Copolymer Melts in Disordered Media

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Copolymer Melts in Disordered Media

S. Stepanow, A. Dobrynin, T. Vilgis, K. Binder

To cite this version:

S. Stepanow, A. Dobrynin, T. Vilgis, K. Binder. Copolymer Melts in Disordered Media. Journal de

Physique I, EDP Sciences, 1996, 6 (6), pp.837-857. �10.1051/jp1:1996245�. �jpa-00247218�

Copolymer Melts in Disordered Media

S.

Stepanow (~,*)

,

A-V-

Dobrynin _(~,~),

T-A-

Vilgis (~)

^and K. Binder

(~)

(~) Martin-Luther-Universitit

Halle-Wittenberg,

Fachbereich Physik, D-06099

Halle/Saale,

Germany

(~) Groupe de Physico-Chimie Th60rique, E-S-P-C-1-, 10 Rue Vauquelin,

75231 Paris Cedex 05, France

(~) Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599, USA (~) Max-Planck-Institut ffir

Polymerforschung,

Postfach 3148, D-55021 Mainz, Germany (~) ^{Instiiut fir} Physik, Johannes-Gutenberg Universitit Mainz, Staudinger Weg 7,

D-55099 Mainz. Germany

(Received

3 July1995, revised 5 January 1996, accepted 5 March 1996)

PACS.05.20.-y Statistical mechanics

PACS.64.60.-I General studies of phase transitions PACS.61.41.+e Polymers, elastomers, and plastics

Abstract. We have considered _a symmetric AB block copolymer melt in _a gel matrix with

preferential adsorption of A _{monomers on} the gel. Near the point of the microphase separation

transition such _a system _can ^{be described} by the random field Landau-Brazovskii model, where

randomness is built into the system during the polymerization of the gel matrix. By using the

technique of the 2-nd Legendre transform, the phase diagram of the system is calculated. We

found that preferential adsorption of the copolymer _on ^the gel results in three effects:

a)

it

decreases the temperature ^of the first order phase transition between disordered and ordered

phase;

b)

there exists _a region _on the phase diagram at some small but finite value of the

adsorption _energy in which the replica symmetric solution for two replica correlation functions

is unstable with respect to replica symmetry breaking; _we interpret this state _{as a} glassy state

and calculate

a spinodal line for this transition; c) _we also consider the stability of the lamellar phase and suggest ^{that the} long _range order is always destroyed by weak randomness.

1. Introduction

The

problem

of the

rigid gel

G immersed in _a

symmetric

fluid mixture

A/B

with

preferential adsorption

of _one component, say

A,

_on the

gel

matrix has been considered

by

de Gennes (1j.

He found that _near the

demixing

transition temperature T~ of the

A/B

mixture there exists

a range of parameters where the behavior of the system ^becomes

dependent

_upon the sample

history

and

interpreted

that _{as a}

glassy-state.

There is _a very

simple physical explanation

of such behavior. With

decreasing

temperature

T, A-particles

will _cover the

gel

matrix in order to decrease the number of

BIG

unfavourable contacts

forming

A-rich domains _near the

gel

rods. The

boundary

of this

regime

_can be found

using simple scaling arguments balancing

the

energetic gain

due to

A/G

contacts, ENg, ^{with surface} energy between A-rich and B-rich

(*) Author for correspondence (e-mail:

stepanow©physik.uni-halle.d400.de)

© Les

(ditions

_{de Physique 1996}

regions, N(/(~,

where

(

=

T~~=((T T~)/T~)~~

with _{v m}

2/3 being

the correlation

length

of the

A/B

mixture and

Ng

^{the number of} monomers on the rods. This

immediately

^results

in a condition for the

strength

of the

preferential adsorption

_{energy E} >

NgT~/~

^{for which}

A-rich domains will form _near the

gel

rods and the characteristics of the system ^will

depend

on the

preparation

condition of the

gel

matrix. One should _note that _at the theoretical level this

problem provides

_an

example

of the Random Field

Ising

Model

(RFIM) [2-7],

where

randomness is built into the system

during polymerization

of the

gel

matrix.

Let _{us now} ask _a

question:

"How will the behavior of the AB mixture in the

rigid gel

matrix

change

if _{we connect} A and B _monomers into

poly.mer

chains?" There _are two different _{ways to}

do it. We _can create either chains

consisting only

of A and B _{monomers or} chains which have

two types ^of monomers. In the first _case the

problem

_can be reduced to the _case considered

by

de Gennes (I]

corresponding

to _a

simple

renormalization of the

coupling

constant _on the

length

N of the

polymer

chains. Another _case when A and B

particles

form for

example

_a

symmetric

diblock

copolymer chain,

AjBj can not be reduced to the _case of _the

A/B

mix- ture and is worth

special

consideration. It is important to note that

already

without the

gel, copolymer

systems behave

differently

from _a mixture of Aj and Bj

polymers (8-12].

