EUROPHYSICS LETTERS

1 January 2001

Europhys. Lett., 53 (1), pp. 46–52 (2001)

Mean-ﬁeld approach to polymeric microemulsions
H. Kodama 1,2 , S. Komura 3 and K. Tamura 3
1
Japan Chemical Innovation Institute, Nagoya University
Nagoya 464-8601, Japan
2
Yokohama Research Center, Mitsubishi Chemical Corporation
Yokohama 227-8502, Japan
3
Faculty of Computer Science and Systems Engineering,
Kyushu Institute of Technology - Iizuka 820-8502, Japan(received 9 November 1999; accepted in ﬁnal form 20 October 2000)
PACS. 64.75.+g – Solubility, segregation, and mixing; phase separation.
PACS. 82.20.Wt – Computational modeling; simulation.
PACS. 82.70.Kj – Emulsions and suspensions.
Abstract. – Equilibrium structure of ternary blends of A and B homopolymers and symmetricAB block copolymer is investigated using an extended lattice self-consistent ﬁeld theory bymeans of numerical calculations. We demonstrate that bicontinuous polymeric microemulsions
result from microphase separation for relatively longer copolymer chain. Detailed free energy
analysis revealed that the stability of the lamellar phase is marginal in the corresponding region.
For relatively shorter copolymer chain, 3-phase coexistence between the lamellar, A-rich andB-rich phases is observed.

Introduction. – Blends of homopolymers and block copolymers have been stimulating aconsiderable amount of experimental and theoretical studies because of the richness in theirphase behavior as well as their technological importance [1]. Such systems are known toform either mesoscopically structured phases with long-range order (LRO) or a homogeneous
phase depending on the temperature or the composition. Recently, Bates et al. found a
new bicontinuous phase without any LRO in a ternary blend of A and B homopolymers ofequal sizes and a symmetric AB diblock copolymer [2, 3]. This phase has been termed as“polymeric microemulsion” (PME) and its physical origin is of great interest and importanceto understand.
In the above experiment, the average volume fraction of the copolymer (denoted by θ)is changed, while the volume fractions of the two homopolymers are kept to be equal. Thissystem exhibits both macrophase and microphase separations into 2-phase and lamellar 1phase for small and large θ, respectively. PME has been observed for intermediate θ but
within a narrow region close to the mean-ﬁeld Lifshitz volume fraction θL [4–6]. Bates et al.
assumed that, due to the strong ﬂuctuations, a singular cusp in the transition temperature is
induced around θL , and PME is regarded as a strongly ﬂuctuating disordered phase [2, 3].
From the theoretical point of view, the appearance of PME in the polymeric ternary
system has not yet been well understood, although various structures with LRO have beensuccessfully explained within the self-consistent ﬁeld theory (SCFT) [7–9]. In these works,
the free energies of several periodic structures were calculated by assuming their mesophasesymmetry a priori. Hence this approach is not suitable for dealing with structures without
any LRO such as PME. On the other hand, there is another approach in which the SCFTc EDP SciencesH. Kodama et al.: Mean-field approach to polymeric microemulsions

47

equations are solved numerically without assuming the symmetry of the structure [10, 11].
Although this method generates the structures corresponding to either stable or metastableequilibrium, it has proved to be a powerful tool in ﬁnding new mesophases.
In this letter, using the extended lattice SCFT method with randomly generated initial potential ﬁelds, we demonstrate that the bicontinuous structure of PME results from microphaseseparation for relatively longer copolymer chain. However, since the obtained PME may correspond to a metastable equilibrium, we have also performed the same numerical calculation
starting from the initial potential ﬁelds with an imposed lamellar symmetry. By examining
the free energy of the lamellar phase as a function of θ, we ﬁnd that there is a region where
the stability of the lamellar phase is marginal. This means that the copolymer monolayers do
not interact each other in the corresponding region.
Lattice SCFT and simulation method. – We ﬁrst describe the lattice SCFT for general
copolymer mixtures [12]. The space coordinate of each segment composing the copolymers is
speciﬁed by a lattice point n = (nx , ny , nz ) in an Lx × Ly × Lz cubic lattice box with a lattice
constant b with periodic boundary conditions. We denote the type of molecule by i, and thetype of segment by J and K. Segments in a chain are indexed by N  = 1, · · · , Ni , where
Ni is the polymerization index of the i-th molecule. The molecules of type i are indexed by
a = 1, · · · , ni , where ni is the total number of the i-th molecule. The conﬁguration space is(i)denoted by Γ ≡ {na,N  }. In order to explain how we extended the lattice SCFT, we start with
the Hamiltonian description. Throughout this letter, all the energies are expressed in units of
kB vT , where kB and T represent the Boltzmann constant and the temperature, respectively.
The Hamiltonian of the system is composed of two parts:
Ĥ(Γ) = Ĥ0 (Γ) + Ŵ(Γ),(1)

