IOP PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 20 (2008) 155107 (10pp)

doi:10.1088/0953-8984/20/15/155107

The dynamics of order–orderphase separation
K Yamada and S Komura
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan
University, Tokyo 192-0397, Japan
E-mail: komura@tmu.ac.jp

Received 29 December 2007, in final form 3 March 2008
Published 25 March 2008
Online at stacks.iop.org/JPhysCM/20/155107
Abstract
In multi-component soft matter, two microphase separated ordered phases such as lamellar andhexagonal phases coexist in equilibrium. We call such a macroscopic phase separation‘order–order phase separation’. Based on a coarse grained single order parameter
Ginzburg–Landau free energy, we investigate the dynamics of order–order phase separation.
For this purpose, the equilibrium phase diagram is determined in three dimensions, including
BCC and gyroid phases. Focusing on the phase separation between the lamellar and hexagonalphases, we study its dynamics using computer simulation in both two and three dimensions.
We show that the microphase separation takes place in the early stage and the macrophaseseparation follows through the breaking and reconnection of microdomains. The observed
epitaxial relation between the lamellar and hexagonal phases is compared with an analyticalestimate of the interfacial energies of different mutual directions.
(Some figures in this article are in colour only in the electronic version)ordered phase to another ordered phase are considered.) For
example, a coexistence between the stripe (two-dimensional
lamellar) and hexagonal phases was theoretically predicted
for insoluble Langmuir monolayers by Andelman et al [2].
In the experiment, it was found that the stripe (labyrinth)and hexagonal phases coexist in binary mixed monomolecular
films containing cholesterol and phospholipid, and that it ispossible to observe the phase separation between these twomodulated phases [3]. The order–order phase separation
between the stripe and hexagonal phases is also anticipated
for two-component lipid bilayer membranes in which thecomposition and the membrane curvature are coupled to eachother [4]. In binary mixtures of non-ionic surfactant andwater, it is known that the lamellar phase coexists with othermesophases such as the hexagonal or gyroid phases [5, 6].
In different systems, Matsen calculated the binary phase
diagrams of AB diblock copolymer and A homopolymer
having similar degrees of polymerization [7, 8]. In his work,
various phase coexistences among the ordered mesophases
have been determined using self-consistent field theory. A
more systematic study by changing the ratio of the chain
lengths was performed by Janert and Schick [9, 10]. Moreover,order–order phase separation occurs also for ternary blends of
A and B homopolymers plus slightly asymmetric AB diblock

1. Introduction
Phase separation plays an essential role in soft matter, because
the resulting interfaces are responsible for the internal mesoscopic structures. The existence of such meso-scale structures
is intimately related to non-linear and non-equilibrium behaviors of the materials. In contrast to mixtures of simple liquids, which typically exhibit macrophase separation for certain
conditions, systems containing amphiphilic molecules such as
block copolymers or surfactants undergo microphase separation at low temperatures. Typical microphase separated structures appear as ordered structures such as lamellar, hexagonal,
body-centered-cubic (BCC), and gyroid phases, in which oneof the components is periodically arranged in space. In particular, the equilibrium structures in diblock copolymer melts
have been extensively studied in the past decades both experimentally and theoretically [1].
In multi-component systems containing amphiphilicmolecules, two different ordered phases can generally coexist
in thermodynamic equilibrium. We refer to such a macroscopic
phase separation and the resulting coexistence between the twoordered phases as ‘order–order phase separation’. (One should
distinguish ‘order–order phase separation’ from ‘order–orderphase transition’. In the latter case, phase transitions from one
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J. Phys.: Condens. Matter 20 (2008) 155107

