct a} self-consistent solution for the

phase-separated

state, discuss its

stability

and examine the

composition

of the free energy. In section 3, _we

further the realistic system ^with distributed end

density profile.

In both sections, _an uniform

configuration along y-direction

is

assumed, only

the

phase separation along

x-direction is considered. In section

4,

_we discuss the

possibility

of the

phase separation

in both _x and y directions. We also examine the weak interaction limit and its

comparison

with the results from other methods. The

possible experimental

_system suitable for observation of the

segregation

is also addressed.

2. Phase

separation

of the

grafted polymers

^with same

height.

We first

study

_a

simplified

system, in which all the

homopolymer

chains stretch _to the _same

height

h

(as

^{shown in}

Fig. I),

to get some

insight

to the process of strong

phase separation.

This

simplified

system, ^known as the Alexander-de Gennes

brush,

is

widely

used and

gives good qualitative

results

[1, 2].

As it will become clear

later,

_we will _use the result of the

simplified polymer

brush _as a

building

block to obtain the

phase segregated

state of the realistic

polymer

brush of _{our concern.}

Fig. I. The sketch of

grafted

A-B

homopolymers

without

demixing

interaction, _{on >.-z}

plane,

A, B

polymer

chains _are

grafted alternatively along

x-direction.

Under _mean field

approximation,

the pressure

profile

is uniform

along

x-direction.

Therefore there is

essentially

no horizontal force

along

x-direction _to deform the stretched chains before the

demixing

interaction is turned _on.

Realistically,

the

grafted polymer

chains may deforrn and _cross each

other,

_so

acquire

a « natural width _» R

[9].

Since horizontal

bending

and

stretching

will increase energy, so the horizontal fluctuation will be small

compared

to vertical

stretching.

There fore without

losing generality,

_we will _assume that the chains stretch

straight

_up and

ignore

^horizontal fluctuation. The idealized

configuration

is shown in

figure

2.

h

x

Fig. 2. -Grafter A, B

homopolymers

without demixing interaction. The solid and dashed lines represent ^{A and B}

homopolymer

chains

respectively.

We consider the _case where A and B

homopolymers composed respectively

of A and B type

monomers are

alternately grafted

on an x-y surface with _a uniform

spacing.

Each chain

displaces

_a volume V. A

polymer

melt is

essentially incompressible,

thus V is

proportional

to the number of _monomers in it. So in _a melt of

density

_p, the volume V is related to the chain's

molecular

weight

M

by

M

=

pV.

The total volume of the

polymer

brush will be the volume V

multiplied by

^{the number of} chains in the brush. The _rms end-to-end distance of _a chain in _a melt is

R~

=

(3 Vla)'~~,

and the radius of

gyration

is

R~

=

R~/6"~ [5].

The _«

packing length

_»

« a »

depends

on chemical

make-up

of the

chains,

and is

usually

or order

101 _[11].

_The

grafting density

_~r is defined _as the number per unit _area of chains

grafted

_on the surface.

Therefore each chain

(A

_or

B) occupies

_{an area} I/~r _on the x-y surface.

Quantity

h is the

height

of the

polymer

brush. Under the melt

condition,

_we have h

=

WV. A chain _can be described

by

its conformation

r(u).

The variable _u represents the volume of the

polymer

^chain from the

grafted point

_ro to _r. We consider the

strongly

^stretched

regime

in which h _w

R~.

In this

regime

there is _a substantial elastic tension in each chain. A chain

uniformly

stretched _to

height

h transmits _a force

Fj

=

3(kT) h/R( _[9].

Here kT is the thermal energy. In the

sequel

we shall express all

energies

in units of kT. Since the force is

proportional

_to the

displacement,

the

polymers

act like ideal

springs

with

spring

constant

3/R(

m a/V. The

grafting

surface exerts _a force F _on each chain at its attachment

point.

If the

layer

is _to be in

equilibrium

there _must be _a

balancing

force. The

balancing

force arises from _a pressure p

throughout

the

layer [2].

Evidently

the force per chain

arising

from this pressure is

p/~r.

Thus _we have _p

=

~rF

=

~rha/V

=

~r~a.

It is this pressure that

pushes

each chain out to the

required elongated

state.

The pressure

p(r) only depends

_on the

height

_z in _our

polymer layer.

