SOFT MATERIALS
https://doi.org/10.1080/1539445X.2017.1354024

Coexistences of lamellar phases in ternary surfactant solutions
Isamu Soua, Ryuichi Okamotoa, Shigeyuki Komuraa, and Jean Wolffb

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Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo, Japan; bInstitut Charles
Sadron, Strasbourg Cedex, France

ABSTRACTARTICLE HISTORY

We theoretically investigate the coexistences of lamellar phases both in binary and in ternarysurfactant solutions. The previous free energy of a lamellar stack is extended to take into accountthe translational entropy of membrane segments. The obtained phase diagram for binary surfactant solutions (surfactant/water mixtures) shows a phase separation between two lamellar phases
and also exhibits a critical point. For lamellar phases in ternary surfactant solutions (surfactant/surfactant/water mixtures), we explore the possible phase behaviors and show that the phase
diagrams exhibit various three-phase regions as well as two-phase regions in which different
lamellar phases coexist. We also find that finite surface tension suppresses the undulation
fluctuations of membranes and leads to a wider three-phase and two-phase coexistence regions.

Received 21 June 2017
Accepted 7 July 2017

1. Introduction
One of the simplest mesoscale structures found in
mixtures of water and surfactant molecules is thelamellar phase in which the bilayers of amphiphilicmolecules form roughly parallel sheets separated bywater [1]. In some surfactant/water binary systems, it
is known that the lamellar phase can be swollen almost
without limit. For example, in the mixture of
C12E5 (n-akylpolyglycolether) and water, the repeat
distance of the lamellar phase can exceed 3,000 Å[2]. The transition from the bound lamellar phase to
the unbound phase is generally called the “unbindingtransition” [3]. Although it is very rare, the coexistence of two lamellar phases in thermodynamic equilibrium has been also reported for binary surfactant/water solutions typically containing DDAB (didodecyldimethylammonium bromide) [4–7]. In this case, ahigher-density condensed lamellar phase is in equilibrium with a lower-density swollen lamellar phase.
However, the reason why such a lamellar–lamellar
coexistence is so rare in binary surfactant solutions isstill a matter of debate and even puzzling [8, 9].
In contrast to binary mixtures, lamellar–lamellar
coexistence is fairly common in ternary systems suchas surfactant/surfactant/water mixtures [10–15] orpolymer/surfactant/water mixtures [16–18]. Here, thesurfactant molecules assemble into stacked bilayers,
while they are organized as coexisting lamellar phases.

KEYWORDS

Lamellar phase; surfactant
solutions; phase behaviors;
Helfrich interaction

The fact that ternary solutions typically exhibit a lamellar–lamellar coexistence can be explained if the bilayerswith different components have different interactions
across the water layer [19]. In such a lamellar–lamellar
phase separation, it is known that a long-ranged repulsive interactions such as electrostatic interaction and/or
steric repulsive interaction play an important role. The
latter interaction is known as the Helfrich steric interaction which arises from the reduced undulation
entropy of fluctuating membranes [20, 21]. In otherwords, the excluded volume of the neighboring membranes limits the configuration of a membrane andhence reduces its entropy.
For example, Harries et al. investigated the phaseseparations of charged surfactants by taking into
account both the electrostatic and the nonelectrostaticinteractions within a mean-field theory [22]. They
found that the lamellar–lamellar phase separation iscontrolled by nonelectrostatic interactions between the
counterions, and also by the interactions between theneutral and charged surfactants. On the other hand,lamellar–lamellar coexistences in charged membranes
were described only by electrostatic interactions in theother work [23].
For electrically neutral bilayer membranes, the combination of the steric repulsive interaction and otherdirect microscopic interactions, such as long-ranged
van der Waals attraction and short-ranged hydration

CONTACT Shigeyuki Komura
komura@tmu.ac.jp
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan
University, Tokyo 192-0397, Japan.
Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lsfm.
© 2017 Taylor & Francis Group, LLC

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I. SOU ET AL.