K Yamada and S Komura

φ(r) = ψ(r) − ψc , where ψc is the composition at thecritical temperature Tc . In the above, τ = (T − Tc )/Tc is the
reduced temperature, and the coefficients of the other terms
are taken to be constant without loss of generality. Spatial
modulations are preferred due to the negative coefficient of
the gradient square term, while the Laplacian square term
plays the role of a homogenizing interaction. The√ modulation
occurs with a dominant wavevector k ∗ = 1/ 2 due to thecompetition between these two gradient terms. Such a freeenergy functional has been used to describe various systems
such as diblock copolymers [1], Langmuir films [2], magnetic
layers [19] and microemulsions [20]. It should be noticed,
however, that the free energy (1) is very generic, and it does
not pertain to any specific system.
With the use of equation (1), the phase diagram in two dimensions showing the relative stability of the lamellar (stripe)and hexagonal phase was obtained previously [2, 4, 17]. Here
we obtain the phase diagram in three dimensions including the
BCC and gyroid phases in addition to these classical phases.
For this purpose, we employ a two-mode expansion method
and compare between the free energies of different orderedphases [21]. We employ a mean-field approximation, and critical fluctuation corrections are not taken into account [22, 23].
In order to express all the ordered structures, we expand
φ(r) ascopolymer [11]. Recently, Mareau et al investigated theinterface between the perforated layer phase and gyroid phaseby 3D transmission electron microscopy in a blend of AB
diblock copolymer and homopolymer [12]. In non-equilibrium
situations, some groups have identified the coexisting states
between distinct structures for block copolymer melts under
external fields [13–15].
The previous studies on order–order phase separation havemainly focused on the equilibrium properties and less attention
has been paid to its dynamical aspect. In this paper, we
discuss the kinetics of order–order phase separation. We shallinvestigate how the two-phase coexistence is macroscopicallyachieved starting from a uniformly disordered state. In otherwords, our primary interest is to clarify the mechanism of
a double phase separation, i.e. the competition between themacrophase and microphase separations. A somewhat similar
question was raised before for copolymer and homopolymer
mixtures by Ohta and Ito, who studied the phase separation
dynamics between the lamellar and disordered phases [16].
For this purpose, we employ a modified Ginzburg–Landau
free energy, which yields several modulated phases and theircoexistences. With the use of a two-mode expansion method,
we first construct a mean-field equilibrium phase diagramin three dimensions including the BCC and gyroid phases.
Based on the obtained phase diagram, we perform computer
simulations of the proposed dynamical equation. In the present
study, we mainly focus on the order–order phase separation
between the lamellar and hexagonal phases in two dimensions.
We also study analytically the equilibrium interfacial profile
between the two ordered phases. Attention is paid to themutual orientation between the lamellar and hexagonal phases
in two dimensions. The result will be compared with that fromour computer simulation as well as the previous studies oninterfaces of modulated phases [17].
This paper is organized as follows. In the next section,
we first introduce a single order parameter Ginzburg–Landau
free energy functional which is used throughout this paper.
Based on this free energy, we present the equilibrium phase
diagram obtained by a two-mode expansion method in three
dimensions. In section 3, we consider a dynamical model ofthe order–order phase separation, and perform its computer
simulation both in two and three dimensions. Various
quantities characterizing the transient structures are calculated
in order to clarify typical features of the order–order phaseseparation. In section 4, we present an analytical calculationof the interface between the lamellar and hexagonal phases fordifferent mutual orientations. Finally, we conclude in section 5.

φ (r) = φ̄ + 2

12


6



a j cos q j · r + 2
bk cos (pk · r) , (2)

j =1

k=1

where φ̄ is the average composition and a j and bk are real
amplitudes. The fundamental reciprocal vectors q j and pk aredefined by
q1 = C Q (2, −1, 1) ,q2 = C Q (−2, 1, 1) ,q3 = C Q (−2, −1, 1) ,q4 = C Q (2, 1, 1) ,q5 = C Q (−1, −2, 1) ,q6 = C Q (1, −2, 1) ,q7 = C Q (−1, 2, 1) ,q8 = C Q (1, 2, 1) ,q9 = C Q (1, −1, −2) ,q10 = C Q (1, 1, −2) ,q11 = C Q (−1, 1, −2) ,(3)

q12 = C Q (−1, −1, −2) ,p1 = C P (2, 2, 0) ,p2 = C P (2, −2, 0) ,p3 = C P (0, 2, 2) ,p4 = C P (0, −2, 2) ,p5 = C P (2, 0, 2) ,p6 = C P (−2, 0, 2) ,√√with C Q = Q/ 6 and C P = P/(2 2), where Q and P are
the absolute values of q j and pk , respectively.
Within this expansion, the lamellar phase is given when
a1 = AL and all the other amplitudes vanish, whereas thehexagonal phase is given when a3 = −a8 = a9 = AH andall the other amplitudes are zero. On the other hand, the BCC
structure is obtained when a j = 0( j = 1, . . . , 12) and bk =BS (k = 1, . . . , 6). Finally, the gyroid phase is constructedby putting |a j | = AG ( j = 1, . . . , 12) and |bk | = BG (k =1, . . . , 6), together with the following condition between theabsolute values of the reciprocal vectors [21]:2. Model and phase diagram
One of the simplest free energies that exhibits various
modulated phases is given by 2 2
τ 2 1 42
F {φ(r)} = dr 2 ∇ φ − 2 (∇φ) + φ + φ ,2
4(1)
which was used in [18]. This mean-field free energy is a
modified Ginzburg–Landau expansion in the order parameterQ 2 = 34 P 2 .
2(4)J. Phys.: Condens. Matter 20 (2008) 155107

K Yamada and S Komura

Figure 1. Mean-field phase diagram of the free energy (1) in the(φ̄, τ ) plane, where φ̄ is the average composition and τ is the
reduced temperature. The red (L), yellow (G), blue (H) and green (S)regions correspond to the lamellar, gyroid, hexagonal and BCC
phases, respectively. The region of the disordered phase is denotedby ‘D’. All the lines are first-order transition lines and the criticalpoint is located at (0, 1).