Before the

demixing

interaction is turned _on, there is _no difference between A and B

monomers. Each chain stretches up due to the pressure within the

layer.

^{The melt} condition

requires

that the _monomer

density

be _constant

everywhere

in the

configuration.

Thus the

monomer

density profile

of all the chains is also uniform

along

the z-direction. Since the chains

are

uniformly

stretched, the local tension

along

the chains should be _{a constant.} Therefore the pressure

profile

should also be uniform

along

the z-direction. Thus _a

simple configuration

results when there is _no

demixing

interaction present both the

density

and pressure

profile

are

uniform

throughout

the whole

region,

and the A and B chains all stretch in the direction normal to the

grafting

surface _as shown in

figure

2.

2.I THE CONFIGURATION OF THE PHASE-SEPARATED STATE. We _now consider the _case

where there is

demixing

interaction between A and B _monomers. This will _{create a}

demixing

interaction energy per unit volume

Edem>x

(r

= A

j

A

(r

_w

B

(r

The A is the interaction

strength

it _can be

readily

measured and is of order kT per thousand

Angstroms,

_e-g- A

=

2 _x

10~~(kT/l~)

_for

polystyrene

and

polybutadiene [12, 13].

The A is related _to the

Flory

x-parameter

[9] by

AV

=

XN,

where N is the

polymerization

index of the chains. The

~b~(r)

and ~b~

(r

are the _monomer

density

fractions of A and B _monomers. The melt condition is

~b~(r)+ ~b~(r)

= I. We _assume the

demixing

interaction is _not strong

enough

to disturb the

density

and pressure

profile along

z-direction _so that

density

and pressure

profile

as well _as the end

density

distribution will remain the _same for the

phase segregated

state.

Upon

the introduction of the

demixing

interaction, the chains stretch and deform _to

minimize _contacts between A and B _monomers in order to minimize the

demixing

_energy.

However,

the

stretching

deformation induce _an increase in the elastic energy. The

competition

between the

demixing

^and

stretching

of the

polymer

chains determines the final

segregated configuration.

As mentioned above, we consider here that the

profile

in the z-direction is

unperturbed

_upon the introduction of the

demixing

_energy and the chains

only

deform in the x-direction- When _a strong

demixing

interaction is present, ^{the above}

competition

results in _a

breaking

of symmetry

between A and B chains. The chains bend and stretch

slightly

in the x-direction _so that chains of the _same type congregate

locally

to form _a state

resembling

the _one shown in

figure

3.

Fig.

3. The sketch of the

phase-separated

A-B

homopolymers

under strong demixing inteaction.

The local

congregation

of the chains from

regions composed

of pure A _or B _{monomers as}

region

I _or II in

figure

3. The A-B mixed

region

III forms

inevitably

_{as a} result since A and B chains _are

grafted alternatively

_on the x-y surface. Phase

separation

_as shown in

figure

3 is _a

result of

minimizing

the total energy which is

composed

of two

competing

factors _: the

increase in the elastic energy due to horizontal

stretching

in the

region

III is _more than

compensated by

the

significant

decrease in the

demixing

_energy ⁱⁿ the

segregated regions

I and II.

The

configurations

within these different

regions

can be examined in _more detail. Since

regions

I and II _are

regions composed

of pure A _or B _monomers, there is _no force induced

by

the

demixing

interaction within. Also _note that since the pressure

profile

is

uniform,

these

regions

_{are «} force free _{» zones.} Thus the

portions

of the chains that lie within these

regions

stretch normal _to the x-y surface. In the mixed

region

III, the chain segments are stretched

straight

_to achieve minimum

stretching

_energy. We also expect ^that _{~b~ ~b~ =} 0 in

region

III, since any local A, B

density

difference will _create local tension due _to

demixing

interaction, and increase energy

density. Furthermore,

to

satisfy

the melt condition, the total _monomer

density

must be uniform

throughout

the

region.

On the boundaries between

region

^I

(or II)

with III, there _are

demixing

forces. These forces stretch the chains

horizontally

so that the

microscopic

mechanical balance of the chains _can be achieved.

Figure

4, _{as an} ideal version of

figure

3, shows the _state which

incorporates

these features.