repulsion, determines whether membranes bind eachother or unbind to have an infinite separation between
them. Lipowsky and Leibler pointed out that a simple
superposition of the Helfrich steric repulsion and otherdirect interactions within a mean-field level gives incorrect (first-order) description of the unbinding transition [24]. An appropriate treatment of this problem
using a functional renormalization-group method
showed that the unbinding transition should be a continuous second-order transition (known as the criticalunbinding transition).
Later, Milner and Roux proposed a theory for theunbinding transition in a bulk of lamellar phase following the spirit of a mean-field theory for polymers [25].
In their argument, the Helfrich estimate of the entropy
is taken into account accurately, whereas the otherdirect microscopic interactions are approximately
incorporated as a correction to the hard-wall result
for the second virial coefficient. Their theory correctly
accounts for the second-order nature of the criticalunbinding transition. Furthermore, it has been used
to predict both the unbinding and preunbinding behaviors of a lamellar stack in binary surfactant solutions [26].
In this article, we investigate the coexistences of lamellar
phases both in binary surfactant solutions (surfactant/water mixtures) and in ternary surfactant solutions (surfactant/surfactant/water mixtures) within a mean-field theory. We consider a situation when the surfactant molecules
are electrically neutral, or the electrostatic interaction is
sufficiently screened in the presence of electrolyte. By taking into account the translational entropy of membranesegments, we extend the mean-field theory by Milner andRoux in order to properly account for the phase behaviors
in binary and ternary surfactant solutions.
Based on the proposed phenomenological free energy,
we first discuss the phase diagrams of binary systems inwhich we find a lamellar–lamellar coexistence that ends ina critical point, as found in one of the experiments [5]. By
considering three interaction parameters (virial coefficients) between different components, we further discuss
ternary mixtures and explore the possible types of ternary
phase diagrams (Gibbs triangles). The phase behavior isvery rich, and three-phase coexistences as well as twophase coexistences between different lamellar phases are
predicted for a certain range of the interaction parametervalues. We investigate both symmetric and asymmetriccases in terms of the two surfactant/water interactions. In
the symmetric case, the interactions (or the virial coefficients) between the same surfactant species are identical,
whereas they are different in the asymmetric case. We also
discuss the effects of finite membrane surface tension on
the phase behavior of ternary surfactant solutions.

In the next section, we explain the extension of themean-field theory by Milner and Roux. Using theextended free energy, we first calculate the phase diagrams of binary surfactant solutions. In Section 3, weconsider the phase behavior of ternary surfactant solutions. Various types of ternary phase diagrams are
obtained for both the symmetric and the asymmetriccases. In Section 4, we also calculate the ternary phase
diagrams in the presence of finite surface tension acting
on the membranes. The summary of our work andsome discussions are given in Section 5.

2. Lamellar phases in binary mixtures
Bilayer fluid membranes experience steric repulsion
arising from their reduced undulation entropy [1].
The corresponding interaction energy per unit areaof membrane was considered by Helfrich and isgiven by [20, 21]vs ð,Þ ¼

bðkB TÞ2:
κð,  δÞ2

(1)

Here kB is the Boltzmann constant, T is the temperature, κ is the bending rigidity, , is the average repeat
distance between bilayers, as shown in Fig. 1, and aconstant δ is the membrane thickness that is used as thesmallest cutoff length. Note that ,  δ in the denominator corresponds to the intermembrane distance inwhich membranes can undergo out-of-plane fluctuations. The numerical prefactor b was calculated to beb ¼ 3π2 =128  0:23 in the original work by Helfrich[20], but its value is debatable in the literatures [27–29].
For example, Monte Carlo simulations in Ref. [29]yielded a lower value of b  0:12, almost a half of theabove value. In the present study, the exact value of b
does not affect the results because we rescale all theenergy densities by including the factor b [see later
Eq. (4)].
In order to describe the free energy of a lamellar stack
in a binary surfactant/water solution, we first introduce
the membrane volume fraction ϕ ¼ δ=,  0. Here, wehave assumed that all the surfactant molecules constitutebilayers. Extending the argument by Milner and Roux[25], we consider the following grand potential per unit
volume of a lamellar stack:f ðϕÞ ¼ kB T ϕðlog ϕ  1Þ  kB Tχ ϕ2
δ3bðkB TÞ2 ϕ3
ϕ:
þ
2  μ
κδ3 ð1  ϕÞ

