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Copolymer Melts in Disordered MediaS. 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 dePhysique I, EDP Sciences, 1996, 6 (6), pp.837-857. �10.1051/jp1:1996245�. �jpa-00247218�

https://hal.archives-ouvertes.fr/jpa-00247218

https://hal.archives-ouvertes.fr

J. Phys. I FYance 6 (1996) 837-857 JUNE 1996, PAGE 837

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 considereda

symmetric AB block copolymer melt in a gel matrix with

preferential adsorption of A monomers on the gel. Near the point of the microphase separationtransition such

a system canbe 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 copolymeron

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; weinterpret this state as a glassy state

and calculatea

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 preferentialadsorption 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 gelrods. 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

838 JOURNAL DE PHYSIQUE I N°6

regions, N(/(~, where (=

T~~=((T T~)/T~)~~ with v m 2/3 being the correlation lengthof 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 energyof 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 symmetricsolution with respect to replica symmetry breaking. Section 5 explores the stability of the

lamellar mesophase exposed to random fields. In conclusion we discussour 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 positionof 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=ifN

N°6 COPOLYMER MELTS IN DISORDERED MEDIA 839

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-

jj~ 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 interactionsl + 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 givenby

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 gyrationradius 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 theLeibler'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 Hamiltonianin 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], achieveshis 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=

/ f4lalq)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-replicafree 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 parametersI, 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 symmetricsolution 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-diagonalelements are f. The inverse of G is given by

~~~~~~~ A~~~ig i) g

/+~i ~~~~


which in the limitn ~

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~a IQ) > "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 afunctional 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

~

igj~~

/~=

° 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~abj IQ) 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 spinodalr 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-orderterms

on (34) is discussed at the end of

Tocalculate

the phase diagram of the system under onsideration, the energies

mogeneousand

orderedphases have to be ompared. Substituting trial functions given

by uations (26) into the pression for the -replica free energy F~ and taking the limit<

n-o

lessriables)

and for the ordered one

The phase agramof

thesystem

inandom 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 emperatureof the

phasetransition is

proportionalto /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 arandom 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 largerthan 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 thefollowing parameter is small

~ljl))~~~ 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 approximationworks

~~~j~~)2/3 ~

~~~~34/581/5 ~qo

~ ~' ~~~~


In the weak random field, /h/r < I, equation (40) reduces to

loo~_

(27r)2/3 1~~~

~~~ ~2Qo ~ ~' (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 then -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 perturbationboab(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~/5we have

r~p = (~c/h)~/~, /h > c~/~ (54)

Substituting the solutions r~p into equation (28) giving r as function of Tfor the disordered

sthte, we get the following boundaries for the replica symmetric solution in the (T, /h) planein 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 usingequations (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. Disordervs. Ordering in the Lamellar Phase

The previous section discussed the behavior of the block copolymer melt in the one phaseregime, I.e., at high temperatures before the micro phase separation transition. The possibilityof 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 wed(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 stabilityof 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 asN~/~.

We argue that this long range order is unstable against arbitrary weak random fields (whichlocally 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 ci4).

Erandomfield CC

~HrfWCC

-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 m2h/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 instabilityalready 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 orV$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 "topologicaldefects" 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 separationtransition. 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)


liiilal

fiblA-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-consistentlyand 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)

2~ lhAaAb eXp(- ) 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 Gab IQ)

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 areassumid 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 parametersassuming 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 Iiicos [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

zz' (jl > I((u(z)-u(z)))

= m, , ,

qo 2~ Inz

z qo, forz z

(j~ <

z~z)

=o (88)

and

((iL(Xl) MIX) ))~) "

j ~~ j~~~ ~ ~~~~ ~~ ~~~ ~ ~~

~ ~~ Iio ~ n xi x ~ qo, or xi x ~ ~

~z-z'~ =o (89)


where fzqo=

t~~/~ and f~qo=

t~~/~are the correlations lengths in z and x, y directions

respectively. The form of the correlation function shows that there are two different regions.