Based _on the above very

general

_{arguments, we} have constructed _a self-consistent model for the

phase-separated

state. The

polymer

chains form _two

large regions

I and II

composed

^of

pure A and B _monomers, within which all the chain segments ^stretch up normal to the _x-y

surface. Inside the A-B mixed

region

III, the chains stretch with _a

specific degree.

^{The chain} segments ^{within each}

region

are

straight

and have uniform

density profile.

Forces

only

exist is at the

boundary

of

region

I

(or II)

^with

region

III. There is _an

abrupt change

^of monomer

composition

from

region

III _to

region

I

(or II).

The

corresponding demixing

force is

perpendicular

to the

boundary, balancing

the elastic forces

along

_x and _z directions. The total

monomer

density

is uniform

everywhere,

melt condition is thus satisfied.

2.2 COMPUTATION _{OF THE FREE ENERGY.} The

stretching

and

demixing

forces

acting

on the

chains and the

corresponding

free

energies

_can be obtained for the

phase-separated

configuration

shown in

figure

4.

The forces

acting

_{on a}

single

chain _are shown in

figure

^5.

Fj, along

=-direction, comes from the pressure

acting

on the

chain,

_as

explained just

above section 2. I. The pressure ^{is measured}

Fig.

4.

Phase-separated

homopolymers under strong

demixing

interaction. The solid and dashed lines represent A and B chains. The regions I and II _are the pure A and B regions, region III is the A-B mixed

region.

z

F

I II

0

~ ^x

Fig. 5. The forces acting _on a single chain,

Fj

is the tension created by _pressure. F~ ^{is the} demixing interaction. The combined force F balances the elastic force acting _on the chain.

by

force per unit _area. Since the force

equals

to the pressure

~7) multiplied by

the _area per chain

(I/~r),

_{so we} have _:

~ _P

1-

In the

original

state before

phase separation (Fig. 2),

the pressure is in balance with the elastic force

along

z-direction :

~

=~h.

Since _{we assume} the pressure remains the _same for the

phase-separated

state, so we have _:

~ ^p ah

i"-~f

The force

F~

^{shown in}

figure

5 is

coming

from the

demixing

interaction. As shown in

figure

6, when _a small

portion

of the B chain with volume As

hi

_{is moved from the A-B mixed}

region

III

(where

_~b~

= ~b~ ₌

1/2)

_to the pure B

region

II, the

demixing

_energy is reduced

by

A.~b~.~b~.AsAi=

A.

.AsAimf~.Ai.

Since each chain

occupies

an area I/~r _on the >.-y

grafting surface,

it

occupies

_{an area} of As

=1/(~r2

sin

a)

on the

boundary

between

region

II and III

(Fig. 6).

With I/~r

=

V/h, _we have _:

~ ~ l

~ ~ l I I AV

~~ 2 2 ^~~ 2

2'~r2sina _~8hsina

II

Fig.

6. Under _a

demixing

force F~, _a portion of _a single B chain (shaded) is moved by a distance hi _{from the A-B mixed} region III _to the pure B region II. The As is the _area of the _cross section of the

chain

on the

boundary.

The condition that the combined force F

=

Fj

₊

F~

^balances the elastic force

resulting

from

stretching requires

that F

points along

the direction of the chain segment ^within

region

Ill

(Fig. 5).