Substituting equation (2) into equation (1) and ignoring
the higher order harmonics, we can rewrite the free energy interms of a j , bk , Q and P . We leave the full expression to theappendix since it is lengthy. The free energies of the lamellar( FL ), hexagonal ( FH ), BCC ( FS ), and gyroid phases ( FG ) are
obtained as

FL = FD + 4 Q 4 AL 2 − 4 Q 2 AL 2 + τ AL 2 + 3φ̄ 2 AL 2
3
+ AL 4 ,2
FH = FD + 12 Q 4 AH 2 − 12 Q 2 AH 2 + 3τ AH 2 + 9φ̄ 2 AH 2
45 4
+ 12φ̄ AH 3 +AH ,2
FS = FD + 24 P 4 BS 2 − 24 P 2 BS 2 + 6τ BS 2 + 18φ̄ 2 BS 2
+ 48φ̄ BS + 135 BS ,3

4

Figure 2. Mean-field phase diagram in the presence of thecoexistence regions between the different ordered phases. The red(L), yellow (G), blue (H) and green (S) regions correspond to thelamellar, gyroid, hexagonal and BCC phases, respectively. The white
regions are the coexisting regions between the two neighboring
phases located on the left- and right-hand sides. (b) The
magnification of (a) close to the critical point.

(5)

(6)
between the lamellar and hexagonal phases, and it ends at(φ̄, τ ) ≈ (±0.10, 0.79). This mean-field phase diagram isvery similar to that of a diblock copolymer melt obtainedby Ginzburg–Landau type theory [21] or self-consistent field
theory [24].
Since we are interested in the order–order phaseseparation, we next calculate the coexistence regions between
the different ordered phases using a common tangent
construction and the result is presented in figures 2(a) and (b)(the latter is a magnification of the former close to thecritical point). The white regions are the coexisting regions
between the two neighboring phases located on the left- andright-hand sides. The lamellar–hexagonal coexistence ends
at τ ≈ 0.67, and it starts again at τ ≈ −0.60 and ends
at τ ≈ −1.56. The lamellar–gyroid and gyroid–hexagonalcoexistences occurs for −0.60  τ  0.67. The hexagonal–
BCC coexistence ends at τ ≈ 0.38. Although a phase diagramwith coexistence regions has been obtained before in 2D
systems in the absence of the BCC and gyroid phases [4, 17],the corresponding phase diagram in 3D has not yet been
calculated. It should be noted, however, that the binodal
lines between the lamellar and hexagonal phases coincide bothin 2D and 3D systems. We shall further investigate thislamellar–hexagonal coexistence both by computer simulation
and analytical treatment.

(7)

FG = FD + 48 Q AG + 24 P BG − 48 Q AG − 24 P BG 2
4

2

2

2

2

2

2

+ 12τ AG 2 + 6τ BG 2 + 36φ̄ 2 AG 2 + 18φ̄ 2 BG 2
+ 306 AG 4 + 216 AG 2 BG 2 + 135 BG 4 − 48φ̄ AG 3
+ 48φ̄ BG 3 − 72φ̄ AG 2 BG ,(8)respectively, whereFD =τ 2 1 4φ̄ + φ̄2
4(9)is the free energy of the disordered phase.
First, we compare the above free energies and look forthe structure which has the lowest free energy. In otherwords, we determined the phase boundaries in figure 1 fromthe crossings of the free energies. Hence we have ruled out
the possibility of any phase coexistence between the orderedstructures. In figure 1, the red (L), yellow (G), blue (H) andgreen (S) regions represent the lamellar, gyroid, hexagonal
and BCC phases, respectively. The region of the disorderedphase is denoted by ‘D’. All the lines are first-order transitionlines. The critical point is located at (φ̄, τ ) = (0, 1), at
which a second-order phase transition between the disordered
and lamellar phases takes place. The gyroid phase appears3

J. Phys.: Condens. Matter 20 (2008) 155107

K Yamada and S Komura

Figure 3. Time evolution of phase separation pattern for φ̄ = 0.1 and τ = 0.8 in two dimensions. (a) t = 0, (b) t = 150, (c) t = 500,(d) t = 5000.