As the temperature

decreases,

the

copolymers

form _one, two _or three dimensional domain patterns

depending

on the

composition

of the

copolymer

chains, I._e. the fraction of A _{monomers on} the

chain. The domain structure appears as a result of competition between

short-range

monomer-

monomer interactions that want to decrease the number of unfavorable contacts between A

and B _monomers and

long-range

correlations due to chemical bonds between those parts ^of the

chains that tend to segregate into domains. The first stage ^{of the domain} pattern formation

can be described in terms of Landau-Brasovskii effective Hamiltonian

(13-15]

when the

ampli-

tude of the local

composition

fluctuations is small in

comparison

with its average value. This Hamiltonian describes

phase

transitions in the systems ^such as

weakly anisotropic

antiferro- magnets

(14],

^the

isotropic-cholesteric

and nematic-smectic C transition in

liquid crystals

_[16]

and pion condensation in neutron stars [17]. ^The

homogeneous

state of this Hamiltonian is unstable with respect to fluctuations of finite _wave number _qo that result in the formation of the domain structure with

period

L ₌

27r/qo

below the transition

point.

The

copolymer

melt in a

gel

matrix discussed in this _paper is another

example

of such _a system.

The _paper is

organized

_as follows. In Section 2 _we discuss _a model and demonstrate that the

problem

of the

copolymer

system in the

gel

_can be reduced to random field Landau-Brasovskii model. After that

using

the 2-nd

Legendre

transformation

[18-21(

_we calculate the free _energy of the system. Section 3 presents the universal

phase diagram

of the

copolymers

in the

gel

for

replica symmetric

solution. In Section 4 _we calculate the

spinodal

of the

replica symmetric

solution with respect to

replica

symmetry

breaking.

Section 5

explores

the

stability

of the lamellar

mesophase exposed

to random fields. In conclusion _we discuss _our results.

2.

Polymer

Formalism

We

begin

with the

assumption

that the

copolymer

melt _can be described in terms of the individ- ual coordinates _r~(s~)

(the

index I ₌ I,...n~

,

counts the

copolymer chains,

_sz

gives

the

position

of the segments

along

the

chain) by using

the Edwards Hamiltonian [22].

Replacing

these individual coordinates

r~(sz)

with the densities of

polymer

_segments

(collective coordinates) according

to

pi

(r)

=

f /~~ ^dsb(r

rz

is))

and

p2(r)

=

f _j~

dsb(r

_r2

Is))

z=i °

z=i fN

where

f

is the fraction of

species

I, say, N the chain

length

of the total

copolymer,

and is the statistical segment

length

of the

polymer,

the

partition

function of _n~

copolymer

chains

can be

represented

_as

Z ₌

/ _{Dr(8)b(i uP(r))} exPl-HllP(r)I)) (1)

with

HllPlr)I)

=

( _{~j /~} d810rz18)/08)~

12)

+

~ / pa(ri)V~p(ri r2)pp(r2)

+

/ p~(r)U~(r),

where

[

=

f

d~r and the _sum convention _over Greek indices is assumed. The delta-function in the r.h.s. of

(I)

takes into account the

incompressibility

condition. The matrix V~p

jr)

describes

interactions between the segments, U~

jr) represents

the interaction of

polymer

segments with the random environment. For

example,

if _we will consider AB

copolymer

that is immersed in

gel

matrix, the random field U~

jr)

describes

adsorption

of _monomers of ath type. ^{In this} case

the random field

U~(r)

is

U~

jr)

=

-~E~4lg(r) (3)

where _E~ is

adsorption

_energy of ath _monomer in units of

kT,

_~ is the excluded volume of the

gel-copolymer interaction,which

is assumed to be the _same for all types ^of interactions,

4lg(r)

is the local

gel

concentration. The

gel

structure _can be characterized

by

two first correlation functions

<

4lg(r)

_>

=

§

=

Ng IRS (4)

and

<

4lg(r)4lg(r')

> ₌

G(r r') (5)

where

Ng

^{is the number of} monomers between cross-links and llm is the mesh size distance which is

proportional

to

Ng

for _a

rigid

network and

-~

N(/~

for _a Gaussian _one.

Taking

into account the relations

(4-5),

_{we can} write two first moments of the random fields

U~(r)

_as

follows

< U~

jr)

> ₌

~Enj (6)

<

Un(r)Up(r)

> =

~~EnEpG(r r')

=

~~EaEpN(Rjj~b(r r'). ii)

One should note that the last

equality

in the r-h-s- of

equation ii)

is valid _as

long

_as the characteristic

length

scale of the

polymer

melt L is

larger

than Rm.

Introducing

_an

auxiliary

field

4l(r)

_{we can} rewrite

(I)

_as follows

z =

/ _{D4l(r) expj-} j

j~

^Am

^{jri )V~p jri}

^r2)Alp

^jr2

¹⁸⁾

/ Ua(r)Ala(r))

<

b(( _P)

_>o

where

V~p(ri r2)

= ~

~ _~

b(ri r2)

is the matrix

representing

interactions

l _{+ x} 1

between _{monomers, x} is the

Flory-Huggins

interaction parameter between the _monomer

species

of the two blocks of the

polymer,

the brackets <>oin

(8)

represent ^the average over the

configuration

of the ideal

copolymer chains,

which is

given by

the first term in the

exponential

of

(I).

