The random phase approximation for polymer melts with quenched degrees of freedom

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The random phase approximation for polymer melts with quenched degrees of freedom

M. Brereton, T. Vilgis

To cite this version:

M. Brereton, T. Vilgis. The random phase approximation for polymer melts with quenched degrees of freedom. Journal de Physique I, EDP Sciences, 1992, 2 (5), pp.581-598. �10.1051/jp1:1992168�.

�jpa-00246520�

Classification

Physics

Abstracts

05.40 61.25H

The random phase approximation for polymer melts with

quenched degrees of freedom

M. G. Brereton

(I)

and T. A.

Vilgis (2)

(1) WC in

Polymer

Science and

Technology,

The

University

of Leeds, Leeds LS2 9JT, G-B- (2) Max Planck Institut fur

Polymerforschung,

Postfach 3148, 6500 Mainz,

Germany

(Received I December I99I,

accepted

5 February 1992)

Abstract. There _are many concentrated

polymer

systems ^{of interest} (e.g. crosslinked rubber networks) where the chains retain their

configurational

freedom but the translational

degree

of

freedom is absent. To

investigate

the effect of the loss of this

degree

of freedom _on the

density

fluctuations _we have considered _a

simple

_case of _a melt of polymer chains where _one end of each chain is anchored to _a fixed

point

in space. ^{The random}

phase approximation (WA)

used for melts cannot be

immediately applied

to this

problem,

since the loss of translational freedom

couples

together density fluctuations

belonging

to different wavevectors. The quenched chain end

variables must first be accounted for

using

a «

replica

_» calculation, then the WA can be used for the annealed

configurational

degrees of freedom. An

explicit analytic

result has been obtained,

which shows that

despite

the loss of the translational

degree

of freedom the

density

fluctuations in the

quenched

system ^remain similar to those in the _true melt out to

spatial

scales of the order of N I/~R, where R is the size of _a

single

chain coil (~ N ~°) and N is the

degree

of

polymerisation.

We have also considered the concentration fluctuations _a binary blend of chains with anchored chain ends. The situation is shown to be very similar to a diblock

polymer

melt in _so far _as when the

Flory

interaction parameter ^exceeds a

given

value

microphase

separation occurs.

Agaul

the loss of the translational freedom of the chains makes little difference _to the

stability

criterium.

I. Introduction.

The random

phase approximation (WA)

was introduced into

polymer physics by

de Gennes

[I]

and has been

responsible

for _a whole series of

important

theoretical and

experimental

advances in the

understanding

of concentrated

polymer systems [2-9].

The

approximation

deals

directly

with the statistical mechanics of

density

and concentration fluctuations in _tennis of _{a mean} field

concept.

^{An essential}

part

of the WA is the

assumption

that there _are many

overlaping polymer

chains and that the chains must be free _to be

anywhere

in space and to

adopt

_any

configuration

consistent with the chain nature of the molecule.

Correspondingly

its

application

is

largely

^restricted to annealed systems ^such as melts _or concentrated solutions.

However there _are many concentrated

polymer

systems ^{of interest where} not all of these conditions _are fullfilled. For

example

in _a crosslinked rubber the chains

overlap

each other and have

nearly complete configurational

freedom but _are constrained

by

^crosslinks to be in

localised

regions

of space. In this _case the crosslinks have

effectively quenched

the

translational

degrees

of freedom associated with the _centre of _mass of the chains. Of

particular

interest _are the cases of networks formed from

polymer blends,

co-block

polymers

or

inter-penetrating

networks

[10,

II

].

Other

examples

of concentrated

polymer systems

^with

quenched

variables which

negate

^{the direct}

application

of the RPA _{are :} flexible side chain

polymers

attached to immobile groups such _as colloidal

particles, liquid crystalline

stems _or surfaces

[12, 13].

In all _cases the location of the

polymer

chain is localised in space and has to be considered _{as a}

quenched

variable. Recent attempts

[14, 15]

have been made _to

apply

the

RPA,

_as used for

melts, directly

to

quenched (network)

_systems without theoretical

justification.

In this paper ^we wish to address the statistical mechanical

problems posed by polymer

melts when the translational freedom of the molecules is

quenched.

To

accomplish

this _we will

consider the

simple

_case of a melt of

polymer

chains where _one end of each chain is fixed in space,

apart

^{from this}

single

restriction the chains will have

complete configurational

freedom. For statistical

systems

^with

quenched variables,

the Partition Sum is

ideally

calculated

by averaging

_over the annealed variables for _a fixed set of values of the

quenched

variables. Then _some

physical quantity

of

interest,

such _as the free energy, is found and

averaged

over the distribution of the

quenched

variables. In

practice

it is _more convenient _to average the

physical quantity

_over the

quenched

variables first

by

_means of the _«

replica

trick _». This

technique

is _now well established since the

early

work of Edwards _on

spin glasses [16, 17]

and networks

[18].

It has

recently

been used

by Panyukov [19]

to

investigate

the effect of

topological

defects _on

density

fluctuations in

polymer

networks.

By using replicas

the

quenched

_average is done before the annealed average and _a variation of the conventional RPA is then used _to

perform

the annealed averages. For the

problems

considered in this

paper the final

replica

_space calculations _can be done

explicitly

and without further

approximations.