(2)Here, the first term represents the translational entropyof membrane segments, which was not consideredSOFT MATERIALS

3

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Figure 1. Schematic representation of two coexisting lamellar phases in binary surfactant/water solutions. , and δ are the lamellar
repeat distance and the bilayer thickness, respectively. For such a lamellar structure, the surfactant volume fraction is given by ϕ
¼ δ=,. The dilute lamellar phase (left) is characterized by a smaller ϕ value, while the condensed one (right) has a larger ϕ value.

before [25]. This term, however, plays an essential role
when we calculate the phase diagrams within equilibrium thermodynamics. In general, the translational
entropy term should be given by ϕ log ϕ þ ð1 
ϕÞ logð1  ϕÞ [21, 30], and the first term in Eq. (2)corresponds to its lowest order expansion in terms ofsmall ϕ  1. However, since such a generalization does
not result in any essential modification, we shall study
the above grand potential in this article. Our approximation is justified because the term ð1  ϕÞ logð1  ϕÞ
vanishes when ϕ ! 1, while the third term in Eq. (2)diverges, as explained below.
The second term in Eq. (2) is the correction to theentropic hard-wall result, and χ is the second virial
coefficient obtained fromð1
χ ¼  2 drð1  exp½Uν ðrÞ=kB TÞ;(3)2ν
where ν  δ3 is the volume of the membrane segment,Uν ðrÞ is the interaction between bits of membrane ofvolume ν, and r is a three-dimensional vector. All the
direct microscopic (van der Waals, hydration, and electrostatic) interactions are taken into account throughUν ðrÞ. The third term in Eq. (2) is due to the Helfrichsteric repulsion, and the factor of ð1  ϕÞ2 comes fromthe finite membrane thickness [see also Eq. (1)] [31].
, is needed for the conFinally, the chemical potential, μservation of the surfactant volume fraction ϕ. A similar
free energy to Eq. (2) was also proposed in other works todescribe the unbinding transition [32–34] but without thetranslational entropy term that we have introduced. We
note again that the first translational entropy term in Eq.
(2) is not accounted for by the Helfrich steric repulsion
term which also has an entropic origin. Without thetranslational entropy term, the behavior of the free energy

is thermodynamically inappropriate around ϕ  0 and
one cannot describe the correct phase behaviors.
It is convenient to rescale all the energy densities by2bðkB TÞ2 =ðκδ3 Þ. Then, Eq. (2) can be presented in adimensionless form asf ðϕÞ ¼ aϕðlog ϕ  1Þ  χϕ2 þ

ϕ3
 μϕ;
2ð1  ϕÞ2(4)

where a ¼ κ=ð2bkB TÞ is a numerical factor of orderunity, and hence can be set as a ¼ 1 in the following
discussion for simplicity, while χ ¼ χ κδ3 =ð2bkB TÞ is adimensionless interaction parameter. The equation ofstate is then given by minimizing the grand potential,@f =@ϕ ¼ 0, and becomesμ ¼ log ϕ  2χϕ þ

3ϕ2
ϕ3
:
2þ
2ð1  ϕÞ
ð1  ϕÞ3(5)

Using the above grand potential density, we canobtain the spinodal from the condition [35]:@ 2 f ðϕÞ
¼ 0:
@ϕ2With the use of Eq. (4), it can be written as
"
#
1 1
3ϕ
6ϕ2
3ϕ3
þ
þ
þ
:
χ¼
2 ϕ ð1  ϕÞ2 ð1  ϕÞ3 ð1  ϕÞ4

(6)

(7)

The conditions for the critical point is given by@ 2 f ðϕÞ
@ 3 f ðϕÞ
¼
0;
¼ 0:
@ϕ2
@ϕ3

(8)

These conditions and Eq. (5) can be numerically solved toobtainthecriticalpointasðχ c ; ϕc ; μc Þ ¼
ð3:16; 0:23; 2:76Þ.

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I. SOU ET AL.

