Soft Materials, 6(2): 85–95, (2008)
Copyright # Taylor & Francis Group, LLC
ISSN 1539-445X print/1539-4468 online
DOI: 10.1080/15394450802077700

DYNAMICAL BRAZOVSKII EFFECT

S. Komura and N. Shimokawa

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Department of Chemistry, Graduate School of Science and Engineering,Tokyo Metropolitan University, Tokyo, JapanA
Based on Brazovskii’s theory for fluctuation-induced first-order transition, we propose adynamical model of ordered phases such as lamellar and hexagonal phases. Analysis of thesuggested nonlinear equations shows that the system is bistable when the temperature is smallenough. Close to the end point of the stable branch, the system can be transiently trapped in theintermediate ordered state.

Keywords Brazovskii effect, Block copolymers, Phase transition, Lamellar phase,
Hexagonal phase, Amplitude equation

INTRODUCTION
Block copolymers are quite useful in many applications such as surfactants, compatibilizers in polymer blends, or adhesives. It is known thatmelts of block copolymers undergo microphase separation transition to
produce various spatially ordered modulated phases (1). In 1980 Leibler
obtained the phase diagram of block copolymer melts in the weak segregation limit within a mean-field theory (2). For a symmetric diblock melt
for which the two blocks have the same length, the system is homogeneously disordered when temperature is higher than a critical value.
Only at the critical point of the phase diagram, Leibler’s mean-field
theory predicts a second-order phase transition to a lamellar phase. On
the other hand, asymmetric diblock melts undergo a first-order phase transition to a body-centered-cubic (BCC) phase by decreasing temperature. Byfurther decreasing temperature, the mean-field theory suggests that there
are first-order transitions from the BCC phase to a hexagonal phase andsubsequently to the lamellar phase. Such a phase behavior coincides with

Received 10 January 2008; Accepted 24 March 2008.
Address correspondence to S. Komura, Department of Chemistry, Graduate School of Scienceand Engineering, Tokyo Metropolitan University, Tokyo 192-0397, Japan. E-mail: komura@tmu.ac.jp85

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S. Komura and N. Shimokawa

that obtained by Matsen and Schick according to numerical solution ofself-consistent field equations without relying on any approximation (3).
However, the above mentioned phase behavior is modified whenfluctuation corrections to Leibler’s theory are taken into account. This fluctuation effect is called “Brazovskii effect” (4), which has been particularly
calculated for diblock copolymers by Fredrickson and Helfand (5). They
predicted that a symmetric diblock copolymer exhibits a first-order transition at a lower temperature than the second-order transition foundwithin the mean-field theory. Moreover, the Brazovskii effect leads towindows in composition through which it is possible to pass from thehomogeneous phase to each of the ordered phases by changing temperature. A similar phase diagram including the so-called “gyroid phase” wascalculated before (6, 7). In the presence of Brazovskii critical fluctuations,a direct phase transition between the disordered and gyroid phases would
be possible.
The onset of the ordering at finite values of wave vector occurs not onlyin block copolymers but also in other systems such as weakly anisotropic
antiferromagnets (4), liquid crystals near the nematic– smectic C transition(8), and fluids near the Rayleigh-Bénard instability (9, 10). In all thesesystems, fluctuation-driven first-order transitions are expected to takeplace. Many efforts have been made to experimentally verify the
Brazovskii effect in liquid crystal mixtures (11, 12), or in block copolymer
melts (13– 16).
Despite many investigations concerning equilibrium properties of
Brazovskii effect, there are few works that deal with its dynamical aspects
except those given in the references (17– 19). In the present article, we
argue the kinetics of ordered phases by considering a dynamical versionof Brazovskii effect within a coarse grained level. We propose nonlinear
amplitude equations for the lamellar and hexagonal phases to discuss
the time evolution of the respective ordered phases. We analyze the properties of the stationary solution of the amplitude equations, which aresolved numerically. We find that the system is bistable when the temperature is low enough. In the next section, we first summarize the equilibrium
properties of Brazovskii effect and present phase diagrams both for meanfield and Hartree approximations. In the third section, we present dynamical models for the lamellar and hexagonal phases. Analysis of themodels and results of numerical simulations are given there. The final
section concludes the article.
EQUILIBRIUM PHASE BEHAVIOR
We consider an order parameter f(r), which vanishes on average inthe homogeneous disordered phase, while it takes non-zero values in theordered phases [see later Eqs. (2) and (3)]. Our starting point is theDynamical Brazovskii Effect87