where the ﬁrst and the second terms represent the ideal and the interaction Hamiltonians,
respectively. The ideal Hamiltonian Ĥ0 describes independent polymer chains represented as
random walks, and is given by   (i)(i) (2)exp −Ĥ0 (Γ) =δ na,N  +1 − na,N   , 1 ,i,a,N where δ[, ] is a Kronecker delta-function. The interaction Hamiltonian Ŵ(Γ) is assumed to
(i)have the form of Ŵ(Γ) = W[{φ̂J (n; Γ)}]. Here φ̂J (n; Γ) = i,a,N  δ[n, na,N  ] δ[J, fi (N  )] is
the microscopic representation of the segment density ﬁeld, where fi (N ) is a “type function”
representing the type of the segment N  in the chain i. The functional form of W is given by
W[{φJ (n)}] =1 
χJK φJ (n) φK (n) ,
2 n

(3)

J,K

where χJK is the Flory-Huggins interaction parameter and X(n) denotes the average ofX(n) over the nearest-neighbor lattice sites, i.e., X(n) = (1/6) n δ[|n − n |, 1] X(n ).
The partition function is calculated by Z = (1/ i ni !) Γ exp[−Ĥ(Γ)]. As a reference
state to compare the energy, we choose a pure system in which the segment interactions
areoﬀ. Hence the partition function of this reference system is given by Z0 =
 switched
Ni −1 ninNz/ni !, where z is the coordination number.
i iiUsing the mean-ﬁeld approximation, we can express the free energy relative to this reference state as a functional of the segment density {φJ (n)} and the segmental potential ﬁelds

48

EUROPHYSICS LETTERS{UJ (n)} conjugate to {φJ (n)},F = − log

Z
Z0

≈−




ni login Gi (n, Ni )ni Ni

+ W[{φJ (n)}] −JUJ (n) φJ (n),(4)

nwhich should be minimized under a suitable constraint. In the above, we have introduced thesingle-chain statistical weight Gi (n, N ) for a chain i under the segment potential {UJ (n)}starting from the free segment N  = 1 and having the segment N  = N at n. Similarly,
the conjugate weight G†i (n, N ) is deﬁned for a chain having the free segment N  = Ni .
Minimizing F withrespect to both {φJ (n)} and {UJ (n)} subjected to the constraint of local
incompressibility, J φJ (n) = 1, we obtain the set of SCFT equations as
UJ (n) =δW[{φJ (n)}]+ u(n) =χJKδφJ (n)φK (n)+ u(n),(5)ni Gi (n, N  ) G†i (n, N  )δ[J, fi (N  )].
exp[−Ufi (N  ) (n)] n Gi (n, Ni )(6)K

φJ (n) =Ni i

N  =1

In eq. (5), u(n) is a Lagrange multiplier which ensures the incompressibility condition. In thepractical numerical procedure, we use the following recursive relations to calculate Gi (n, N  )and G†i (n, N  ):Gi (n, N  ) = exp[−Ufi (N  ) (n)] Gi (n, N  − 1) ,(7)

G†i (n, N  ) = exp[−Ufi (N  ) (n)] G†i (n, N  + 1) ,(8)with the initial conditions Gi (n, 1) = exp[−Ufi (1) (n)] and G†i (n, Ni ) = exp[−Ufi (Ni ) (n)].
We now extend the present lattice SCFT formulation in order to avoid the geometricconstraints associated with a lattice. For this purpose, we redeﬁne the ideal Hamiltonian ineq. (2) as [13]
exp −Ĥ0 (Γ) =
i,a,N 

2

 (i)
(i) 
δ na,N  +1 − na,N   , 1 +2
2
1  (i)1  (i)
(i) 
(i) 
+ δ na,N  +1 − na,N   , 2 + δ na,N  +1 − na,N   , 3
2
6

.