Before

presenting

these calculations _we will consider _a very

simple quenched

system ^{where the}

density

fluctuations _can be found

directly

without

using

the

replica

trick.

This system ^{consists of}

point impurities

fixed in space and surrounded

by

the

polymer

melt.

The

impurities

are assumed to interact with the _monomers

through

_an

arbitrary

interaction

potential.

The form of this result, obtained without

using replicas,

is

important

for the later work and

provides support

^{for the later results where} use of the

replica

method is essential.

2.

Polymer density

fluctuations in the presence of

quenched impurities.

The

system

^{of fixed}

impurities interacting

with _a

polymer

melt is shown

schematically

in

figure

I

together

with the notation to be used.

An

impurity

fixed at the

position R~

^{interacts with the} s segment ^of a

polymer

chain

a located at

R)

with _an interaction

strength U(R~ R[ ).

The

polymer

chains

comprising

the melt interact with each other

through

_a

potential

V

(R) R().

A

generalisation

to chains of

different

species

is

quite straightforward

and will _not be

pursued

here. The

polymer-impurity potential U(R)

is

expressed

in _terms of Fourier components as

U(R)= n~l£U~expiq.R (2,I)

q

where n is the volume of the system.

The total

polymer-impurity

interaction energy can be

expressed

_as

z U(R~ R) )

= n ^l

z _U~

_exp

iq. (R~ R)

a. n wn

=

z _U~

_p

~ exp

iq R~ (2.2)

qn

yawn «cMh

«

'°'"'~~

($~~nti

_at

h

g_a vj

i~

_R~

S ~~

~~

l' /~

jJ(t~-11')

~

'

~m ant

@ ~

R,'

f~t$~~~$~n ^{~ ~}

Fig,

I.

Impurities

located at fixed

points

R~

interactblg

with _a

polymer

melt

through

an interaction

potential

U. The polymer molecules _are

interacting

with each other

through

_a

potential

V.

where

p_~

^{is the Fourier} component ^{of the}

polymer density

fluctuations

given by

p_ ~ ⁼

n ^l

z

_exp

iq. R) (2.3)

In tennis of the collective

density

variables

(p~)

the interactions between the chains is

given by

z

v

~ p

~ p_

~

(2.4)

q

The

partition

sum Z

(R~)

and hence the free energy In Z

(R~), depends explicitly

on the

position

of the

impurities (R~)

and is

given by

Z

(R~)

=

exp

_~

z _{V~ p~}

^~ ₊

_pq

U

~

£

^{~~~ ~~}

^~"

_{+ c,c.}

(2.5)

kT

q «

Jl

o

The

averaging (.. )~

is _over the

unperturbed

^chain

configurations (R)),

which _are contained in the collective

density

variables

(p~).

The RPA for

polymer

melts consists in

directly averaging

over these variables

by treating

them _as Gaussian random variables. In

general

_a Gaussian distribution function is deternlined

by

the second moment correlation

function,

which in this _case is

given by (p~ p~)

~, where

(p~ p~)~

=

z (exp (q. R[

+ k.

R$))

~

«flss'

=

£ (exp

I

(q

R ^~ ₊ k.

Rfl ))

~

(exp

I

(q r)

+ k

r$ ))

~

(2.6)

aflss'

Here

R)

=

R~ ₊

r)

and the

position

vectors

(R")

of the chain ends _are

independent

of the

configurational

vectors

(r~)

In _a melt the system ^is

translationally invariant,

where each chain _can be

anywhere,

hence the first ternl on the

right

hand side of

(2.6) gives

(expi(q.R~ +k.Rfl))~= 8(k+q)/n. (2.7)

This result

(2.7)

is the crucial

point

of

departure

when _{we come to} consider chains where _one

end is fixed.

Only

when the _{average over} the

quenched

variables

(denoted by

_a

bar)

is

perforated

does the

system

^show translational

invariance,

I-e-

exPi(q.R" +k.Rfl)= n-iexpi(q.~Ra -Rfl))&(k+q). (2.8)

In the

present

case of _an annealed melt the distribution of

density

fluctuations P

(p~)

is

given by

the Gaussian distribution _:

P

(p _~)

_exp

l~ ^{z (~~ (2.9)}

q q

where

S(

is the structure factor for

non-interacting chains,

I-e- V

= 0.

S(

=

£ (exp iq (R[ R) ) (2,10)

~

ass'

~

Using

the distribution function

(2.9)

and

(2.5)

for the

partition function,

the free energy functional F

(p~

;

R~)

for the

system

^{in the} _presence ^{of the}

impurities

_can be identified _as

F

(p~

_;

R~)

_m ^~

£ i

p~ ~~

+

~~

+

p~ U_~ £

^{~~~ ~~}

^~"

₊

.c,j

(2,I1)

2 s

~

n

The

density fluctuations,

which

depend

on the

positions R~

^of the

impurities,

_are then calculated in the usual way as

fl ldp~[p~[~exp

^F

^{(p~ R~)} llPqlRni _l~)

=

~

(2.'2)

fl dp~

_exp F

(p~

_;

R~)

~

The

integrals

_are standard Gaussian

integrals

and

give

for the full structure factor

S~(R~) =Sf+ (SfU~(~n~l£expiq. _{(R~-R~) (2.13)}

where

~o

Sf

₌ ^~

~

(2, 14)