The average ^<

bill pi

_{>o can} be written _as

[8,9]

exPl-) / b4lal-qlsi[plqlb4lplq) Rut lb4lll,

so that the

partition

function takes the form

z =

/ D~(rj exp(- ~~ j-qjv~p jq

_j4~p

jqj (9

/ b4~al~q)Silfll~)b~fllql / _{U*l~q)~* lql}

_ant

_lb~))1

where f~ ⁼

f d~q/(27r)~, b4ln(q)

is the Fourier transform of the function

b4ln(r),

the matrix

Sp(p(q)

is the inverse of the matrix with elements

being

the structure factors of _a Gaussian

coiolymer

_chain

_(see

for

example

_[8]

),

the functional Rnt

(ill)

is _a series in powers of the order parameter ill. In the _case of _an

incompressible copolymer

melt

(~pm

-J

I),

which _we will consider in this _paper, b4l2 ₌

-b4li,

_so that in this _{case we} have _a scalar order parameter

density

_Ah

(r)

_{+ Ah}

(r) fp,n. However~

since the

ordering

does not _occur at wave vector k

= 0

in

reciprocal

_space, this remark should be taken _{as a} statement _on the classification of the order parameter in the _sense of

universality

classes in

phase

transitions. The effective Hamiltonian reads in this _case

Hill)

=

j/All-q)Gi~lq)4llq)

+

~ / / / 4llqi)4llq~)4llq314ll-qi

_-q~

q3)

₊

/hl-q)4llql> ^(lo)

where _uo is the fourth-order _vertex function

computed

at

qRg

= qo,

h(r)

=

Ui(r) U2(r),

_pm

is the average monomer

density being

for melt _pm

-~ I

In,

N is the

degree

of

polymerization

of

copolymer chains,

and the Fourier transform of the inverse propagator Go

(ri r2)~~

is

given by

Gi~lq)

"

lNPm)~~121xs xlN

₊

~llql ~0)~li Ii ii

where _we have

approximate Go(q) by

its

Taylor

series _up to the second order terms, ^and x~

is the value of

Flory-Huggins

parameter _on the

spinodal computed by

Leibler

(xsN

= lo.495 at

f

₌

1/2).

The constant e in equation

ill

is

given by

_{e ci}

R(,

^with

Rg being

^the

^gyration

radius of the

copolymer chain,

_qo

It 1.94/Rg

at

f

=

1/2)

is the

peak

position ^{of the}

scattering

factor. Because of the unique scale of the

problem,

_qo

°~

Rj~,

it is useful to introduce _new dimensionless variables

q =

qR~, ~2jq)

=

~2jq) /jp~NRj),

>

=

uoN/jp~Rj),

_~

=

2jx~ x)N j12)

In these _new variables the effective Hamiltonian reads

Hill)

=

/lllql

^{qo)~ +}

T)4llq)All-q) l13)

+

( _fl / _{4llq~)bl~j qi)}

+

/ ^~@/2 _~l-q)4llq)

~i

Q~

~i

q g

~

One _{can see} from

equation (13)

that the order parameter

4l(q) couples linearly

to the _ran- dom field

h(q). Thus,

the

copolymer

melt in _a disordered medium is described _{in terms} of the

Leibler's free _energy

coupled linearly

to the random external field

h(q).

This

problem

is analo- gous to the random field

Ising (RFIM)

model [2]. ^{The difference} of the

copolymer

Hamiltonian in

comparison

to that of the RFIM consists of the

following.

While the propagator in the

RFIM achieves the maximum for _zero momentum, the Leibler propagator,

Go (q)

_(8], achieves his maximum at the _some finite _{momentum qo.}

The average over the random

potential h(q)

_can be carried _out

by using

the

replica

trick.

As _a result the

multireplica

effective Hamiltonian is obtained _as

Hnl14lll

=

/ ^f 4lalq)lbabGi~lq) ~)4lbl-q) l14)

+

( ^f ij / _{4lalqz)blf qz),}

a=iz=i Q~ ~=i

where _n is number of

replicas,

and /h

=

Npm~2

_(El

E2)~Nj IRS.

To calculate the free _energy Fn _{we use a} variational

principle

based _on the second

Legendre

transformation

(18,19].

In accordance with these references

(see

also

Appendix A)

the

n-replica

free _energy is obtained within the framework of the second

Legendre

transformation _as

Fn = min

Wnl14lalq)1, lGablq)1) lis)

with

Wn _"

-(Sp / _lnjGabiq))

+

I f

/ibabGp~io) ^/h)Gabi-q)

~

a,b=1 ^~

+

f /

<

4~ai~q)

>

i~abGi~io) ~)

_<

_4~biqi

_>

+Snii< 4~ai~) >ii i~abi~)i)1 (16)

where the minimum is to be

sought

with respect to functions _<

Ala(q)

> and the renormalized Green's function

Gab(q)

that _are considered _as

independent

variables at the fixed parameters I, _T, /h. The

quantity Sn((< Ala(q) >), (Gab(q)))

is the so-called

generating

^functional of all

2-irreducible

diagrams

that _cannot be cut into two

independent

parts

by removing

_any two lines between vertices I

(see Appendix

A for

details).

