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Glass Transition Versus Microphase Separation: a Phenomenological Replica-Field Theory for AB
Copolymer Systems
Andrey Dobrynin
To cite this version:
Andrey Dobrynin. Glass Transition Versus Microphase Separation: a Phenomenological Replica-Field Theory for AB Copolymer Systems. Journal de Physique I, EDP Sciences, 1995, 5 (6), pp.657-669.
�10.1051/jp1:1995158�. �jpa-00247091�
Classification Pbysics Abstracts
36.20Hb 64.70Pf 64.75+g
Glass Transition Versus Microphase Separation:
a Phenomenological Replica-Field Theory for AB Copolymer Systems
Andrey V. Dobrynin
Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627-
00ll, U-S-A-
(Received 3 August 1994, revised 29 November 1994, accepted 16 February1995)
Abstract. We present a phenomenological replica-field theory for symmetric AB copolymer systems with the A.block
near its glass transition point. Using a variational principle based on the second Legendre transformation we prove that the interaction between composition fluctu-
ations, due to the loss mobility of the A monomers, results in the elimination of microphase separation. Instead of a microphase separation there is a third-order phase transition between the homogeneous state and a glass-like state with broken replica symmetry of the two-replica
density-density correlation function. We show that near the point of the glass transition two-
replica correlation function Qoefl(q) is proportional to trie square of the one-replica correlation function QoeflG$~(q), where Qoefl is a matrix describing trie metric properties of trie replica space.
The phase diagram for this class of copolymer systems is calculated.
1. Introduction
Numerous theoretical and experimental studies il-?] during the last 20 years have been devoted to the investigation of microphase separation in block copolymer systems (macromolecules
consisting of sequences or blocks of chemically distinct repeat units). The simplest block copolymer structure is the diblock copolymer AnBm containing two chemically different blocks
of A and B monomers. The behavior of this type of system is determined by trie degree
of polymerization N, trie architectural structure of trie macromolecule, and trie value of trie
Flory-Huggins parameter X (which characterizes trie strength of the short- range segregating
interaction between different sorts of monomers).
Above trie critical value of trie Flory-Huggins parameter, x~, these systems microphase- separated into domains structure il-?]. It is caused by trie interplay between two competing factors: 1) a short range segregation interaction of copolymer monomers and ii) a long-range
correlation due to chemical bonds between different blocks of monomers. Trie first stage of microphase separation the weak segregation limit, in which the local composition fluctuations
are smaller than the average composition, can be understood in terms of the Landau phase
transition theory. Within the framework of the Landau approach, trie homogeneous state of
© Les Editions de Physique 1995
the copolymer loses stability with respect to composition fluctuations of finite wave number qo 18-10]. The period of the microphase structure below the transition is proportional to the
Gaussian block size L
=
~~
+w
/Ù.
%
Consider the following interesting question: "How do the properties of the copolymer melt
change if we frustrate trie macromolecular chemical structure, or change trie properties of trie block copolymer constituents?"
Trie first type of frustration is realized in trie copolymer materais with ill-defined macro- molecular architecture. Trie simplest example is copolymers consisting of randomly distributed
A and B monomers iii,12], or randomly connected monomer blocks [13]. In trie region x > xc
trie A and B monomers tend to segregate into A and B-ricin domains. But in these systems,
there is no long-range correlation in trie monomer sequences along trie chain. Redistribution of these monomers results in a strong deformation of trie macromolecule as a whole with respect
to usual case of regular block copolymers and results in a significant loss of the conformational entropy of the copolymer chain. This additional entropy cost can be larger than trie ener-
getic penalty due to monomeric repulsion and it stabilizes trie homogeneous state against trie microphase separation [14,15].
However, a more interesting example not previously investigated theoretically is trie subject of this paper. We consider trie behavior of AB copolymer system near trie glass transition
point of trie A block. With increasing X Parameter, trie mobility of trie A monomers in trie A.ricin domain decreases. This results in a loss of mobility of the chain and a frustration of the domain structure on trie scales larger than trie correlation length. In this paper we present a
phenomenological model describing this phenomena and construct a phase diagram of such a
copolymer system.