So _we have the

following

condition

sin a sin

(w/2

2 _a

f

_F

Using

the above results for forces F and

F~,

we obtain the

following expression

for

angle

_a :

~~~

j8ajhj2

$

_V ^~

Note that Ala is

proportional

to the square of the interfacial tension

[10].

As

expected,

the

angle

a is

completely

determined

by

the interfacial tension and the

grafting density

« =

h/V.

Now _{we can} calculate the

composition

of the free energy of the

phase-separated

state.

Firstly,

the elastic energy is

composed

^of contributions from

stretching along

_z and horizontal direction. For _a

single

chain, the elastic energy due _to

stretching along

z-direction is

Eo

₌

~

_h~

Since there _are

(2

d

« chains

(A

or B within _a unit of

length

2 d

(Fig. 5),

_so the elastic energy of all the chains within _a unit due _to

stretching along

z-direction is _:

(2d)«.~~h~=~d.

(II-])

2V

V~

The additional elastic energy of _a

single

chain

grafted

at

position

_xo due _to horizontal

stretching

is

given by

~

2d-,roj2

~

_{~ ~}

~°

cotg a

~ ~

2 h

In which the second factor is the

spring

constant _« a/AV _» of the chain segment of volume AV

(the

_segment inside

region

III,

Fig. 5).

The third factor above is the square of the

stretching

distance of the chain segments. ^{So the additional elastic} energy from all chain segments ^within

region

III due _to horizontal

stretching

is

given by

:

2d

~

2d-Xo

2

~ h 2

~

« d~ro =

d tg _a

(II-2)

~

°

~ ~ ~°

cotg a

~ ~ ~ ~

2 h

Adding (II-I), (II-2)

and

dividing by

the number of chains in the interval 2 d, _we have the average total elastic energy of _a

single

chain

~~'~ (2d)« V~~~2 ~V

^~

~"~

2V~4V~~~"' ^~~~~~

Secondly,

the

demixing

_energy contains _two parts. ^One comes from

region

III

(volume

V~)

where A and B _are mixed

uniformly together

_:

A.~b~.~b~.V~=A.

.2d. ~

=

~~~

(II-4)

2 2 2 tga

4tga

The other part comes from the

boundary

between

regions

I and II, where A and B chains _are

adjacent

and mixed into each other

(Fig. 4).

This is _{a narrow} A-B mixed

boundary region

with

a small interfacial width

[10]

~

~ Aa

The

demixing

_energy from this _narrow

region (volume Vi)

is

given by

:

i i d

j

i

jAj~

^d

(IIS)

A.~bA'lbB'Vi~A'f'j'~~(

Aa 4 _a tga

The above is the part ^of

demixing

_energy that is

responsible

for the

phase separation

since it tends to increase the

periodic

size 2 d. Without this part ^{of the} energy, the size d will decrease to zero since other parts ^{of the free} energy tend to reduce the size of the

regions.

Therefore this part ^{of the}

demixing

_energy

essentially

is the

driving

froce of

phase separation. Combining (II- 4)

and

(II-5),

the average

demixing

_energy of _a

single

chain is

~ l

~

d~ jA

^{~ d}

~"

(2 d)

_« 4 tg a

~ 4 _a tg a

~ _tla

~

g~a

~~~

~~

All

together,

_we have the average total free energy of _a

single

chain _:

~

~~'

_~

~~" ~~

~ ~ ~~ " ~

~ _tla

~

§/~ ^g~a

^{~~~ ~}

Single angle

_a ^is

already

determined

by

the

microscopic

balance of the

forces,

the

periodic

size d is determined

by minimizing

the above total free energy. We have _:

~~~ ~~2jl/4

~

§/~' ~~21/2

10+~

ah~

The

height

H of the

triangular region

III is thus

given by

_:

~v2

3/4

~ ~

8

+

q

~

tg £Y

$ Av2

1/2

' 10 ₊

ah~

Since _{we assume} the strong

demixing

interaction is still very weak

compared

to the

pressure :

F~

w F

j, or tg _a w I. At the _same time, the

demixing

interaction per chain is strong

compared

to kT _: AV _w I

(in

unit of

k7~.

So _we have relation _:

WA «

£. (II-8)

V V

All the results _can be

simplified using

the above relation.

Firstly,

the

angle

is

simplified

_{as :}

~~"

8a h

j'

Secondly,

the size of the

region

III is

simplified

_as :

d~ 2

jI/4 jV

^{2 11}

~5 a 25 ~

(II-9)

Where R

=

(Vla)'~~

is the _« natural width _{» or} radius of

gyration

of the

polymer

chains

[9].

Thirdly,

the

height

of the A-B mixed

region

III is

simplified

_as

4 _x 2'~~ _h

(II-10

~

,fi ,$

Since AV _w I, so

according

_to the above

equation,

the

height

of the

region

III is much smaller than

h,

and it decreases _as the

demixing

interaction gets stronger. ^{The size d}

essentially

remains

independent

^{of the}

demixing

interaction

strength

A. Since the

segregation

_structure in

figure

4 is formed

by

the horizontal

stretching

of the short chain segments ⁱⁿ

region III,

and the

fluctuations of these short segments are much smaller than the radius of

gyration

R for the whole

chains,

_so the

segregation

pattern ^will not be

significantly

disturbed

by

the fluctuations _even

though

we have d ~R in

equation (II-9).