Downloaded By: [Komura, S.] At: 06:15 13 February 2011Landau-Brazovskii (LB) free energy functional, which can be written(in units of kBT) (2, 5, 20)

ð  4
j0
t
m
l
2
3
4
2
2
2
F½fðrÞ ¼ dr
fðr þ q0 ÞfðrÞg þ ½fðrÞ þ ½fðrÞ þ ½fðrÞ ; ð1Þ
2
2
3!
4!
where j0 is the correlation length, q0 is the critical wavenumber, t is the
reduced temperature, m and l are higher order vertex functions thatare taken as constants (local approximation) (5). With the above mentioned
free energy, the Fourier transform of the two point correlation functions
shows a pronounced peak at jqj ¼ q0. Hence the dominant fluctuationsin the system are plane waves with wave vectors of magnitude q0, or superpositions of such plane waves. The third order coefficient m measures theintrinsic asymmetry, and vanishes for symmetric systems. Without loss ofgenerality, one can assume that m is positive because the LB free energy
is invariant under the simultaneous transformation of f ! 2 f andm ! 2 m. The fourth order coefficient l is positive for the stability reason.
Mean-Field Approximation
We first discuss the mean-field theory for the formation of the lamellar
and hexagonal phases. For simplicity, we do not take into account the BCC
phase nor the gyroid phase in this article. In the weak-segregation limit,
the one-dimensional lamellar phase along the z-axis is expressed asfðrÞ ¼ 2a‘ cosðq0 zÞ;

ð2Þ

and the two-dimensional hexagonal phase in the y-z plane can be given by
pﬃﬃﬃ



 pﬃﬃﬃ 3y  z
3y  zfðrÞ ¼ 2ah cosðq0 zÞ þ cos q0;
ð3Þ
þ cos q022
where a‘ and ah are the lamellar and hexagonal amplitudes, respectively.
Inserting the above-mentioned forms of the order parameter in the LB
free energy (1) and taking the average over one period, we obtain (2, 21)lF‘ ¼ ta2‘ þ a4‘
4

ð4Þ

for the lamellar phase andFh ¼ 3ta2h þ 2ma3h þ

15l 4
a
4 h

ð5Þ

for the hexagonal phase. By the minimization of these free energies andcomparison of their respective values at the minimum, one can determine
the phase transition lines between the disordered–lamellar, disordered–hexagonal, and hexagonal– lamellar phases.

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S. Komura and N. Shimokawa

As shown in Fig. 1(a), the mean-field phase diagram can be
represented in the (m/l, t/l) plane. The disordered, hexagonal, lamellar,
and inverted hexagonal phases are denoted by “D,” “H,” “L,” and “IH,”
respectively. The phase diagram is symmetric with respect to m ¼ 0 as
discussed before. When t/l is large and temperature is high, we havean homogeneous disordered phase. For symmetric cases with m ¼ 0,
the system exhibits a second-order phase transition from thedisordered phase to the lamellar phase at t ¼ 0. Hence the point (m/l,t/l) ¼ (0, 0) is a critical point. For asymmetric cases with m = 0, both
the disordered– hexagonal and hexagonal– lamellar transitions are firstorder, and the amplitude changes discontinuously at the transition line.
By lowering temperature from the disordered phase, the hexagonal(or the inverted hexagonal) phase appears first at the line given byt/l ¼ (4/45)(m/l)2  0.089(m/l)2. As the temperature is further
lowered, then there is again a first-order phase transition from hexagonal
to lamellar. The correspondingtransition line is determined by theprelation t/l ¼ 2[(7 þ 3 6)/5](m/l)2  22.87(m/l)2.

Hartree Approximation
Brazovskii considered the role of the critical fluctuations on the phasetransition that occurs in systems described by LB free energy (4). This canbe done by the standard diagrammatic formalism, but attention must be
paid to those integrals that diverge as t vanishes. These divergences
occur near the spherical shell jqj ¼ q0. Brazovskii showed the way to

FIGURE 1 (a) Mean-field phase diagram of Landau-Brazovskii free energy functional (see Eq. (1)) as afunction of m/l and t/l. There are four different phases: the disordered phase (D), the hexagonalphase (H), the lamellar phase (L), and the inverted hexagonal phase (IH). These phases are separated
by first-order transition lines. The filled circle at (m/l, t/l) ¼ (0, 0) indicates the critical point at which
the transition is second-order. (b) Hartree phase diagram of Landau-Brazovskii free energy functional
as a function of m̃ and t̃ defined in the text. The four phases are separated by first-order transition lines.
The filled triangles at (m̃, t̃) ¼ (+0.564, 22.03) indicate the triple points.

