JOURNAL OF CHEMICAL PHYSICSVOLUME 117, NUMBER 21

1 DECEMBER 2002

Real-space mean-field approach to polymeric ternary systems
Shigeyuki Komuraa)
Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan

Hiroya Kodamab)
Science and Technology Research Center, Mitsubishi Chemical Corporation, Yokohama 227-8502, Japan

Keizo Tamura
Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan

共Received 22 April 2002; accepted 4 September 2002兲
Phase separated structure of ternary blends of A and B homopolymers and symmetric AB diblock
copolymer is investigated using a lattice 共real-space兲 self-consistent field theory. This paper includes
the detailed description of our published results 关Kodama, Komura, and Tamura, Europhys. Lett. 53,
46 共2001兲兴 as well as more extended calculations. We consider the symmetric case, namely, 共i兲 both
A and B homopolymers have the same degree of polymerization N A ⫽N B ; 共ii兲 AB diblock
copolymer of length N AB is symmetric; 共iii兲 average volume fractions of A and B homopolymers areequal. We looked into the influence of relative chain lengths ␣ ⫽N A /N AB on the phase separated
structure. Our numerical simulations are performed in the real space without assuming the symmetryof the structure a priori. For the fixed copolymer length and ␣ ⬍1, the typical length scale of the
microphase separated structure become smaller for relatively shorter homopolymer chains 共small ␣兲.
In other words, the homopolymers becomes more efficient to swell the microphase separated
structure for longer homopolymer chains 共large ␣兲. Detailed free-energy analysis revealed that thestability of the lamellar phase is marginal for small block copolymer volume fraction. For ␣ ⬎1, on
the other hand, three-phase coexistence either between the disorder, A-rich and B-rich phases or
between the lamellar, A-rich and B-rich phases is observed. © 2002 American Institute of Physics.
关DOI: 10.1063/1.1517038兴

I. INTRODUCTION

into account the microphase separation of the ternary blends
including block copolymers.8,9 For the symmetric ternary
case, the line of macrophase separation into A-rich andB-rich phases is connected to the line of microphase separation to the lamellar phase at the isotropic Lifshitz point 共seeSec. II兲.8,9
On the basis of self-consistent field theory 共SCFT兲,
Noolandi and Hong showed that the copolymer chains lower
the interfacial tension.10,11 They showed that the reduction ininterfacial tension increases with copolymer molecularweight and concentration. Their theory was extended by
Shull and his co-workers to investigate the interfacial properties of ternary blends by performing the numerical treatment of the SCFT.12–14 Banaszak and Whitmore calculated
phase diagrams of ternary blends which undergo both microphase and macrophase separation on the basis of a fourthorder expansion.15 Israels et al. examined the effect of
diblock size relative to the homopolymers on the compatibilization of homopolymer blends.16 They showed that interfacial tension can be reduced to zero if the blocks in the
diblock are longer than the corresponding homopolymer. The
Fourier-space SCFT calculations of ternary blends have been
performed by Janert and Schick.17,18 They investigated theinfluence of relative chain length on the phase behavior of
the system.17 The calculated phase diagrams include disordered, lamellar, hexagonal, and cubic phases, as well as twoand three-phase coexistence region. In their later paper, theunbinding nature of the system was discussed.18 Upon dilu-

A. Polymeric ternary systems

Traditional microemulsions being mixtures of oil, water,and surfactant, are known to exhibit various interesting microstructures depending on the temperature or thecomposition.1,2 When the concentration of surfactant is relatively large, they show a rich variety of regularly orderedstructures such as the cubic 共gyroid兲 phase, the hexagonalphase, or the lamellar phase. By lowering the concentrationof surfactant and if hydrophilic and lipophilic natures of thesurfactant are balanced, microemulsions form a bicontinuous
structure where a randomly oriented monolayer of surfactants separates oil-rich and water-rich subvolumes. A direct
observation of the randomly intertwined structure by using
the freeze-fracture microscopy has been reported.3
It has been widely recognized that mixtures of high molecular weight homopolymers and diblock copolymers areanalogous to traditional microemulsions.4,5 Similar to thelow molecular weight surfactant molecules, AB diblock copolymers tend to locate at the interface between the twophases rich in A and B homopolymers. The emulsifying effect of copolymer added to blends of homopolymer was considered using Flory–Huggins theory and random phase approximation 共RPA兲 by Leibler.6,7 His theory was used to takea兲Electronic mail address: komura@comp.metro-u.ac.jp
Electronic mail address: kodama@rc.m-kagaku.co.jpb兲

0021-9606/2002/117(21)/9903/17/$19.00

9903© 2002 American Institute of Physics

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9904

J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

Komura, Kodama, and Tamuraa singular cusp in the phase transition temperature is assumed as ␪ is varied about ␪ L , and the region of the disordered phase is considered to extend to very low temperatures
within this cusp.
Various theoretical approaches for PME have been addressed. Kielhorn and Muthukumar extended the mean-field
theory by including composition fluctuations, and concluded
that the Lifshitz point is destroyed due to the fluctuations.25
Later, Kielhorn and Muthukumar derived a mean-field timedependent Ginzburg–Landau equation to model the spinodal
decomposition of polymeric ternary systems.26 Matsen calculated the elastic properties of a diblock copolymer monolayer within SCFT,27 and showed that the calculated bending
modulus and saddle-splay modulus satisfy the conditions required to melt the lamellar phase. Using SCFT and strongsegregation theory, Thompson and Matsen have calculated
the effective interactions between the copolymer
monolayers.28 From the condition that the attraction becomes
sufficiently small, they concluded that the optimum size ofthe homopolymer molecules is about 80% that of the copolymer molecule. In a more recent paper, Thompson and Matsen
predicted that copolymer polydispersity can improve PME
by forming flexible and nonattractive monolayers.29
C. The present work
FIG. 1. The phase prism for a ternary blend of two homopolymers and a
block copolymer. ␾ A and ␾ B are homopolymer volume fractions, and ␪ isthe copolymer volume fraction. The vertical coordinate is the incompatibility degree ␹ N A . The shaded plane cut through the prism at ␾ A ⫽ ␾ B is theisopleth.

tion by the homopolymers, the symmetric lamellar phase
does not unbind, whereas it always becomes unstable to theasymmetric lamellar phases which do unbind.
B. Polymeric microemulsions

One of the most interesting experimental findings by
Bates et al. in a polymeric ternary system is the existence ofa bicontinuous structure without any long-range order
共LRO兲.19,20 The investigated system consists of nearly equal
size poly共ethylene兲 共PE兲 and poly共ethylene propylene兲 共PEP兲homopolymers and a symmetric PE–PEP diblock copolymer.
They focused on the symmetric case in which the volume
fractions of the two homopolymers are equal 共see Fig. 1兲. Bychanging the average volume fraction of the copolymer ␪,transmission electron microscopy 共TEM兲 images revealed
that a lamellar structure is observed for ␪ ⬎ ␪ L and a twophase structure appears for ␪ ⬍ ␪ L , where ␪ L is the Lifshitz
volume fraction. Images obtained from the specimens closeto ␪ L bear a striking resemblance to bicontinuous microemulsions in mixtures of oil, water, and surfactant. Such a
new phase has been termed as ‘‘polymeric microemulsions’’
共PME兲. For different types of ternary systems, it is demonstrated that there is a common region in phase space overwhich bicontinuous PME is stable and the general phase behavior is rather universal.21–24 In their works, the bicontinuous PME has been regarded as a disordered phase which isdriven by fluctuations of lamellar phase.19–21 To explain this,In this paper, using a lattice 共real-space兲 SCFT method,
we investigate the phase separated structure of the symmetricmixtures of A and B homopolymers plus AB diblock copolymers. Our purpose is to study the morphological change ofthe microphase separated structure when the total copolymer
volume fraction or the relative chain lengths between thehomopolymers and the diblock copolymer is varied systematically. We numerically solve the SCFT equations in a real
space without assuming the symmetry of the structure a priori according to the method proposed by Hasegawa.30 Application of this method for the polymeric ternary system andsome of the preliminary results have been reported in ourprevious works.31–33 This paper provides the detailed description of our results as well as more extended calculations. Independently, the real-space SCFT method has beenused by other authors such as for linear ABCA tetrablock
copolymers,34,35 or star and linear ABC triblock
copolymers.36 In general, this method proved to be a powerful tool in finding new mesophases either with or without
LRO.37
Starting with randomly generated initial potential fields,
we demonstrate that a bicontinuous structure similar to PME
results from the microphase separation for relatively longer
copolymer chain. For the fixed copolymer length, we findthat the typical length scale of the microphase separated
structure becomes smaller for relatively smaller homopolymer chains. For relatively longer homopolymer chains, weobserved the three-phase coexistence either between the disorder, A-rich and B-rich phases or between the lamellar,
A-rich and B-rich phases depending on the strength of theinteraction parameter.
Since the obtained bicontinuous structure may correspond to metastable equilibrium, we have also performed thesame calculation starting from the initial potential fields with

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J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

Mean-field approach to ternary systems

an imposed lamellar symmetry. By calculating the free energy of the lamellar phase as a function of the average volume fraction of the copolymer, we discuss the effective interactions between the copolymer monolayers. We find that
there is a region where the stability of the lamellar phase ismarginal.
This paper is constructed as follows. In the next section,
the mean-field phase diagrams of the polymeric ternary system are explained within the RPA for various relative chain
lengths. In Sec. III, we summarize the general formulation of
the real-space SCFT for a system including copolymers, anddescribe the numerical procedure based on the lattice version
of SCFT. Section IV gives the analysis of our simulation
results. Summary and conclusion are provided in Sec. V. Details of the general formulation and some technical remarks
are given in Appendix A. Appendix B describes the iteration
scheme to solve the set of self-consistent equations.