With the size d obtained

ab'ove, using equation (II-7),

the lowest total free energy per chain

E~,~

is

given by

_:

E~,~

i

$

+

~~~~ ,,'$. _(II-

₎

4 _x 2

The above free energy of the

segregated

state contains two terms. The first term, which is _not

demixing related,

is in fact the free energy per chain

Eo

of the

unsegregated

state. The second term, which is

phase separation

related, ^is the direct result of the strong

demixing

interaction _:

AE~j~

_m

E~j~ Eo

m

~'~

~

,$

4

'~'~

With AV w

I,

the

demixing

related energy component ^{is much} greater ^{than kT} as

expected.

It is

proportional

to A'~~, rather than A

corresponding

to the strong

layering segregation [7].

In conventional

units,

the

phase separation produces

a

change

in energy per chain

AE~,~

that

depends only

on the

demixing strength XN

_:

AE~j~

₌

~~~~

,/XN

4 _x

where N is the

degree

of

polymerization.

The

height

H of the

triangles

relative _to the total

height

h also

depends only

_on

XN

~ ~ _~ ~ ⁱⁱ j

I ,fi ,@

The width d of the

triangles

is

independent

of the

demixing strength

_Xi

~ ^IN

~

,@ ^~~

2.3 THE STABILITY OF THE SOLUTION. The

stability

of the solution under small

perturbations

is determined

by

the behavior of the chains

undergoing

small

deformation,

_as shown in

figure

7a. Since the

single

chain deformation has

negligible

effect _on the

density profile,

the A- B

composition

in

regions I,

^II _or III will _not be

changed. Therefore,

there is _no force _to sustain

the

bending

of the deformed chain within these

corresponding regions.

So the chain segments

Dawnwcww

la)

16)

Fig.

7. The

phase-separated

state under _an arbitrary but small

perturbation.

The curved line in (a) is _an

arbitrarily

deformed _test

polymer

chain. The

corresponding

line in (b) is the

only possible

deformation since _no force exists within

region

II _or III, _so the chain segments ^{within these} regions must be straight.

within these

regions

will _{return to}

straight.

Thus the

only possible

deformation is the

sliding

of the

bending position

of the chain

along boundary

of

regions

II and

III,

_as shown in

figure

7b. In this _case the force F

acting

_on the

chain,

_no

longer aligned

with the deformed chain segment ⁱⁿ

region

III, will

push

the

bending position

_to its

unperturbed original position.

^{So the}

configuration

described in the

previous

section is stable and has resistance to an

arbitrary

but small

perturbation.

3. Phase

separation

of

grafted polymers

^with distributed end

density profile.

The

phase-separation

model introduced

previously

is for _a

simplified

system ^{where all the} chains end _at the _same

height.

We have obtained _a self-consistent solution and _a

general picture

of the

phase separation

_process under strong

demixing

interaction. Now _we consider _a

more realistic system ^{in which the chains end} at different

heights,

with _a definite distribution of

probability.

The

probability

that _a

single

^chain has its free end located at

height

z~ is

[5]

y~

h h~

/~ z(

For any local _{area on} the x-y

grafting surface,

there _are chains with

length

_range from 0 _to h. If

we still _assume that the

phase separation

_occurs

according

_to the process shown in

figure 4,

short chains will

completely

fall inside the A-B mixed

region.

^{These chains} are too short to reach the

boundary

of

region

III. With _no horizontal force

acting

on them,

they

will stretch

straight

_up, instead of

stretching

with _an

angle

_a

along

with the deformed

long

chains. Then the total _monomer

density

is _no

longer

uniform within

region

III,

violating

the melt condition. To

avoid

this,

the chains will deform and the A-B relative

density

distribution

~b~(r)-

~b_~

(r)

will be

changed.

Thus _extra local tension will be created within

region III,

which in turn will further deform the A, B chains. So the model shown in

figure

4 fails _to

provide

_us with _an

adequate

solution for this realistic system. Instead, the

phase-separated

blocks of chains may

no

longer

be

straight,

clear-cut pure A, B

regions (I

and

II)

_may not be formed, and the A-B relative

density

in A-B mixed

regions (III)

_may be non-uniform.