Dynamical Brazovskii Effect89

handle these divergences using a self-consistent Hartree approximation
for the renormalized propagator. In this approximation, only a particularclass of diagrams are summed in the perturbation series. The final result of
Brazovskii’s theory can be represented by thermodynamic potentials forthe lamellar and hexagonal phases with respect to the disordered phaseas given by
1 2
l 4
1=2
ðr  r2d Þ þ aðr1=2‘  rd Þ  a‘ ;
2l ‘
4

ð6Þ

1 2
3l 4
1=2
3
a ;
ðrh  r2d Þ þ aðr1=2
h  rd Þ þ 2mah 
2l
4 h

ð7Þ

F‘ ða‘ Þ ¼
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Fh ðah Þ ¼

respectively (4, 5), with a constant a ¼ q0/(8pj20) (20). In the above
mentioned equations, the renormalized inverse susceptibilities of thedisordered (rd), lamellar (r‘), and hexagonal (rh) phases are determinedby the following self-consistent equations (4, 5):alrd ¼ t þ 1=2 ;
rd

ð8Þ

al
r‘ ¼ t þ 1=2 þ la2‘ ;
r‘

ð9Þ

al
rh ¼ t þ 1=2 þ 3la2h :
rh

ð10Þ

For our later purpose, we rescale the quantities as F̃i ; l(al)24/3Fi,r̃i ; (al)22/3ri, ãi ; l1/2(al)21/3ai (“i” stands for either “d,” “‘,” or “h”),
and the Landau coefficients as t̃ ; (al)22/3t, m̃ ; l21/2(al)21/3m.
As in the case of the mean-field theory, the phase diagram in the presence of critical fluctuations can be obtained by minimizing the above
mentioned thermodynamic potentials and comparing their values at theminimum. The resulting phase diagram is presented in Fig. 1(b) on the(m̃,t̃) plane. The same phase diagram was previously presented in Ref.
(21). Compared to the mean-field phase diagram shown in Fig. 1(a),there are two important differences. First the horizontal disorder– lamellar
phase transition line is shifted to negative values of the temperature parameter at t̃  22.03. Moreover, the transition is first-order even for symmetric cases with m̃ ¼ 0. Hence a fluctuation-induced first-order transition
takes place. Secondly, there exists a finite region of m̃ for which the hexagonal phase can never be stable. In fact the hexagonal phase appears only
when the asymmetry is larger than jm̃j  0.564. Notice that the three
first-order transition lines meet at the triple point denoted by the filled

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S. Komura and N. Shimokawa

triangles. For large values of m̃, the transition lines approach their meanfield values as given in Fig. 1(a).
DYNAMICS OF ORDERED PHASES
Lamellar Phase
In this section, we discuss the dynamics of the lamellar and hexagonalphases. Phenomenologically, we assume that the time evolution of themodulation amplitude is driven by the external force. For the lamellar
phase, the proposed coarse grained amplitude equation has a relaxation
form:

Downloaded By: [Komura, S.] At: 06:15 13 February 2011d~a‘ ðtÞ
¼ h~ ‘ ;dt

ð11Þ

where the kinetic coefficient is taken to be unity, for simplicity, and h̃‘ is theexternal field defined by the derivative of the lamellar potential F̃‘ withrespect to the lamellar amplitude ã‘:
~‘
1 dF
1
h~ ‘ ¼¼ r~‘ a~ ‘  a~ 3‘ :2 da~ ‘2

ð12ÞThis is the Brazovskii’s equation of state for the lamellar phase. In theabove, the rescaled inverse susceptibility r̃‘ satisfies
1r~ ‘ ¼ t~ þ 1=2 þ a~ 2‘
r~‘