On the other hand, the thermodynamic equilibrium
between the two coexisting phases denoted as “1’’ and“2’’ and characterized by ϕ1 and ϕ2 satisfies the following conditions [35]:Downloaded by [220.147.193.142] at 17:45 01 September 2017@f ðϕÞ @f ðϕÞ 
¼
¼ 0;
@ϕ 1
@ϕ 2

f ðϕ1 Þ ¼ f ðϕ2 Þ:(9)We have numerically solved the above set of conditions
to obtain the phase diagrams.
The calculated phase diagrams in the ðχ; ϕÞ andðχ; μÞ planes are shown in Fig. 2. The red solid lines
are binodal lines, and the black dashed lines are spinodal lines. When χ > 3:16, the binary mixture separates
into two lamellar phases characterized by different ϕvalues indicated by the horizontal tielines (see also
Fig. 1). Notice that there is a critical point at
ðχ c ; ϕc ; μc Þ ¼ ð3:16; 0:23; 2:76Þ where the two lamellar
phases become identical. For χ < 3:16, on the other
hand, the complete unbinding of the lamellae occurs

upon swelling with excess water [36, 37]. In the ðχ; ϕÞphase diagram, such a transition occurs when we takethe limit of ϕ ! 0.
The calculated phase diagram in Fig. 2(a) resembles
that obtained for DDAB/water binary mixtures [4–7] if
we assume that the interaction parameter χ is inversely
proportional to the temperature. In these experiments,
the existence of a critical point and associated criticalphenomena were experimentally evidenced by various
scattering methods [5].

3. Lamellar phases in ternary mixtures
Next, we consider the lamellar phases in ternary surfactant/surfactant/water solutions in which bilayer
membranes are composed of two different types of
surfactant, say surfactant A and surfactant B, as
shown in Fig. 3. Let us define the volume fractions ofsurfactant A and B by ϕ and ψ, respectively. Then, the

5

5

4

4

3

3

2

2
0

0.2

0.4

0.6

0.8

1

-4

-3

-2

Figure 2. Phase diagrams of a binary surfactant solution as a function of (a) the surfactant volume fraction ϕ and the (scaled) virial
coefficient χ and (b) the surfactant chemical potential μ and the virial coefficient χ. The red solid lines are binodal lines and the blacksolid lines are tielines separating two distinct lamellar phases. The black dashed lines are spinodal lines. The black circles represent
the critical point at (Xc, ϕc, μc) = (3.16,0.23, - 2.76).

Figure 3. Schematic representation of lamellar phases in ternary surfactant solutions, i.e., surfactant/surfactant/water mixtures. Eachbilayer consists of surfactant A (blue) and surfactant B (red) whose volume fractions are ϕ and ψ, respectively. (a) A single lamellar
phase having a unique repeat distance that is fixed by the total surfactant volume fraction ϕ + ψ. (b) An example of three coexistinglamellar phases characterized either by different repeat distances or by A/B relative compositions.

SOFT MATERIALS

volume fraction of water (solvent) is automatically fixedby 1  ϕ  ψ due to the incompressibility condition. As
a generalization of Eq. (4), we consider the following
dimensionless grand potential per unit volume for ternary mixtures:gðϕ; ψÞ ¼ ϕðlog ϕ  1Þ þ ψðlog ψ  1Þχ ϕϕ ϕ2  χ ψψ ψ 2  χ ϕψ ϕψ

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ðϕ þ ψÞ3
 μϕ ϕ  μψ ψ:þ2ð1  ϕ  ψÞ2(10)

In the above, the first two terms represent the translational entropy of each surfactant component, the next
three terms describe the different interactions characterized by the three dimensionless virial coefficientsχ ϕϕ , χ ψψ , and χ ϕψ which are assumed as independent
parameters. The first term in the third line corresponds
to the Helfrich steric repulsion acting between mixed
membranes. Notice here that the total surfactant
volume fraction ϕ þ ψ corresponds to the volume fraction of membranes. Several other assumptions that leadto this expression are separately discussed in Section 5.
Furthermore, μϕ and μψ are the chemical potentials forthe two surfactants. When, for example, surfactant B isabsent and hence ψ ¼ 0, Eq. (10) reduces to Eq. (4), as
it should.
In order to discuss the stability of the above grandpotential with two independent variables, we consider
the 2 × 2 Hessian matrix of gðϕ; ψÞ given by [35]
1
@2g
@2g
B @ϕ2 @ϕ@ψ C
C:
H¼B
@ @2g
@2g A
@ψ@ϕ @ψ 25@gðϕ; ψÞ @gðϕ; ψÞ
¼
¼ 0;
@ϕ 1
@ϕ 2