Figure 4. Time evolution of phase separation pattern for φ̄ = 0.12 and τ = 0.7 in two dimensions. (a) t = 0, (b) t = 70, (c) t = 500,(d) t = 5000.

and (0.12, 0.7), respectively. Notice that these points arelocated inside the coexistence region between the lamellar andhexagonal phases in figure 2(b). In the early stage, many
circular spots start to appear simultaneously, as shown infigure 3(b) at t = 150. Some of the spots are separated
while the others are connected to each other to form stripes.
Then we see in figure 3(c) that the breaking and reconnectionof the stripes take place so that large domains are graduallyformed, although it is still hard to identify clear domains.
Finally, we obtain the coexistence between the lamellar andhexagonal phases as shown in figure 3(d). Because this patternpersists until t = 20 000, we believe that the system hasreached almost the equilibrium (or at least metastable) state. A
qualitatively similar time evolution can be observed in figure 4
with a different choice of parameters. Hence we expectthat the kinetics of order–order phase separation between thelamellar and hexagonal phases is rather universal, at least in
two dimensions.
The above sequence of patterns tells us that the order–order phase separation initially starts with a microphaseseparation, and a macrophase separation follows later throughthe rearrangement of microdomains. To see this behavior morequantitatively, we present in figure 5 the Fourier transformed
patterns of figure 3 in the two-dimensional reciprocal space.
As seen in figure 5(b), a circular ring pattern is seen in
the early stage at t = 150. This means that the structureis almost isotropic on average and is characterized by thetypical wavelength k ∗ . Hence the microphase separation
dominates in this time region. Then the six spots (Bragg peaks)
become pronounced in the course of time as in figure 5(c),indicating that the structure becomes anisotropic with sixfold
symmetry. The Fourier pattern of the final state is given by
figure 5(d), which is very similar to that of the hexagonal
structure. However, it should be noted that the two peaks

3. Dynamics
3.1. Dynamical model and simulation method
In this section, we discuss the dynamics of order–order phaseseparation by means of numerical simulations. For the time
evolution of the relative composition φ , which is a conserved
quantity, we consider the following equation:
∂φ
δF
= ∇2
∂t
δφ
 4
2
= ∇ 4∇ φ + 4∇ 2 φ + τ φ + φ 3 .
(10)
For simplicity, the kinetic coefficient is taken to be unity. In thepresent dynamical model, we neglect both the hydrodynamic
effect and thermal fluctuations.
In the numerical simulation, we have used the implicit
method with the FFTW algorithm.
Compared to theEuler’s method, this algorithm is useful in solving differential
equations with higher order derivatives such as equation (10).
We have employed the periodic boundary condition and thesimulation starts from a disordered state with a small random
noise around the average composition φ̄ . Hereafter we focus
on the coexistence region between the lamellar and hexagonalphases (see figure 2(b)). We performed numerical simulations
both in two and three dimensions, although the latter result
is still preliminary. In 2D simulations, the mesh number is
256 × 256, the time increment is t = 0.01, and the space
increment is x = 0.59. In 3D simulation, on the other
hand, the mesh number is 64 × 64 × 64, the time increment
is t = 0.01, and the space increment is x = 0.65.
3.2. 2D simulation
In figures 3 and 4 we show typical time evolutions of theorder parameter φ in two dimensions when (φ̄, τ ) = (0.1, 0.8)
4

J. Phys.: Condens. Matter 20 (2008) 155107

K Yamada and S Komura

Figure 5. Two-dimensional Fourier patterns of figure 3 in the reciprocal space. (a) t = 0, (b) t = 150, (c) t = 500, (d) t = 5000.

explained above. In the late stage, the lamellar peak intensity
is twice as large as that of the hexagonal phase. This ratio isdependent on the average composition φ̄ . According to thelever rule, the lamellar peak intensity will increase as φ̄ isdecreased in the coexistence region. Figure 6(b) is the time
evolution of the free energy density. It decreases monotonicallyas a function of time to reach the final structure.
In order to further quantify the evolution of the phaseseparated patterns, we calculate the so called Minkowski
functionals. These quantities are used in digital picture
analysis for the characterization of black-and-white discrete
images. For the details of Minkowski functionals, see [28–30].
In the patterns of figure 3, we first reset the gray value ofeach lattice to either white or black depending on whether
the original value is larger or lower than the thresholdvalue φ = 0, respectively. (The result is insensitive tothe choice of the threshold value as long as φ is close to
zero.) Then three quantities are calculated: (i) relative whitearea v , (ii) length of the boundary line s between black
and white regions, and (iii) Euler characteristic χ , whichmeasures the connectivity of the black and white regions [28].
These quantities in the simulation of figure 3 are plotted infigure 7 as a function of time. The relative white area v
increases rapidly in the early stage and saturates after t ≈800, as seen in figures 7(a) and (b). The length of theboundary line s takes a maximum value at around t ≈ 120,
when many circular domains are formed. In figures 7(e)and (f), the absolute value of the Euler characteristic χ
takes large values in the early stage, since many circulardomains appear. Then it decreases because they connect
to form long stripes. These time evolutions of Minkowski
functionals are consistent with the analysis of the Fourier
patterns.