A. Notations

In general, we consider a ternary blend of A and B homopolymer of polymerization indices N A and N B , respectively, and AB diblock copolymer of polymerization index
N AB of which a fraction f consists of A monomer. Then, we
introduce two dimensionless parameters defined by
NB
␣ B⬅
,
N AB

共2.1兲

respectively. The volume fractions of A and B homopolymers are denoted by ␾ A (r) and ␾ B (r), respectively, and thevolume fraction of AB block copolymer by ␾ AB (r). In addition, ␾ a (r) and ␾ b (r) represent the volume fractions of A
and B blocks in the AB block copolymer, respectively. The
average volume fraction of each component is indicated with
¯A, ␾¯ B , or ␾¯ AB ⬅ ␪ . We assume that thea bar, such as ␾
system is incompressible, i.e.,␾ A ⫹ ␾ B ⫹ ␾ a ⫹ ␾ b ⫽1.

共2.2兲

Due to this constraint, we define the following three quantities as independent order parameters:⌽⬅ ␾ A ⫺ ␾ B ,共2.3兲

⌿⬅ ␾ a ⫺ ␾ b ,共2.4兲

⌬⬅ ␾ A ⫹ ␾ a ⫺ ␾ B ⫺ ␾ b .

共2.5兲

The last order parameter represents the difference in the volume fractions of all A and B monomers. The Flory–Huggins
interaction parameter between A and B segments is denotedby ␹ AB ⬅ ␹ .
The phase behavior of a polymeric ternary blend can be
represented in the form of a phase prism shown in Fig. 1.
Within the large parameter space, we focus on the symmetriccase, namely 共i兲 both A and B homopolymers have the samelength (N A ⫽N B and hence ␣ A ⫽ ␣ B ⬅ ␣ ); 共ii兲 the AB block
copolymer contains A and B monomers in equal proportions( f ⫽1/2); and 共iii兲 the average volume fractions of A and B
homopolymers are equal and determined by

1⫺ ␪
.
2

共2.6兲

In other words, we consider the isopleth 共shaded plane in
Fig. 1兲 in the phase prism.

B. Correlation function

Here, we discuss the phase behavior of polymeric ternary systems within the random phase approximation 共RPA兲
for various values of ␣. Mean-field phase diagram on theisopleth for ␣ ⫽1 was considered by Leibler,6,7 and later calculated for ␣ ⫽1 by Broseta and Fredrickson.8 The essential
part of the phase diagram can be obtained by looking at themonomer concentration correlation function 共structure factor兲. Within the RPA, Leibler calculated the correlation function for ⌽ as7
S ⌽⌽ 共 x 兲 ⫽II. RANDOM PHASE APPROXIMATION

NA
␣ A⬅
,
N AB

¯ A⫽ ␾
¯ B⫽
␾

9905

2S A 共 x 兲 兵 1⫺ ␹ 关 S AA 共 x 兲 ⫺S AB 共 x 兲兴 其.
1⫺ ␹ 关 S A 共 x 兲 ⫹S AA 共 x 兲 ⫺S AB 共 x 兲兴

共2.7兲Here, x⫽(qR) 2 ⫽q 2 N A b 2 /6, where q⫽ 兩 q兩 is the absolutevalue of the wave vector q, R is the radius of gyration, and b
is the statistical segment length. In the above, S A (x) is thecorrelation function of independent A homopolymer, S AA (x)and S AB (x) are A – A and A – B monomers correlation functions of independent copolymer chains. These are given by1⫺ ␪
N A g 共 1,x 兲 ,
2

共2.8兲

S AA 共 x 兲 ⫽␪NA 1 x
g , ,
␣
2 ␣

共2.9兲

S AB 共 x 兲 ⫽

␪NA 1
1 x
x
g 1, ⫺g ,
␣ 2
␣
2 ␣

S A共 x 兲 ⫽冉 冊
冋 冉 冊 冉 冊册,共2.10兲

where g(h,x) is the Debye function defined byg 共 h,x 兲 ⫽2
共 hx⫹e ⫺hx ⫺1 兲 .
x2

共2.11兲

The limit of stability of the disordered phase occurs forthe smallest value of ␹ (x) 共highest temperature兲 at which
共2.7兲 diverges. From 共2.7兲 to 共2.11兲, this condition is writtenas11⫽ 2 关 ⫺1⫹x⫹e ⫺x
␹共 x 兲NA x⫹ ␪ 兵 1⫺e ⫺x ⫹ ␣ 共 ⫺3⫹4e ⫺x/2␣ ⫺e ⫺x/ ␣ 兲 其 兴 .
共2.12兲
When ␣ ⫽1, this equation reduces to Eq. 共3.2兲 in Ref. 9.
Minimizing ␹ (x) with respect to x, we see that for 0⬍ ␪
⬍2 ␣ 2 /(1⫹2 ␣ 2 ), the minimum occurs at x⫽0 and the corresponding value is
1⫺ ␪
1
⫽
.
␹NA
2

共2.13兲

This is called the ‘‘Scott line’’ 共consolute line兲 along which
the disordered phase makes a continuous transition to coexisting uniform (q⫽0) phases. The system exhibits the mac-

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9906

J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

rophase separation between A-rich and B-rich phases. For
2 ␣ 2 /(1⫹2 ␣ 2 )⬍ ␪ ⬍1, the instability occurs at a nonzero
value of x. This case corresponds to the microphase separation to a lamellar phase. When the transition to the lamellar
phase (q⫽0) occurs, the density of the copolymer is givenby␪ 共 x 兲 ⫽ 关 ⫺2⫹x⫹ 共 x⫹2 兲 e ⫺x 兴 / 关 6 ␣ ⫺2⫹ 共 x⫹2 兲 e ⫺x
⫹ 共 x⫹2 ␣ 兲 e ⫺x/ ␣ ⫺2 共 x⫹4 ␣ 兲 e ⫺x/2␣ 兴 ,共2.14兲

and the transition line is located according to 共2.12兲 with
共2.14兲. It is important to note that 共2.14兲 holds only for 0⬍x⭐3.785. 9 The point which connects the macrophase andmicrophase transition lines is known as an isotropic (m⫽d, where m is the number of dimensions in which wave
vector instability occurs, and d is the space dimension兲 Lifshitz point8,38

␪ L⫽

2␣2
,
1⫹2 ␣ 2

共 ␹ N A 兲 L⫽2 共 1⫹2 ␣ 2 兲 .

共2.15兲

It is instructive to derive the Ginzburg–Landau 共GL兲
free energy from the obtained correlation function. By ex⫺1
in 共2.7兲 for small q, we obtain the free energy
panding S ⌽⌽F 关 ⌽ 兴 asF GL关 ⌽ 兴 ⬇

␳0
2N A

冕

dr关 c 共 ⵜ 2 ⌽ 兲 2 ⫹g 共 ⵜ⌽ 兲 2 ⫹t⌽ 2 兴 ,共2.16兲

where ␳ 0 is the monomer density and the coefficients are

冋

册

c⫽

1
1
9共 ␹ N A 兲2␪ 共 ␹ N A 兲3␪ 2 4⫹⫺
R ,36 1⫺ ␪16␣ 3
8␣4

共2.17兲

g⫽

1 1
共 ␹ N A 兲2␪ 2⫺
R ,3 1⫺ ␪
8␣2

共2.18兲

t⫽

1
␹NA
⫺
.
1⫺ ␪
2

共2.19兲

冋册

Throughout this paper, all the energies are expressed in unitsof k B T, where k B and T represent the Boltzmann constantand the temperature, respectively. Notice that the Scott line
共2.13兲 is determined from the condition t⫽0. On the other
hand, the phase boundary between the lamellar and the twophase coexisting region is determined from the relation39
2 共 2⫺ 冑6 兲 a⫽g 2 /c,