Trie paper is organized as follows: Section 2 discusses trie model and denves a replica Hamiltonian for trie symmetric AB block copolymer system m trie vicinity of trie A block glass
transition. Using a variational approach in Sections 3 and 4 we calculate a free energy of trie system and show that trie homogeneous state is unstable with respect to trie off-diagonal
perturbations of density-density correlation function m rephca space. Section 5 presents trie phase diagram for trie system.
2. Model and Effective Hamiltonian
The behavior of a concentrated copolymer system m trie region of microphase separation is
described by trie effective Hamiltonian [2,4-6]:
H(14Î(q)1)
= Ho(14Î(q)1) ) / 4Î(q)4Î(-q)) + j là
lÉqz) 4Î(qz)~ (i)
where
Ho(ii~(q)1)
"
/ )9~~(q)i~(q)i~(-q) (2)
where il(q) is trie Fourier transformation of trie order-parameter describing trie local compo- sition fluctuation of monomers. In trie block copolymer system trie homogeneous state loses
stability with respect to trie fluctuations of finite wave number. Trie bare propagator in equa- tion (2) bas trie typical form for weak crystallization [8-10]:
g~~(q)
= g~~ (qo) + ((q( qo)~ + ° (((q( qo)~) (3)
The constants À, qo, g~~(qo) are related to trie structural characteristics of trie copolymer
macromolecule. Here we consider only trie case of trie symmetric AB block copolymer system for which trie third-order terni in trie Landau expansion (1) is equal to zero (see Refs. [2-7]).
Near trie glass transition point, trie A monomer mobility in trie A.domain decreases and results in non-thermodynamic composition fluctuations on lime scales larger than trie char- acteristic monomer relaxation lime Tm. On these lime scales trie A-part of copolymer chains will be frozen into an A.domain that can be interpreted as an effective cross link between trie
different chains. To describe ibis effect we suppose that interaction parameter x is not constant
throughout trie system volume, but depends on trie distribution of the random variable ((x) describing the A.monomer distribution in trie system. Trie random function x(((x)) cari be
determined by two first distribution moments;
lx(f(x)))av
= x (4a)
(x((l'~I))x((l'~2))lav lxl(l'~I)))av lxl(l'~2))lav
" ~f ~~~
ç
~~~ 14b)
where trie bracket )av denotes averaging over trie distribution of random variable ((x). ~
is a phenomenological pararneter, ( is trie scale characterizing trie spatial correlations of trie random variable ((x). When ( is smaller than trie scale qo of trie composition fluctuations
il(q) (qo( < 1), one can substitute a delta function à(x) for function f(x) in equation (4b).
Below, we examine how this type of disorder in the monomer-monomer interaction changes trie statistical properties of the system.
In accordance with the general theory of disordered systems [16,17] the physically observable quantifies (free energy and ils denvatives) are averaged over the quenched disorder probability
P(1((x)1):
lflav " £ P(ii(X)i)F(ii(X)1) (5)
where F((((x))) is trie free energy of trie system with given disorder distribution (((xi). To average trie free energy over quenched disorder trie replica approach is used [16,17]. One considers n copies of trie saine system, averages over all distribution of (((xi) first and then takes trie limit n ~ 0:
(F)av
= lim ~~~~ ~~ ~~
(6)
n-o n
where Fn is trie average n-rephca free energy and can be represented in terms of a functional
integral over all possible spatial composition fluctuations in o~~ replica ~oe(xa)
exP (-Fn)
=
/ exP - £ H (14Îa qa)1)) fl ô4Îa (qa) (7)
The summation on r-h-s- of equation (7) is taken over n replicas and we also average trie term connected with trie distribution of Flory-Huggms parameter:
exp / x(((x)) £ il$(x)dV
= exp ~ £ / il( (x)dV
+
~ ~j / l$(x)il)(x)dvl
~
oe
av
~
a
~ a,p
(8) Substituting equation (8) to trie r-h-s- of expression (7) we can rewrite Fn m trie following
form:
~~P (~Fn) " /
~XP (~Hn (fila(~n)i)) flôila(~a) (~)
~
where trie n-rephca effective Hamiltonian is
~~~~~~~~~ ~ Î Î ~° ~~~~~"~~~~"~ ~~(ÎÎ3 ~ ! Î ~~ jj
~°~~~~ ÎÎÎ3~
1/ à il
~Ya(~i)~Pa(~2)~P~(~3
~Pô1~4
i
(10)
and = 3K is a new vertex of trie one replica interaction, G[~(q)
= ((q( qo)~ + T is trie
bare propagator and T
= g~~(qo) X is trie reduced temperature.