3.

Replica Symmetric

Solution

In this section _we will consider

minimizing

the free _energy

by taking

into account

only

the

one-loop

contribution to

Sn(...).

One _can

easily

_see

that,

in this case,

only

_a

replica

_symmetric solution exists. In the

replica

symmetric _case, the effective propagator ^is

expected

to have the

following

form

Gab(q)

"

9(q)bab

+ (1

nab)flq)i (ii)

where

g(q)

and

f(q)

_are

arbitrary

functions of the form

given

below. The

representation (17)

is

orthogonal

in the _sense that the

diagonal

elements of the matrix Gab _{are g} and the

off-diagonal

elements _are

f.

The inverse of G is

given by

~~~~~~~ A~~~ _{ig i)}

g

/+ ~i

^~~~~

which in the limit _{n ~} o

simplifies

to

Gabio)~~

₌

i) ^f

~~~

ibab f

~~~

il

bob).

i19)

In the ordered

phase,

there is _{a nonzero} average value of the order parameter <

4l(q)

>, the Fourier transform of which has the form

< 4~aIQ) > _"

Ai~io

_{Q0) +}

~i~

+ Q0)).

12°)

Here _we will consider

only

the ordered

phase

with the lamellar type ^of symmetry ^{of the} mi-

crophase

structure because it has the lowest free _energy for the effective Hamiltonian

(14)

with- out the cubic term.

Substituting

the trial functions

(19-20)

into

expression

of the free _{energy we}

can write the free _{energy as a} functional of

A, f,

and _g. In order to find the values A,

f,

_{g cor-}

responding

to the extremum of Fn _we have to take variational derivatives of Fn with respect to these functions. In the

one-loop approximation

for the functional

Sn((< Ala(q) >), (Gab(q)))

this results in the

following

extremum

equations A(T

+

'

/

G~aa~(q)

⁺ '~

= o,

(21)

2

~

2

~

ig

j~~

/~ ₌ ° 122)

We _note that the equations ^for _g and

f

_can be also obtained

by inserting (19)

into

(A.3)

and

comparing

the coefficients in front of (bob

I)

and bob.

Equation (21)

_can also be obtained

directly

from equation

(A.20) using

equation

(20).

For the disordered phase

(A

₌

o)

this system _can ^be

simplified

to

19

f)~~

"

Gi~lq)

₊

/

G(aa)lq)1

124)

f

₌

/hig f)~. ₁₂₅₎

One _{can see}

that,

in the

one-loop

approximation, the function

(g f)~~

has to have the form

((q(

qo)~ + r, because the

integral

in the r-h-s- of equation

(24)

results in renormalization of the effective temperature T.

Taking

into account the equations

(17,25)

_{we can} write the

two-replica

correlation function G~abjIQ) as

G(ab)IQ) " 98(Q)bab +

iig~

_IQ)

(26)

with

gB(q)

₌

I/(((q(

qo)~ +

r).

It is convenient to introduce _new reduced variables t ₌

T/(lq( _/27r)~/~,

b

=

/h/(lq( _/27r)~/~,

z =

r/ (lq( _{/27r)~/~ (27)} Substituting

the function

given by (26)

into

(24)

_we

get

~ ~ ~

47r@~~

^~ 2r~' ~~~~

which,

_on

substituting

^with dimensionless

variables, gives

z = t +

)z~~/~(l

⁺

)). ₍₂₉₎

Repeating

the _same calculations for the ordered

phase (A # o),

for which A~

= 2

/(1(g ii),

we

finally

obtain

~

~ ~ ~

4~$~~

^{~ ~' ~~~~}

which

gives

in dimensionless variables

t ₌ z +

~z~~/~(l

₊

)). ₍₃₁₎

First _we consider the _case of weak field /h

IT

< I.

Analyzing equation (30)

one can see that the solution of this

equation

_appears

only

below _some critical temperature T~ that _can be

considered _as the

"spinodal"

of the ordered

phase.

The critical value of _r at which

(30)

has _a solution

corresponds

to the minimum of the r-h-s- of this

equation.

For _a weak random field the

spinodal

_r is obtained _as

~

r~ =

l'~° _{)~/~, (32)}

47r

which

gives

the

following

critical value for the effective temperature T~

Extrapolating

equation

(30)

to

large

fields

(/h IT

>

I)

_{we get} the critical value of _{r~ as}

~~

l~7r~'~~~~~~

The effect of

the

next-order _terms ^on (34) ^is

discussed

at

the

end of

To

_calculate the

phase

diagram of the

system under

onsideration, the _energies

mogeneous

and

ordered phases have

to

^be ompared. Substituting trial functions _given

by ^uations (26) into thepression

for the

^{-replica free} energy F~ andtaking

the limit

<

n-o

less riables)

and for the ordered _one

The

phase

agram of

the

system

in

_{andom media is}

ketched in Figure I in

the plane

16

= /h/((lq]/27r)2/~), t =

In

_{the limit of} a weak

the irst-order phase ransition occurs

at

Ttr Cf -1.3(lq(/27r)~/~. At

the

_strong random field

limit

_the emperature

of the

phase transition

is

proportional to _/h~/5 The latter _behaves

lo

8

6

DIS GLASS

LAM

0 -1 -2 -3 -4

t

Fig. 1. The phase diagram of the copolymer melt in

a random field environment obtained within

the one-loop replica symmetric solution. The continuous _curve is the coexistence _curve between the ordered

(lamellar)

and disordered _states. It is computed by setting equal equations

(35-36

). The dashed line is the stability line of the replica symmetric solution. It is derived by solving equations

(52,29).

Imry-Ma

arguments (23]. ^The statistical fluctuation of the random field

h(q)

will

destroy

the domain _{structure as}

long

_as the amplitude of the

4l(q)

due to this random field

h(q)

is

larger

than the

amplitude

of the dense _wave A. In other

words,

_as

long

_as the energy of the domain

structure (T(~

IA

is lower than the fluctuation _energy

/hq( / @~,

^the

following

criteriuIn holds

true:

l~trl ~

(iql~h)~/~

₁₃₇₎

In the

approach

used above, _we

neglected

^the contribution of the second-order

diagrams

in powers of the vertex I in the

Dyson

_equation

(24).

The contribution of this

diagram

is

1(2)

=

j>2 /48gr)r-3/2jji

₊

_z~/j2r))3 jz~/j2r))3), j38) Comparing

^I12) with the contribution of the

one-loop diagram,

_{one can} write

In the _case of strong

fields,

/h

IT

> I, we can

neglect

the second order

diagram

_as

long

as the

following

parameter is small

~lj

l))~~~ ^b/Z~

^I1.

^j40)

The first-order

phase

transition to the ordered

phase

_occurs at

Zc "

~b)~/~

(41) Substituting

the latter into

equation (40)

_we get an

inequality

for b where _our approximation works