共2.20兲

within the GL expansion. It important to note that this relation holds only close to the critical point where the expansion is justified. We comment here that when c,t⬎0 and g⬍0, 共2.16兲 is the free energy of traditional microemulsions
proposed by Teubner and Strey.40 In this case, ⌽(r) is defined as an order parameter which measures the local difference in volume fractions between oil and water.
In addition to these phase boundaries, one can locate the‘‘disorder line’’ and the ‘‘Lifshitz line’’ in the disordered region of the phase diagram.9 The disorder line is the locus ofpoints at which the asymptotic behavior of the correlation
function changes from a monotonic exponential decay to an
exponentially damped oscillatory decay. This oscillatory behavior reflects the tendency of the copolymer to order the

Komura, Kodama, and Tamura

homopolymers. The disorder line can be calculated from
共2.12兲 and 共2.14兲 by setting x⫽⫺y, where y ranges over all
positive real numbers (y⬎0). On the other hand, the Lifshitz
line is the locus of points at which the oscillations are sufficient to produce the dominant peak in the associated structure function at nonzero wave vector. The Lifshitz line forthe ⌽–⌽ structure function S ⌽⌽ 关see 共2.7兲兴 is obtained by
setting g⫽0 in 共2.18兲, and one obtains

冋册

1
共 1⫺ ␪ 兲 ␪ 1/2
⫽
.
␹NA
8␣2

共2.21兲

However, another situation occurs as ␪ is increased. The Lifshitz line is connected to the ‘‘equimaxima line’’ at which thetwo peaks 共one at q⫽0, and the other at q⫽0) in the structure function become of equal height.9
Phase diagrams on the isopleth for ␣ ⫽0.25, 0.5, and 1
including several different lines are shown in Fig. 2. Noticethat Fig. 2共c兲 is the same as Fig. 3 in Ref. 9. The Scott lines
start from ( ␪ , ␹ N A )⫽(0,2) 共point C兲, whereas the lines of
microphase separation end at ( ␪ , ␹ N A )⫽(1,10.5␣ ) 共pointG兲. Disorder lines start from 共0,0兲 共point D兲 except for ␣⫽1. The transition line between the lamellar and the twophase coexistence region is drawn only near the Lifshitz
point where the GL expansion is valid. According to Fig.
2共c兲 ( ␣ ⫽1), Holyst and Schick concluded that the copolymer is inefficient in the organization of homopolymers since
the Lifshitz line 共line LE兲 is far from the disorder line 共lineLD兲. However, this tendency changes as ␣ is decreased from
1. We see that the Lifshitz line appears even in the small
copolymer volume fraction region such as in Fig. 2共a兲. In theexperiment by Bates et al., the parameter ␣ was chosen as␣ ⬃0.2 in order to ensure that both the points C and G in thephase diagram can be explored over the similar temperature
window.19,21,22

C. Multiphase coexistence

It is important to notice that the above discussion is valid
only for ␣ ⭐1. For ␣ ⬎1, the Lifshitz point is preempted bya tricritical point8

␪ TCP⫽

2␣
,
1⫹2 ␣

共 ␹ N A 兲 TCP⫽2 共 1⫹2 ␣ 兲 .

共2.22兲

Notice that ( ␹ N A ) TCP⬍( ␹ N A ) L for ␣ ⬎1. The special case of␣ ⫽1 corresponds to a ‘‘Lifshitz tricritical point,’’ as has
been pointed out in Refs. 8 and 9. The tricritical point indicates a continuous segregation into three phases; twohomopolymer-rich phases and a copolymer-rich lamellar
phase.
A typical phase diagram for ␣ ⫽2 is calculated in Fig. 3.
For small ␪, the disordered phase undergoes a continuoustransition to A- and B-rich uniform phases at the Scott line
共2.13兲. This line ends at a tricritical point 共2.22兲, beyond
which is a line 共triple line兲 of three-phase coexistence between A-rich, B-rich, and disordered block copolymer
phases. The region of three-phase coexistence can be calculated within the Flory–Huggins approximation where the reduced free energy is given by8

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J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

Mean-field approach to ternary systems

9907

FIG. 3. Phase diagram of the symmetric ternary mixture for ␣ ⫽N A /N AB⫽2. In addition to the notations given in Fig. 2, the region labeled 3P is theregion of three-phase coexistence between A-rich, B-rich, and disorderedblock copolymer phases. T is the tricritical point.

f 共 ⌽, ␪ 兲 ⫽1⫺ ␪ ⫹⌽
log共 1⫺ ␪ ⫹⌽ 兲
2⫹

1⫺ ␪ ⫺⌽
␹ N A⌽ 2
log共 1⫺ ␪ ⫺⌽ 兲 ⫹ ␣␪ log ␪ ⫺.
2
4
共2.23兲

The compositions ⌽ and ␪ of the three coexisting phases are
obtained by equating the chemical potentials of each species

⳵f
,
⳵⌽

共2.24兲

⳵f,⳵␪

共2.25兲␮ ⌽⫽␮ ␪⫽and the osmotic pressure⌸⫽ f ⫺ ␮ ⌽ ⫺ ␮ ␪ .

共2.26兲

The triple line terminates at a four-phase point at which
the lamellar phase coexists with the other three phases.41
Beyond this point, there is a direct transition line between thedisordered phase and the lamellar phase. These regions are
not shown in Fig. 3.
III. LATTICE SELF-CONSISTENT FIELD THEORY
A. Self-consistent field theory

FIG. 2. Phase diagrams of the symmetric ternary mixture for 共a兲 ␣⫽N A /N AB ⫽0.25; 共b兲 ␣ ⫽0.5; and 共c兲 ␣ ⫽1 as obtained from the random
phase approximation. ␪ is the total volume fraction of the copolymer. The
regions labeled LAM, DIS, and 2P are, respectively, regions of lamellar,
disordered, and two-phase coexisting phases. L is the Lifshitz point. The line
CL is that of continuous transition from the disordered phase to coexisting
A-rich and B-rich phases, LG is the line of continuous transitions from thedisordered phase to the lamellar phase, LD is the disorder line, and LE and
EG are the Lifshitz and equimaxima lines for the ⌽–⌽ structure function,
respectively.

Self-consistent field theory 共SCFT兲 in polymer science
originates from the early works by Edwards42 or by Helfand
and Tagami.43 The basic idea of SCFT is as follows: polymers are represented as random walks in a positional dependent segment potential, which depends in a self-consistent
way on the distribution of segments. SCFT has been applied
to various problems and its recent achievements are summarized such as in Refs. 37 and 44. In fact, the RPA used in theprevious section is one of the approximated formulations of
SCFT.

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9908

J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

Komura, Kodama, and Tamura

There are, basically, two possible ways to formulate the
SCFT: Fourier-space and real space representations. Recently, the Fourier-space SCFT method proved to be quite
useful to predict the phase diagram of block copolymer
melts.45,46 In this method the set of self-consistent equations
is solved by using a restricted Fourier basis appropriate foran assumed mesophase symmetry. Later, various periodic
structures with LRO were successfully explained within the
SCFT for polymeric ternary systems.15,17,18 In these works,
the free energies of several periodic structures were calculated by assuming their mesophase symmetry a priori.
On the other hand, the set of self-consistent equations
共see the next subsection兲 can be solved also in a real space.
This real-space method was initiated by Scheutjens and Fleer
as a lattice version of SCFT,47 and has been further developed for surface problems with specific boundary
conditions.48,49 More recent works using the real-space 共bothlattice and off-lattice兲 SCFT are given in Ref. 50.
Furthermore, it is advantageous in the real-space methodthat one need not assume the symmetry of structures apriori.31–37 In order not to bias the symmetry of patterns thatemerges, one numerically solves the self-consistent equations by starting with random potential fields. Although thismethod may generate the structures corresponding to either
stable or metastable equilibrium, it has proved to be a powerful tool in finding new mesophases. A similar approach
was taken in Ref. 16, in which they observed droplet microemulsions. In the present paper, we employ the real-spacelattice SCFT with a slight extension of the previous method.

G i 共 n,1兲 ⫽exp关 ⫺U f i (1) 共 n兲兴 ,共3.3兲

G †i 共 n,N i 兲 ⫽exp关 ⫺U f i (N i ) 共 n兲兴 .