3. Variational Principle
To calculate the free energy Fn we use a variational pnnciple based on trie second Legendre transformation [18]. In accordance with reference [21] within the framework of trie second Legendre the ii-rephca free energy (9) is
Fn = min Wn (ililalq)li, iGaô lqi>q2)1)> Ill) Wn = -jTr Ln (Gaô (qi, q2)) + Sn (ll4În(q)11 lGnô (qi, q211)
f / ~Gaa(q)Gi~(q) + / )Gi~q)4Înlq)4În(-q)j ~~~~
where trie minimum is to be sought with respect to functions ila(q) and trie renormalized Green function Gnô(qi,q2) are considered as mdependent variables at trie fixed parameters À, K. Trie quantity Sn ((ila(q))), (Gnô(qi, q2))) is trie so-called generating functional of all 2-irreducible
diagrams that can not be cut into two mdependent parts by removing any two lines between ver- tices À, ~. In trie one-loop approximation we can wnte the sum Sn (((il~(q))), (Ga~(qi, q2)))
in the followmg form (see iii]
Sn (ililalq))i, iGaôlqi> q2)1)
= (13)
( là ~ à j~ à
j~
a=1 8 ~ 4 ~ 4!
ùÎ8~j'~4~~~4~~2~~81~~'Î
In equation (13) trie solid lines denote trie renormalized Green function and symbols
G- x «Y-t1
are the average value of trie composition fluctuations ila(q) and trie vertices of one-rephca
and interreplica interaction respectively. One should note that the functional derivative of equation (11), with respect to renormabzed correlation function Gap (qi, q2), leads to the Dyson equation for effective Hamiltonian (10). Expression (12) for the free energy m combination
with trie one-loop approximation (13) for Sn (((il~(q))), (G~p(qi,q2))) coincides with trie diagram expression for trie free energy denved by Mezard and Parisi (MP) [19] using trie Feynman variational principle. However, within trie MP approximation trie replica symmetric
solution [20] for our model is stable, whereas corrections to trie former (shownbelow) result in
a broken replica symmetry solution [21].
We search for a minimum Wn with respect to trie tr1alfunctions:
~~~~~~ ~~~ ÎÎI~~ÎÎÎÎ
'
~~~~~
~Pa(x) =
j L
exP (iqJx) + C.C. (14b)
j
~
i
The summation in equation (14b) is implied over ail k vectors qj, with modulus (qj( = qo pertaining to trie first coordination sphere of the corresponding Bravais lattice. Substituting equation (14) into equations (11)-(13) and going to the limit n ~ 0 we arrive at the expression for the free energy
~ / ~~ (Î + ~)s~ (î + ~) sA~
~~~~~~~~ ~~~
2 ~ 2@ ~ 8r ~ 2@ ~ ~~~ ~ ~~~~~ ~~~~
here s
=
) and the factor Bk
is determined by the symmetry type of the inverse lattice
(for lamella type of microphase structure Bk = ~). The equilibrium values of r and A are 4
determined as the solutions of the extremalequations:
~~)~~~~
=
~~(~~~~
=0 (16)
that can be rewritten for disordered phase in trie form
and for ordered one as follows:
~
~~ÎÎ~~
~ (1 -~~~ 2~' ~~
(1 2~k) 2~ ~~~~
Note that solution (18) exists only when trie inequality À(1- 2Bk) 2~ > 0 holds.