~~~

j~~)2/3

~

~ ~~~

34/581/5 _~qo ^{~ ~' ~~~~}

In the weak random

field, /h/r

< I,

equation (40)

reduces to

loo

~_

(27r)2/3

1

~~~

~~~ ~2

Qo _{~ ~'}

(43)

4.

Stability

of the

Replica Symmetric

Solution

In this section, _we consider the

stability

of the

replica symmetric

solution

Gab(Q)

_"

/hf(q)

with respect to

replica

symmetry

breaking.

With this _purpose let _us write the renormalized

two-replica

correlation function Gab_IQ) in the

following

form

~abi~)

" 9BiQ)~ab +

~911Q)~i~b

₊

boabio) i~~)

where _{e =}

(I,..,I)

and

boab(q)

is _a function that

equals

_zero for _a

= b.

Substituting

the function

Gab(q)

into the r-h-s- of the _n -replica free _energy

(A.21)

and

expanding SpIn(Gab)

in powers of the function

boab(q)

_one finds

~Fn ₌

11 lib

~ele/boablq)bocalq)1

¹⁴⁵⁾

( / / / boab(k)boab(P)flq)flq

k P)I

where in the r-h-s- of equation

(45)

the summation _over all repeat ^{indices is assumed. The} derivative of

Sp In(Gab

^with respect to

boab

_IQ) is easy to obtain

using

the relation

bGjj _/bGbc

"

-Gj/ Gj/

and

neglecting

terms

proportional

to n~. One should note that the last _term in equa- tion

(45)

is the second-order term in powers of the vertex I. The form of the

perturbation boab(Q)

_can be found

by analyzing

the

Dyson

equation for the

off-diagonal

part ^{of the} two-

replica

correlation function

including

the second-order

diagram

_powers of the _vertex I. This

analysis

shows that the fluctuations with (q( = qo

give

the main contribution to the renormal- ization of the bare characteristic of the system, so we can choose

boab(Q)

in the form

~oabio)

"

Qab9iiQ)1

1~6)

where

Qab

is _{a n x n} matrix.

bitroducing

Parisi's function

q(x)

_(24] defined in the interval (o,

I]

and connected to

Qab by

~ Q~ix)dx

=

it ^jn~

i~

j Qi~

v k

14i~

a,

the

quadratic

_part of the free _energy in power of

Qab

is obtained _as

~F ₌

-(1 _{/~ /~llli} -13)blx

_Y)

+12)qlx)qlY)dxdYl, 148)

~~~~~

ii "

/9~(Q) ~~

~~~' _~~~~

q

~~

1~7r2'~~~~~~

~'

_~~~~

12 " 2~l

/ _g((q)

=

~

q(/hr~~/~ ₍₅₁₎

~

87r

The

replica symmetric

solution is unstable in the

region

of parameter values where the matrix

(Ii -13)b(x y)

+12 has _a

negative eigenvalue

1-=11+12-13 <0

or in terms of reduced variables

1-

-~

(1+

^~ _c

~~ § o

(52)

where _c

=

((27r)2/~/64)(1/qo)~/~

_< l. Equation

(52)

determines the

spinodal

line of the

replica symmetric

solution. In the _case of the weak random field

/h/r

< I _{we can}

neglect

the second term in the r-h-s- of

(52)

and write

r~p =

(c/h~)~/~,

/h < c~/~

(53)