共3.4兲

Here, f i (N ⬘ ) is the ‘‘type function’’ representing the type ofthe segment N ⬘ in the ith chain, and 具具¯ 典典 denotes a neighborhood average defined later by 共3.9兲. The segment potential is given by
U J 共 n兲 ⫽u 共 n兲 ⫹兺K ␹ JK 具具␾ K共 n兲 典典,

共3.5兲

where ␹ JK is the Flory–Huggins parameter between J and K
segments. Physically, u(n) represents an entropic hard-corepotential which is independent of the segment type, whereas
the second term is the interaction part. The segment distribution function in 共3.5兲 is expressed asNi␾ J 共 n兲 ⫽ 兺兺 ␦ 关 J, f i共 N ⬘ 兲兴 ␾ i,N ⬘共 n兲 ,共3.6兲N ⬘ ⫽1iwith␾ i,N ⬘ 共 n兲 ⫽n i G i 共 n,N ⬘ 兲 G †i 共 n,N ⬘ 兲exp关 ⫺U f i (N ⬘ ) 共 n兲兴 兺 n⬘ G i 共 n⬘ ,N i 兲

,

共3.7兲

where ␦ 关 J, f i (N ⬘ ) 兴 is a Kronecker delta with value 1 if segment f i (N ⬘ ) is of type J, and 0 otherwise. Equation 共3.7兲 iscalled the ‘‘composition law.’’ Finally, the incompressibility
condition requires that

B. Self-consistent equations in the lattice formulation

Here, we give the set of self-consistent equations in thelattice formulation of SCFT which is essentially based on
Refs. 48 and 49. While Ref. 48 treats only surface adsorption
problems, we apply the lattice SCFT to higher space dimensions. Although there are many derivations of SCFT, we givethe details of the lattice formulation in Appendix A for thecompleteness of our paper. Notice that the present lattice
formulation is not a simple discretization of the continuum
model.
The space coordinate of each segment composing thepolymers is specified by a lattice point n⫽(n x ,n y ,n z ) in an
L x ⫻L y ⫻L z (⬅⍀) cubic lattice box with a lattice constant a.
We denote the type of molecule by i, and the type of segment by J and K. 共In the present case, i⫽A,B for A and B
homopolymers, and i⫽AB for AB copolymer, whereas J or
K points to either A or B segment.兲 Segments in a chain areindexed by N ⬘ ⫽1,2,...,N i , where N i is the polymerizationindex of the ith molecule. The molecules of type i are indexed by p⫽1,2,...,n i , where n i is the total number of ith
molecule.
We introduce two Green’s functions G i (n,N ⬘ ) andG †i (n,N ⬘ ) for walks starting at N ⬘ ⫽1 and N i , respectively.
These functions obey the following recursive relations:
G i 共 n,N ⬘ 兲 ⫽exp关 ⫺U f i (N ⬘ ) 共 n兲兴 具具G i 共 n⬘ ,N ⬘ ⫺1 兲 典典,

共3.1兲

G †i 共 n,N ⬘ 兲 ⫽exp关 ⫺U f i (N ⬘ ) 共 n兲兴 具具G †i 共 n⬘ ,N ⬘ ⫹1 兲 典典,

共3.2兲

together with the initial conditions

兺J ␾ J共 n兲 ⫽1,

共3.8兲

holds at any site. From 共3.1兲 to 共3.8兲 forms a set of selfconsistent equations to be solved for 兵 ␾ J (n) 其 and 兵 U J (n) 其 .
Up to now, the formulation is rather general and standard. In order to avoid the geometric constraints associatedwith a lattice, the neighborhood average 具具¯ 典典 in 共3.1兲, 共3.2兲,or 共3.5兲 is taken according to51

具具X 共 n兲 典典⫽ 兺
n⬘

⫹

冋

6
3
␦ 关 兩 n⫺n⬘ 兩 2 ,1兴 ⫹ ␦ 关 兩 n⫺n⬘ 兩 2 ,2兴
80
80

册

1
␦ 关 兩 n⫺n⬘ 兩 2 ,3兴 X 共 n⬘ 兲 .
80

共3.9兲

This type of neighborhood average has the desirable property
of being more generally isotropic, and is a natural extensionof the standard nearest-neighbor average given by 共A5兲. See
also the corresponding ideal Hamiltonian 共A29兲. On theother hand, 共3.9兲 has been successfully used in the celldynamical simulations.51 A related discussion is provided inRef. 52 in which the coefficients in 共3.9兲 are carefully determined in order to be consistent with the Gaussian chain
model in a continuum treatment.
After solving these equations, we calculate the mixing
free energy F m with respect to the unmixed pure reference
system where the segment interactions are switched off.
Then, we have

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J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

Mean-field approach to ternary systems

9909

TABLE I. Characterization of the symmetric polymeric ternary systems for the fixed copolymer length case( ␣ ⭐1).
SystemN A ⫽N B

N AB

␣

␹

␹ N A⫽ ␹ N B

␹ N AB( ␪ L ,( ␹ N A ) L)

2
4
8

8
8
8

0.25
0.5
1

2
2
2

4
8
16

16
16
16

共1/9, 9/4兲共1/3, 3兲
共2/3, 6兲system 1 共S1兲system 2 共S2兲system 3 (S3⫽S6)

F m关 ␾ J ,U J 兴 ⫽a well-defined periodicity. We chose different parameters for
1D and 2D calculations in order to maximize the computational efficiency for large 2D systems.

nNi
兺i n i log 兺 nG ii共 n,N
i兲

⫹

1
2 n

⫺

兺n 兺J U J共 n兲 ␾ J共 n兲 .

␹ JK ␾ J 共 n兲 具具␾ K 共 n兲 典典
兺兺J,K

IV. RESULTS AND DISCUSSION

共3.10兲

C. Numerical procedure

The above set of self-consistent equations is solved in
the following way.30,48,49 We first give an initial guess for
兵 U J (n) 其 which can be either random or periodic in space. In
general, we do not need to assume any symmetry of thestructure a priori, which differs from the Fourier-space treatment of SCFT such as in Refs. 45 and 46. Then, we computethe single chain statistical weights G i (n,N ⬘ ) and G †i (n,N ⬘ )from 共3.1兲 and 共3.2兲, respectively. Using these quantities, we
obtain the distribution 兵 ␾ i,N ⬘ (n) 其 from 共3.7兲. Then, ␾ J (n) is
calculated by 共3.6兲. At this point, we examine whether theincompressibility condition 共3.8兲 is satisfied, and whether thesegment potential gives the same u(n) for each segment
type. If these constraints are satisfied, the initial estimate
corresponds to equilibrium. Otherwise, new estimates for
兵 U J (n) 其 are made until the conditions are met. Notice thatthe present algorithm is not intended to mimic real polymer
dynamics, but is an efficient artifice to evolve a system as
fast as possible to a free-energy minimum which can be either stable or metastable. The details of the Picard-type iteration scheme used is presented in Appendix B.
First, we numerically solve the set of SCFT equations
starting with randomly generated potential fields. This simulation is performed in 2D (L x ⫽L y ⫽256, L z ⫽1) and 3D(L x ⫽L y ⫽L z ⫽64) systems. In 2D cases, we neglect any gradients in the z-direction and put53
X 共 n x ,n y ,n z ⫺1 兲 ⫽X 共 n x ,n y ,n z ⫹1 兲 ⫽X 共 n x ,n y ,n z 兲 .
共3.11兲
Furthermore, the periodic boundary condition is imposed. As
in Sec. II, we only consider the interaction between the A
and B segments

␹ AB ⫽ ␹ BA ⫽ ␹ .

共3.12兲

Notice that although the present method may not be suitable for determining the fully equilibrated structures, it enables us to generate a structure which does not possess any
LRO such as PME as a metastable structure. Since the obtained patterns may depend on the initially prepared potential
fields, we also study 1D systems starting with 1D periodic
potential fields. The latter case yields lamellar structures withIn this section, we present our results of simulations described previously. We focus here on the symmetric case as
in Sec. II, and examine the influence of the relative chain
length between homopolymers and copolymer. We did it in
two ways; one by fixing the copolymer length and changing
the homopolymer length, and the other by fixing the homopolymer length and changing the copolymer length. Somepreliminary results for ␣ ⬎1 case and 3D system are also
reported. All the results from Sec. IV A to IV D are obtained
starting with randomly generated potential fields without assuming any symmetry, whereas Sec. IV E treats the casestarting from 1D periodic initial conditions.
A. Fixed copolymer length „ ␣ Ï1…

We first discuss the case where the length of AB block
copolymer is fixed to A4B4 (N AB ⫽8), whereas lengths ofboth A and B homopolymers are changed. Three different
homopolymer lengths are considered, which we refer to assystems 1, 2, 3 共or S1, S2, S3兲 as listed in Table I. For eachhomopolymer length, the average volume fraction of the co¯ AB ) is varied between 0 and 1. The interacpolymer ␪ (⫽ ␾tion parameter between A and B monomers is fixed to ␹
⫽2. Hence, the degree of incompatibility is fixed to ␹ N AB
⫽16, which corresponds to the temperature below the microphase separation temperature ( ␹ N AB ) MST⬇10.5 for ␪⫽1.54 The typical patterns obtained from 2D simulations are
depicted in Figs. 4 and 5 for S1 and S3, respectively. The
volume fractions ␾ A , ␾ B , ␾ a , and ␾ b as defined in Sec. II
are drawn in gray scale for five different copolymer volume
fractions ␪ ⫽0.1, 0.3, 0.5, 0.7, and 0.9. Here, the values 0
and 1 correspond to white and black, respectively. In Fig. 6,
we plotted the part of cross-section profiles of ⌽ and ⌿ 关see
共2.3兲 and 共2.4兲兴 for ␪ ⫽0.1, 0.5, and 0.9 obtained from S3.
For small ␪, there are large regions of A and B homopolymer rich domains. As can be clearly seen from Fig.
6共a兲, the profile of ⌽ changes sharply from ⫺1 to 1 at theinterface between the A-rich and B-rich domains. Most of
the copolymers are located at the interface, and form saturated monolayers without any LRO. As ␪ is increased, thedomain size becomes smaller and the amount of the interface
increases, although the LRO still does not exist. Moreover,the difference between the maximum and the minimum values of ⌽ becomes smaller. This is due to the fact that thehomopolymers are excluded by the copolymer brushes which