+

'~q Sq

is the _structure factor for the pure melt. The average over the

position

of the

quenched

variables

(R~)

is

straightforward

to do and

gives S~ (R~

=

Sf

₊

Sf _U~

^~ C

(q) (2, 15)

where

C(q)

is the structure factor of the

impurities C(q)

=

n

£

_exp

iq. (R~ R~ ) (2.16)

To discuss the result

(2,15)

_{we use a} random distribution of

impurities

_so that

C(q)

= the

impurity

concentration

= q~ and _an

approximation [4]

for

Sf

which is valid for Gaussian chains of N segments ^and step

length

b. When N is

large

_:

~

£~

12 _c

(2,17)

~~ b~q~ f~+

where

I

is the Edwards

screening length f~

=

b~/(12 cV) (2,18)

and _c is the concentration of

segments.

For

simplicity

we take the

impurity polymer

interaction

U~

^{to be the same as the}

polymer polymer

interaction

V~

^{and consider the}

particular

_case when

i~

=

b~/12

_{then the result}

(2,15)

for the

density

fluctuations in the

polymer melt/quenched impurity

system can be

written _as

~~ i~ni

q2 ~l

_{+ i} ⁺

_q2 ~'

+ i

~ (2. '9)

The

impurities

contribution

according

to their number but

acquire

_a

scattering

structure

(form factor)

due to the interactions with the

polymer

chains. This is reflected in the

spatial

variation of the

density fluctuations, S~(R

;

R~),

which is

given by

the fourier transform of

(2,19)

_as

S~

(R

R

~)

=

f _I

exp

(- R/f )

₊

I

exp

(- R/f )j ^(2.20)

f

^R

f

The

polymer density

fluctuations associated with the

impurities

_are different from those in the pure melt.

In the _next section _we will consider the _case of _a melt of

polymer

chains where _one end of each chain is anchored in space, I-e- the

position

variables R~ for each chain end _{are now} the

quenched

variables. We will show that the fixed ends behave

similarly

to the frozen

impurities

we have

just considered,

however there _are also

significant

differences. The situation of the anchored chains is

sufficiently

_more

complicated

that _a

replica

calculation is necessary.

3. A melt of anchored chains

In the

simple

_case of

non-interacting

chains there is

already

_a

major

difference in the

density

fluctuations of _a

single

anchored chain

compared

to one which _can be

anywhere

in space.

Before

dealing

with the effect of interactions and the method of

handling quenched

chain end variables it will be useful _to consider this _case first.

3, I DENSITY FLUCTUATIONS FOR A NON-INTERACTING ANCHORED CHAIN. We will denote

by lt$

the anchored end of each chain _a and let

r[

label the _s segment ^{of the chain from this}

end,

hence

R[

₌

R$

+

r[. (See

also

Fig. 2).

For _a

non-interacting chain,

with _one end

anchored _at the

origin,

the

unperturbed

structure factor

A(

^is

^{given by} Al

_{= w}

(ll~ql~lo- l~q>ol~)

where

~~

=

£

_exp

iq r[ ^{(3, I)}

s

and _q~ is _now used to denote the number

density

of chains.

For the annealed _case, where the

single

chain that _can be

anywhere

in space, the second tern

(~~)~

₌ ⁰ for q # 0 as a result of translational invariance of the

system.

^The non zero

value of this _term for the

unperturbed

anchored chain will account for the

principal

difference between the melt and the anchored

« melt _» of

interacting

chains. It will be useful to have

explicit

model

expressions

for

(~~)~

^and

([~~[~)~.

^These are

readily

found

using

the Gaussian chain model _as follows _:

z (exp iq r))

~ ⁼

z

_{exp q}

~(r)~) _~/2

s s

N

= ds exp q^~b^~ _s /6

o

~

~2 ~2

p~

= j

(I

^exp

^(3.2)

q b 6

and

similarily

z (exp iq (r[ r[, )) N

~ ⁼

ds ds' exp

q~ b~ Is s'[

/6

ss. o

=

~~

ii

~

~ (l

^exp

^{~~~~~ (3.3)}

q~b

_{q b N} 6

Using (3.2)

and

(3.3)

ⁱⁿ

(3,I)

for

A( ^gives

o

12Nq~

6

j2j ^q2b~N q~b2Nj

~q

~

2

~2

^{+ 9'}

fi

^{~ ~ ~~P} _~ ^{+ eXP} _~

^(3.4)

The

principal

features of this result _are

given by

the

following

limits _: at q values

large enough

to

explore

the intemal structure of _a

single

coil I,e,

q~b~N _WI,

there is _no

significant

difference between the fixed chain and _one which is free to be

anywhere

_:

o 12

Nq2 ~o _{(3 5)}

~q

"

~2 ~2

^q

The difference between

A(

and

S(

first appears ^at

larger

scales

beyond

the size of _a

single

coil

(I,e,

_q values _:

q~ b~

N «

I).

The first two

leading

terms in

(3.4)

cancel and

A°

_{tends to zero as}

A(

_m ^cN

^{(q~ b~N/9 (3.6)}

whereas in the melt

S(

= NC at q = 0

(3.7)

c is the concentration of _{monomers, c}

=

Nq2.

For later _use it will be necessary ^to have _a

simple analytic

forms that

parameterise

^both

A(

^and

(~~)~.