Extrapolating

equation

(52)

to

large /h,

/h

IT

> I _or, /h > c~/5

we have

r~p =

(~c/h)~/~,

^{/h > c~/~}

⁽⁵⁴⁾

Substituting

the solutions r~p ^into

equation (28) giving

_{r as} function of _T for the disordered sthte, we get ^the

following

boundaries for the

replica

symmetric solution in the (T,

/h) plane

in the limit of small and

large fields, respectively

Tsp =

c2/9z~4/9 )c-1/9z~2/911

⁺

)iz~/c2/5)5/9),

^{/~ <}

^{c2/5 155)}

Tsp =

(c/h)~/~

^~

/h~/~(2c/3)~~/~(l

₊

2((2c/3)~/~llh)~/~),

/h > c~/~

(56)

4

However,

in order that this

region

with the RSB solution exists _on the

phase diagram,

the effective temperature _{T as} function of /h has _to be

higher

than the effective temperature ^of the first-order

phase

transition _Ttr. The

boundary

of the RS

solution,

which is

computed by

using equations

(52,29)

for the value _c

= o.5, is also

plotted

in

Figure

I. It follows that RSB _occurs

already

in the disordered

phase.

5. Disorder _vs.

Ordering

in the Lamellar Phase

The previous section discussed the behavior of the block

copolymer

melt in the _one

phase regime, I.e.,

at

high

temperatures ^{before the} micro

phase

separation transition. The

possibility

of the existence of the

glass phase

before the micro

phase

separation transition _was shown. In the remainder of the _{paper we} discuss the effect of random fields at temperatures below the micro

phase

separation transition. For

simplicity,

_we consider

again only

symmetric diblock

copolymers.

In this _case it is well known that the structure of the melt consists of the lamellar

phas~jy,

^{which can be viewed as}

^periodic

arrangements ^of

interphases

between thin films of A-rich _a d B-rich domains. The

interesting

question that emerges is what is the effect of the random field _on the

phase boundaries?,

_I-e-, the interfaces between the domains. In the

following

_we

d(tinguish

_{between the weak} Segregation

lin~it

^{and the} so called strong segregation limit

deep

^below

th~,MST

transition temperature. ^In the latter _case the

phase

boundaries _are well formed and

the'density profiles

_are

sharp,

whereas in the _case of weak segregation the

density profiles

_can be diodelled

by trigonometric functions,

at temperatures ^below but close to the MST transition temperature. The

following

consideration of the

stability

of the lamellar

phase

in the strong

segregation

limit is based _on

Imry-Ma

arguments (23]. ^The

stability

of the lamellar

phase

in the weak

segjegation

limit is

cinsidered _{by mapping}

_the

_copolymer

Hamiltonian onto the Hamiltonian of the random field XY model [25].

interfaces

o

B-rich -rich B-rich A-rich

P

--z h

A-rich

i

' flit@rfQC@S

' al bl

'

Fig. 2. The distortion of lamellas due to random fields.

5.I. COPOLYMER MELTS: STRONG SEGREGATION LIMIT. Consider block

copolymer

melts

with

f

₌

1/2

in the strong

segregation

limit. The

ordering

consists of

parallel

interfaces

separating

thin films of A-rich and B-rich composition,

arranged

_as a one-dimensional lattice of

spacing

D

(see Fig.2).

The interfacial free _energy is

independent

of chain

length N,

while D scales _as

N~/~.

We argue that this

long

_range order is unstable

against arbitrary

weak random fields

(which locally prefer A,

_or

B, respectively.

This

instability

_occurs

against

weak

long

_range distortions of the interfacial pattern relative to the ideal pattern, ^which acts _as a local deviation bD of

the thickness of the lamella. These local deviations add up in _a random-walk-like fashion _over

a

large

distance _z

perpendicular

to the local tangent

plane

to the interfaces. Hence, there is

no

long

_range order _over

large

distances.

Consider first the energy balance for _a distortion of _one interface

by

_a small distance h

(compared

to

D)

_{over a} radius _p

parallel

to the interface

(in

d dimensional

space)

_[26]. The volume of the shaded

region

in

Figure

2 is

V =

constp~-~h. (57)

The

gain

in random field _excess energy scales like the square ^root of this volume

(Hrf

is the

strength

of the random field and is

given roughly by

the root of its variance, I.e., Hrf ci

4).

Erandom

field ^CC

~HrfW

CC

-HrfPi~~~~/~/l~/~ j58)

We estimate the energy cost for distortions of the interface from the

capillary

_wave

description (assuming

_a small

angle

8 for the local

bending

^of the interface: 8 _m

2h/p

_<

I)

H~w ₌

a~l~~~~ / _{~c(Vh)~df. (59)}

2

Here df is the element of the

id I)-dimensional interface,

_a the

underlying microscopic length

(size

of

polymer segment),

_~c the interface stiffness

(which

is

equal

to the interfacial free energy

for block

copolymers).