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9910

J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

FIG. 4. Final patterns of system 1 in Table I for different values of ␪ drawn
in gray scale. The values 0 and 1 correspond to white and black, respectively. ␾ A and ␾ B are volume fractions of A and B homopolymers, and ␾ aand ␾ b are the volume fractions of A and B blocks in the copolymer,
respectively.

start to overlap. When ␪ is relatively large, the system exhibits a lamellar structure 共with many defects兲 which is swollen
by the small fraction of homopolymer chains. The lamellar
structure can also be clearly seen in Fig. 6共c兲.
Both in Figs. 4 and 5, bicontinuous structures without
any LRO appear below the transition temperature. Noticethat in the present simulation, the effect of fluctuation is notincluded except for the initial random distribution of segment potential. We remark that there is a resemblance between the sequence of patterns shown here and those of PME
obtained from the TEM micrographs19 as ␪ is varied. In order
to discuss the stability of PME, one needs to know, for example, the entropic contribution due to the random configuration of interfaces, which is balanced with their bending
energy.27 Of course, this is beyond the scope of the present
mean-field approach. However, our result with random initialconditions shows, at least in the static sense, that PME withrandom configuration of interfaces results from microphaseseparation as metastable equilibrium. Notice that even theexperimentally obtained PME may not necessarily be fully
equilibrated structure. Interestingly, many holes observed in
the A-rich and B-rich domains in the experiment 关Fig. 3共d兲 inRef. 19兴 are also reproduced such as in ␪ ⫽0.1 in Fig. 4 or
␪ ⫽0.3 in Fig. 5. These holes should be due to the emulsification of homopolymers by block copolymers.
In order to analyze the obtained patterns quantitatively,

Komura, Kodama, and Tamura

FIG. 5. Final patterns of system 3 in Table I. Notations are the same as thosegiven in Fig. 4.

we pay attention to the behavior of ⌬ given in 共2.5兲. First, we
define a discrete Fourier transform of ⌬(n) by⌬ 关 k兴 ⫽兺n ⌬ 共 n兲 exp共 ik•n兲 ,共4.1兲

with k⫽2 ␲ m/L, m苸 兵 0,1,...,L⫺1 其 2 , and L is the systemsize. Then, the structure factor is given by
S 共 k兲 ⫽ 具 ⌬ 关 k兴 ⌬ 关 ⫺k兴 典 ,共4.2兲

where the average is over the ensemble of systems. Figure 7
shows the circularly averaged structure factor S(k) (k⫽ 兩 k兩 ) of S3 for several values of ␪ ranging from ␪ ⫽0.2 to 1.
Structure factors for ␪ ⫽0 and 0.1 are not shown here because their peak heights are much larger than those plotted in
the graph. The peak height decreases as ␪ is increased at least
up to ␪ ⫽0.5. This reflects the fact that the contrast between
overall A and B segments becomes smaller. Above thisvalue, the peak height gradually increases up to ␪ ⫽1.
By increasing ␪ from 0 to 1, the peak position shifts
systematically to higher values of the wave vector. The peak
position characterizes the typical length scale of each pattern.
To see this in more detail, we calculate the 共inverse兲 characteristic length scale defined by51

具k典⫽

兺 k⫽0兩 k兩 ⫺1 S 共 k兲
兺 k⫽0兩 k兩 ⫺2 S 共 k兲

.

共4.3兲

In Fig. 8, 具 k 典 is plotted as a function of ␪ for S1, S2, and S3
simultaneously. Although the homopolymer lengths are dif-

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J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

Mean-field approach to ternary systems

9911

FIG. 7. Circularly averaged structure factor for different values of ␪ ofsystem 3 in Table I. ␪ ⫽0.2, 0.3,...,1.0 from left to right peaks.

cient to swell the microphase separated structure by choosing
larger ␣. In other words, the homopolymer chains penetrate
into the copolymer brushes for small ␣.14 However, it should
be noted that this tendency is true as long as the system is not
too close to the unbinding transition point at which thelamellar spacing diverges upon dilution by homopolymers.
Since we do not know of a systematic experimental
study on the polymeric ternary system, we compare our results with the experiment on the traditional microemulsions.
The structure of one-phase AOT/D2 O/decane microemulsions containing equal volumes of water and oil and a variable concentration of surfactant was investigated by Kotlarchyk et al. using SANS.55 They found that as the volume
fraction of AOT is increased from 0.18 to 0.42, the peak
position exhibits an approximately linear shift to larger values of wave vector, while the peak height rapidly diminishes.

FIG. 6. Part of the cross-section profiles of ⌽ and ⌿ for 共a兲 ␪ ⫽0.1; 共b兲 ␪⫽0.5; 共c兲 ␪ ⫽0.9 of system 3 in Table I.

ferent for these three systems, the values of 具 k 典 at ␪ ⫽0 are
the same. 共In principle, macrophase separation takes placefor 具 k 典 ⫽0. Finite value of 具 k 典 for ␪ ⫽0 in our simulation isdue to the finite system size effect.兲 For 0⬍ ␪ ⬍1, 具 k 典 islarger for relatively longer copolymer chain 共smaller ␣兲. This
result indicates that the homopolymers becomes more effi-

FIG. 8. The inverse characteristic length as a function of ␪ for systems 1, 2,
and 3 in Table I which have the same copolymer length but different homopolymer length.

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9912

J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

Komura, Kodama, and Tamura

TABLE II. Characterization of the symmetric polymeric ternary systems for the fixed homopolymer length case( ␣ ⭐1).
SystemN A ⫽N B

N AB

␣

␹

␹ N A⫽ ␹ N B

␹ N AB( ␪ L ,( ␹ N A ) L)

system 4 共S4兲system 5 共S5兲system 6 (S6⫽S3)

8
8
8

32
16
8

0.25
0.5
1

2
2
2

16
16
16

64
32
16

共1/9, 9/4兲共1/3, 3兲
共2/3, 6兲The latter observation is in accord with our results in Fig. 7
for ␪ ⬍0.5. However, the behavior of 具 k 典 depends on ␣ inthe polymeric systems 共see Fig. 8兲.
B. Fixed homopolymer length „ ␣ Ï1…

Next, we fix both of the homopolymer lengths to N A⫽N B ⫽8, and change the copolymer length N AB . Three different copolymer lengths are considered as in Table II, which
we refer to as systems 4, 5, 6 共or S4, S5, S6兲. Notice that S6
is identical to S3. The interaction parameter is fixed to ␹
⫽2, i.e., the degree of incompatibility is set to ␹ N A ⫽16.
The sequences of patterns of S4, S5, and S6 ( ␣ ⭐1) as afunction of ␪ are essentially similar to those of S1, S2, or S3.
However, the lamellar phase contains many defects forlonger copolymer chains. According to the SANS
experiments,56 the intermediate-segregation regime was
identified as ␹ N AB ⬇5 – 29. Hence, S4 and S5 belong to thestrong-segregation regime, whereas S6 (⫽S3) belongs to theintermediate-segregation regime. In the strong-segregation
regime, the defects are hardly eliminated once they are created. The 共inverse兲 characteristic length 具 k 典 is plotted as afunction of ␪ for S4, S5, and S6 in Fig. 9 as before. The
typical length scale is smaller for systems containing shorter
copolymer chains. This means that the copolymer length
dominates the overall structure.
For ␣ ⫽1, Janert and Schick predicted a three-phase region at which the symmetric lamellar phase coexists with theA- and B-rich phases for large values of ␹ N A . 18 However,

FIG. 9. The inverse characteristic length as a function of ␪ for systems 4, 5,
and 6 in Table II which have the same homopolymer length but different
copolymer length.

we have not observed this three-phase coexistence for ␣⫽1 in our simulation. This is because the energy difference
between the homogeneous lamellar phase and the threephase coexisting state is too small to detect within our numerical accuracy.
C. ␣ Ì1

We now show some results for the case of ␣ ⬎1. As
described in Sec. II, the Lifshitz point is preempted by thetricritical point beyond which there is a region of three-phasecoexistence between A-rich, B-rich, and disordered phases8
共see Fig. 3兲. The typical obtained patterns for system 7 共S7兲and system 8 共S8兲 are depicted in Figs. 10 and 11, respectively. Also see Table III. The difference between these twosystems is the value of Flory–Huggins interaction parameter␹. Compared to the case of ␣ ⬍1, the patterns in these cases
exhibit completely different sequences and the microphaseseparated bicontinuous structure does not appear. For intermediate ␪ ( ␪ ⫽0.3, 0.5, and 0.7 in Fig. 10兲, the system isFIG. 10. Final patterns of system 7 in Table III. Notations are the same as
those given in Fig. 4.