(9)

This is an important extension in our work to deal with isotropic structures such as PME.
According to eq. (9), the deﬁnition of the neighborhood average should be modiﬁed as
X(n)

=


n

6
3
1
δ[|n − n |2 , 1] +
δ[|n − n |2 , 2] +
δ[|n − n |2 , 3] X(n ).
80
80
80

(10)

The above self-consistent set of equations are solved numerically by employing the iteration
scheme used in refs. [10, 11, 14]. We ﬁrst give an initial guess for {UJ (n)} which can be either
random or periodic in space. Then by using eqs. (7) and (8), the single-chain statisticalweights Gi (n, N  ) and G†i (n, N  ) are calculated, and the density ﬁelds {φJ (n)} are evaluatedby eq. (6). We ﬁnally update {UJ (n)} according to both eq. (5) and the incompressibility
condition. We repeatthese steps until the variance of errors both in the multiplier u(n) andin the local density J φJ (n) becomes less than 10−8 . Notice that the present algorithm is49

H. Kodama et al.: Mean-field approach to polymeric microemulsions

Table I – Characterization of the polymeric ternary systems. The parameter values in parentheses
are for 1D systems whereas the others are for 2D systems (see the text).
SystemαχN

System 1 (S1)0.25

2 (0.2)2 (20)

System 2 (S2)0.5

2 (0.2)4 (40)

System 3 (S3)1

2 (0.2)8 (80)

System 4 (S4)22 (0.2)16 (160)NABθL(χN )L

8 (80)

1/9

9/4

8 (80)

1/3

3

8 (80)

2/3

6

8 (80)––

not intended to mimic the 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.
We consider symmetric ternary systems composed of equal-size A and B homopolymers
and a symmetric AB diblock copolymer. The common degree of polymerization of A and B
homopolymers is denoted by N and that of the block copolymer by NAB . Such systems arecharacterized by two parameters, i.e., α = N/NAB and χNAB (χ = χAB ). We change thecopolymer volume fraction θ, keeping the volume fractions of the two homopolymers equal.
Within the mean-ﬁeld level, the isotropic Lifshitz point is located at θL = 2α2 /(1 + 2α2 )and (χN )L = 2(1 + 2α2 ) [5]. As listed in table I, we consider four diﬀerent values of α, but
keeping χNAB = 16 which corresponds to the intermediate-segregation regime of pure diblock
copolymers. We also ﬁx the copolymer length, whereas the A and B homopolymer lengths
are changed systematically. Notice that the Lifshitz point appears only for α ≤ 1 since it ispre-empted by the tricritical point for α > 1 [5].
First, we numerically solve the set of SCFT equations starting with randomly generated
potential ﬁelds. This simulation is performed in 2D systems (Lx = Ly = 256, Lz = 1). Since
the obtained patterns may depend on the initially prepared potential ﬁelds, we next study 1D
systems starting with 1-dimensional periodic potential ﬁelds using the parameters written in
the parentheses of table I. (We chose diﬀerent values for 1D and 2D calculations in order tomaximize the computational eﬃciency for large 2D systems.) The latter case yields lamellar
structures with a well-deﬁned periodicity.
Random initial conditions. – Let us denote the volume fractions of A and B homopolymers by φA and φB , respectively, and that of AB block copolymer by φAB . Moreover, thevolume fractions of A and B blocks in the copolymer is represented by φa and φb , respectively. We then introduce three order parameters deﬁned by Φ = φA − φB , Ψ = φa − φb , and∆ = φA + φa − φB − φb . Typical equilibrium patterns of Φ and Ψ starting with randomly
generated potential ﬁelds are shown in ﬁg. 1 for S2 (α = 0.5) and S4 (α = 2). The patterns
obtained for S1 and S3 are qualitatively similar to those of S2 although they are diﬀerent
from the patterns of S4. For θ = 0.1 of S2 most of the copolymers are located at the interface
between A-rich and B-rich domains, and form saturated monolayers without any LRO. As θ
is increased to 0.4 in S2 the amount of interface increases, but LRO still does not exist. Notice that this pattern which results from microphase separation resembles the experimentallyobserved PME structure [2]. For θ = 0.9 of S2, the system exhibits a lamellar structure (withdefects) which is swollen by the small fraction of homopolymer chains. On the other hand,
S4 (α = 2) undergoes macrophase separation and we observe 3-phase coexistence between
the copolymer-rich lamellar phase and the A-rich and B-rich homogeneous phases. This isclearly seen for θ = 0.4 of S4. Such a dependence on α is entirely consistent with the previous
theoretical predictions in refs. [5, 8].

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EUROPHYSICS LETTERS

Fig. 1 – Typical 2D equilibrium patterns of System 2 (α = 0.5) and System 4 (α = 2.0) for three
diﬀerent values of θ. Φ = φA − φB and Ψ = φa − φb . The dark area denotes the region of higher
values.

For systems with α ≤ 1, we have calculated the following (inverse) characteristic length ofeach pattern:

−1
S(k)
k=0 |k|
k =
.
(11)
−2 S(k)|k|k=0Here S(k) = ∆[k]∆[−k] is the structure function of all the A and B monomers, and ∆[k] =k ∆(k) exp[ik · n] is the discrete Fourier transform of ∆(k) with k = 2πn/L and n ∈{0, 1, · · · , L − 1}2 . In ﬁg. 2, k is plotted as a function of θ for diﬀerent systems. We seethat k is larger for relatively shorter homopolymer chain by which the penetration of thecopolymer brushes becomes more eﬃcient [15].