It follows from trie dependence T(r) for an ordered phase that solutions of form (14) appear only below trie critical temperature
~
~~~ 2BkÀ ~~~
T < Tcrit " ~@~~~~~~ ~~~ (19)
~(Î ~~~~ ~~
One can see that trie value Tcrit tends to minus infinity when trie parameter À(1- 2Bk) 2~
tends to zero. In ibis case, trie region T > T~rjt extends over trie whole region -co < T < co, where homogeneous phase described by equation (17) is trie thermodynamic equilibrium state.
4. Instability of the Diagonal Correlation Function (Replica Symmetry Broken)
Let us consider trie stability ofthe diagonal correlation function go(q)ônpô(qi+q2) with respect to small off-diagonal perturbations Qnp(q). With this purpose we represent trie renormalized
two-replica correlation function G~p(qi,q2) in trie following form
Gap (qi, q2)
" (go (qi ônP + (1 ônfl)Qnp (qi )1à (qi + q21 (20j
where
~°~~~ "
((q( -Îoi
+ r
This perturbation Qnp(q) is an interreplica correlation function
Qa4(q)
= li~nlq)1~4(-q)) li~n(q)) l1~41-q)) (21)
that describes trie correlation between fluctuations of trie order parameter in different rephcas.
Substituting equation (20) into trie r-h.s. of equation (12) and expanding Tr Ln Gap (qi, q2))
m trie power of function Qn4(q) one can find Fn (Qap(q)) = ( L Il )
))~~(Î ~ / )Qap(q)) j
a#4 ~ 9° ~) ~
-( L / ) (90(q))~~ Qa4(q)Qp~(q)Q~n(q) (~~)
n,4,~
where trie first and trie last term in trie r-h-s- of equation (22) appear from expansion of trie Tr Ln (G~p(qi,q2)) and trie second term is from 6~~ diagram of trie expansion equation (13). Trie solution for trie two-rephca correlation function Qn4(q) can be found from extremal equation:
ôF» (Qn4(q))
~ ~ 123)
ôo~p(q)
that is useful to rewrite m trie following form
Qa4(q) = ~(90(q))~ / )Qn4(k)
+ L
~°~~](j~~~~ (24)
One can see that for trie case ~
= 0 trie equation (24) bas only zero solution. In other words, the diagonal correlation function in trie replica space (Eq. (14a)) is stable. For the case ~ # 0 the first approximation of the function Qap(q) is
Qap(q)
= ~(go(q))~ / ~oap(k) (25)
It is important to note that trie function Qap(q) is equal to trie square of trie diagonal cor-
relation function times some constant that depends on replica indexes. In accordance with
equation (25) a two-replica correlation function can be defined in trie followmg form
Qap(q)
= Qapf(q)
= Qap ~~
~~
(26)
where Qap is a n x n matrix. Substituting equation (26) into equation (24) and integrating
bath parts of ibis equation with respect to q, we get
Equation (27) bas a symmetric solution for trie matrix Q~p in trie region
~=l-$<0 (28)
that, for n = 0, bas trie form
Qap " ~~~ (29)
In order to find a solution with broken replica symmetry for matrix Qap we should include
higher order ternis into trie expansion of equation (22).
New we can obtain a criterion to approximate function Qap(q) with trie triai function given by equation (26). Substituting this trial function into r-h-s- of equation (24) we find that trie second term m r-h-s of equation (24) is insignificant at wave numbers (q( m qo, that give general
contribution to trie integrals, as long as pararneter ~~~ tz (A((s~)~/~ is small. One should note r
that relation (25) between two-replica correlation function Qap(q) and trie renormalized one-
replica correlation function is a general result for multi-replica effective Hamiltonians in which interaction between fluctuations of trie order parameter in two different rephcas is described
by a local fourth-order terni (see Eqs. (8),(10)).
After expanding Fn near trie point Qap
= 0 correct to trie fourth order, trie averaged free
energy is expressed as sum of two ternis
ifiav " llllll tif(Ù)iav + Fgl(Q)) (3°)
The first terni in r-h-s- of equation (30), (F(0))av is given by expression là) and trie second one,
Fgi(Q), is trie free energy connected vith trie non-zero order parameter Qap and is determined
by
Fgiio) = ii SI Il~Y Q~ j~~Y Q3 (( L Qip + Î~ lO Qi~oi~
"#P "#P>~
(~i)
~~~ ~j QOEpQp~Q~ôQôOE + ÎpQÎô) ~~ ~ QOEpAaAp +..