Figure 6. (a) Time evolution of the Bragg peak intensities forφ̄ = 0.1 and τ = 0.8, as shown in figure 5. The vertical axis is thepeak intensity I normalized by its value at t = 20 000, denoted as I0 .
The solid line corresponds to the peak of the lamellar pattern while
the dashed line is that of the hexagonal pattern. (b) Time evolution of
the free energy density for the same parameter values.

corresponding to the lamellar pattern (the upper and lower
spots) have stronger intensities than the other four peaks. Thismeans that there is an epitaxial relation between the lamellar
and hexagonal phases in the macroscopically phase separated
system. A similar epitaxial relation has been identified in
the kinetic pathways of order–order transitions in diblock
copolymer melt [25–27].
In figure 6(a), we plot the time evolution of the Bragg
peak intensities I in figure 5. We follow the peak intensityof the upper spot corresponding to the lamellar structure (solidline), and that of the other four peaks reflecting the hexagonal
structure (dashed line). Here the respective peak intensities
are normalized by the lamellar peak value I0 at t = 20 000.
Both intensities increase almost in the same manner up tot ≈ 100 and then they grow separately. Some cusps in the
curves reflect breaking and reconnection of microdomains as

3.3. 3D simulation
Based on the phase diagram of figure 2(b), we have also
carried out the numerical simulation in three dimensions. In
figure 8, we show the time evolution of the 3D pattern for(φ̄, τ ) = (0.1, 0.8). Notice that this choice of the parametersis the same as that in figure 3. As mentioned before, thecoexistence lines between the lamellar and hexagonal phasesin this region are identical between the 2D and 3D cases. The
surfaces in figure 8 represent the iso-planes at φ = −0.2
and we do not see any pattern in figure 8(a). In the earlystage, a random network appears, as seen in figure 8(b).
This may correspond to the phase separated patterns observed
5

J. Phys.: Condens. Matter 20 (2008) 155107

K Yamada and S Komura

Figure 7. Time evolution of Minkowski functionals for φ̄ = 0.1 and τ = 0.8, as shown in figure 3. (a), (b) The relative black area v , (c), (d)the length of the boundary line s between black and white regions, (e), (f) the Euler characteristic χ . (b), (d) and (f) are the expansions of thesmall-time regions of (a), (c) and (e), respectively.

Figure 8. Time evolution of phase separation pattern for φ̄ = 0.1 and τ = 0.8 in three dimensions. (a) t = 0, (b) t = 150, (c) t = 1000,(d) t = 5000. The surfaces represent the iso-planes at φ = −0.2.

in the 2D simulations such as in figures 3(b) and 4(b).
Through the breaking and reconnection of the network, we
finally obtain an almost periodic structure as presented infigure 8(d).
Due to the small system size, we could not observe
any macroscopic phase separation between the lamellar andhexagonal phases in this simulation. Instead, we obtained the
Bragg peak intensities of figure 8(d), as presented in table 1.
We show here only the peaks whose intensities are larger
than 0.02. It should be noted that this structure was nottaken into account when we calculated the phase diagram insection 2.
6

J. Phys.: Condens. Matter 20 (2008) 155107

K Yamada and S Komura

Introducing a phase factor x 0 and writing as k1 ·r = k ∗ (x −x 0),
we carry out the integration by parts. The other terms of thefree energy are not presented here because they are lengthy.
In the case of configuration (a), we postulate that theamplitudes ci (x) and the average composition φ̄(x) are
expressed as




1
1
x
x
c1 (x) =1 + tanh
1 − tanh
aH +
aL ,
2
ξ
2
ξ
(14)


1
x
ci (x) =1 + tanh(i = 2, 3),
(15)
aH
2
ξ




x
x
1
1
φ̄H +
φ̄L ,
φ̄(x) =
1 + tanh
1 − tanh
2
ξ
2
ξ
(16)
where aH and aL represent the equilibrium amplitudes of
the hexagonal and lamellar phases, while φ̄H and φ̄L are
the average compositions in these phases, respectively. The
length ξ determines the interfacial width, which is assumed
to be much smaller than the spatial period of the modulated
structures, i.e. k ∗ ξ  1. In the case of (b), on the other hand,the amplitudes ci (x) can be written as




1
1
x
x
c2 (x) =1 + tanh
1 − tanh
aH +
aL ,
2
ξ
2
ξ
(17)


1
x
ci (x) =1 + tanh(i = 1, 3),
(18)
aH
2
ξ

Table 1. The Bragg peak intensities of the structure shown infigure 8(d). We list only the peaks whose intensities are larger than
0.02.
Reciprocal vectors