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J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

Mean-field approach to ternary systems

9913

from S2 for different ␪. Shown is configuration of ⌬ in grayscale where values ⫺1 and 1 correspond to white and black,respectively.
To characterize the geometrical properties of the surfaces, we measured various area averaged curvatures. So far,several methods have been proposed for the curvature determination of the bicontinuous structure.57–59 For the surface
defined implicitly by⌺ 共 x,y,z 兲 ⫽0,

共4.4兲

we use here convenient expressions for the mean and Gaussian curvatures60
H⫽1
关 ⌺ 共 ⌺ 2 ⫹⌺ z2 兲 ⫺2⌺ x ⌺ y ⌺ xy ⫹perm兴 ,2⌼ 3 xx yK⫽1关 ⌺ ⌺ ⌺ 2 ⫺⌺ 2xy ⌺ z2 ⫹2⌺ xz ⌺ x 共 ⌺ y ⌺ yz ⫺⌺ z ⌺ y y 兲⌼ 4 xx y y z⫹perm兴 .

共4.5兲

共4.6兲

Here
⌼⫽ 兩 ⵜ⌺ 兩 ,共4.7兲

and ⌺ i ⫽ ⳵ ⌺/ ⳵ r i , where r⫽(x,y,z), and ‘‘perm’’ indicatesthat two additional permutations of each term should be considered, i.e., one where (x,y,z)→(z,x,y) and another with
(x,y,z)→(y,z,x).
The results for the average mean curvature squared
具 H 2 典 , and for the average Gaussian curvature 具 K 典 for S1, S2,
and S3 in Table I are shown as a function of ␪ in Fig. 13.
Average over four different equilibrium configurations has
been taken for each data point. The average mean curvature
具 H 典 vanishes due to the symmetry of the system. We see that
the Gaussian curvature is negative, corresponding to a surface which is dominated by saddle-shaped configurations.
For small ␪, both 具 H 2 典 and 具 K 典 are small, corresponding to
the macrophase separated structure in the symmetric system.
For large ␪, on the other hand, the system is dominated by
the lamellar structure, and both curvatures take small values.
It is interesting to note that there exist maximum and minimum of 具 H 2 典 and 具 K 典 , respectively. Although it was difficult
to distinguish between the three systems clearly in this analysis, we can at least say that the value of ␪ which gives the
maximum of 具 H 2 典 or the minimum of 具 K 典 is smaller forsmaller ␣ as in S1.

FIG. 11. Final patterns of system 8 in Table III. Notations are the same as
those given in Fig. 4.

now in three-phase coexistence between A-rich, B-rich, andblock copolymer rich phases, although the copolymer region
is not microphase separated. For large ␪ ( ␪ ⫽0.9 in Fig. 10兲,the homogeneous disordered phase is obtained. Notice that
␹ N AB ⫽8 in S7 is smaller than ( ␹ N AB ) MST⬇10.5. These behaviors are consistent with the predicted phase diagram
given in Fig. 3.
By increasing ␹, the disordered AB copolymer region
starts to microphase separate, and the system is in anotherthree-phase coexistence between A-rich, B-rich, and lamellar
phases. This situation can be observed for the intermediate ␪in Fig. 11. It is interesting to note that the lamellar structurealigns parallel to the interface between other homogeneous
phases, and the shape of domain is polygonal.

E. Periodic initial conditions

Since it is probable that the patterns obtained so far may
correspond to metastable equilibrium, one should also pay
attention to the existence of equilibrium phases with LRO.
To check this point, we performed the same numerical cal-D. 3D case „ ␣ Ï1…

Here, we present some preliminary results from 3D
simulations. Figure 12 shows typical configurations obtained

TABLE III. Characterization of the symmetric polymeric ternary systems for the same ␹ N A and ␣ case ( ␣⬎1).
SystemN A ⫽N B

N AB

␣

␹

␹ N A⫽ ␹ N B

␹ N AB( ␪ TCP ,( ␹ N A ) TCP)

system 7 共S7兲system 8 共S8兲

8
8

4
4

2
2

2
4

16
32

8
16

共4/5, 10兲
共4/5, 10兲

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9914

J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

Komura, Kodama, and Tamura

FIG. 13. 共a兲 Average mean curvature squared 具 H 2 典 and 共b兲 average Gaussian curvature squared 具 K 典 for systems 1, 2, and 3 in Table I which have thesame copolymer length but different homopolymer length.

FIG. 12. Typical 3D final configuration for 共a兲 ␪ ⫽0.1; 共b兲 ␪ ⫽0.6; 共c兲 ␪⫽0.9 of system 2. Shown is configuration of ⌬ in gray scale where values⫺1 and 1 correspond to white and black, respectively.

culations starting with the assumed 1D periodic potential
fields and generated artificial lamellar structures. The parameters used in this case are listed in Table IV. Four different
systems are considered as given from system 9 共S9兲 to sys-

tem 12 共S12兲. Notice that ␣ ⬎1 in S12. We then obtained thefully equilibrated lamellar phase by minimizing the free energy F m in 共3.10兲 with respect to the lamellar spacing.61,62
The resulting free energy F m is shown as a function of ␪ in
Fig. 14共a兲. In Fig. 14共b兲, we plotted the deviation of the freeenergy from the tangent at ␪ ⫽0.05 for each curve in order tomagnify its curvature. By carefully examining Fig. 14共b兲, wefind that there are three distinct regions for each curve; region 1 共R1兲 where the profile is flat, region 2 共R2兲 where thecurvature of the profile, ⳵ 2 F m / ⳵ ␪ 2 , is negative, and region 3
共R3兲 where the curvature is positive. These different regions
are indicated in Fig. 14共b兲 for S12. One should note that only
R3 is locally stable.
Physically speaking, the loss of configurational entropyof the homopolymer confined between the monolayers
causes an attraction, while its translational entropy produces
a repulsion. From the curvature of the free energy, we candiscuss the effective interactions between the copolymer
monolayers. The flat part of the free energy in R1 indicatesthat the local stability of the lamellar phase is marginal. In
this region, arbitrary number of phases with different ␪ cancoexist without any free-energy cost. This means that the

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J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

Mean-field approach to ternary systems

9915

TABLE IV. Characterization of the symmetric polymeric ternary systems starting with 1D periodic initialconditions. * is the value for the tricritical point.
SystemN A ⫽N B

N AB

␣

␹

␹ N A⫽ ␹ N B

␹ N AB( ␪ L ,( ␹ N A ) L)

system 9 共S9兲system 10 共S10兲
system 11 共S11兲
system 12 共S12兲

20
40
80
160

80
80
80
80

0.25
0.5
1
2

0.2
0.2
0.2
0.2

4
8
16
32

16
16
16
16

共1/9, 9/4兲共1/3, 3兲
共2/3, 6兲
(4/5,10) *copolymer monolayers do not interact with each other, andany separation distance between the monolayers within thecorresponding range is allowed. Similar behavior has been
predicted for a highly swollen lamellar phase in binary
blends.63,64
For a certain region of small ␪, we also find that ␾ AB in
the bulk region is independent of ␪ (⬇0.01 for all the systems in Table IV兲 in R1. This value of ␾ AB should coincidewith the value of ␪ at which the lamellar periodicity diverges
共unbinding transition兲17,18 and the free-energy curve issmoothly connected to that of two-phase state. Since we areconsidering the intermediate-segregation regime ( ␹ N AB⫽16), ␪ ⬇0.01 is much smaller than the Lifshitz volume
fraction ␪ L . We also point out that R1 is larger for relatively
shorter copolymer chain. This is consistent with the result in
Fig. 8, where the typical length scale increases for larger ␣.
Negative curvature in R2 results from the attractive interactions between the monolayers.63,64 In this region, thelamellar phases in R1, R2 and a part of R3 are thermodynamically unstable, and macrophase separation into thelamellar, A-rich and B-rich phases 共determined by a common
tangent construction兲 should take place.8,17,18 This unstableregion hardly exists for S9 and S10, whereas it is remarkablefor S12. This fact is consistent with the occurence of thethree-phase coexistence shown in Fig. 11. Notice that theattractive interaction between the monolayers is present in
the patterns of Fig. 5.
Similar to our work, Thompson and Matsen calculated
the effective interactions between the monolayers by using
the Fourier-space SCFT.28 From the condition that the attraction becomes sufficiently small, they concluded that there isan optimum value of ␣ ⬇0.8 in order to emulsify the homopolymers. They further insist that emulsification efficiency is enhanced by introducing the copolymer
polydispersity.29
V. SUMMARY AND CONCLUSION

In this paper, we have investigated the real-space structure of ternary blends of A and B homopolymers and symmetric AB diblock copolymer using an extended lattice 共realspace兲 SCFT by means of numerical calculations. We
restricted ourselves to the symmetric case where 共i兲 both A
and B homopolymers have the same length; 共ii兲 the AB
block copolymer is symmetric; 共iii兲 the average volume fractions of A and B homopolymers are equal. We performed asystematic study to see how the copolymer fraction ␪ and therelative chain lengths ␣ ⫽N A /N AB affect the microphaseseparated structure. Starting with randomly generated initialpotential fields, i.e., without assuming any symmetry of the

FIG. 14. 共a兲 The free energy F m as a function of ␪ for four different systems,
and 共b兲 its deviation from the tangent at ␪ ⫽0.05. 共c兲 is the same plot with共b兲 magnified in the vertical axis.