To

study

the

problem,

consider the system as one

composed

^of

infinitely

_{many «}

layers

_».

Each

layer

contains all the chains with the _same

height

_z~

(ranges

^from 0 to

h).

Then the whole system ^is

just

_a combination of all the

layers

of different

heights,

with _a definite

weight

of

probability.

Since all the A, B chains within each

layer

^have the _same

height

_z,,, our

previous

results _can be

applied

for each

layer

^with h

replaced by

_z~. Thus if the

demixing-related

interactions between the

layers

_are

neglected,

the

phase-separated configuration

_can be considered _{as an}

superposition

of all the

phase-separated layers,

each

resembling

that shown in

figure

4. From _our

previous

result, the width 2 d of the

periodic region

III is

independent

of the

height

and _so is the _same for all the

layers.

While the

height

H of the

region

is

proportional

to the

height

of the

corresponding layer.

^But as we will _see

below,

the interactions between

layers

in fact will

change

the

density profile

of each chain

along z-direction, making

^the

density profile

non-uniform. The pressure

p(z)

will also be

changed.

The horizontal stretch of the chains will be disturbed _as well. But _we will find _out that the

demixing-related

interactions

between

layers

have

only

small

impact

on the

segregation

patterns. ^{This make it}

possible

to find the

phase-separated

solution based _{on our}

previous

results, ^with the

major non-demixing-

related

layer-layer

interaction included. We can first consider a

two-layer

system, ^then

generalize

the results to a

multi-layer configuration [3].

3. Two-LAYER _SYSTEM. We first consider _a

simple two-layer configuration.

As shown in

figure 8,

the

grafted polymer

chains _are

separated

into two

layers layer-

and

layer-2.

Each

layer

contains

polymers

with the _same

height.

We further _assume that the ratio of the number of chains in

layer-2

to the number of chains in the whole system ^is A.

(So

if the total number of

chains is I, there will be chains in

layer-2,

A chains in

layer-I.)

Before the

demixing

interaction is turned _on,

layer-2

is allowed _{to contract}

freely

to a new

height

^h' if that is

z h

Pi

h'

P~

0

x

Fig.

8. The two-layer system without

demixing

interaction. A chain in

layer-

I has _two blocks. The upper block has volume

Vi

and

experiences

_{pressure pj.} The lower block shares the _same

region

with chains in

layer-2,

has volume V~ ^and pressure p~. The _two blocks of _a

layer-I

chain have different

monomer

density,

satisfying ^{the melt} condition.

energetically

favorable. Then since the melt condition is still

satisfied,

the lower

portion

of the

layer-

I chains will be _«

squeezed

_» and stretch _more

extensively.

At the _same

time,

the upper

portion

of the

layer-I

chains will contract,

keeping

the overall

density

distribution _constant.

Thus when the

polymer

chains in

layer-2

contract to a new

height h',

the

density profile

of the chains in

layer-I

^is no

longer

constant. We denote the volumes of the upper and lower

portions

of _a

layer-I

chain

by Vi

and

V~ respectively. They certainly satisfy Vi

₊ V~ =

V. The pressure

profile along

z-direction is also

changed

in order to

keep

the mechanical balance of the chains. We _assume the pressures are p~ and p~

respectively (Fig. 8).

Note that the upper

portion

^of a

layer-

I chain has volume

Vi

and

length

^{h h'.} Its cross-section

occupies

an area of

Vj/(h h').

While the lower

portion

^of a

layer-

chain has volume V~ ^and

length

h'. Its cross-section

occupies

an area

V~/h'. Similarly,

_a

layer-2

chain will have _area

V/h'. Thus the melt conditions for the upper and lower

regions

_{are :}

Vi

_j

~~

~~h-h'

«

A

~+

(l -A)~)

=

With I/«

=

V/h,

the _two

equations

_are in fact

equivalent

_as

expected.

Thus _we obtain the volumes _:

~ ~ l h-h'

~ ~ l h'-Ah

' I -A h ^{' ~~} l -A h

The mechanical balance of all

portions

of the chains

requires

that the force

coming

from the pressure

gradient

be balanced

by

the force

coming

from the vertical

stretching.