Fig. 2 – The inverse characteristic length k as a function of θ for diﬀerent systems.

H. Kodama et al.: Mean-field approach to polymeric microemulsions

51

Periodic initial conditions. – Since it is probable that the patterns in ﬁg. 1 may correspond to metastable equilibrium, one should also pay attention to the existence of equilibriumphase with LRO. To check this point, we performed the same numerical calculations staring with the assumed 1-dimensional periodic potential ﬁelds and generated artiﬁcial lamellar
structures. We then obtained the fully equilibrated lamellar phase by minimizing the freeenergy F with respect to the lamellar spacing [16]. The resulting free energy F is shown as
a function of θ in ﬁg. 3(a). In ﬁg. 3(b), we plotted the deviation of the free energy from thetangent at θ = 0.05 for each curve in order to magnify its curvature. By carefully examining
ﬁg. 3(b), we ﬁnd that there are three distinct regions for each curve; Region 1 (R1) where
the proﬁle is ﬂat, Region 2 (R2) where the curvature of the proﬁle (∂ 2 F/∂θ2 ) is negative, andRegion 3 (R3) where the curvature is positive. These diﬀerent regions are indicated in ﬁg.
3(b) for S4.
One should note that only R3 is locally stable. On the other hand, the ﬂat part of the freeenergy in R1 indicates that the local stability of the lamellar phase is marginal. In this region,an arbitrary number of phases with diﬀerent θ can coexist without any free energy cost. Thismeans that the copolymer monolayers do not interact with each other, and any separation
distance between the monolayers within the corresponding range is allowed. Similar behavior
has been predicted for a highly swollen lamellar phase in binary blends [17]. We also ﬁnd thatφAB in the bulk region is constant (≈ 0.01 for all the systems in table I) in R1. This constantvalue should coincide with the value of θ at which the lamellar periodicity diverges (unbindingtransition) [8] and the free energy curve is smoothly connected to that of the 2-phase state.
Since we are considering the intermediate-segregation regime (χNAB = 16), θ ≈ 0.01 is muchsmaller than the Lifshitz volume fraction θL . We also point out that R1 is larger for relatively
shorter copolymer chain. This is consistent with the result in ﬁg. 2 where the typical lengthscale (∼ 1/ k ) is larger for larger α.
Negative curvature in R2 results from the attractive interactions between the monolayers [17]. In this region, the lamellar phases in R1, R2 and a part of R3 are thermodynamically unstable, and macrophase separation into the lamellar, A-rich and B-rich phases(determined by a common tangent construction) should take place [5,8]. This unstable region
hardly exists for S1 and S2, whereas it is remarkable for S4 and this is consistent with the
3-phase coexistence shown in ﬁg. 1.

Fig. 3 – (a) The free energy F as a function of θ for four diﬀerent systems, and (b) its deviation fromthe tangent at θ = 0.05. The inset in (b) is the same plot magniﬁed in the vertical axis.

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EUROPHYSICS LETTERS

Discussion. – 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 balancedwith their bending energy [18]. Of course, this is beyond the scope of the present mean-ﬁeld
approach which showed the marginal stability of the strongly swollen lamellar phase. However,
our result in ﬁg. 1 with random initial conditions shows, at least in the static sense, that PMEwith random conﬁguration of interfaces results from microphase separation as a metastableequilibrium. We comment that even the experimentally obtained PME may not necessarily
be a fully equilibrated structure.
In summary, we have investigated the equilibrium structure of ternary blends of A and B
homopolymers and symmetric AB block copolymer by numerically solving the set of extended
lattice SCFT equations. We showed that PME results from microphase separation for α ≤ 1 atleast as a metastable equilibrium. Detailed free energy analysis revealed the marginal stabilityof the strongly swollen lamellar phase in the corresponding region. Further developments ofthe present study, such as 3D simulations, asymmetric ternary system, binary system, or
diﬀerent copolymer structure (e.g., ABC triblock copolymer [19]) are in progress.
∗∗∗
HK acknowledges support from the national project, which has been entrusted to the Japan
Chemical Innovation Institute (JCII) by the New Energy and Industrial Technology Development Organization (NEDO) under MITI’s Program for the Scientiﬁc Technology Development
for Industries that Creates New Industries. We thank T. Kawakatsu, D. Andelman and
H. Orland for useful discussions.
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