^Based on

((3.5), (3.6)

and

(3.2))

suitable forms _are

~~

^{2 cN}

(q~Nb~/6)

~

[(4/3)~~~

₊

q~Nb _~/6]~

(~~)~

₌

~

(3.8)

1 ₊ q Nb /6

In order to facilitate the discussion of _our results it is

slightly

inaccurate but

qualitatively

convenient to

replace

the numelical factor

(4/3)~'~-

_{l and to define}

a scaled variable

Q

^~

= q^~N b

~/6,

then

~o

~

2

CNQ

^~

~

li

₊

Q~l~

and

(~~)~

=

~

~.

(3.9)

+

Q

In the _next section _we will consider the effect _on the

unperturbed

structure factor

A(

^{of both the} interactions between the chains and the presence of anchored chain ends. In this _case the chain ends must be handled

by

the

replica

method first before the usual random

phase approximation

_can be

applied.

3.2 REPLICA CALCULATION FOR THE DENSITY FLUCTUATIONS OF ANCHORED CHAINS.

The situation is shown

schematically

in

figure

2.

« chain end fixed at

it

R~

V(R) R)

ii,

s

j

chain end fixed at

R)

Fig. 2. A melt of

interacting

polymer chains with each chain _a having one end anchored to _a fixed

point R(.

The free energy for the

interacting

chains is

given by

F

(R()

= In Z

(Rl)

where the

partition

_sum Z (ROY) ^is

given by

n ^I

Z

(R()

= exp

z z V~

exp ik.

(R(

+

r[ R( r( ) (3.10)

~

k _a, s 0

fl,s'

The

averaging represented by (. )~

is done _over the

unperturbed

chain

configuration

variables

(r)),

while the

(Rl)

remain _as

quenched

variables. The

(r))

_{averages are}

evaluated

using

a method devised

by

Edwards

[3, 4], whereby

_a transformation is made from the

polymer

chain variables

(r))

_to the collective variables

(p~) representing

the

polymer density fluctuations, given by (2.3).

This is

accomplished using

the

identity

element

I

=

fl ldp

_~ ⁸

_(p

_~

^£

exp ik.

(R~

+

r) )) ^(3.

^II

⁾

~ ~ ~

Furthermore in order to find the free energy

fl averaged

over the

quenched position

of the anchor

points (RY )

,

fl

= F

(RY)

the

replica identity

is used for the In term tern, I,e.

fl

=

In Z

(RY)

=

lim

jj _{[Z "(RY) Ii.}

(3,12)

n~o ~

«=i

The details of the calculation _are

presented

in the

Appendix

where it is shown that the

quenched

free energy F _can be written _{as :}

I ₊ nF

=

j _{fl dpf}

exp

~

z pf(Qij,(k)) pi' (3.13)

k«

2

k««,

where

Qij,(q)

is _a

replica

_space matrix with the

following

structure :

Qil,(q)

=

[Aj

l

C~ T~ U]~~> (3,14)

1is _a unit matrix in the

replica

_space and U is _a matrix with all the matrix elements set

equal

to I. The coefficient

Aj

^is

given by

Aj

₌

A(~

₊

_V~ (3,15)

A(

^{is the}

unperturbed

structure factor for _an anchored

chain, given by (3.I), V~

^{is the} interaction

potential

between the

chains, C~

is the

quenched

structure factor of the chain ends

given by

Cq

" ~l

i

_eXP

iq [(R~ R~')]. (3,16)

(For

_a random distribution of chain ends

C~

= q2) ^and

T~

=

(~~)~/A(. (3.17)

From

(3,13)

the structure factor

S~ (R~)

for the melt of anchored chains is

given by

the relica method as

S~

(R~)

=

n

p~ ~)

=

lim

z Q

««

(q ) (3,18)

n~0 ~

~i

The

replica

_space matrix

Q

is

readily

inverted

(the

details _are

given

in the

Appendix)

to

give S~ (R~ )

= A~ ⁺ A~

T~

^{~ C}

~

(3. 19)

The result

(3,19)

is _a

major

result of this paper and will be discussed in detail in the next section.

3.3 DENSITY FLUCTUATIONS IN A MELT OF ANCHORED CHAINS. The result

(3,19)

for the

structure factor of _a melt of anchored chains is

formally

similar to that obtained in the last

section for the molten

polymer

in _an environment of

quenched impurities, through

the

replacements

_:

S(

_~

A(, Sf

~ A~ ^and

U~

~

T~ (3.20)

In the result

(3,19)

there _are

again

two kinds of

density

fluctuations _: those associated with the

« molten _»

degrees

of

freedom,

described

by

the _tern A~, ^{and the others}

directly

related to the

distribution of the

quenched

chain ends

through

the chain end structure factor

C~

^but

enhanced

by

the _ternl

[A~T~[~.

^The denominator of the term A~ ^{has more structure} as a

function of q than the

corresponding

melt _case

given by Sf

and

gives

rise to two correlation

lengths.