The

following

three

equations

result from the balance of the _upper

portion,

lower

portion

of _a

layer-

I chain, and a

layer-2

chain :

Vi

a

~'h-h'~

,

V~ ^~~

~ _~~~

~P2-Pi)j+Pi _{~_~~,} =)h'

₍₁₁₎

~P2 PI

)

~

)

~' _~~~~~

We define

height

ratio _7J

w h'/h.

Using

the results for

Vi

^and

V~,

we have the

following

solution for the pressure

profile

:

pi = (I )~

~~~,

p~ =

[(I

_{A )~ + 7J~]}

~~~.

v2 v2

In which _7J is _a function of A _;

where

f _m ²⁷ 54 A + 25 A ^~ ₊ 3~~~(1 A

)(27

54 A ₊ 23 A ~)'~~

JOURNALDEPHYS>QUE>I -T1, N'7 JULY1991

We have the

limiting

behavior of _7J

A-0:

7~mA'~~

~ _~ '~ ~~ ~ _~~

The relation of

7~ vet-sus A is shown in

figure

9.

h' /h

0.

o.6 o.a 1 "~~°

Fig. 9. The height ratio _~ h'/h of the _two layers i>e;sas the ratio A. As A decrease~

(decreasing

number of

polymers

in layer-2), ^the chains in layer-2 contract very

slowly

with the wide range of value, but shrink

rapidly

to zero when there _are

only

a few

polymers

left in layer-2.

For A =0.5

(equal

number of chains in

layer,I

and

layer-2),

we have _7J

=

0.877,

pi =

0.25

ah~/V~

_and

p~ =

1.02

ah~/V~.

_{Note that}

we have _p

=

ah~/V~

_{for the}

single-layer

system. ^{The results confirm}

that,

when

released,

the chains in

layer-2

indeed _contract to a new

height

in order _to achieve lower free energy. While the pressure in the upper

region decreases,

the pressure in the lo,ver

region

increases _as

expected.

As shown in

figure 9,

when A is very

small

(only

a few

polymer

chains _are

released),

_7J is also very small. The few released chains

contract almost _to the bottom. But for A in _a wide range from 0.4 _to I, we have

7J m 0.8. Therefore, in _{most cases,} the chains in

layer-2

^will

only

contract a little. This allows

us to construct the

two-layer configuration

from the

single-layer

results. For

=

I, _we have 7J = 1, returns to the

single-layer

model considered before.

Note with the proper results for

density

and pressure

profiles

obtained above, we

already

include the interaction

(not demixing related)

between the two

layers.

We _now need to

investigate

what kind of effect the

demixing-related

interaction between the _two

layers

^will have _on the final

segregated configuration.

When the

demixing

interaction is tumed _on,

phase separation happens

in both

layers.

If _we

assume the two

layers phase

_separate

independently,

_we will have _a

configuration

shown in

figure

10

according

to our

previous

discussions in section 2. Each

layer

^{forms its} pure A and B

regions

_as well _{as an} A-B mixed

region.

As _we have shown in section 2, the size d is the _same for both

layers.

While the

angle

_a

depends

_on the

height

of the

chains,

_so is different for two

Fig.

IO- Illustration of the

phase separation

of _a two-layer system ^without considering the demixing- related interaction between the _two layers.

layers.

^{As shown} in

figure

10, each chain goes

through

two boundaries of the _two A-B mixed

regions

of

layer-I

and

layer~2.

So the

bending

force will be _«

split

_» into forces exerted at

slightly

different locations, instead of accumulated at _a

single position.

If the

angles

and the

heights

of the _two

triangular regions

do _not differ very

much,

the

demixing

interactions between _monomers from _two

layers

will be small and

negligible.

Then the method used for the

single-layer

_{case can} still be

employed

here to estimate the free energy for this

two-layer system.

^{So in the}

following

section _we will show that the

chain-bending regions

_{are narrow, so}

we can make _use of the

single-layer

results.