Intensity(−2, −1, 4)(−3, 3, 2)(−1, 2, 4)(−4, 0, 2)(1, −4, 2)(−4, 2, 1)(−3, −2, 3)(2, 4, 1)(3, 0, 3)(2, −3, 3)

0.122 064
0.119 915
0.071 968
0.066 277
0.062 359
0.056 203
0.048 956
0.039 712
0.032 639
0.023 958

4. Interface between the lamellar and hexagonalphasesIn this section, we study the mutual orientation between
the lamellar and the hexagonal phases at the interface whenthese two phases coexist in equilibrium. We shall determine
analytically the most stable interfacial profile between thetwo phases. For simplicity, we compare between the four
different interfaces shown in figure 9, where we choose
the horizontal and vertical axes as the x - and y -directions,
respectively. Here the directions parallel to the stripe and theinterface make angles (a) 0, (b) π/3, (c) π/2, and (d) π/6,
respectively.
In the cases of figures 9(a) and (b), the spatially varying
order parameter φ(r) can be expressed as

φ(r) = φ̄(x) + 2

3ci (x) cos(ki · r),and φ̄(x) is identical to equation (16).
Next we perform the variable transformation x/ξ → x .
From the assumption k ∗ ξ  1, we see that the third-orderterms in ξ −1 are dominant in the free energy functional. Hence,in the cases of figures 9(a) and (b), the free energy can beapproximated as

2
3



∂ 2 φ̄
∂ 2 ci−3F φ̄(x), ci (x) ≈ 2 L y ξ
dx
+2
∂x2
∂x2
i=1

∂c1 ∂ 3 φ̄
∂ 2 φ̄ ∂ 2 c1
∂ 2 c2 ∂ 2 c3
+ 8 L y ξ −3 dx − 2−−2∂x ∂x2∂x2 ∂x2∂x ∂x3(11)i=1where the three reciprocal vectors are defined by√
k2 = k ∗ (1/2, 3/2),
k1 = k ∗ (1, 0),√
k3 = k ∗ (1/2, − 3/2).

(12)

Notice here that the average composition φ̄(x) and theamplitudes ci (x) are functions of x .
Substitution ofequation (11) into equation (1) yields the equilibrium freeenergy in terms of φ̄(x), ci (x), and k ∗ . In the calculation, we
use an approximation such that all the higher order harmonics
are ignored, which is valid in the weak segregation limit. For
example, the second term of equation (1) becomes



2
3 

∂ φ̄
∂ci 2
2
dr(∇φ) = L y dx
+ 2 L y dx
∂x
∂x
i=1


∂c2 ∂c3k∗2∂ φ̄ ∂c1+++ k ∗ 2 ci 2 + 4 L y d x
c2 c3
∂x ∂x
∂x ∂x
2


k ∗ ∂c3∂ φ̄− c2
× cos(k1 · r) + 4 L y dx −k ∗ c1
∂x
2 ∂x

∗
k
∂c2
− c3
sin(k1 · r),
(13)
2 ∂x
where we have assumed the periodic boundary condition in
the y -direction, and L y is the system size in the y -direction.

−

 ∗ 
∂c2 ∂ 3 c3
∂c3 ∂ 3 c2
k x0 .
−cos∂x ∂x3∂x ∂x3(19)

Using equations (14)–(16), we obtain for case (a) asFa ≈ 2 L y ξ −3 dx φ̄H2 − 2φ̄H φ̄L + φ̄L2 + 6aH 2 − 4aH aL+ 2a L 2

×
+
≈

7

dx

sinh2 xcosh6 x

+ 8 L y ξ −3 cos k ∗ x 0 sinh2 x−4 φ̄H − φ̄L (aH − aL ) − 4aH 2
cosh6 xφ̄H − φ̄L (aH − aL ) + aH 2

1
cosh4 x8L y ξ −3 φ̄H2 − 2φ̄H φ̄L + φ̄L2 + 6aH 2 − 4aH aL + 2aL 2
15

 

32
+
L y ξ −3 cos k ∗ x 0 φ̄H − φ̄L (aH − aL ) + aH 2 .
15(20)J. Phys.: Condens. Matter 20 (2008) 155107

K Yamada and S Komura

Figure 9. Four possible interfaces between the coexisting lamellar and hexagonal phases.