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9916

J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

mesophase a priori, we obtained various bicontinuous structures for ␣ ⭐1. For the fixed copolymer length, the typicallength scale of the microphase separated structure becomes
smaller for relatively shorter homopolymer chains 共small ␣兲.
This means that the homopolymers becomes more efficientto swell the microphase separated structure for longer homopolymer chains 共large ␣兲. Some preliminary results of 3D
simulations indicate that the average Gaussian curvature isnegative corresponding to a surface which is dominated bysaddle-shaped configurations. The bicontinuous patterns obtained for ␣ ⭐1 resemble the TEM images of PME foundexperimentally by Bates et al. Our result with random initialconditions shows, at least in the static sense, that PME withrandom configuration of interfaces results from microphaseseparation as metastable equilibrium.
The situation is different for ␣ ⬎1. In this case, the Lifshitz point is preempted by the tricritical point within themean-field theory, and the three-phase coexistence between
either the disorder, A-rich and B-rich phases or between thelamellar, A-rich and B-rich phases occurs. Here, the block
copolymer rich phase can either be disordered or microphase separated into the lamellar structure depending on thestrength of the interaction parameter.
Since the obtained bicontinuous structures may correspond to metastable equilibrium, we have also performed thesame numerical calculation starting from the initial potential
fields with an imposed lamellar symmetry. Detailed freeenergy analysis revealed that there is a region where thestability of the lamellar phase is marginal. This means thatthe copolymer monolayers do not interact each other in thecorresponding region.
We finally comment that the metastable irregular defectstructures obtained in 2D or 3D cases are due to the initiallyprepared random potential fields. In order to avoid such irregularities in real-space mean-field calculations, one can apply various tricks, such as by adding noise, adding external
fields, or using a more realistic dynamics.35 These methods
will be included in our future calculations especially for 3D
case.
Further developments of the present study, such as,asymmetric ternary system 共asymmetry in ␣ A and ␣ B , asym¯ A and ␾¯ B , asymmetry in the block copolymer,
metry in ␾
etc.兲, homopolymer/copolymer binary system,63,64 copolymer/copolymer binary system,65,66 or different copolymer structure 共e.g., ABC triblock copolymer67兲 are in
progress.
ACKNOWLEDGMENTS

This work is supported by the Ministry of Education,
Science and Culture, Japan 共Grant-in-Aid for Scientific Research No. 09740318兲. S.K. thanks the Ministry of Education, Science and Culture, Japan for providing financial support during his visit to Weizmann Institute of Science, Israel.
Support from the exchange program between the Japan Society for the Promotion of Science 共JSPS兲 and the Israel
Ministry of Science and Technology is also gratefully acknowledged. H.K. acknowledges the support from the Ministry of Education, Science and Culture, Japan 共Grant-in-Aid

Komura, Kodama, and Tamura

for Scientific Research No. 13554001兲. We thank T.
Kawakatsu, D. Andelman, and H. Orland for useful discussions.

APPENDIX A: EXTENDED LATTICE FORMULATIONOF SCFTIn this Appendix, we give the detailed formulation of theextended lattice SCFT for the completeness of the present
paper. All of the self-consistent equations in Sec. III B are
derived here.
Some of the notations are repeated first. We denote thetype of molecule by i, and the type of segment by J and K.
Segments in a chain are indexed by N ⬘ ⫽1,2,...,N i , whereN i is the polymerization index of the ith molecule. The molecules of type i are indexed by p⫽1,2,...,n i , where n i is thetotal number of ith molecule.
We start to write the total Hamiltonian of the system asĤ共 ⌫ 兲 ⫽Ĥ0 共 ⌫ 兲 ⫹Ŵ共 ⌫ 兲 ,

共A1兲

(i)

where ⌫⬅ 兵 np,N ⬘ 其 represents the configuration space. The
first term is the ideal Hamiltonian describing independent
polymer chains represented as random walks, and is given byni

exp关 ⫺Ĥ0 共 ⌫ 兲兴 ⫽N i -1

兿i p⫽1
兿 N兿⫽1 ␦ 关 兩 n(i)p,N ⬘⫹1 ⫺n(i)p,N ⬘兩 ,1兴 ,⬘共A2兲

where the Kronecker delta function is defined in Sec. III B.
Next, the interaction Hamiltonian Ŵ(⌫) is assumed to to be
writtenasadensityfunctional
form
Ŵ(⌫)
ˆ J (n;⌫) 其 兴 , where⫽W关 兵 ␾ni

␾ˆ J 共 n;⌫ 兲 ⫽Ni

兺i p⫽1
兺 N兺⫽1 ␦ 关 n,n(i)p,N ⬘ 兴 ␦ 关 J, f i共 N ⬘ 兲兴⬘

共A3兲is the microscopic representation of the segment density
field. The ‘‘type function’’ f i (N ⬘ ) is also introduced in Sec.
III B. The functional form of W is given byW关 兵 ␾ J 共 n兲 其 兴 ⫽12

␹ JK ␾ J 共 n兲 具具␾ K 共 n兲 典典.
兺n 兺
J,K

共A4兲

In the above, ␹ JK is the Flory–Huggins parameter between J
and K segments, and 具具X(n) 典典 denotes the average of X(n)over the nearest-neighbor lattice sites

具具X 共 n兲 典典⫽

1z␦ 关 兩 n⫺n⬘ 兩 ,1兴 X 共 n⬘ 兲 ,
兺
n
⬘

共A5兲

where z is the number of the nearest-neighbor sites (z⫽6 fora cubic lattice兲. At the end of this Appendix, we will extend
this definition.
Under the incompressibility condition兺J ␾ J共 n兲 ⫽1,

共A6兲

the partition function is calculated by

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J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

Z⫽

Mean-field approach to ternary systems

1exp关 ⫺Ĥ共 ⌫ 兲兴
兿 in i! 兵⌫其

兺

⫽

冕 ␾ 冕
D关

J 共 n 兲兴冋兺 冉
u 共 n兲

nni
␾ i,N ⬘ 共 n兲 ⫽
␦ 关 n,nN(i)⬘ 兴
Zi 兵 ⌫ i 其

兺

册

冋兺兺

冊册

1
exp关 ⫺Ĥ0 共 ⌫ 兲 ⫺Ŵ⬘ 共 ⌫ 兲兴 ,
兿 in i! 兵⌫其

兺

共A8兲

Q i 共 n0 ,N 0 ;n1 ,N 1 兲
⫽

ni

⬘

1

再兿

N 1 -1兺⍀z 兩 N 1 ⫺N 0 兩 兵 ⌫ i 其N ⬘ ⫽N 0冋兺兺Ni

兺J 兺i p⫽1
兺 N兺⫽1␦ 关 兩 nN(i)⬘ ⫹1 ⫺nN(i)⬘ 兩 ,1兴

N1

␦ 关 J, f i 共 N ⬘ 兲兴 U J 共 n(i)p,N ⬘ 兲 . 共A9兲

⫻exp ⫺

From 共A7兲 we obtain the mean-field free energy asN ⬙ ⫽N 0J␦ 关 J, f i 共 N ⬙ 兲兴 U J 共 nN(i)⬙ 兲册冎⫻ ␦ 关 n0 ,nN(i) 兴 ␦ 关 n1 ,nN(i) 兴 .
0F 关 ␾ J ,U J 兴 ⫽⫺log Ze关 U J 兴 ⫹W关 ␾ J 兴⫺册␦ 关 J, f i 共 N ⵮ 兲兴 U J 共 nN(i)⵮ 兲 .