For the _case

considered,

Vh _c~

8,

and hence

(we

^henceforth measure

all

lengths

in units of

a)

(H~w)

c~ ~cp~~~8~ c~ ~cp~~~h~.

(60)

For

suppressing

such distortions of the

interface,

_we would need

lHcw)

>

lErandomfieldl

~tP~~~/l~ >

HrfPi~~~~/~/l~/~ 161)

P~~~~~/~h~/~ >

Hrf/~t. 162)

We _see that for d _< 5 this

inequality

is violated for _an

arbitrarily

small

strength

of the random

field,

for

large enough wavelengths

_p at a finite distortion h. The thermal fluctuations would

only

induce _a

divergence

^{of the local} interfacial width for d < 3

(in

d

= 3 (h~)~~~~~~~ ^c~ In p, due _to the

thermally

excited

capillary waves).

The random field creates such _an

instability already

in d < 5 dimensions. In

particular,

for d

= 3 the condition that interfacial distortions must

satisfy

in order to _occur is

p~~h~/~

_<

Hrf/~c

or

V$8

< Hrf

IN. (63)

The balance between the random field _energy and the

bending

_energy

gives

_a connection between the distortion h and the

length

scale _p

along

the interface

/l _Cf

(H

f/lQ)~/3

p(5-d)/3 j6~)

The lamellar

mesophase

will become instable, when the distortion h in

(64)

will become _com-

parable

with the lamella

spacing

D.

Equation (64)

with h

= D

-~

N~/~ gives

^the

length

scale at which distortion of the lamellar structure is relevant

id

=

3)

pm~x ^cf N~C

/Hrf. (65)

The

instability

of the lamella due to random fields should also exist in

mesophases

of _cross- linked

polymers

such _as

interpenetrating polymer

networks [27].

It _must be

emphasized

that the considered distortions of the interfacial pattern constitute

only

_one class of defects in the order stabilized

by

the

randomness,

but there _may be other distortions that also _are worth

considering.

One is the fluctuation in the direction of

q(x),

over distances

ix x'(

_» D. This defect needs consideration of the

"bending rigidity'

of the pattern ^of

parallel

interfaces _{as a} whole. Also it _may be of interest to consider

"topological

defects" in the interface pattern, ^such as shown in

Figure

3, which

possibly

also could be stabilized

by

random field _energy

gains.

These

problems

_are left for future works, because _even if it is found that such defects _are also

stabilized,

this result would

only strengthen

and not

weaken _our conclusion that lamellar

phases

_are destabilized

by

random fields.

5.2. COPOLYMER MELT: WEAK SEGREGATION LIMIT. In the weak

segregation

limit for

which the fluctuations in the local

composition

_are small in comparison with its average

value,

we can use the Hamiltonian

(13)

to describe the system ^below the

microphase separation

transition.

Keeping only gradient

and random field terms, the effective Hamiltonian is

H

ii~i IT)))

₌

( / _iiv~

+

Qii

^~

ir))~ ~li/l~~ / _hit)~

IT) 166)

For the _case of _a

periodic

structure in the _z

direction,

the _average value of the order parameter (4l

IT))

^is

ill jr))

= 2A _cos (qo

(z

₊

u(r)))

,

(67)

liiil

al

fibl

A-r,ch

Fig. 3. Example of topological defects.

where the scalar function

u(r)

describes the deformation of the

layers

in the z-direction. Sub-

stituting equation (67)

into the r.h.s. of equation

(66),

after

averaging

_over all

possible

_con-

figurations

of the random field

h(r),

_{one can} write the effective Hamiltonian for the function

~(T)

H

ilUa

IT)

I)

=

[ ) /

(ili~,yUo ir))~

⁺

4qi iV~Uo ir))~)

~ _/hAaAp /

CDS iqo

iUoir) Upir)))

₁₆₈₎

a,p ^r

where the first term in the r.h.s. of

equation (68)

describes the deformation of the lamellar

layers

and the second _one

couples

this deformation in different

replicas.

It is

interesting

to note that the cosine-like

coupling

term between fluctuations of the order parameter ⁱⁿ two different

replicas

_appears in disordered

physical

systems such _{as an} array of flux-lines in type ^II

superconducting films,

in

magnetic

films [28], ⁱⁿ

crystalline

surfaces with disordered substrates (29] ^{and in the random field-XV model} [25]. ^{In all these} systems ^the cosine term results in the

breaking

of

long-range

order and spontaneous

replica

symmetry

breaking (30, 31].

In the framework of the Gaussian variational

principle

_{(31, 32]} the contribution to the free energy of the system due to fluctuations of the

displacement ua(r)

in different

replicas

is

Fvar "

~jSP /'~lGab(q))

⁺

lH11"a(q)1) _H0)0

₁₆₉₎

where _we have introduced

~0

~ / ~a/(~)~a(Q)~b(~Q) (I°)

a,b

and

Gj/(q)

is the _two

replica

trial function whose form has to be defined

self-consistently

and the brackets

(...)~

^{denote the thermal}

^averaging

^with

weight

_exp

(-Ho ).