Consider two separate

layers

with

heights

h and h',

following

the same

procedure

used in section

2,

_we have the forces for

layer-1

:

Fj=ih'=~.(l-A)

'~

V2

^V 7J-A

l

V2

_{AV I}

7J A

~~~~

2 2 h' 2sina

~8hsina

_1-A

7J

Subsequently,

the

angle

_a is determined

by

mechanical balance of the chains _:

The forces for

layer-2

:

~,

a~,

^ah

i~p ~f"J

,

I I V I AV I

~~~~'2'2

h' 2sina' 8hsina' _~

Similarly,

_we obtain the

angle

_:

~~~' j8ajh~j2 _$

^~

V

'~~~

The

angles depend

_on the ratio A _as

expected.

The

plot

of fractional difference of the two

angles (a'-a)la'

_versus ratio A is shown in

figure11 (with «rigidity

_parameter»

r

m

(h/V)~

8 a/A

=

10).

For A

= I, we have a'-

a =

0,

_as

expected

for the

single-layer

case. When decreases, the fractional difference increases

slightly.

When A decreases to 0.5.

the

angle

fractional difference is still below 22 fb.

In

figure

12, the

angle

fractional difference _vet-sus

rigidity

_parameter r is shown for A

=

0.8. The

change

in the fractional difference is less than 9 fb when rruns from 5 to 100. So r is not an

important

factor here and the result shown in

figure

I I is valid for all r values.

Fractional Angle Difference

00

0.

~ ~ ~ , ~ ~ Ratio

-0

-o

-I

Fig,

I. The angle fractional difference (a'~a )la _i,eisas ratio A for r

=

lo- With greater ^than O.5, the angle fractional difference is below 22 %.

Fractional Angle Difference

o

0.

~ ~ Parameter

Fig.

12. The angle fractional difference (a'-a)la _i,ei,uis rigidity parameter ^{r with A 0.8. The} angle fractional difference is almost _a constant with _a wide range of r values.

The

limiting

behavior of the

angle

fractional difference _can also be obtained _:

~ _~'~ a' _« 4 4 4

~jj

^{~ 2/3 4 ~ ~ ~ l}

a'

«fi~ _«fi aT' «,I

^{~ 'r}

A-1:

"'~"- ~-

^~

)(l-A)- ^~-

^~

)(l-A)~+O((I-A)~, ).

a' 2 6r 8 6r r

We conclude from the above results that the

angle

^{fractional difference} is very small and increases very

slowly

with _a wide range of A value. This _means when the _two

layers

are put

together,

there is

only

a very narrow «

stripe

_»

along

the

boundary

between

region

I

(or II)

with III

(Fig. 10).

Within the

stripe,

the chains bend and _can take

complicated shapes.

^{But since} the

stripes

_{are very narrow} and contribute very little energy, we can assume that the _two

triangular regions

coincide without

causing

much deviation _to the free energy estimation and the overall

phase separation picture.

So with _a not too small A

(m 0.5),

_{we assume} the

angles

for both

layers

_are the _same, which _we denote

again

_{as a.}

As _we have discussed in section

2,

for _a

single

chain in

layer~l,

the elastic energy due _to vertical

stretching

is _;

~i(h-h')~+~~h'~=~~h~(I-A)((1-7J)+ ^'~~ j.

2Vj 2V2

2V 7J-A

For chains in

layer- I,

the elastic energy within

region

III due to horizontal

stretching

is

given by

j

j2d

^~

2d-xo I j2

~

2d-xo

V~ ^{2 ~~}

~~" ~°

2

~~~

h'

=

~

^{~ ~}

d~

_tg

a

(I

_)~ ^'~

2 V _7J ~

For _a

single

chain in

layer-2,

the elastic energy due _to vertical

stretching

is

given by

:

I _a

~,2 1ah~

I

_{V 2 V} ^'~2

For chains in

layer-2,

the elastic energy within

region

III due _to horizontal

stretching

is

given by

_:

li~~

^a

~2d-X( _j2

^{~ ~ j~ 2 ~}

~

~

~~~ ~~tg0)

^~'

~

~~~~

~

~~

^{~ ~~~ ~~}

2 h'

So the average total elastic energy of _a

single

chain is _:

E~j=(I-A).~~h~(I-A)((1-7J)+

^'~~

j+A.~ih~7J~

2V

J-~A

2V

+ _2d~r

(~

_{2 V}^~

)~d~tga ^{(I -A)~}

^'~

^+~

^~

)~d~tga .A~j

~ -A 2 V

=))~(l-A)~((l-~)+jj+A~~)+(~dtga.((I-A)~

^{~ '~}

_+A~j.

'l- V

~-A