(14)



@gðϕ; ψÞ @gðϕ; ψÞ
¼
¼ 0;
@ψ 1
@ψ 2

(15)

gðϕ1 ; ψ 1 Þ ¼ gðϕ2 ; ψ 2 Þ:

(16)

Similarly, for a three-phase coexistence between phases“1,” “2,” and “3”, the following set of conditions should
be satisfied [35]:@gðϕ; ψÞ @gðϕ; ψÞ @gðϕ; ψÞ
¼
¼
¼ 0;
(17)
@ϕ 1
@ϕ 2
@ϕ 3



@gðϕ; ψÞ @gðϕ; ψÞ @gðϕ; ψÞ
¼
¼
¼ 0;
@ψ 1
@ψ 2
@ψ 3
gðϕ1 ; ψ 1 Þ ¼ gðϕ2 ; ψ 2 Þ ¼ gðϕ3 ; ψ 3 Þ:

(18)
(19)

For lamellar phases under consideration, an example ofthree-phase coexistence is schematically presented inFig. 3(b). In the following, we present the numericallycalculated ternary phase diagrams (Gibbs triangles) for
different interaction parameters.
We first consider the symmetric case between the two
surfactants A and B, i.e., χ ϕϕ ¼ χ ψψ . Figure 4 shows aternary phase diagram when χ ϕϕ ¼ χ ψψ ¼ 3 (symmetric)and χ ϕψ ¼ 3. This is the case when each surfactant/water
binary solution does not exhibit lamellar–lamellar phase

0

(11)

At the spinodal, the Hessian defined as the determinantof the matrix, H ¼ det H, vanishes, i.e., H ¼ 0. The
critical point can be obtained by considering another2 × 2 matrix
0

@2g
B @ϕ2
H0 ¼ B
@ @H
@ϕ

1
@2g
@ϕ@ψ C
C;
@H A
@ψ

(12)

and its determinant H 0 ¼ det H0 . Then, the conditions
for the critical point are given by [35]H ¼ 0; H 0 ¼ 0:

(13)

For ternary mixtures, the thermodynamic equilibrium between the two coexisting phases denoted as“1” and “2” and characterized by ðϕ1 ; ψ 1 Þ and ðϕ2 ; ψ 2 Þsatisfies the conditions [35]:Figure 4. Phase diagram of a ternary surfactant solution when
the interaction parameters are χϕϕ = χψψ= 3 and Xϕψ ¼ 3. “A”
and “B” stand for surfactant A and surfactant B, respectively,
while “S” indicates solvent such as water. The phase diagram issymmetric with respect to the equal A/B relative compositions.
The red solid lines are binodal lines, and the black solid lines
are tielines separating two distinct lamellar phases. The blue
triangle represents the region of three-phase coexistence, andthe black dashed line is a spinodal line. The upper white region
corresponds to the one-phase lamellar region. The black circles
represent the critical points at (ϕc ; ψc Þ ¼ ð0:026; 0:23Þ andð0:23; 0:026Þ.

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I. SOU ET AL.

separation because χ ϕϕ ¼ χ ψψ < 3:16 (see Fig. 2). The
obtained phase diagram is always symmetric with respect
the line ϕ ¼ ψ. Due to the strong repulsion between the A
and B components (χ ϕψ ¼ 3), there is a wide region of
two-phase coexistence (red solid lines) with horizontal
tielines (black solid lines). In the upper part of the triangle,
there is a region of three-phase coexistence (blue triangle)
associated with two wings of two-phase coexistence. These
two-phase coexistence regions end in two corresponding
critical points (black circles). The black dashed lines indicate the spinodal lines that appear inside the coexistence
regions. When we make χ ϕψ larger such as χ ϕψ ¼ 3 (notshown), only the two-phase coexistence region remains in
the lower part of the triangle with horizontal tielines, andthe three-phase region disappears.
In Fig. 5, we present the ternary phase diagrams
when χ ϕϕ ¼ χ ψψ ¼ 4 (symmetric), while the A/B interaction is changed as (a) χ ϕψ ¼ 5, (b) χ ϕψ ¼ 3, and (c)χ ϕψ ¼ 0. These are the cases when each surfactant/water binary solution exhibits lamellar–lamellar phaseseparation because χ ϕϕ ¼ χ ψψ > 3:16 (see Fig. 2), as
represented on the two S-A and S-B sides of the triangles. In the case of Fig. 5(a), these two-phase regions
merge to form a single two-phase region with tilted
tielines in the upper part of the triangle. In the lower