^After thermal

averaging

^the variational free _energy reads

~f~

^"

)sp / illjGab(q))

+

jsp / _{(Gi~ lq)bab G&/}

IQ))

Gab(q)

~ _lhAaAb

_eXp

(- )

2

^{ab) Iii)}

b#a

Where

Bab _"

/ _iGaaio)

+

Gbbio) Gabio) Gbaio)) 172)

and the inverse bare propagator

Gp~ (q)

is

(~ ((q(

⁺

^q()~

⁺

4q(q])

^{The trial function GabIQ)}

can be found from extremal equations

Gj/ (q)

₌

Gp~ (q)

+ 2

~j _/hq(AaAb

_exp ^~~_Bab

,

(73)

a#b

~

~a/

_IQ)

_~21~Q~~a~b

_eXP

(~j

^~

_~abj (14)

Analyzing

the last equation _one can conclude that function

Gj/(q)

_{does not}

_depend

on the

momentum q for _a

#

b.

So,

_{we can} define

Gj/(q)

= -aab

la _{# b).}

In the _case of the one-step

replica

symmetry

breaking

for which the elements of the matrix _{aab are}

assumid

_{to have two}

different values _ao and _al

depending

_on whether the _two indices _a and b

belong

to the _same blocks of

length

_{m or} not, one can rewrite the extremal equations

(see

_(31] ^for

details).

al " Y exp

(~

^ln

t)

,

(75)

ao = Y exp ~ ln^t

(2~

In

Lqo

+ _~ ln

t)

,

76)

m

where L is the linear size of the system and _we have introduced the

following

parameters

assuming

Aa to be the _same in all the

replicas

and

equal

to A

~

16~A2'

~~~2q(~~~'

~'

~~~~~~' ₍₇₇₎

In the

thermodynamic

limit L _{~ cc} equation

(76) gives

_ao

" o.

Taking

this fact into account

the trial function

Gab(q)

is

~~~~~~

p~(q~+

mar

~

Gj~(q) _(G ~(q)

₊

_mail

^~~~~

Gab_{IQ) =}

~ ⁱ

(

_for

a, b E

diagonal

blocks _{m x m}

(79) I

IQ)

(Gi

_{IQ) +}

mail

Gab(Q)

₌ o, for _a, b E

offdiagonal

blocks _{m x m}

(80)

Substitution of the solution

(78-80)

for the trial function

Gab(Q)

into the

expression

for the variational free energy yields (31]

f~~r ₌ lim ^~

(F~~r(t) F~~r(o))

₌

'f~

_~°

(l

)~t

₊

Y'(I )t~)

,

(81)

"-° n

~ ~

m

where Y'

=

2Y/(A~q()

=

4/h/q(.

The

equilibrium

values of the parameters _m and t _can be found from the system of the

equations

~ ~t Y't~

= o,

(82)

(1 +

Y'(I m)t~~~

= o.

(83)

For ~ > l this system has

only

the trivial solution _m

= I and t ₌ o that

corresponds

to the

replica symmetric

^{solution with all}

off-diagonal

elements of the matrix

Gab(Q) equal

to _zero.

The nontrivial solution for _~ < l is

given by

m = ~,

(84)

1

t ₌

(Y'~)1-~ (85)

In other

words,

at ~ = l the system

undergoes

a

phase

transition at which the correlation function Gab_IQ)

changes

form. We can rewrite the condition _~

= l in terms of the parameters

of the system. ^{In the} mean field

approximation

^A~ _{= 2 (T(}

IA,

which

gives

the effective temperature ^{of the}

phase

transition (T( =

qol/327r. Comparing

this temperature ^vJith the temperature ^{of the first order}

phase

transition (T( re

(q(1/h)~/~one

_{can see} that for /h >

q(/~l~/~

there is _a first order

phase

transition between the disordered state

IA

₌

o)

and the ordered

phase IA # o)

with _one step

replica

_symmetry

breaking

for the correlation function

I"al~)"b(~~))

To

complete

the

analysis

_{we now} calculate the correlation function

(4l(r)4l(r'))

^{in the} or-

dered

phase

below the

phase

transition

(~

<

l).

Due to fluctuations of the

displacement u(r)

the correlation function is:

14l(r )4l(r'))

^{c~ A~} cos

(qo(z z'))

exp

~-

^~~

^{((u(r) ~t(r')} _)~))

,

(86)

2

with

(jujr) ujr'))2j

= 2

Iii

_cos

_{[qjr r')))} jujq)uj-q)). j8i) Substituting

the

expression

for the correlator

(u(q)u(-q)),

which is

given by

the

diagonal

part of the two

replica

correlation

function,

_one can write

, ~

2 (1

~)

In _{t +} 2 In

z '~

qo, for _z

z' (jl

_> I

((u(z)-u(z)))

= m _, , ,

qo 2~In

z

z qo, for _{z z}

(j~

<

z~

z)

= o

(88)

and

((iL(Xl) ^MIX) ))~)

"

j

^~~

^j~~~

^{~ ~~}

^{~~ ~~}

^~~~

^~

^~~

^{~ ~~} _Ii

o ~ n xi x

~ qo, or xi _{x ~}

~

~z-z'~

= o

(89)