ð13Þ

according to Eq. (9). Eqs. (11), (12), and (13), describe the nonlinear time
evolution of the lamellar amplitude ã‘(t). The only control parameter inthe equation is t̃, corresponding to temperature.
We first argue the steady state condition (nullcline) of the abovementioned equation. This is equivalent to consider the zero field condition,that is, h̃‘ ¼ 0. Then we have r̃‘ ¼ ã2‘ /2 from Eq. (12). Inserting this relationin Eq. (13), we obtain
pﬃﬃﬃ
1 2
2
t~ ¼  a~ ‘ :
ð14Þ2
j~a‘ j
The above-mentioned equation is plotted in Fig. 2(a) in the (t̃, ã‘) plane.
We see that the system is bistable when t̃ , 21.89 (denoted by the filled
circles). Notice that this value is larger than the equilibrium disorder –lamellar transition temperature at t̃ ¼ 22.03 (see Fig. 1(a)). In
Fig. 2(a), the stable branches are drawn by the solid lines. The dashedlines correspond to the threshold values of the initial amplitude ã‘0.
The stationary state is the homogeneous disordered phase with ã‘ ¼ 0

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Dynamical Brazovskii Effect

91

FIGURE 2 (a) Nullclines of the dynamic equation for the lamellar phase (see Eq. (11)) as a function of t̃
and ã‘. The solid lines correspond to the stable branches, whereas the dashed lines give the thresholdvalues of the initial amplitudes ã‘0.The filled circles at (t̃, ã‘) ¼ (21.89, +1.12) indicate the end pointsof the stable branches. Six different filled squares at t̃ ¼ 23 are the initial values of ã‘ used in the calculations of Fig. 3. (b) Nullclines of the dynamic equation for the hexagonal phase (see Eq. (15)) as afunction of t̃ and ãh. The asymmetry parameter is set to be m̃ ¼ 0.5. The meaning of the solid anddashed lines is the same as in (a). The filled circles at (t̃, ãh) ¼ (22.01, 20.515) and (27.06, 1.23) indicate the end points of the stable branches. Six different filled squares at t̃ ¼ 210 are the initial values ofãh used in the calculations of Fig. 5.

when jã‘0j is smaller than these threshold values. In the opposite case, on
the other hand, the system relaxes to the lamellar phase with ã‘ = 0.
When t̃ . 21.89, on the other hand, the disordered phase is always
the stationary state.
In Fig. 3, we plot the time evolutions of the lamellar amplitude ã‘ forsix different initial values of ã‘0 while the temperature parameter is fixed
to t̃ ¼ 23. The set of equations are solved numerically. Because different
initial values of ã‘0 are realized at different temperatures, such a simulation corresponds to a temperature jump experiment in the lamellar
phase. With this choice of parameter, the stationary value of the lamellar
amplitude is jã‘j ¼ 2.17 (see the cross points of the solid and dotted lines
in Fig. 2(a)). In all the cases, the lamellar amplitude changes monotonically and approaches to its stationary value. Compared to the relaxation
to the lamellar phase, it takes much longer time to decay to the disordered phase with ã‘ ¼ 0. (Note that the time scale is different in
Fig. 3(b).) In Fig. 4(a), we plot the time evolutions of the lamellar amplitude starting from the same initial value (ã‘0 ¼ 2) but with different
temperature parameter values t̃. As mentioned earlier, the disorderedphase is the stationary state when t̃ . 21.89 and the amplitude ã‘
decays to zero. For the values close to t̃ ¼ 21.89, however, the amplitude decreases in two steps. This means that the system is transiently
trapped in the intermediate lamellar state. The trapping time becomes
longer as the temperature is approached to t̃ ¼ 21.89. For largeDownloaded By: [Komura, S.] At: 06:15 13 February 2011

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S. Komura and N. Shimokawa

FIGURE 3 Time evolution of the lamellar amplitude ã‘ for six different initial values ã‘0 shown in
Fig. 2(a) when the temperature parameter is fixed to t̃ ¼ 23. The initial amplitudes are (a) ã‘0 ¼ 2.5
and 1.5, (b) 0.3 and 20.3, (c) 21.5 and 22.5. The steady state values are (a) ã‘ ¼ 2.17, (b) 0, (c) 22.17.

FIGURE 4 (a) Time evolution of the lamellar amplitude ã‘ for different temperature parameters. The
initial value is fixed to ã‘0 ¼ 2. The temperature parameters are t̃ ¼ 21.89, 21.88, 21.87, 21.86,
21.85, 21.8, 21.7 from top to bottom. (b) Time evolution of the hexagonal amplitude ãh for different
temperature parameters when m̃ ¼ 0.5. The initial value is fixed to ãh0 ¼ 21. The temperatureparameters are t̃ ¼ 22.01, 22.00, 21.99, 21.98, 21.97, 21.90, 21.80 from bottom to top.