part of the triangle, on the other hand, there is a regionof two-phase coexistence with horizontal tielines. This
two-phase region ends in a critical point. When χ ϕψ ismade smaller as in Fig. 5(b), the upper two-phasecoexistence region separates into distinct two-phaseregions that also end in two corresponding criticalpoints. For even smaller χ ϕψ as in Fig. 5(c), the three
two-phase regions meet each other forming a threephase coexistence region similar to Fig. 4. As a result,
all the three critical points disappear.
In Fig. 6, we show the ternary phase diagrams for an
asymmetric case of χ ϕϕ ¼ 4 > 3:16 and χ ψψ ¼ 3 < 3:16,
while the A/B interaction is chosen as (a) χ ϕψ ¼ 4 and(b) χ ϕψ ¼ 3. In this case, only the binary A/S mixture
exhibits the phase separation, while the binary B/S doesnot. Here, Fig. 6(a) should be compared with Fig. 5(b).
The two two-phase regions end in the respective criticalpoints. When χ ϕψ is made smaller as in the case of
Fig. 6(b), the two two-phase coexistence regions areconnected to each other with the appearance of athree-phase coexistence region. This three-phase region
accompanies another small two-phase region and acritical point on the S-B side of the triangle. In thisasymmetric case, the tielines are not horizontal andtilted especially in the upper part of the phase diagram.

Figure 5. Phase diagrams of a ternary surfactant solution when the interaction parameters are Xϕϕ ¼ χ ψψ ¼ 4 (symmetric) and (a)Xϕψ ¼ 5, (b) Xϕψ ¼ 3, (c) Xϕψ ¼ 0. The meanings of different lines are explained in Fig. 4. One and three critical points exist in (a)and (b) (black circles), respectively, while a three-phase coexistence region appears in (c) (blue triangle).

SOFT MATERIALS

7

Figure 6. Phase diagrams of a ternary surfactant solution when the interaction parameters are (a) Xϕϕ = 4, χ ψψ ¼ 3, Xϕψ ¼ 4 and (b)Xϕϕ = 4, χ ψψ ¼ 3, Xϕψ ¼ 3. These phase diagrams are asymmetric with respect to the equal A/B relative compositions. The
meanings of different lines are explained in Fig. 4. Two and one critical points exist in (a) and (b) (black circles), respectively, while athree-phase coexistence region appears in (b) (blue triangle).

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4. Effects of surface tension
In this section, we consider lamellar phases in ternarysurfactant solutions in which surface tension, σ, is acting
on membranes. It is known that finite surface tension
significantly suppresses membrane undulations, and therange of fluctuation-induced interaction between tense
membranes becomes shorter. Although the calculation ofthis interaction is complicated in general, Seifert provided asimple self-consistent calculation [38]. He showed that theenergy per unit area of membrane in the presence of surface tension is given by2bðkB TÞ2
ð,  δÞ=
vs ð,; Þ ¼;
(20)
κð,  δÞ2 sinh½ð,  δÞ=
where  ¼ ð2kB T=πσÞ1=2 is the characteristic length
arising from the competition between the thermalenergy and the surface energy. When the surface tension σ is small ( ! 1), the above expression reduces
to Eq. (1) for a tensionless membrane and recovers thelong-range algebraic decay. When the surface tension σ
is large ( ! 0), on the other hand, Eq. (20) decays
exponentially with distance ,, consistent with therenormalization-group result [3] and Monte Carlosimulations [29]. The reduction of membrane undulations in the presence of surface tension was experimentally observed in Ref. [39].
Using Eq. (20) in the presence of surface tension, we
consider a modified grand potential per unit volume forternary surfactant solutions as [31]gðϕ; ψ; xÞ ¼ ϕðlog ϕ  1Þ þ ψðlog ψ  1Þχ ϕϕ ϕ2  χ ψψ ψ 2  χ ϕψ ϕψ
x2 GðxÞ