Here we considered that the integral ranges from −∞ to ∞,and used the following integrals: ∞sinh2 x
4
dx
,
(21)
=
6
15
cosh x
−∞
 ∞
1
4
dx
(22)
= .
4
3
cosh
x
−∞

For the parameter values φ̄ = 0.1 and τ = 0.8, as
we chose in the 2D and 3D simulations, we can numericallyestimate aH ≈ −0.128 86, aL ≈ −0.243 46, φ̄H ≈ 0.113
and φ̄L ≈ 0.086 from the phase diagram in figure 2. With
these values, one can evaluate the minimum of the above freeenergies as Fa ≈ 0.0078 L y ξ −3 when k ∗ x 0 = π , Fb ≈0.0109 L y ξ −3 when k ∗ x 0 = 0, and Fc ≈ Fd ≈ 0.0498 L x ξ −3 .
The difference between cases (c) and (d) appears only in theorder of ξ −1 , and it is estimated as Fc − Fd ≈ 0.014 L x ξ −1 > 0.
Summarizing all these results, we find that Fa < Fb < Fd <Fc holds when L x = L y . Hence the case of figure 9(a) is
most stable among the other orientations within the present
approximation.
In the 2D simulation results of figures 3(d) and 4(d), wemostly observe the interfaces corresponding to the case of
figure 9(b) rather than case (a). Hence it is likely that thesystem is trapped in a metastable state in the simulation. Oneof the possibilities to avoid such a metastable state in thesimulation is to include a thermal noise term in equation (10).
However, we leave this possibility as our future task, because itis expensive in computer time. It could be also possible that thecondition k ∗ ξ  1 which is assumed in the above calculation
may not be satisfied in the simulation. We also note that theestimated energy difference between cases (a) and (b) is rather
small, at least compared to either case (c) or (d).
The interface between the lamellar and hexagonal phases
was previously studied by Netz et al based on a free energy
similar to equation (1) [17]. They investigated the effectof temperature on the interfacial energy using a numericalconjugate-gradient method.
In their work, the mutualorientation between the two phases was assumed to be that offigure 9(a). This is in agreement with our analytical argument
presented above. We note that the interface of figure 9(a) was
indeed observed in some amphiphilic systems [31, 32].

Similarly, we obtain the free energy for case (b) as


8
Fb ≈
L y ξ −3 φ̄H2 − 2φ̄H φ̄L + φ̄L2 + 6aH 2 − 4aH aL + 2aL 2
15


32
+
L y ξ −3 cos(k ∗ x 0 ) φ̄H − φ̄L + aH − aL aH .
(23)
15
In the cases of figures 9(c) and (d), the amplitudes varyalong the y -direction and are given by




1
1
y
y
c1 (y) =1 + tanh
1 − tanh
aH +
aL ,
2
ξ
2
ξ
(24)


y
1
aH
ci (y) =1 + tanh(i = 2, 3),
(25)
2
ξ
and





1
1
y
y
1 + tanh
1 − tanh
aH +
aL ,
2
ξ
2
ξ
(26)


1
y
ci (y) =1 + tanh(i = 1, 3),
(27)
aH
2
ξ

c2 (y) =respectively. In both cases, the mean composition φ̄(y) is also
a function of y , and is given by




y
y
1
1
φ̄H +
φ̄L .
φ̄(y) =
1 + tanh
1 − tanh
2
ξ
2
ξ
(28)
By repeating similar calculations to the cases of (a) and(b), the free energy of cases (c) and (d) is obtained as
 2 2

∂ φ̄
−3
F{φ̄(y), ci (y)} ≈ 2 L x ξ
dy
∂y 2

2
3

∂ 2 ci
+2
(29)
,
∂y 2
i=1

5. Conclusion
In the present paper, we have investigated both the equilibriumand dynamical properties of order–order phase separation.
Based on a single order parameter Ginzburg–Landau freeenergy, we first obtained the phase diagram in three dimensions
within the two-mode expansion method. Various phasecoexistences between the different ordered phases have been
identified, including the BCC and gyroid phases. Using
computer simulation, we have next focused on the phaseseparation kinetics between the lamellar and hexagonal

where L x is the system size in the x -direction. Finally, the freeenergies of the interfaces (c) and (d) are given by8
Fc ≈ Fd ≈L x ξ −3 φ̄H2 − 2φ̄H φ̄L + φ̄L2 + 6aH 215
− 4a H a L + 2a L 2 .
(30)
8J. Phys.: Condens. Matter 20 (2008) 155107

K Yamada and S Komura+ a1 a7 a10 b1 + a1 a8 a9 b4 + a1 a8 a11 b1 + a1 a8 a11b3

phases both in two and three dimensions.
We findthat the microphase separation takes place in the earlystage and the macrophase separation follows through the
breaking and reconnection of microdomains. This fact
has also been confirmed by analyzing the time evolutionsof the Fourier patterns and various Minkowski functionals.
Finally, we have analytically determined the most stableorientation at the interface between the lamellar and hexagonalphases.
Further study of the order–order phase separation is now
in progress. In this paper we have discussed only the dynamics
of the lamellar–hexagonal phase separation. Certainly othercombinations of phase coexistence are possible in 3D systems,
including the BCC and gyroid phases (see figure 2). We areinvestigating the kinetics of other types of order–order phaseseparation as well as the equilibrium interfacial properties. It
is also interesting to see how one coexistence state between
the ordered phases transforms to the other coexistence stateby, for example, lowering the temperature. In real systems,
critical fluctuations can affect the phase behavior close to thecritical point [22, 23]. Investigation to see how the various
phase coexistences are modified by critical fluctuations is left
for a future problem.