Inside the domains of size ~xi xj~ < (~ andz '~

< (z the system has smectic A-like

properties (33]. On larger length scales the fluctuations u(r) of the layer displacement wash

out the long-range modulated order and result in the formation of the highly anisotropicitranslationally incoherent domains with (z/(~

-J

/h~+ These domains first appear at

very weak random field /h-J

L~~(~~~J

6. Conclusion

In the previous sections we have investigated the behavior of the symmetric AiBi copolymers in

the gel matrix. We have found that the weak preferential adsorption of A monomers results in

three effects: I) it decreases the temperature of the first order phase transition between ordered

and disordered states; for strong external random field, there is a new power dependence forT

as a function of the strength of the random field /h on the line of the first-order phase transition:

T -J

(q(1/h)2/~; 2) there exists a region on the phase diagram of the system where the replica

symmetric solution for two replica correlation functions is unstable with respect to replica

symmetry breaking; we have interpreted this region as a glassy state; it is well known (24] that

the solution with broken replica symmetry for matrix Qab is related to the appearance of a

hierarchical set of valleys in the phase space of the system; this results in the dependence of

the properties of the system on the sample history; in our case it could be demonstrated as a

sample to sample fluctuation of the scattering intensity or as an exponential slowing down of

the dynamics near the first order phase transition into the ordered phase; however, in order to

give a quantitative answer to the question whether the system will really be trapped in one of

the metastable minima, we have to estimate the barrier between different metastable states; we

hope to consider this question in future publications; 3) already in the ordered phase (A # o),

a very weak random field /h-~

L~~(~~~) destroys the long-range modulated order, resulting in

the formation of highly anisotropic translationally incoherent domains.

One should note that the theory presented here is not limited to the case where the size of

the copolymer is larger or of the order of the mesh size of the gel matrix. In the limit L < Rm

we would have to include the q -dependence of the gel density-density correlation function. It

could be done for example as G(q)=

Go/(qRm)~f + I where df is the fractal dimension of

the network: df=

I for rigid networks and df=

2 for Gaussian ones. The q-dependence of

the correlation function would result in the renormalization of the adsorption energy and the

substitution of /h in all formulae of this paper by the value ~l~

/h/((qoRm/Rg)~f +1).

Acknowledgments

SS acknowledges partial support from the Deutsche Forsch~ngsgemeiTtschajt (grant Bi 314/6,Schu 934/1-2 and SFB 262). AD acknowledges useful discussions with L. Leibler, T. Garel,

H. Orland, J.-F. Joanny, A. Ajdari, and P. Lammert. TAV acknowledges financial support bythe Fonds den chemischen Industne.


Appendix A

The Technique of the lst and the 2nd Legendre Transformations

Having in mind readers who are not familiar with the functional methods of quantum field

theory and to make the paper self contained, we will repeat the main steps of the derivation

of the technique of Legendre transformations [18-21]. The partition function associated with

the Hamiltonian (14) is given by

e~~'"=

Z~=

/ D4lae~~", IA-1)

where W~ is the generating function for connected diagrams. To simplify the notation we

introduce ri % 1, r2 =2, etc. It is easy to see from (A.I) that the functional derivative of Wn

with respect to rob(1, 2)=

Gj)~(l, 2) yields

bwn/blab(1, 2)= < Ala(1)4lb(2) >

=~Gab(1, 2) + < Ala(1) >< 4lb(2) >, (A.2)

where the definition of Gab(1, 2) is obvious from equation (A.2). The functional derivative of

Wn with respect to the external field h is

bw~/bh~ji)= < ~~ji) > jA.3)

The second derivative of Wn with respect to h at h=

o is

bha)~)~b(2)~

~~~

~~~~~'~~' ~~'~~

Considering Fab(1, 2) and h as independent variables we can write the variation of Wn as

bwn= )Gab(1, 2)blab(1, 2) + < Ala(1) > blab(1, 2) < 4lb(2) > (A.5)

+ < Ala(1) > bha(I),

where the sum convention is used. Performing in equation (A.5) the Legendre transform of

Gab(1,2) and < Ala(1) > as independent variables we obtain

bwn b()F(1, 2)G(1, 2) + < 4l(1) > F(1, 2) < 4l(2) > + < 4l(1) > h(I))=

~F(1, 2)bG(1, 2) h(I)b < 4l(1) >, (A.6)

where in (A.6) the integration over the repeated "indices" I and 2 is implied. The replicaindices in (A.6) are suppressed. Introducing instead of the potential Wn the potential Fn

=

Wn )F(1, 2)G(1, 2) ) < 4l(1) > F(1, 2) < 4l(2) > < 4l(1) > hi I) we arrive at

br~=

-jrji, 2)bGji, 2) hji)b < ~ji) > jA.I)


0~+

S/+

Fig. 4. Example of diagrams contributing to the self-energy.