ðϕ þ ψÞ3
 μϕ ϕ  μψ ψ:
ð1  ϕ  ψÞ2(21)

Here, the dimensionless quantity x is defined by δ 1ϕψ
x¼
;

ϕþψ

(22)

whereas the scaling function GðxÞ is given byGðxÞ ¼ 

1:2sinh2 x(23)

We note that x depends also on ϕ and ψ, while δ= is adimensionless parameter that can be given externally.
Figure 7 shows the calculated ternary phase diagrams when χ ϕϕ ¼ χ ψψ ¼ 4 (symmetric) and χ ϕψ ¼ 5,
while the parameter controlling the surface tension is(a) δ= ¼ 10 and (b) δ= ¼ 20. These phase diagrams
should be compared with that in Fig. 5(a) for which
δ= ¼ 0. As shown here with finite surface tension, thephase separation is dramatically enhanced and theupper two two-phase coexistence regions extend down
to the middle part of the triangle. At the expense of thecritical point in Fig. 5(a), there appears a large threephase coexistence region in the middle part. This threephase coexistence region is connected to the lower twophase region with horizontal tielines. As we see in
Fig. 7(b), the three-phase coexisting region further
expands when δ= is made larger. Therefore, surface
tension promotes the phase separation between different lamellar phases. The fact that dense lamellar phases
coexist with an excess water on the two S-A and S-B
sides of the triangles is in accordance with the experiment [39]. In the presence of finite surface tension, theyobserved membranes merging one by one into bundles
at mutually adhering membranes. Also, the water
between the membranes was driven into a small number of compact water pockets [39].

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I. SOU ET AL.

Figure 7. Phase diagram of a ternary surfactant solution in the presence of membrane surface tension. The interaction parameters
are Xϕϕ ¼ χ ψψ ¼ 4 (symmetric), Xϕψ ¼ 5, while the tension parameters are (a) δ= ¼ 10 and (b) δ= ¼ 20. These phase diagrams
should be compared with that in Fig. 5(a) for which δ= ¼ 0. The wide three-phase coexistence region (blue triangle) further
expands when δ= becomes larger.

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5. Summary and discussion
In this article, we have investigated the coexistences oflamellar phases both in binary and in ternary surfactant
solutions. To calculate the phase diagrams, we haveextended the previous free energy of a lamellar stack[25] by taking into account the translational entropy ofmembrane segments. The obtained phase diagrams fora binary surfactant solution show a phase separation
between two lamellar phases and also exhibit a criticalpoint. For lamellar phases in ternary surfactant solutions, we have further extended the free energy to take
into account the different interactions between three
species and explored the possible phase behaviors.
The calculated phase diagrams include various coexistences between three different lamellar phases (threephase regions) or between two lamellar phases (twophase regions). A systematic change of the phase behavior has been observed by changing the interaction
parameter between the two surfactant species. Finally,
we have looked at the effects of finite surface tension
which suppresses membrane fluctuations and leads to awider three-phase coexistence region.
We stress again that the addition of translational
entropy terms in the free energies [see Eqs. (2) and(10)] is essential in calculating the correct phase diagrams. Without these terms, one cannot obtain thecoexistence between two lamellar phases having different repeat distances. The original free energy by Milner
and Roux was considered in order to explain theunbinding transition in surfactant solutions [25].
Although their phase diagram exhibits a coexistence
between a lamellar phase and excess water (i.e.,
unbound lamellar phase), a coexistence between two
distinct lamellar phases does not occur. We consider
that these translational entropy terms should be
included in addition to the Helfrich steric interaction
which also has an entropic origin.