+ a2 a3 a9 a10 + a2 a3 a11 a12 + a2 a4 b1 b2 + a2 a4 b1 b6+ a2 a4 b2 b5 + a2 a4 b5 b6 + a2 a5 a9 b1 + a2 a5 a9 b4+ a2 a5 a11 b3 + a2 a6 a12 b1 + a2 a7 a9 b2 + a2 a7 a9 b3+ a2 a7 a9 b6 + a2 a7 a11 b4 + a2 a7 a11 b5 + a2 a8 a10b5+ a2 a8 a12 b2 + a2 a8 a12 b6 + a3 a4 a5 a8 + a3 a4 a6 a7+ a3 a5 a10 b1 + a3 a5 a10 b4 + a3 a5 a10 b6 + a3 a5 a12 b3+ a3 a5 a12 b5 + a3 a6 a9 b5 + a3 a6 a11 b1 + a3 a6 a11b6+ a3 a7 a10 b2 + a3 a7 a10 b3 + a3 a7 a12 b4 + a3 a8 a11 b2+ a4 a5 a9 b2 + a4 a6 a10 b3 + a4 a6 a12 b2 + a4 a6 a12b4+ a4 a7 a9 b1 + a4 a7 a9 b5 + a4 a7 a11b6 + a4 a8 a10 b4+ a4 a8 a10 b6 + a4 a8 a12 b1 + a4 a8 a12 b3 + a4 a8 a12 b5+ a5 a6 a7 a8 + a5 a6 a9 a12 + a5 a6 a10 a11 + a5 a7 b1 b2+ a5 a7 b1 b3 + a5 a7 b2 b4 + a5 a7 b3 b4 + a6 a8 b1 b2+ a6 a8 b1 b4 + a6 a8 b2 b3 + a6 a8 b3 b4 + a7 a8 a9 a12+ a7 a8 a10 a11 + a9 a10 a11a12 + a9 a11 b3 b4 + a9 a11 b3 b5+ a9 a11 b4 b6 + a9 a11 b5 b6 + a10 a12 b3 b4 + a10 a12 b3 b6+ a10 a12 b4 b5 + a10 a12 b5 b6 + b1 b2 b3 b4 + b1 b2 b5 b6
+ b3 b4 b5 b6 .
(31)

Acknowledgments
References

This work is supported by KAKENHI (Grant-in-Aid for
Scientific Research) on Priority Area ‘Soft matter physics’ andgrant No 18540410 from the Ministry of Education, Culture,
Sports, Science and Technology of Japan.

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Appendix. General form of the free energy
Substituting equation (2) into equation (1) and ignoring thehigher order harmonics, we can rewrite the free energy in termsof a j , bk , Q and P as follows:12F = 4 Q 4 − 4 Q 2 + τ + 3φ̄ 2
aj2

j =1
6


+ 4 P 4 − 4 P 2 + τ + 3φ̄ 2
bk 2


+3

12


j =1



aj2 +

6


k=1

2
bk 2

k=1

− 32

12


j =1

a j 4 − 32

6


bk 4

k=1

+ 12φ̄ a1 a3 b4 + a1 a7 a12 + a2 a4 b3 + a2 a6 a10 + a3 a8 a9+ a4 a5 a11 + a5 a7 b6 + a6 a8 b5 + a9 a11 b2 + a10 a12b1
+ b1 b3 b6 + b1 b4 b5 + b2 b3 b5 + b2 b4 b6
+ 12 a1 2 b2 b5 + a2 2 b2 b6 + a3 2 b1 b6 + a4 2 b1 b5 + a5 2 b1 b4+ a6 2 b2 b4 + a7 2 b2 b3 + a8 2 b1 b3 + a9 2 b3 b6+ a10 2 b4 b6 + a11 2 b4 b5 + a12 2 b3 b5 + a1 a2 a3 a4+ a1 a2 a5 a8 + a1 a2 a6 a7 + a1 a3 b1 b2 + a1 a3 b1 b5+ a1 a3 b2 b6 + a1 a3 b5 b6 + a1 a4 a9 a10 + a1 a4 a11 a12+ a1 a5 a10 b2 + a1 a5 a10 b5 + a1 a5 a12 b6 + a1 a6 a9 b3+ a1 a6 a9 b6 + a1 a6 a11 b2 + a1 a6 a11 b4 + a1 a6 a11b5
9

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