~
OE#~>ô#P
~ ~
OE#P where we introduce trie following designations:
~ sÎÉ Il (ÎÎ3 ~~°~~~~~~~~~~ ~
Î ÎÎ3~~~~~~j ~ ~
21~Î2~ ~~~~
"~ sÎÉ Î
(~Î3 ~~° ~~~~~~~~~~ ~~ ~~~~
~ Î ~~ff4j ~Î Î j~~)
s@ Î
~ x2r2 s@
©~
a J y à
"
~
a ~ i
Fig. l. The second-order diagrams in trie perturbation expansion of the free energy
on the number of interaction vertices result in
an mstability of the symmetric solution for two-rephca correlation function. The notation is the
same as in equation (13).
where function f(x) is image of function f(q) in coordinate space.
In trie r-h-s- of equation (31) trie terms of fourth order on trie power of Qa4 bave been denved from trie second-order diagrams presented in Figure 1 and result in replica symmetry breaking (RSB) for trie order parameter Qa4. One should note that the last term on r-h-s- of equation (31) is similar to that for the externat magnetic field in a spm glass [22].
Trie solution with Qap = 0 of equation (31) is unstable for A < 0. In this region we use trie Parisi solution with broken replica symmetry for matrix Qap. [16,17,19,22]. Pansi mtroduced
a function q(x) defined on trie interval and connected to Qa4 by
~ ~~~x)dx ii (n~
ii i Q~~ vk ~351
In terms of ibis new continual order parameter, q(z), trie free energy (16) is rewntten in trie
form
Fgi(Q) - ~fmax ll~ dz l'Î'~~(xi + ~i~~~(xi "i~~~(xi bJ3~(xi£~ v~~(Y)1
~"~
/~ xq~(x)) + h~/~ dxq(x)
4
o o
(36)
where
w5 =
~~~
~3
~~°~ (39)
The only difference between trie functional free energy (36) and that of a spin glass functional [16,17] is trie appearance of trie fifth term m r-h-s- of equation (36). One should note that
at non-zero externat field, h # 0, there are two solutions of trie extremal equation ~'~
ôoap
one with broken replica symmetry and another symmetric m trie rephca space. However, trie second solution bas a much higher free energy compared to trie first. Solving trie equation for
q(x) q(i1
qlo)
~ ~0 X,
Fig. 2. Trie function q(~) given by equation (33).
trie stationary point of trie functional Fgi (29), in accordance with reference [22], one finds
q(0), 0 < X < X0
q(x) = ~'°~z,
zo < z < xi (40)
9 W4
q(1), Xl < X <
see Figure 2. Trie values of xo, xi, q(0), q(1) are given by:
(3
~~~~
314~ ~~~
~~~~' ~~~~ ÎÎÎ ~ ~~ ÎÎ~~ ~~~~ ~
~Î ~~
(41)
~° ÎÎ~~~~' ~~
ÎÎ~~~~
Solution (40) with broken replica symmetry only exists in a weak externat field h < hc until zo < xi For higher external field h < h~ the replica symmetric solution is possible. Trie point h = h~, where
~~
'~
~Î~ ~~~
ÎÎÎ
~~~
' ~~~~
is a point of trie second order phase transition characterized by rephca symmetry breaking.
Substituting trie RSB solution (40) m r-h-s- of equation (29), one can derive trie expression for trie contribution to the free energy of trie system under consideration due to the non-zero
interreplica correlations:
'~ ~~ ~ÎÎÎ( ~ ÎÎ~ ~ ÎÎÎ ~~~~~~ ~~~~~~ ~~ ~~~~
The third power law dependence of the free energy (43) for effective temperature A at zero
externat field h shows that the phase transition between trie homogeneous state and glass-like
state with non-zero interreplica correlations is a third order phase transition. To zero~~-
order approximation we can neglect trie last term in trie r-h-s- of equation (43) because it is
important only for large external fields. Combining equations là), (29) and (43), we obtain