In the present work, we have assumed that thelamellar phase is the only lyotropic liquid crystaline
phase that is formed for any temperature (interaction) and composition. In real surfactant solutions,however, this is certainly not the case, because typical
phase diagrams contain other phases such as the
micellar phase, the hexagonal phase, and the cubic
phase [2]. Rather than reproducing realistic phase
diagrams by considering all the possible phases insurfactant solutions, our purpose is to investigate indetail the competition between the Helfrich steric
repulsion and other direct microscopic (van der
Waals, hydration) interactions especially in ternary
mixtures. This is why we have extended the freeenergy of a lamellar stack by Milner and Roux andcalculated various coexistences only between thelamellar phases. A similar theoretical approach wasmade by Noro and Gelbart who also discussed lamellar–lamellar phase separations in surfactant solutions[19]. We also note that the steric repulsive interaction acting between neighboring cylinders in the hexagonal phase is discussed in Ref. [1], which can beused to extend our treatment.
Another simplification in our work is that we havenot taken into account the composition dependence ofthe bending rigidity κ or the surface tension σ when amembrane is composed of two surfactants A and B.
Usually, a membrane made of an A/B surfactant mixture will show an intermediate bending rigidity
between those of pure membranes. In principle, sucha change in the bending rigidity affects the Helfrichsteric interaction. One of the possible ways to describe
the intermediate behavior is to linearly interpolate
between the two pure limits. This approximation canbe justified when the bending rigidities of the purecomponents are not so different. When they are very
different, such as in membranes composed of surfactant

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SOFT MATERIALS

and amphiphilic polymer, a nonlinear effect on thebending
rigidity
becomes
important[40].
Furthermore, a detailed discussion on the surface tension in a mixed membrane was recently given by someof the present authors [41].
Although the phase diagrams calculated in this article may not be simply compared with experimentallyobtained ones because of the previously mentioned
reason, it is still useful to discuss the ternary phase
diagrams of glycolipid/cationic surfactant/water mixtures at room temperature [14]. In this ternary mixture,they found the two-phase regions between the coexisting lamellar phases on both sides of the Gibbs triangle.
More interestingly, they further identified two corresponding critical points and also the region of thethree-phase coexistence according to the phase rule.
Such a phase behavior is very reminiscent of thephase diagrams in Fig. 5(b) and (c). Although not yet
done, we expect that one can reproduce the experimentally obtained phase diagram by further tuning thethree interaction parameters in our model.
In this article, we have considered a situation inwhich the surfactants are electrically neutral or theelectrostatic interaction is sufficiently screened. As isclear from Eq. (1), the Helfrich steric repulsion isimportant only when the membrane is flexible,
κ ’ kB T. For strongly charged and unscreened membranes (no electrolyte), on the other hand, the dominant repulsion originates from the electrostaticinteractions between flat membranes. The interplay
between the electrostatics and fluctuations of a stackof membranes (without van der Waals and hydration
forces) was studied before by Pincus et al. [42]. Whenelectrostatic interactions are strong enough (Gouy–
Chapman regime) compared with the Helfrich steric
repulsion, they showed that out-of-plane membrane
fluctuations become smaller than the intermembrane
separation ,  δ. In the other weaker electrostatic
regimes, on the other hand, the suppression of membrane fluctuations is less important, and in some cases,
the screened electrostatic interactions can be completely neglected [42]. Our assumption for charged membranes is justified for such situations.
Our results indicate that the lateral phase separationin mixed membranes causes different intermembrane
distances. Recently, much efforts have been made to
study the statics and dynamics of multicomponent lipid
membranes [43], mainly using giant unilamellar vesicles in the experiments [44]. In the future, it is interesting to study the phase behaviors of multilamellar
vesicles composed of more than two types of lipid bytaking into account the interactions between

9neighboring membranes. The present work would provide us with a useful theoretical guide for such a
research direction.

Acknowledgments
We thank T. Kato and Y. Kawabata for useful discussions.

Funding
S.K. and R.O. acknowledge the support from the Grant-inAid for Scientific Research on Innovative Areas “Fluctuationand Structure” (Grant No. 25103010) from the Ministry of
Education, Culture, Sports, Science, and Technology ofJapan, and the Grant-in-Aid for Scientific Research (C)(Grant No. 15K05250) from the Japan Society for the
Promotion of Science (JSPS). J.W. acknowledges the support
from the C.N.R.S. (Centre National de la Recherche
Scientifique), the French ORT association, and ORT schoolof Strasbourg.

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