To _see this the

analytic

fornl

(3.9)

is used for

A(

^{and from}

^(3.15)

_A~ ^is

given

_as

~ 2

CNQ~

(3.21)

~

jl

₊

Q~]~

₊ 2

cV~ ^NQ

^~

The denominator _can be factorised _as

(Q~+

a

~)(Q~+ _{p ~),}

_{where for}

_cV~

_{N »1}

a

~

= 2

cV~

^{N and}

p

^~

=

l/(2 cV~ ^N)

= a ~

(3.22)

In _a melt the

screening leigth

_f

(Eq. (2,18))

is of the _same order _as the _monomer

length

b. Hence 12

cV~

m

I,

and

a

~

m N and

p

^~

m N For the range

Q

^~ _~

p

^~

m N A~ ^{can be}

written _as

A ₌ ~ ~~

=

~~ ~~~~

(3 23)

~

(Q~

₊

« 2)

(q2

₊

f~~)

This is the _{same as} the melt _structure factor

Sf (Eq.(2,17))

where f,

given by

equation (2,18),

is the usual

screening length

found in

polymer

melts

[4].

^{The condition}

Q~~ _p

²

_implies

_q

~

(Nb )~

^i,e, the interactions between the anchored chains establish _near melt conditions

(A~

m

Sf)

out to scales of the order of the

length

of _a

fully

extended chain.

This is in _{contrast to} the

unperturbed density

fluctuations

A(

^and

S(

which differed _on scales of the order of _or greater ^{than the coil radius}

(~

N

~'~b).

The

specific

contribution of the anchored chain ends _to the

quenched

structure factor S~

(R~)

is

given by

the second ternl of

(3.19),

I,e.

C~

_A~

T~

^{~. For} a random

distribution,

the

chain end _structure factor

C~

= q2, while

T~

^is

given by (3,17),

hence

C

~ A~

T~

^~ =

~ ~~~

~

(3.24) (1

+

V~

~

Using

the

parameterisations (3.9)

for

~~

^and

A(

~q'~q~q'~ ~~/2)~~))j~~~p2)j2

~~'~~~

as

Q~0 C~[A~T~[~~Nc

whereas the first ternl of S~

(R~)

I,e. A~ - 0 in the limit _q

- 0. Hence the

specific

contribution

(2.25)

from the anchored chain ends dominates the

density

fluctuations in this limit. This is in

contrast to the

quenched impurities

in _a

polymer

melt. The _two contributions _to

S~(R~)

are

comparable, using (3,19)

when

A~ =

C~[A~ T~[~

This _occurs for _a q value q~

~~ ~~~~2

~~~

i~~2

~

fi~

~2

~2 ^~~'~~~

L0Gfli

N=100

ANCHORED CHAIN

MELT 0

extended chain

)

single

call

<~

-2

4 0 4

LOG(

q~b2N16

q ~

b1

~ ~

l/l

b)-1

~~3/4 _~~-1

q~ixui-i

Fig.

3. The structure factor S~

(R~)

of _a melt of

interacting

anchored chains calculated

using (3.21)

with N 100. For comparison the

corresponding

melt _case is also shown.

q~ defines _a scale internlediate between the size of _a

single

chain coil

(N

^~'~

b)~

and the

fully

extended chain

~

(Nb)~

The full behaviour of S~

(R~),

based _on

(3,19)

and

using (3.21), (3.25)

for N

=

100 is shown _{on a}

log-log plot

in

figure

3 and confirnls that the melt-like conditions for the

density

fluctuations in _an anchored chain system ^extend a factor of N ^~'~

beyond

the size of _a

single

chain _even

though

each chain is anchored _{at one} end.

The

spatial

^{behaviour of the}

quenched density

fluctuations

S(R R~)

is

given by

the Fourier transfornl of S~

(R~) using (3.21),

^the

leading

tennis can be written _as

S

(R

_;

_R~)

=

~~

) ⁱ _{exp(- R/f}

+ c exp

(-

6

Rf/(Nb~)) (3.27)

b

~

R N b

If,

_as in the

quenched impurity problem

of section

2,

_we choose

f~= b~/12,

_{and set}

c ₌

Nq2,

where _q2 is the concentration of chains/chain

ends,

then

The result is very similar to that obtained for

quenched impurities

in _a

polymer

melt

(cf. (2.20)).

The first ternl is the pure melt ternl, while the second _ternl is due to the fixed chain ends instead of the

impurities.

In this _case the form factor associated with _a chain end has _a range of the order of the

length

of the

fully

extended

chain,

whereas in the

impurity

_case

it _was of the order of the

screening length f.

^The full

spatial

behaviour of S

(R

_;

R~)

based _on

equation (3.28),

is shown in

figure 4,

LOG( SIR_;R~

lf3Jc)

~ =joo 4

1 _~

i-1

°

~ R~l

exp(- R/j

-I ANCHORED CHAINS

~

exp(- R/lNf)

MELT

-4

-1 -1 o i 1

I

L0©(

1/f)

~

i =

Xl

g ₌

j

fl z fil~l

( )

coil

) extended chain

Fig. 4. The _structure factor S

(R

_;

R~)

of _a melt of

interacting

anchored chains calculated using

(3.28) with N

= 100. For

comparison

the

corresponding

melt _case is also shown.

This

completes

the discussion of the

homopolymer

melt. In the next section _we consider the

case of _a blend of two distinct

polymers (A/B).

The unusual structure of the A~ ^{term and the} appearance of two correlation

lengths

manifests itself in this case with the

possibility

of

microphase separation

_over

spatial

scales with _an upper and lower bound. This can be

anticipated

since for

incompatible polymers

the fixed chain ends will prevent a

macroscopic phase decomposition.

We will show that the

stability

criterium is very similar to that of _an

unrestricted melt of _an A-B diblock

polymer.

4. A blend of anchored chains.

The methods of the

preceeding

sections _are

easily adapted

to an A/B

polymer

blend where

one end of each of the chains is fixed in space. Collective

density

variables

pA~,

p~~

^{for the two}

species

of

polymer

chains _can be defined in _a similar way to

(2.3)

and the interactions between the chains written _as

( _{Z Iv}

~~q p~q p~

q +

vBBq pBq

pB

-q + 2

v~~q _p~q

p~

qi (4. i)

q

The free energy of the system ^is calculated

using

the

previously developed

methods and is

given by

a

generalisation

of

(3.13)

_as

' +

nfl

=

In _{dPi~ dpi~}

exp

( _{Z iPi~ toil« (k)) pit}

k« k««'

+

Pik (QB~«'(~)) Pii

~ ~'AB

P~k Pik ~««'l (4.2)

where

QA~«'(k )

_"

(d~q )

₊

~AA ~««'

~

Ak

~Ak

^~

~««'

and

similarily

for

Qj/~,(k).

For

simplicity

_{we can} consider _an

incompressible system

^and set

p(~

_{= p}

~j.

Then

+

nfl

=

lfl _{dp (~}

exp

~

£ (p (~ (Qjj>(k))

_p

it

k«

2

k««'

where

~"~'~~~ _~~~~q)

₊

_(dBq)

2

XF/C) ^8~~,

~~

Ak

~Ak

^~ +

~Bk TBk ~l U««, (4.3)

and

X~

is the usual

Flory

interaction parameter

given by

X~/c

=

V~~ (V~~

+

V~~)/2

and _c is the total concentration of _monomer units _{: c}

= c~ + c~. The

quenched

structure factor

l~

for the blend is determined _as

~~ j~ ^~j~ ~""~~~

j(A(~ )~

₊

Aj~)~

2

X~/c)

CAk

~Ak

^~ + ~

Bk

~Bk

^~

(

~_

~)

~

(A(~ ^)~

+

(A(~ ^)~

²

X~/c)

^~

The limits of

stability (compatibility)

are then determined

by (A(~ ^)~

+

(A(~)~ ~X~

= 0.

(4.5)

c

The

simple

_case where

N~

₌

N~

=

N will be considered

together

with the

approximate

fornl

(3.21)

for

A(. ^{Equation (4.5)}

^becomes