Dynamical Brazovskii Effect93

enough t̃ such as t̃ ¼ 21.7, the amplitude ã‘ decays to zero almost
monotonically.
Hexagonal Phase
Next we discuss the dynamics of the hexagonal phase. Similar tothe case of the lamellar phase, the amplitude equation for ãh can be
written as
d~ah ðtÞ
¼ h~ h ;dt

ð15Þ

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where h̃h is the external field given by
~h
1 dF
1
¼ r~h a~ h þ m~ a~ 2h  a~ 3h :h~ h ¼
6 da~ h2ð16ÞThis is again the Brazovskii’s equation of state for the hexagonal phase. In
Eq. (16) the rescaled inverse susceptibility r̃h satisfies the relation
1r~ h ¼ t~ þ 1=2 þ 3~a2h
r~h

ð17Þ

from Eq. (10). Eqs. (15), (16), and (17) describe the time evolution of thehexagonal amplitude ãh(t). The control parameters are now t̃ and m̃.
The stationary state condition of ãh(t) is again the zero field conditionh̃h ¼ 0, or equivalently, r̃h ¼ ã2h/2 2 m̃ãh. By substituting this relation into
Eq. (17), we obtain
pﬃﬃﬃ
2
5 2
t~ ¼  a~ h  m~ a~ h  2;
ð18Þ
2
ð~ah  2m~ a~ h Þ1=2
when ã2h 2 2m̃ãh . 0. The above-mentioned relation is plotted inFig. 2(b) for m̃ ¼ 0.5. When m̃ = 0, the two nullclines become asymmetric. The meaning of the solid and dashed lines is the same as inFig. 2(a). The two end points of the stable branches are located at(t̃, ãh) ¼ (22.01, 20.515) and (27.06, 1.23). When the absolute valueof the initial amplitude ãh0 is smaller than the dashed lines, thesystem decays to the homogeneous disordered phase with ãh ¼ 0.
The two dashed lines asymptotically approach ãh ¼ 0 and 1 for smallenough t̃.
In Fig. 5, we plot the time evolutions of the hexagonal amplitude ãh forsix different initial values of ãh0 when t̃ ¼ 210 and m̃ ¼ 0.5. With theseparameters, the stationary values of the hexagonal amplitude areãh ¼ 1.78 and 22.05. As in the case of the lamellar phase, the relaxation
time for the disordered phase is longer than that for the hexagonal

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FIGURE 5 Time evolution of the hexagonal amplitude ãh for six different initial values ãh0 shown in
Fig. 2(b) when the temperature and the asymmetry parameters are fixed to t̃ ¼ 210 and m̃ ¼ 0.5,
respectively. The initial amplitudes are (a) ãh0 ¼ 2.2 and 1.2, (b) 0.5 and 20.5, (c) 21.2 and 22.2.
The steady state values are (a) ãh ¼ 1.78, (b) 0 and 22.05, (c) 22.05.

phase. In Fig. 4(b), we show the time evolutions of the hexagonal amplitude starting from the same initial value (ãh0 ¼ 21) but with different
values of t̃. For the values close to t̃ ¼ 22.01, the relaxation occurs in
two steps so that the system is transiently trapped in the intermediate hexagonal state.
CONCLUSION
In this article we have discussed the dynamics of ordered phases byextending Brazovskii’s theory for fluctuation-induced first-order transition. We have proposed self-consistent nonlinear equations both forthe lamellar and hexagonal phases. Analysis of the stationary solutions of
the equations shows that the system is bistable for small enough temperature parameter. This result has been confirmed by performing numericalsimulation of the time evolution equations. Furthermore, we found thatthe system can be transiently trapped in the intermediate ordered stateDynamical Brazovskii Effect95(either lamellar or hexagonal) close to the end point of the stable branch.
These phenomena are expected to be observed in the future experiments.
ACKNOWLEDGMENTS
This work is supported by KAKENHI (Grant-in-Aid for Scientific
Research) on Priority Areas “Soft Matter Physics” and Grant No.
18540410 from the Ministry of Education, Culture, Sports, Science, andTechnology of Japan.

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