N ⵮ ⫽1J冎Up to now, the problem has been reduced to that of asingle chain statistics under the external potential U J (n)共A11兲 with the incompressibility condition 共A6兲. This problem should be solved in a self-consistent manner. In the ith
chain, let us define the following two-point correlation function for the positions of the segments N ⬘ ⫽N 0 and N 1 :withŴ⬘ 共 ⌫ 兲 ⫽N ⬙ ⫽1␦ 关 兩 nN(i)⬙ ⫹1 ⫺nN(i)⬙ 兩 ,1兴

共A16兲
共A7兲

.

Here, u(n) is a Lagrange multiplier to incorporate the constraint of the incompressibility condition 共A6兲, and Ze关 U J 兴 isa partition function of the chains interacting with the externalpotential field 兵 U J (n) 其 , i.e.,Ze关 U J 兴 ⫽再兿N i -1Ni⫻exp ⫺

兺 ␾ J共 n兲 ⫺1
J

共A15兲

where the single segment distribution is

D关 U J 共 n兲兴兺n 兺J U J共 n兲 ␾ J共 n兲

⫻exp ⫺兺 ␦ 关 J, f i共 N ⬘ 兲兴 ␾ i,N ⬘共 n兲 ,N ⬘ ⫽1i⫻exp log Ze关 U J 兴 ⫺W关 ␾ J 兴⫹Ni

␾ J 共 n兲 ⫽ 兺

冋

9917

兺n 兺J U J共 n兲 ␾ J共 n兲 ,

共A10兲

共A17兲

1

Notice that ⍀z N⫺1 is the partition function of a single ideal
chain of polymerization index N. We next define the forward
and backward end-segment statistical weights as
G i 共 n,N ⬘ 兲 ⫽⍀together with the saddle-point conditions

Q i 共 n0 ,1;n,N ⬘ 兲 ,
兺
n

共A18兲

Q i 共 n,N ⬘ ;n1 ,N i 兲 ,
兺
n

共A19兲

0

U J 共 n兲 ⫽u 共 n兲 ⫹兺K ␹ JK 具具␾ K共 n兲 典典,

共A11兲

and
G †i 共 n,N ⬘ 兲 ⫽⍀and1

␾ J 共 n兲 ⫽⫺1 ␦ Ze关 U J 兴.
Ze ␦ U J 共 n兲

共A12兲

Inserting 共A2兲 and 共A9兲 into 共A8兲, we get
Ze关 U J 兴 ⫽兿i1
共 Z 关 U 兴 兲 ni,
n i! i J

共A13兲

再N i -1

兺 N兿⫽1 ␦ 关 兩 nN(i)⬘⫹1 ⫺nN(i)⬘兩 ,1兴

兵⌫i其

⬘

冋 兺兺
Ni

⫻exp ⫺JN ⬙ ⫽1冎␾ i,N ⬘ 共 n兲 ⫽n i G i 共 n,N ⬘ 兲 G †i 共 n,N ⬘ 兲exp关 ⫺U f i (N ⬘ ) 共 n兲兴 兺 n⬘ G i 共 n⬘ ,N i 兲

.

共A20兲

It is straightforward to see that end-segment statisticalweights satisfy the following recursive relations:where Zi 关 U J 兴 is the single chain partition function
Zi 关 U J 兴 ⫽respectively. The single segment distribution ␾ i,N ⬘ (n) in
共A16兲 can be rewritten in terms of the end-segment statisticalweights as

G i 共 n,N ⬘ 兲 ⫽exp关 ⫺U f i (N ⬘ ) 共 n兲兴 具具G i 共 n⬘ ,N ⬘ ⫺1 兲 典典,

共A21兲

G †i 共 n,N ⬘ 兲 ⫽exp关 ⫺U f i (N ⬘ ) 共 n兲兴 具具G †i 共 n⬘ ,N ⬘ ⫹1 兲 典典, 共A22兲册␦ 关 J, f i 共 N ⬙ 兲兴 U J 共 nN(i)⬘ 兲 ,共A14兲

with ⌫ i ⬅(nN(i) ¯nN(i) ) representing the configuration space of01single ith molecule. By substituting 共A13兲 and 共A14兲 into
共A12兲, ␾ J (n) can be expressed as

together with the initial conditions
G i 共 n,1兲 ⫽exp关 ⫺U f i (1) 共 n兲兴 ,共A23兲

G †i 共 n,N i 兲 ⫽exp关 ⫺U f i (N i ) 共 n兲兴 .

共A24兲

By comparing from 共A17兲 to 共A19兲 with 共A14兲, we see thatthe single chain partition function can be rewritten as

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9918

J. Chem. Phys., Vol. 117, No. 21, 1 December 2002

Zi 关 U J 兴 ⫽z N i

兺n G i共 n,N i 兲 ⫽z N 兺n G †i 共 n,1兲 .

共A25兲

i

The unmixed pure system is taken as a reference system
in which all the segment interactions are switched off. The
free energy of such a reference system is
F 0 ⫽⫺log Z0 ,Komura, Kodama, and Tamura

APPENDIX B: ITERATION SCHEME

Here, we describe the iteration scheme proposed by Hasegawa to solve the set of self-consistent equations.30 A similar Picard-type iteration scheme has been used by others.34,68
The new estimates for 兵 U J (n) 其 are calculated from theprevious ones according to the following rule:

共A26兲

U (T⫹1)
共 n兲 ⫽U (T)J
J 共 n 兲 ⫹c 1

with the partition function
Z0 ⫽

兿i

共 ⍀ iz Ni兲ni
.
n i!

共A27兲

Here, ⍀ i ⫽n i N i is the total number of lattice sites occupiedby the pure system consisting of the ith molecule. From
共A10兲 and 共A26兲, we finally calculate the mixed free energy
F m with respect to the reference system as
F m关 ␾ J ,U J 兴 ⫽F 关 ␾ J ,U J 兴 ⫺F 0
⫽

兺i

⫹c 2

冉

n

␹ JK ␾ J 共 n兲 具具␾ K 共 n兲 典典
兺兺
J,K

ni

exp关 ⫺Ĥ0 共 ⌫ 兲兴 ⫽⫹兿i p⫽1
兿 N兿⫽1 ␦ 关 兩 n(i)p,N ⬘⫹1 ⫺n(i)p,N ⬘兩 2 ,1兴
⬘

1
(i)
(i)
⫹ ␦ 关 兩 np,N ⬘ ⫹1 ⫺np,N ⬘ 兩 2 ,2兴
2
⫹

冊

1
(i)
2
␦ 关 兩 n(i)p,N ⬘ ⫹1 ⫺np,N ⬘ 兩 ,3兴 .
6

共A29兲

This is an important extension in our work to deal with isotropic structures. According to 共A29兲, the definition of theneighborhood average should be modified as51

具具X 共 n兲 典典⫽ 兺
n⬘

⫹

冋

6
3
␦ 关 兩 n⫺n⬘ 兩 2 ,1兴 ⫹ ␦ 关 兩 n⫺n⬘ 兩 2 ,2兴
80
80

册

1
␦ 关 兩 n⫺n⬘ 兩 2 ,3兴 X 共 n⬘ 兲 ,
80

⫺

共A28兲

冉

N i -1

共A30兲

which is used in our simulations.
Equations 共A6兲, 共A11兲, 共A15兲, 共A20兲, 共A21兲, 共A22兲,
共A23兲, 共A24兲, and 共A30兲 form a set of self-consistent equations given in Sec. III B.
We finally comment that a different definition of theideal Hamiltonian gives rise to different Kuhn’s statisticalsegment length b K . For 共A5兲, we have b K⫽a, whereas for
共A29兲, we have b K⫽(33/20)a.

册兺 L 共 U L(T) 共 n兲 ⫺ 兺 M ␹ LM 具具␾ (T)M 共 n 兲 典典兲
兺 L1

冋冏 兺
冏
兺冏
冉 兺

⑀⫽ 兺

We now extend the present lattice SCFT formulation inorder to avoid the geometric constraints associated with alattice. For this purpose, we redefine the ideal Hamiltonian in
共A2兲 as␾ (T)J 共 n 兲 ⫺1冊册兺K ␹ JK 具具␾ K(T)共 n兲 典典 ,共B1兲where c 1 and c 2 are positive constants and T⫽0,1, . . . .
This iteration scheme is continued until the quantity ⑀ defined by

n iN in i log兺 nG i 共 n,N i 兲兺n 兺J U J共 n兲 ␾ J共 n兲 .

J

⫺ U (T)J 共 n兲 ⫺

1
⫹
2 n
⫺

冋

冋兺

J

2

␾ (T)
J 共 n 兲 ⫺1

兺 L 共 U L(T) 共 n兲 ⫺ 兺 M ␹ LM 具具␾ (T)M 共 n 兲 典典兲
兺 L1

J

U (T)
J 共 n兲 ⫺K

␹ JK 具具␾ K(T) 共 n兲 典典

冊冏 册
2

,

共B2兲

becomes less than 10⫺6 . The time step required for the convergence strongly depends on the choice of c 1 and c 2 .
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