~~~f~~

₊

-~x~=o

NQ

~A CB C

or

(1+ Q~)~-4 q2(1- q2)NX~Q~=

0

(4.6)

where _q2

=

c~/(c~

₊

c~)

The roots

Q~(± )

of this

equation

deternline the

spatial

_extent of the

stability

of the

quenched system. They

_can be written _as

Q~(± ₎

=

(xl 1)

±

~ _(4.7)

where

X/=2q~(1-q~)NX~.

Various ranges of

Xl

must be considered _:

(a) Xl

_< 0 _: in this _case

Q~(± ₎

_are

negative

and the fluctuations _are

damped

and the

system

^{is stable} on all scales.

(b)

0

<

Xl

_< 2 _: the

Flory

interaction _energy is

repulsive,

however

Q~(± ₎

_are

_complex

and the

quenched

blend will show

damped

but

periodic

fluctuations of concentration but will still be stable _on all scales.

(c) Xl

~2 _:

Q~(± ₎

_are real and

positive

and for

Q~(-

<

Q~<Q~(+ ₎

the

quenched

blend will be unstable and

microphase separation

will _occur over the

rangle

set

by

Q(±

_{)_}

Microphase separation

first _{occurs on a} scale

given by

the size of _a

single chain,

since when

Xl

₌

2, Q~(± ₎

=

q~Nb~/6

=

1.

This situation and the criterium

(4.6)

_{are very} similar to an annealed A-B diblock

polymer

melt.

Briefly,

if

S(~

_and

_{S(~ represent}

_the

unperturbed single

chain structure factors of the two separate blocks and

S(B

the inter-block structure

factor,

then the random

phase

approximation

deternlines the full structure factor

S~~

of _one block

[20, 21]

_as

~Al

~

~~~

~

~~B

+ 2

S(

~~A S(~

SO

~ 2

X~

~~~

(4.8)

For _a diblock

polymer

of blocks A and B with identical _structures I,e.

S(A

=

SIB,

then

(4.8)

can be written _as

Sil

=

2

(Dp X~) (4.9)

~'~~~~~

~~

"

S~A S~B

D°

has

exactly

the _{same structure as}

At

_{the structure factor of}

a

single

anchored chain. The

stability

criterium

(I X~

D

°)

= 0 is then identical to

(4,5)

obtained for the

quenched

blend.

5. Anchored chains in _a melt.

For

completeness

the

following

situation will also be considered _{: one}

species (A)

of

polymer

chains _are anchored in space while the other

species (B)

^fornl an unconstrained melt. In this

case all the tennis

relating

_to the annealed B chain variables will be

diagonal

in

replica

_space.