The functional derivatives Fn with respect to Gil, 2) and < 4l(1) > give

~bfn/bG(1, 2)=

-F(1, 2), (A.8)

br~/ < b~ji) > =-hji). jA.g)

Equation (A.9) gives for h=

o an equation for the spontaneous magnetization

brn~ (A.10)

b < 4l(1) >

In order to establish the diagrammatic structure of the potential Fn we use the Dyson

equationG

=

F~~ + F~~LG, (A.ll)

where we did not write explicitly the replica indices and the space variables in (A.ll). The

diagram expansion of the self-energy L is given by in Figure 4. Inverting the Dyson equation(A.ll) we get

F=

G~~ + L. (A.12)

Combining (A.8) and (A.12) yields

bfn/bG(1, 2)=

~F(1, 2)=

(G~~(1, 2) + ill, 2)). (A.13)

There are two different possibilities for partial summations of the diagram series of the self-

energy L. The first one includes all summations enabling one to replace the bare propagatorsF~~(1, 2) in L by the exact one Gil, 2). Due to this summation, the diagrams of L become

two-particle irreducible, I.e. they do not separate into two pieces by cutting two lines. The

second type of summation enables one to replace the magnet field h by the average value of the

order parameter < 4l >. Due to this summation, the diagrams of L will become one-particleirreducible (or amputated), I,e. they do not separate into two parts by cutting one line. Thus,

the right-hand side of equation (A.13) is expressed in terIns of the full propagator Gil, 2) and

the average value of the order parameter < 4l >, and can therefore be considered as an equation

for the exact propagator Gil, 2). Equation (A.13) can be written as the stationarity condition

of thefunctional

Fn = Fn +)F(1, 2)G(1, 2).

We will now show that the otential Fn (withrespect

to <4l


(bh(I)/b < 4l(3) >)16 < 4l(3) > /bh(2))=

b(1- 2) (sum convention over repeated indices)and using equations (A.3, A.9)

we get

b < 4l(1)~~<

4l(2) >~bh)I(I(2) ~ ~~'~~~

For h=

o the expression in the bracket of the r,h.s. of (A.15) taken at h=

o is the exact

propagator G(1,2),so that we obtain

l~2r~~

b < 4l(1) > b < 4l(2) >~~~~~

~ ~~'~~' ~~'~~~

Differentiating the I-h-s- of (A.15) with respect to < 4l > we can express the derivatives of

Fn with respect to < 4l > through the derivatives of Wn with respect to h. The latter are

the connected vertex functions. As an example we give only the expression for the four-vertex

function

§4r~~~~'~'~'~~

b< 4l(1) > b < 4l(2) >

j< 4l(3) > b <

l(4)~

~~~~~

~~'~~~

§4w"

~~~~~> ~~~~~~~~'~~~~~~~> ~3~~~~~~>~~~ ~bhjxi)bhjx~)iijx3)ihjx4))~=o

The vertex functions Fz(I,.. ,I) are obtained from the connected ones by amputating the ex-

ternal lines from the connected vertices. Going to the skeleton diagrams enables one to replacethe bare propagator by the exact one in the diagram expansion of the latter.

Taking into account that the potential F~ is a generating function of the amputated vertex

functions, equation (A.9) gives

G~~(1, 2) < 4l(2) > + ~F4(1, 2,3,4) < 4l(2) >< 4l(3) >< 4l(4) > +... =0. (A.18)

The potential F~ can be obtained by integrating equations (A.14, A.18). The interaction

part of F~ is represented by the so-called 2~~-irreducible diagrams, which do not fall into two

pieces by cutting two lines. Inserting L (given up to two-loop order) into equation (A.14) we

get

jlbabGi~ l~) ~h G&/ l~) + dab/

Gabl~) + bab/

< 4~a(~~') >< 4~b(~') >

q'/ /Gab(~l)Gab(~ + ql + ~2)Gab(~2) + 1' lA.19)

Qi Q2

Inserting G~~(1,2) computed up to linear order in the coupling constant I and the bare value

of the vertex function F4(qi,q2,q3, q4) It 1(27r)~b(qi + q2 + q3 + q4)) into (A.18) we get

lbabGi~l~) ~h +/ Gablq')) < 4~lq) > lA'2°)

q'

+/ / /

< 4llqi) >< 4~lq~) >< 4~l-qi q~ q) > =0.

qi q2 q3


The potential Fn can be obtained by integrating equations (A.14) and equation (A.10). Re-

stricting ourselves to the two-loop approximation of the self-energy L we get

Fn"

jsp fill Gablq) +f

/lbabGi~l~) ~h)Gab(~q)Q a,b=I

+

f /< 4~al-q) > lbabGi~lq) ~) < 4~blq) >

+

f /< ~ai-~) >

i( /, ~aai~')) < 4~ai~) > i~.21)

" ~

~~ ~~~ ~ ~~~~~ ~~ ~ (~ / ~aa(q))~

f / d~rG(~(r),

~ ~ a= ~,~~~

The last two lines of equation (A.21) are the low-order terms of the expansion of the functional

Sn((< Ala(q) >), (Gab(q))) (see Eq. (16)) in powers of the fourth vertex I.

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