The method is

entirely

similar to the last section _on it is

only

_necessary here to quote ^{the result} for the

quenched

free energy of _an

incompressible

blend _:

I ₊

nfl

=

lfl _{dp ~j}

exp

~

z (p ~j (Qpj>(k )) ^p11 (5. I)

k«

~

k««>

where

Qp/,(k)

is _a

straightforward

modification of

(4.3),

I,e.

Qpj,(k)

=

(A(~ ^)~

+

(S(~)~

²

cX~) 8~~> C~~ T~~

_{[~ U~~>}

(5.2)

The structure factor is

deternlined,

_as

before, by

§q

# lint

£ Q~~(q)

#

n~0 ~

~l

j(A(~)-

^{i +}

1(~)-

i

~x~j

~

j(A(~)-

i[iii ^l

~x~j

^~'

~~'~~

c c

The limits of

stability (compatibility)

are then deternlined

by

(diq)

₊

_(S~q) XF

" °

(5.4)

Again

the

simple

_case where

N~

₌ N

B = N will be considered

together

with the

approximate

form

(3.21)

for

A(. ^{Equation (5.4)}

^becomes

('+Q~)~ (l+Q~)

₂

CAfifQ~

^~

C~N C~~

^~'

^(5.5)

The roots are determined

by

_an

equation

_very similar to

(4.7) Q~(*

"

~~Nb16

=

(Xl iPB)

±

~/(X/ _iPB)~

_{4 q2B}

_(5.6)

Recall that _q~B is the concentration of the unrestricted melt

component. Again

the blend shows three

regimes

of behaviour

depending

_on the value of

Xl (or X~)

:

(a) Microphase separation

_occurs when the square root in

(5.6)

is real I,e.

XI~q~B+2q~('~

(b)

The blend is

everywhere

stable but shows

damped periodic

concentration fluctuations when

iPB~~9'#~*~/*iPB+~9'~~~

(c)

The fluctuations _are

completely damped

when

X/<q2~-2q2('~

6. Discussion.

In _a

polymer

melt _or blend where each chain has full

configurational

and translational freedom the

density

and concentration fluctuations _are

satisfactorily

accounted for

by

the

random

phase approximation (RPA).

The translational freedom of the chains

plays

an

important simplifying

role in

ensuring

that

density

_or concentration fluctuations

belonging

to different wavevectors _are not

coupled.

However there _are many

polymer systems

^of interest where the translational freedom is absent. To

investigate

the effect of the loss of this

degree

^of

freedom,

_we have considered a

simple

_case of _a melt of

polymer

chains where one end of each chain is anchored to a fixed

point

ⁱⁿ _space. The average over the distribution of the

quenched

chain end variables _was achieved

using

a «

replica

_{space »} calculation

together

^with the usual RPA for the annealed

configurational degrees

of freedom.

An

explicit analytic

^{result has} been

obtained,

which shows that there _are two distinct contributions to the

density

fluctuations _{: one,} which does _not

depend

_on the distribution of the fixed chain

ends,

is identical to the pure melt situation out to scales of the order of the

length

of _a

fully

extended chain I,e. Nb. The other contribution

depends explicitly

_on the distribution of the chain ends and is similar _to the

scattering

fornl factor in the

quenched impurity polymer

melt

problem.

The

density

fluctuations from this ternl dominate those of the first ternl _on scales q <

(N ~'~b)~

_~.

We have also considered _a

binary

blend of chains with anchored chain ends and shown that when the

Flory

interaction

parameter

exceeds _a critical value

(similar

to that in the nornlal

blend) microphase separation

_{occurs on a} scale of the order of R.

Again

the situation _was

shown to be very similar _{to an} unconstrained melt of _an A/B diblock

copolymer.

The

major

conclusions from this work _are that the

density

and concentration fluctuations in the

quenched systems

we have considered _are

virtually

identical to those in the unconstrained

melts, despite

the loss of the translational

degree

of freedom. Whilst this

might

have been

anticipated

on scales of the order of the size R of _a

single coil,

the

surprising

result from this work is that the conclusion extends to much greater ^{scales N}

~'~R. Only

_on these scales does the loss of the translational freedom become

apparent.

Appendix.

Using

the

identity (3. ii) parameterised

as

=

fl ldp

_~ dq~~ _{exp I}

£

_q~

~ p_~

£

_exp ik.

(R~

+

r[ ) (Al)

k k _{a, S}

in

expressions (3,10),

the free energy for _a

given

distribution

(RY)

of chain ends _can be written _as

F

(RY)

₌ In

lfl ^dp~

^dq~~ exp I

£

_{q2~ p~}

£ _{V~ p~ p_~}

_x

k k

~

q

x

exp

I

£

_q2~

£

_exp ik.

(R"

+

r)) (A2)

k

a,

s

~

JOUR~AL DE PHY~IQLE 2 ~ 5. MAh 1992

In order to average the free energy

F(RY)

_over the

quenched position

of the anchor

points (RY),

I.e.

fl

= F

(RY) (A3)

the

replica identity

is used for the In ternl, I-e- In Z

=

(Z" I)

In _{as n -} 0 to

give

+

nfl

=

fi lfl ^{dpi dq2i}

^exp

^{z pi pi z V~ pi p5~}

^x

«=I k k

~

k

exp

I

£ pi £

_exp ik

(R~

₊

r)") .

(A4)

~

~

~

The second _moment

approximation

is used to evaluate both the annealed and

quenched

averages in

(A4).

This consists of

replacing

(exp

iX

)

_exp

j _{(X~) (A5)}

where we

identify

X with

X ₌

£ q2f(£

^{exp ik.}

^(R~

⁺

r[~)) ^(A6)

~~ ~ ~

Then

(i~)

=

z q2f q2l' z exp[ik.R~

₊ ik'. R~

](exp

ik.

r)"

ik'.

r$'"') _(A7)

k« a,s

k'«' a',s'

Translational invariance of the

quenched

system ensures that

only

the ternl k

=

k' in the _sums in

(A7)

will contribute. The annealed average is taken _over

non-interacting chains,

the

single

chain, single replica

term a =

al,

_«

=

«' is

separated

out and the average written _as

(exp

ik.

(r)" r) "'))

~ ⁼

(exp

ik,

r)") _~(exp

ik.

rl'"

~ +

+

((exp

ik.

(r[" rl'"'))~- _(exp ik,r)")~(exp

ik.

rl'"')~) _8~

~

8~

~.

(A8)

The averages in

(A8)

do _not

depend

on either the chain

(a )

_or

replica («)

labels.

Using (A8)

in

(A7) gives

(i~)

=

z

_q2

~

~

In ^~z

^exp

^{[ik. (R" R"')] z (exp}

^ik

^r~)

_~ ^{~ +}

k««' aa' s

+ q2

z (exp

ik

(r~ r~,))

~

(exp

ik

r~)~ (exp

ik

r~,) _~)

8

~ ~

(A9)

ss,

n is the volume of the

system

and _q2 is the concentration of chain ends.

The

density

fluctuations

At

of _a

single

chain fixed with _a fixed end _are

given by

At

=

z (exp

ik.

(r~ r~,))

~

(exp

ik,

r~)~ (exp

^ik

r~,) _~) (A10)

ss,

The

quenched

structure factor C

(k)

of the chain ends is

given by

C

(k)

=

fl~

~fiik. _{(R~ R~')i. (Al I)}

Finally

set

£ (exp

ik

r~)

= ~k

~~~~~

~

Then

(Xi

_can be written from

(A9)

_as

(X~)

=

£ [q2~[~[C(k)[~~[~U~~,+A(8~ _{~] (A13)}

k««'

U~~

is _a

replica

_space matrix with each element set

equal

to I.

Using

the results

(A5)

and

(A13)

in

(A4),

the free energy ^{for the}

quenched system

^is

given

as

1+nfl=- fi lfl dpfexp-~£V~pfps~

x

«=i k

2

k

i l~~'i

~XP

i 9'k'~~~ (k)

_~k

'~ ~««'

+

di _{~«, al}

+ I

I Pi Pi (J~14)

k k««' k

The

integration

_over the _q2~ is _a standard Gaussian and

gives

[det

M

]~

^~'~ _exp

z pf Mp/, pi' _(A15)

~

k««'

where M is the

replica

_space matrix

M~~, (k )

₌ C

(k )

_~~ ^~

U~~,

₊

At _8~

~

(Al 6)

The matrix U

=

(U~~ )

has the property

U~

=

nU,

which enables matrices of the form h4=Al+BU

to be

inverted,

I.e.

_, , B

~

(Al?)

~~'~ "

I iA

+ nB In the limit _n

- 0

~i«~'

~

d(

_~

~, «'~ C

(k )

_~

k

~

~~

^~

~««'

and

[det

M

]~

^~'~

= l

(Al 8)

Finally ^fl

=

fl ldPi

_exp

Z Pi iQp1,(k)j pi'

k« k««'

Wh~~~

Q al'

~

(d~

₊

_{~k ) ~«,}

«' C

(k)

_~k ^~

Al

^~

_U~~, (Al 9)

Which is the result used in section 3.2.

References

[ii DE GENNES P. G.,

Scaling Concepts

in

Polymer Physics

(Comell

University

Press, Ithaca, 1979).

[2] FLORY P. J.,

Principles

of

Polymer Chemistry

(Comell

University

Press, Ithaca, 1971).

[3] EDWARDS S. F., Proc.

Phys.

Soc. 88 (1966) 265.

[4] DOI M., EDWARDS S. F., The

Theory

of

Polymer Dynamics

(Oxford

University

Press, Oxford, 1986).

[5] FISCHER E. W., Frontiers of Macromolecular Science, T. Saegusa et al., Eds. Blackwell Science Publications, Oxford (1989).

[6] STEPANEK P, et al., J. Chem.

Phys.

94 (1991) 4171.

[7] METER G., MOMPER B., FISCHER E. W.,

preprint.

[8] SARABAN A., BINDER K., Macromolecules 21

(1988)

711.

[9] DEUTSCH H. P., BINDER K.,

preprint.

[10] FELISBERTI M. I., STADLER R., Mackro. Chem.

Symp.

44

(1991)

75.

[11] BINDER K., FRISCH H. L., J. Chem.

Phys.

81 (1984) 2126.

[12] MILNER S. T., WITTEN T. A., CATES M. E., Macromolecules 21(1988) 2610.

[13] RAPHAEL E., DE GENNES P. G.,

preprint.

[14] EDWARDS S. F., MCLEISH T. C. B., J. Chem.

Phys.

92

(1990)

6857.

[15] BASTIDE J., BUzIER M., BouE F.,

Polymer

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